Search Results for Chinese History of Mathematics


Biographies

  1. Xu Guang-qi biography
    • He rose in importance to eventually become the leading minister in the Imperial Court of the Ming Dynasty.
    • Before this he had studied Western culture under Matteo Ricci but, before we explain their work together, we should set the scene by quoting from [',' R Mei, The mathematical works of Xu Guangqi (Chinese), in Collected papers in honour of Xu Guangqi (Peking, 1963), 143-161.','6] concerning the background:- .
    • In spite of the dismal political history of the Ming Dynasty (1368-1644), China progressed in many fields including trade and industry, science and technology, philosophy and literature, mainly owing to the wisdom and effort of the people.
    • It was therefore a natural consequence that figures like Xu Guang-qi and others appeared during this period of 'Renaissance'.
    • [Xu Guang-qi] led a long, but politically rather futile, ministerial life for a quarter of a century in the Imperial Court of the Ming Dynasty.
    • It was also during these last decades of the Ming Dynasty that the Chinese first came into contact with European science through the Jesuits.
    • The Jesuits were intent on spreading the Catholic faith in the old Empire, and in order to win over the people, they endeavoured first to gain the favour and the following of the educated class.
    • As an expedient means they brought in various new technological gadgets and apparatus unknown to China, as well as scientific theories which were, though not all of them up-to-date knowledge at the time in Europe, nonetheless of a sufficient novelty and attraction to some educated Chinese.
    • In fact Chinese mathematics had been in a period of decline for some time.
    • Xu Guang-qi was well aware of this and attributed the decline to academics neglecting practical learning and also to a confusion between mathematics and numerology.
    • The brilliant "tian yuan" or "coefficient array method" or "method of the celestial unknown" for solving equations which had been expounded with such skill by Li Zhi in the 13th century was no longer understood in China.
    • The remarkable progress which the Chinese had made in algebra had been largely forgotten, and practical problems which had been solved by algebra were by this time solved by ad hoc means.
    • Even the Nine Chapters on the Mathematical Art was almost unknown, and Xu Guang-qi himself had never read the brilliant Chinese classic, while The Ten Classics were thought to have been lost.
    • Into this weak period in Chinese mathematics came European mathematics brought by scholars such as Ricci.
    • He arrived at Macau on the east coast of China in 1582.
    • He settled in Chao-ch'ing, Kwangtung Province, and began his study of Chinese.
    • He also worked at acquiring understanding of Chinese culture.
    • In 1589 Ricci moved to Shao-chou and began to teach Chinese scholars the mathematical ideas that he had learnt from his teacher Clavius.
    • Together with one of his students, Qu Rukui, he translated the first book of Euclid's Elements.
    • There he presented his essay on the first book of the Elements but this preliminary work is now lost.
    • He taught mathematics to Chinese students and one of these was the high-ranking public official Xu Guang-qi.
    • Xu Guang-qi became the first native of China to publish translations of European books into Chinese.
    • Collaborating with Ricci he translated Western books on mathematics, hydraulics, and geography.
    • The first six books of Euclid's Elements were translated into Chinese in 1607 by Ricci and Xu Guang-qi.
    • As an amusing note, we remark that their translation of "Clavius" into Chinese meant "nail" so they referred to Clavius as "Mr Ding"! Their method of translation is described in [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','3]:- .
    • The translation technique involved Ricci explaining the contents of the original text orally to [Xu Guang-qi] who would then write down what he had understood.
    • Ricci and Xu Guang-qi's translation respects the order of Clavius's work completely; however, it is much less verbose ..
    • The approach to mathematics in these books must have seemed totally alien to Chinese readers whose approach to the subject had been so radically different.
    • Clearly Xu Guang-qi was a total convert to Western thinking but most other Chinese mathematicians stuck to their traditional way of thinking, questioning what to them was absurd such as "a point has no part".
    • The Chinese approach to mathematics had been highly practical and to try to fit the Elements into that tradition Xu Guang-qi explained in his preface how the contents had application to the problem of the calendar, to music and to technology.
    • However the new Chinese terminology which Xu Guang-qi had to invent for point, curve, parallel line, acute angle, obtuse angle etc.
    • (these concepts being alien to Chinese mathematics, there were no Chinese words for them) soon became part of Chinese mathematics, as did the style of the geometric figures, in particular the characters Xu Guang-qi chose to label them.
    • In one sense Xu Guang-qi did a disservice to Chinese mathematics.
    • He was converted to Christianity by Ricci and adopted the position that Chinese culture was inferior to that of the West, in particular, as we indicated above, in their mathematical tradition.
    • This was a great shame, for although clearly much had to be learnt from the transmission of knowledge, there was no need to talk down the fine achievements of the Chinese through a different approach.
    • He predicted that soon everyone in China would be studying the Elements and in this he was largely correct since Western schools were set up in China in which the study of the elements was a compulsory topic.
    • Ricci's main aim in China was to convert the Chinese to Christianity.
    • Many, both in China and in the European Christian Church, felt that he used Western mathematics, science and technology as an unreasonable means to achieve conversions.
    • Ricci himself felt that reforming the Chinese calendar would be the most effective step that could be taken, so "proving" the power of Christianity.
    • Indeed the question of calendar reform had occupied the Chinese for 200 years but, despite various proposals being made, the Bureau of Astronomy had been cautious and done nothing.
    • The Western approach to astronomy and the calendar scored a major success shortly after Ricci died when it accurately predicted the eclipse of 15 December 1610.
    • Another eclipse was predicted for 1629 and a competition was held by the Chinese government to determine who could give the most accurate prediction of its timing.
    • Three different predictions were made, one by the Da Tong traditional Chinese school, one by the Islamic calendar school, and one by the New Method School led by Xu Guang-qi which used European methods.
    • The most accurate prediction for the eclipse of 21 June 1629 was made by Xu Guang-qi and the emperor then appointed him take charge of calendar reform.
    • During the last few years of his life Xu Guang-qi was an extremely influential figure at the Imperial Court of the Ming Dynasty.
    • The Ming were under attack by from the Manchu who were descendants of the Juchen tribes who had ruled North China as the Chin dynasty in the 12th century.
    • Xu Guang-qi, with his strong belief in the superiority of all things European, persuaded the Ming emperor to have his army adopt advanced European artillery against the Manchu.
    • List of References (7 books/articles) .
    • A Poster of Xu Guang-qi .
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Xu_Guangqi.html .

  2. Mei Wending biography
    • Mei Wending was born into a family of considerable mathematical talents.
    • He had three younger brothers, two of whom, Mei Wennai and Mei Wenmi, were both skilled mathematicians and astronomers.
    • To avoid confusion, from now on let us adopt the system of referring to Mei Wending as Mei, while referring to all other members of the Mei family by their full names.
    • He went to Peking in 1630 to undertake a reform of the Chinese calendar.
    • Because the Board of Mathematics, a group of around 200 Chinese mathematicians, had made an error in calculating the calendar in 1611, the government had enlisted the help of missionaries in the task.
    • Schall, rather than just assisting, was put in charge of the production of a new calendar which was adopted in 1645.
    • However this was controversial; many Chinese did not like a Westerner in charge of the calendar, other missionaries were upset that the new calendar still contained references to "lucky" and "unlucky" days.
    • These issues regarding the Chinese calendar were important ones during the years that Mei and his brothers were growing up, and they studied the mathematical and astronomical topics necessary for calendar design under the Daoist teacher Ni Guanghu.
    • The Ming dynasty had come to an end in 1644 when the Manchus, having invaded from the north, took control of Peking.
    • They set up the new Qing regime and the native Chinese were not allowed to hold the highest offices in the administration, these being all taken by Manchus.
    • Mei's family remained loyal to the old Chinese Ming dynasty keeping themselves independent of the Manchu led administration.
    • However, the Qing rulers tried to promote Chinese culture and the Emperor Kangxi, who came to power in 1661 when only seven years of age, worked hard to promote learning.
    • He was keen on both Chinese learning and the new European learning brought to China by the missionaries.
    • Mei also tried to steer a course between the best of the old Chinese learning and the new European learning.
    • Mei tried to situate the new European knowledge properly within the historical framework of Chinese astronomy and mathematics.
    • In his view, Chinese astronomical knowledge had advanced following the adoption of the new, more accurate Jesuit calendar following the reform initiated by Xu Guangqi in 1629.
    • In his historical studies, Mei stressed that Chinese astronomy had improved from generation to generation, progressing from coarseness to accuracy.
    • He gave precisely the same description for the development of Western astronomy.
    • This was Mei's historical rationale for synthesizing Western and Chinese knowledge.
    • In fact arguments about the calendar were reaching a head in China around this time for only two years after this work appeared the Jesuits were accused by Yang Guangxian of using calendar reform as a means of covering up their work on converting the Chinese to Christianity and of subverting the Empire.
    • What did Mei argue in his 1662 work? Pingyi Chu writes [',' P Chu, Remembering our grand tradition: the historical memory of the scientific exchanges between China and Europe, 1600-1800, Hist.
    • Errors in the astronomical texts, he argued, jeopardized the development of the discipline; he then developed an evidential method to analyse traditional Chinese mathematics and astronomy.
    • Mei argued that all sorts of errors in the ancient mathematical and astronomical texts had seriously impaired their transmission regardless of whether they were the corruption of printing boards, or mistakes in coping with a text, or commentating on a text without a proper understanding.
    • In reviving the Chinese mathematical and astronomical traditions, collecting and collating ancient texts was a crucial first step.
    • Mei Wending argued, moreover, that calendrical studies were at the core of the Confucian pursuit of 'gewu qiungli' (investigating things so as to fathom the principle thoroughly).The eternity of 'li' guaranteed the status of the ancient sages, a status profoundly important to the cultural identity of Confucian scholars.
    • The innate capacity of the heart/mind was such that the ultimate 'li' could be attained through mathematical investigations of each ancient calendar.
    • From this angle, Mei Wending suggested the possibility of integrating calendrical study into the newly emerging evidential scholarship and contended that the investigation of ancient calendars and ancient remains were of equal importance for understanding 'li'.
    • In addition, he claimed that the new Western calendar was only a variant of the Chinese calendar, anticipated by the wisdom of the ancient sages.
    • This emphasis on the great importance of astronomy led Mei to reject the claims of Confucian scholars such as Yang Guangxian who were satisfied with understanding the 'li' of astronomy without bothering with detailed calendrical calculations.
    • Yang Guangxian's arguments against the astronomers who were Jesuits, either Europeans or Chinese converts, was so successful that at one stage all were condemned to death.
    • They were saved due to an earthquake hitting not long before the time set for their execution, but later Mei's arguments against Yang Guangxian succeeded since his lack of ability to make complicated calendrical computations became clear.
    • In fact this dispute led to the Emperor Kangxi becoming an enthusiast for mathematics, something which helped Mei in the later part of his career.
    • Kangxi said [',' Q Han, Emperor, Prince and Literati: Role of the Princes in the Organisation of Scientific Activities in Early Qing Period.
    • You only know that I am versed in mathematics.
    • But you do not know why I study mathematics.
    • When I was young, the Chinese officials and the Westerners at the Board of Mathematics were on unfriendly terms with each other.
    • Yang Guangxian and Adam Schall von Bell measured the length of the sun's shadow in front of the Wu Men gate.
    • So I was eagerly determined to study mathematics.
    • Joseph Dauben and Christoph Scriba write [',' J W Dauben and C J Scriba, Writing the history of mathematics: its historical development (Birkhauser, 2002).','2]:- .
    • 'Fangcheng' is one of the 'jiushu' (Nine Subjects Concerning Number) emphasised in Confucian education in the pre-Qin period (before 211 B.C.).
    • When he finished writing this book, Mei Wending wrote to one of his friends saying that "I am disgusted by those Western missionaries who exclude traditional Chinese mathematics, and therefore I wrote this book about which even Matteo Ricci could not possibly say a bad word".
    • Mei Wending clearly wished to demonstrate the superiority of early Chinese mathematics over the methods Western scholars had brought to China, and at least in this case, the example of simultaneous linear equations was an excellent one to stress.
    • He continued throughout his career to argue strongly for the acceptance of the new mathematical ideas coming from Europe and also for preserving the Chinese approach to mathematics [',' J W Dauben and C J Scriba, Writing the history of mathematics: its historical development (Birkhauser, 2002).','2]:- .
    • In his various works, Mei Wending compiled ancient mathematical material and studied a number of almost forgotten topics.
    • For example, the gougu theorem (referred to in the West as the Pythagorean Theorem) was a well-known and important focus of ancient Chinese geometry, but since the time of Liu Hui and Zhao Shang, two brilliant mathematicians of the 3rd century, no proof of the gougu theorem had been given in any mathematical books.
    • Mei Wending, however, proposed two proofs, along with other applications of the theorem in his 'Gougu juyu' (Illustration of the Right-Angled Triangles) (written before 1692).
    • In 1693 Mei again made clear his approach to Western mathematics writing (see for example [',' C Jami P M Engelfriet and G Blue, Statecraft and intellectual renewal in late Ming China: the cross-cultural synthesis of Xu Guangqi (1562-1633) (Brill, 2001).','3]):- .
    • Mei used traditional Chinese methods in Jihe bubian (Complements of Geometry) Mei to calculate the volumes and relative dimensions of regular and semi-regular polyhedrons.
    • The Jihe tongjie (Complete Explanation of Geometry) contains Mei's approach to Euclidean geometry.
    • In 1700, in the Qiandu celiang (The Measurement of a Prism with Two Right Triangular Bases), he gave a trigonometric interpretation of the celestial coordinate transformation introduced by Guo Shoujing in 1280.
    • Although this was the first meeting between Mei and the Emperor, earlier Mei had taught mathematics to the son of Li Guangdi, the Emperor's mathematical advisor, along with his own grandson Mei Juecheng at Baoding.
    • Mei was too old by this time to serve the Emperor but the discussions between Mei and the Emperor led eventually to the establishing of the Mengyangzhai (the Academy of Mathematics) in 1713.
    • Its main aim was to supervise the compilation of mathematical and astronomical works, and many of the mathematicians trained by Mei, including his grandson Mei Juecheng, were chosen to work at the Academy.
    • The ancient Chinese calendar makers had used a method of interpolation in their work and Mei explained their methods in his 1704 work Pinggliding sancha xiangshuo Ⓣ.
    • In 1710 he produced a text Fangyuan miji Ⓣ in which he gave his methods for finding the formula for the volume of a sphere.
    • List of References (12 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Mei_Wending.html .

  3. Li Shanlan biography
    • He was the greatest Chinese mathematician of the 19th century.
    • There are two slightly different versions of how Li first came upon mathematics.
    • One suggests that when he was eight years old he found a copy of the Nine Chapters on the Mathematical Art in the library of a private school.
    • It does seem certain then that he in love with mathematics through a study of the Nine Chapters.
    • In 1825, when he was fourteen years old, Li studied the first six books of Euclid's Elements which had been translated into Chinese in 1607 by Xu Guangqi and Matteo Ricci.
    • He continued his studies of the classics and of mathematics and several years later travelled to Hangzhou to sit examinations.
    • Although he failed the examinations the trip to Hangzhou was extremely valuable, for there he purchased two further mathematics texts.
    • One was the Ce yuan hai jing Ⓣ, originally written by Li Zhi in 1248, which was of fundamental importance in the development of Chinese algebra.
    • The other was Gougu geyuan ji Ⓣ which was a trigonometry text written by Dai Zhen, who is famous as an editor of the Nine Chapters on the Mathematical Art, and had been appointed as an editor for compiling an encyclopaedia of knowledge by the Emperor Qianlong in 1773.
    • The "tian yuan" or "coefficient array method" or "method of the celestial unknown" of setting up equations, which Li learnt from Li Zhi's famous text, had a huge influence on him and he began to push these algebraic techniques forward solving a whole variety of new problems.
    • Although he had been largely self educated in mathematics, Li now made contact with others having the same interests.
    • He wanted to make a career for himself in mathematics but at this time in China the subject was not considered of sufficient importance that one could make a living as a professional mathematician.
    • This is the route Li was forced to take and he was fortunate to be employed in 1845 by a family who had a deep love of education.
    • He now moved in circles which included others with similar interests in mathematics to his own and exchanging ideas proved fruitful as he pushed forward his research.
    • It had been a time of wars in China, however, for in 1841 British warships had arrived in Hong Kong and a war broke out.
    • The reasons were complex and relate to the huge profits that the British were getting through the opium trade, while the Chinese government was trying to stop Chinese use of the drug.
    • The British claimed that they were supporting the Chinese people against their leaders (always a good policy!).
    • Certainly the Chinese government had not been popular and after this its prestige declined further.
    • He would remain there for eight years working with these missionaries translating Western science works into classical Chinese.
    • With Alexander Wylie, Li translated Elements of Analytical Geometry and of the Differential and Integral Calculus which had been written by Elias Loomis and published in New York in 1851.
    • Their Chinese translation was published in 1859 and became the first book to introduce Newton's calculus into China.
    • Although this is the first time that the principles of algebraic geometry have been placed before the Chinese, in their own idiom, yet there is little doubt that this branch of the science will commend itself to native mathematicians, in consideration of its obvious utility ..
    • A spirit of enquiry is abroad among the Chinese, and there is a class of students, by no means small in number, who receive with avidity instruction on scientific matters from the West.
    • Li also worked with the protestant missionary Joseph Edkins on a translation of W Whewell's An elementary treatise on mechanics.
    • It was the first introduction of Newtonian mechanics into China.
    • Another major translation which Li undertook with Alexander Wylie was the translation of the final nine books of Euclid's Elements.
    • Other mathematics books translated by Li include De Morgan's Elements of algebra but Li did not just translate mathematics book, however, for with Alexander Williamson and Joseph Edkins he translated John Lindley's Botany.
    • He also translated John Herschel's Outlines of astronomy.
    • For eight years Li worked with the missionaries of the London Missionary Society in Shanghai.
    • During this period the Taiping Rebellion had at first had great success with the capture of Nanking in 1853.
    • Li left Shanghai, probably before the attack on the city, and moved to join the staff of Xu Youren who was Governor of Jiangsu province.
    • They financially supported the publication of Li's complete mathematical works which were published in Nanking in 1867.
    • Li was recommended to the newly established T'ung-wen-kuan (College of combined learning) in 1864.
    • The T'ung-wen-kuan had opened in Peking in the previous year as an initiative of the Chinese government who aimed to train Chinese students in foreign languages so they might gain Western knowledge more easily.
    • However Li did not take up the appointment until 1866 for he did not wish to be part of a translating school.
    • However the government realised the importance of mathematics in the development of the country and in 1868 the T'ung wen-kuan was upgraded to a college and a department of mathematics and astronomy was added.
    • After this, in July 1869, Li was happy to accept the position of Professor of Mathematics.
    • There, Li worked with William A P Martin (1827-1916), who served as president of the college from the time of its upgrading in 1869 until 1882, teaching mathematics and preparing translations of scientific works.
    • There was an eight year degree course which contained the study of foreign languages and Western sciences.
    • There was also a shorter five year degree course which combined traditional Chinese mathematics with a study of European sciences.
    • For example the first of the five years was spent on a study of the Nine Chapters on the Mathematical Art.
    • Although Li Shanlan is important as a translator of Western science texts, it is not in this capacity that he is most famous.
    • These were remarkable in that although he became familiar with Western mathematics he actually based his research on ancient Chinese mathematics.
    • He produced his own versions of logarithms, infinite series, and combinatorics but he did not follow the style of western mathematics but made his research naturally develop out of the foundations of Chinese mathematics.
    • One has to praise him very highly for this approach for his neither threw away the heritage of Chinese mathematics nor did he live in the past by ignoring the progress which had been made in the West.
    • Li wrote Duoji bilei Ⓣ (published in 1867 as part of his collected works) where, in Chapter 4, he gave fascinating formulae relating binomial coefficients, Stirling numbers, Eulerian numbers and many others.
    • The summation of series constitutes a branch of Chinese mathematics called Short Width [Chapter 4 of the Nine Chapters on the Mathematical Art.
    • The works of the great astronomer Guo Shoujing concerning the inequalities of the solar and lunar motion, Wang Lai's iterated sums, Dong Fangli's cyclotomical computations, and lastly the summation of series which appear in the algebra and the differential calculus of the Westerners constitute the major part of this chapter.
    • The usefulness of the mathematical techniques contained is indeed multifarious.
    • Zhu Shijie from the Yuan dynasty is the only one who has made use of the prescriptions relating to summation of series.
    • But his intention was only to expound the algebra and for that reason he presents the summation of series neither precisely nor methodically.
    • Both [5], [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','2] and [',' J-C Martzloff, Histoire des mathematiques chinoises (Paris, 1987).
    • ','3] discuss some of the wonderful summation formulae that Li discovered.
    • ','3] Li's method of writing the sum of the pth powers of the first n natural numbers as sums of binomial coefficients is given.
    • His use of a generalised version of Pascal's triangle is also explained.
    • Li worked out his own form of integration to compute volumes.
    • The following quotation from Fang yuan chanyou Ⓣ (published in 1867 as part of his collected works) shows how he thought:- .
    • A book one chi thick is made up of sheets of paper, whereas taffeta one zhang long is made up of threads of silk.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (6 books/articles) .
    • A Poster of Li Shanlan .
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Li_Shanlan.html .

  4. Loo-Keng Hua biography
    • Loo-Keng Hua was one of the leading mathematicians of his time and one of the two most eminent Chinese mathematicians of his generation, S S Chern being the other.
    • He spent most of his working life in China during some of that country's most turbulent political upheavals.
    • If many Chinese mathematicians nowadays are making distinguished contributions at the frontiers of science and if mathematics in China enjoys high popularity in public esteem, that is due in large measure to the leadership Hua gave his country, as scholar and teacher, for 50 years.
    • Hua was born in 1910 in Jintan in the southern Jiangsu Province of China.
    • The family was poor throughout Hua's formative years; in addition, he was a frail child afflicted by a succession of illnesses, culminating in typhoid fever that caused paralysis of his left leg; this impeded his movement quite severely for the rest of his life.
    • Hua's formal education was brief and, on the face of it, hardly a preparation for an academic career - the first degree he would receive was an honorary doctorate from the University of Nancy in France in 1980; nevertheless, it was of a quality that did help his intellectual development.
    • The Jintan Middle School that opened in 1922 just when he had completed elementary school had a well-qualified and demanding mathematics teacher who recognized Hua's talent and nurtured it.
    • In addition, Hua learned early on to make up for the lack of books, and later of scientific literature, by tackling problems directly from first principles, an attitude that he maintained enthusiastically throughout his life and encouraged his students in later years to adopt.
    • Next, Hua gained admission to the Chinese Vocational College in Shanghai, and there he distinguished himself by winning a national abacus competition; although tuition fees at the college were low, living costs proved too high for his means and Hua was forced to leave a term before graduating.
    • By the time Hua returned to Jintan he was already engaged in mathematics and his first publication Some Researches on the Theorem of Sturm, appeared in the December 1929 issue of the Shanghai periodical Science.
    • Hua's lucid analysis caught the eye of a discerning professor at Quing Hua University in Beijing, and in 1931 Hua was invited, despite his lack of formal qualification and not without some reservations on the part of several faculty members, to join the mathematics department there.
    • He began as a clerk in the library, and then moved to become an assistant in mathematics; by September 1932 he was an instructor and two years later came promotion to the rank of lecturer.
    • By that time he had published another dozen papers and in some of these one could begin to find intimations of his future interests; thanks to his natural talent and dedication, Hua was now, at the age of 24, a professional mathematician.
    • At this time Quing Hua University was the leading Chinese institution of higher education, and its faculty was in the forefront of the endeavour to bring the country's mathematics and science abreast of knowledge in the West, a formidable task after several hundred years of stagnation.
    • During 1935-36 Hadamard and Norbert Wiener visited the university; Hua eagerly attended the lectures of both and created a good impression.
    • Wiener visited England soon afterward and spoke of Hua to G H Hardy.
    • By now he had published widely on questions within the orbit of Waring's problem (also on other topics in diophantine analysis and function theory) and he was well prepared to take advantage of the stimulating environment of the Hardy-Littlewood school, then at the zenith of its fame.
    • Hua lived on a $1,250 per annum scholarship awarded by the Culture and Education Foundation of China; it is interesting to recall that this foundation derived its funds from reparations paid by China to the United States following wars waged in China by the United States and several other nations in the previous century.
    • The amount of the grant imposed on him a Spartan regime.
    • Hardy assured Hua that he could gain a PhD in two years with ease, but Hua could not afford the registration fee and declined; of course, he gave quite different reasons for his decision.
    • During the Cambridge period Hua became friendly with Harold Davenport and Hans Heilbronn, then two young research fellows of Trinity College - one a former student of Littlewood and the other Landau's last assistant in Gottingen - with whom he shared a deep interest in the Hardy-Littlewood approach to additive problems akin to Waring's.
    • They helped to polish the English in several of Hua's papers, which now flowed from his pen at a remarkable rate; more than 10 of his papers date from this time, and many of these appeared in due course in the publications of the London Mathematical Society.
    • His argument rested on the deployment of intricate algebraic identities and yielded rather poor admissible values of s(k).
    • In 1918 Hardy and Ramanujan returned to the case k = 2 in order to determine the number of representations of an integer as the sum of s squares by means of Fourier analysis, an approach inspired by their famous work on partitions, and they succeeded.
    • This encouraged Hardy and Littlewood in 1920 to apply a similar method for general k, and they devised the so-called circle method to tackle the general Hilbert-Waring theorem and a host of other additive problems, Goldbach's problem among them.
    • During the next 20 years the machinery of the circle method came to be regarded about as difficult as anything in the whole of mathematics; even today, after numerous refinements and much progress, the intricacies of the method remain formidable.
    • This is the background against which Hua set to work as a young man, and it is probably fair to say that it is for his contributions in this area that Hua's name will remain best remembered: notably for his seminal work on the estimation of trigonometric sums, singly or on average.
    • Hua might well have remained in England longer, but home was never far from his thoughts and the Japanese invasion of China in 1937 caused him much anxiety.
    • However, Quing Hua University was no longer in Beijing; with vast portions of China under Japanese occupation, it had migrated to Kunming, the capital of the southern province of Yunan, where it combined with several other institutions to form the temporary Associated University of the South West.
    • There Hua and his family remained through the World War II years, until 1945, in circumstances of poverty, physical privation, and intellectual isolation.
    • Despite these hardships Hua maintained the level of intensity of his Cambridge period and even exceeded it; by the end of 1945 he had more than 70 publications to his name.
    • During this time he studied Vinogradov's seminal method of estimating trigonometric sums and reformulated it, even in sharper form, in what is now known universally as Vinogradov's mean value theorem.
    • This famous result is central to improved versions of the Hilbert-Waring theorem, and has important applications to the study of the Riemann zeta function.
    • Hua wrote up this work in a booklet that was accepted for publication in Russia as early as 1940, but owing to the war, did not appear (in expanded form) until 1947 as a monograph of the Steklov Institute.
    • Hua spent three months in Russia in the spring of 1946 at Vinogradov's invitation.
    • Mathematical interaction apart, he was impressed by the organization of scientific activity there, and this experience influenced him when later he reached a position of authority in the new China.
    • His instinct for what was important and his marvellous command of technique make his papers on number theory even now virtually an index to the major activities in that subject during the first half of the twentieth century.
    • In the closing years of the Kunming period Hua turned his interests to algebra and to analysis, as much as anything for the benefit of his students in the first instance, and soon began to make original contributions in these subjects too.
    • Thus Hua became interested in matrix algebra and wrote several substantial papers on the geometry of matrices.
    • In September 1946, shortly after returning from Russia, Hua did depart for Princeton, bringing with him projects not only in matrix theory but also in functions of several complex variables and in group theory.
    • At this time civil war was raging in China and it was not easy to travel; therefore, the Chinese authorities assigned Hua the rank of general in his passport for the "convenience of travel." .
    • According to his biographer, Hua's "most significant and rewarding research work" during his stay in the United States was on the topic of skew fields, that is, on (non-commutative) division rings, of which the quaternions are a classic example.
    • There was much else, of course, to distinguish this last major creative period of his life.
    • Hua wrote several papers with H S Vandiver on the solution of equations in finite fields and with I Reiner on automorphisms of classical groups.
    • Much of his algebraic work later provided the basis for the monograph Classical Groups by Wan Zhe Xian and Hua (published by the Shanghai Scientific Press in Chinese in 1963).
    • On the personal side, in the spring of 1947 Hua underwent an operation at the Johns Hopkins University on his lame leg that much improved his gait thereafter, to his and his family's delight.
    • In the spring of 1948 Hua accepted appointment as a full professor at the University of Illinois in Urbana-Champaign.
    • There he directed the thesis of R Ayoub, later a professor at Pennsylvania State University; continued his work with I Reiner; and influenced the thinking of several young research workers, L Schoenfeld and J Mitchell among them.
    • His stay in Illinois was all too brief, exciting developments were taking place in China, and Hua watched them eagerly, wanting to be part of the new epoch.
    • He was then at the peak of his mathematical powers and, as he wrote to me many years later, the 1940s had been to him in retrospect the golden years of his life.
    • Back in China, Hua threw himself into educational reform and the organization of mathematical activity at the graduate level, in the schools, and among workers in the burgeoning industry.
    • In July 1952 the Mathematical Institute of the Academia Sinica came into being, with Hua as its first director.
    • The following year he was one of a 26-member delegation from the Academia Sinica to visit the Soviet Union in order to establish links with Russian science.
    • At this time Hua entertained doubts whether the Communist Party at home trusted him, and it came as an agreeable surprise to him to learn in Moscow that the Chinese government had agreed to a proposal by the Soviet government to award Hua a Stalin Prize.
    • Following Stalin's death the prize was discontinued, and Hua missed out; in view of later developments, he told me, he had a double reason to be satisfied! .
    • (The preface to the 1975 Chinese edition was excised by government order because Hua was out of favour during much of the Cultural Revolution); later this was published by Springer in English translation and is still in print.
    • Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains came out in 1958 and was translated into Russian in the same year, followed by an English translation by the American Mathematical Society in 1963.
    • In connection with the last of these, the study of the Monte Carlo method and the role of uniform distribution led them to invent an alternative deterministic method based on ideas from algebraic number theory.
    • Their theory was set out in Applications of Number Theory to Numerical Analysis, which was published much later, in 1978, and by Springer in English translation in 1981.
    • The newfound interest in applicable mathematics took him in the 1960s, accompanied by a team of assistants, all over China to show workers of all kinds how to apply their reasoning faculty to the solution of shop-floor and everyday problems.
    • Whether in ad hoc problem-solving sessions in factories or open-air teachings, he touched his audiences with the spirit of mathematics to such an extent that he became a national hero and even earned an unsolicited letter of commendation from Mao, this last a valuable protection in uncertain times.
    • Hua had a commanding presence, a genial personality, and a wonderful way of putting things simply, and the impact of his travels spread his fame and the popularity of mathematics across the land.
    • When much later he travelled abroad, wherever he stayed Chinese communities of all political persuasions flocked to meet him and do him honour; in 1984 when he organized a conference on functions of several complex variables in Hangzhou, colleagues from the West were astonished by the scale of the publicity accorded it by the Chinese media.
    • A pronouncement of Mao dated as early as June 26, 1965, sent a dire signal of things to come to the intellectuals:- .
    • Hua spent many of these years under virtual house arrest.
    • He attributed his survival to the personal protection of Chou En-lai.
    • Even so, he was exposed to harassing interrogations, some of his manuscripts (on mathematical economics) were confiscated and are now irretrievably lost, and attempts were made to extract from his associates and former students damaging allegations against him.
    • (In 1978 the Chinese ambassador to the United Kingdom described one such occasion to me; Chen Jing-run, then probably the best known Chinese mathematician of the next generation, was made to stand in a public place for several hours, surrounded by a mob, and exhorted to bear witness against Hua.
    • Chen, present at this conversation, chimed in to say that, actually, he had quite enjoyed the occasion, since no student could trouble him with silly questions and he had had time, uninterrupted, to think about mathematics!) It is surely no accident that the flow of Hua's publications came to an untimely end in 1965.
    • He continued to work, of course.
    • With the end of the Cultural Revolution in 1976 Hua entered upon the last period of his life.
    • Honour was restored to him at home, and he became a vice-president of the Academia Sinica, a member of the People's Congress and science advisor to his government.
    • In addition, Chinese Television (CCTV) produced a mini-series telling the story of Hua's life, which has been shown at least twice since then.
    • In 1980 he became a cultural ambassador of his country charged with re-establishing links with Western academics, and during the next five years he travelled extensively in Europe, the United States, and Japan.
    • In 1979 he was a visiting research fellow of the then Science Research Council of the United Kingdom at the University of Birmingham and during 1983-84 he was Sherman Fairchild Distinguished Scholar at the California Institute of Technology.
    • For much of this time he was tired and in poor health, but a characteristic zest for life and a quenchless curiosity never deserted him; to a packed audience in a seminar in Urbana in the spring of 1984 he spoke about mathematical economics.
    • In his last letter to me, dated 21 May 1985, he reported that unfortunately most of his time now was devoted to:- .
    • He died of a heart attack at the end of a lecture he gave in Tokyo on 12 June 1985.
    • Hua received honorary doctorates from the University of Nancy (1980), the Chinese University of Hong-Kong (1983), and the University of Illinois (1984).
    • He was elected a foreign associate of the National Academy of Sciences (1982) and a member of the Deutsche Akademie der Naturforscher Leopoldina (1983), Academy of the Third World (1983), and the Bavarian Academy of Sciences (1985).
    • Professor Wang Yuan has written a fine biography [',' Y Wang, Hua Loo-Keng : A biography (Singapore, 1999).','1] of Hua, and I am indebted to it for some of the information I have used.
    • Article by: Heini HalberstamClick on this link to see a list of the Glossary entries for this page .
    • List of References (15 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Hua.html .

  5. Mei Juecheng biography
    • Mei Juecheng was the grandson of Mei Wending.
    • He learnt mathematics at Baoding, a city near Beijing which was a provincial capital and also a centre of culture.
    • The Emperor Kangxi had come to power in 1661 when only seven years of age.
    • He was keen on both Chinese learning and the new European learning brought to China by the Jesuit missionaries.
    • Two key players in the development of mathematics in China around this time are Yunzhi, the Emperor Kangxi's third son, and Li Guangdi who was a minister and mathematical scholar who the Emperor Kangxi had appointed to teach his sons.
    • Mei Juecheng and Li Guangdi's son had both been taught mathematics at Baoding by Mei Wending and so Mei Juecheng's mathematical skills were known to the highest officials in the land.
    • The Emperor Kangxi himself had studied mathematics from 1689 to 1692 and realised that there was a lack of talented Chinese mathematicians at this time.
    • He asked Li Guangdi to find the best mathematics books and in 1703 Li gave the Emperor Kangxi a copy of Lixue yiwen (Inquiry on Mathematical Astronomy) written by Mei Wending in 1701.
    • From this time on Mei Juecheng played a major role in the compilation of mathematical and astronomical works.
    • This became an important project under the Emperor Kangxi who had been advised that both Chinese and European mathematics texts should be compiled into a major encyclopaedia.
    • Li Guangdi and the Emperor's son Yunzhi were both part of an editorial team comprising largely of men trained by either Li Guangdi or Mei Wending.
    • Of course there was much resistance to the new European learning brought by the Jesuits and the Chinese looked to be able to accept this material without making China feel inferior to Europe in learning.
    • The acceptance of European learning was eased by the hypothesis that it was all of Chinese origin.
    • Lately I served at the Imperial Court, receiving from His Majesty the Emperor Kangxi, a work on the 'Jie-fang-gen' [algebra], together with an Imperial Edict, saying that "the people from the Western Ocean name this method as 'A-er-re-ba-da' which can be translated into Chinese as 'Tong-lai-fa' (Method from the East)." Respectfully I read it and found its method extraordinary, capable of serving as a guide to mathematics.
    • However, realizing its method to be very similar to that of 'Tian-yuan-i', I reexamined the 'Shou-shi-li-cao' ..
    • During the Yuan dynasty, scholars, whether they were composing books on mathematics, or whether they were regulating mathematics, were all dealing with this subject of algebra.
    • In the following year the Emperor Kangxi established the Mengyangzhai (the Academy of Mathematics) and Mei Juecheng joined the team of people working on the compilation Yuzhi shuli jingyun (Treasury of Mathematics) which was published in 1723.
    • Unlike earlier efforts in compiling encyclopaedias of mathematical knowledge, no Jesuits were involved in this work.
    • Mei Juecheng and Chen Houyao (1648-1722) were the chief editors and they were assisted by He Guozong, Ming Antu and, in the early stages of the project, by Mei Wending.
    • reapportioned credit to Chinese scholars for many discoveries that earlier Jesuit-Chinese compendiums had credited to Europeans.
    • In particular, studying Western algebra enabled Mei Juecheng to decipher older Chinese mathematical treatises from the Song (920-1279) and Yuan (1206-1368) dynasties whose methods had been lost.
    • This led him to expound a theory of the Chinese origin of Western knowledge.
    • While now acknowledged as grossly overstated, his views helped to revive interest in traditional Chinese mathematics and remained highly influential for many decades.
    • to join the Academy of Mathematics.
    • In addition to those in the Academy of Mathematics, who studied mathematics, astronomy, and music, a large number of instrument makers were hired for the technical needs of the new academy.
    • A team of fifteen calculators verified the computations based on the theoretical notions, mathematical techniques and applications, and numerical tables in the first part of the Treasury.
    • Although Briggs's work had been introduced in 1653, the 'Treasury' explained the use of logarithms in greater detail, and it also included tables for sines, cosines, tangents, cotangents, secants, and cosecants for every ten seconds up to ninety degrees, as well as a list of prime numbers and a log table of integers from 1 to 100,000 calculated to 10 decimal places.
    • In fact the Treasury of Mathematics was part of a larger project, the Luli yuanyuan (Sources of Musical Harmonics and Mathematical Astronomy).
    • Also included in the Sources was the Compendium of Observational and Computational Astronomy.
    • Again Mei Juecheng was the leading academic in this project which, like the Treasury, followed the style of European works.
    • The first part was a general introduction to mathematical astronomy but then Mei Juecheng was able to make use of his grandfather Mei Wending's study of the motion of the moon to provide improved predictions of eclipses of the moon.
    • By accepting the best of European and Chinese astronomical data, the Sources surpassed both.
    • In fact the Jesuit missionary Pierre Jartoux (1669-1720) (known in China as Du Demei) introduced the infinite series for the sine into China in 1701 and it was known there by the name 'formula of Master Du'.
    • In fact Pearls recovered from the Red River was one of two chapters that Mei Juecheng appended to the works of Mei Wending that he was editing and republishing.
    • In this chapter, Jean-Claude Martzloff writes [','J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','4]:- .
    • [Mei Juecheng] reflects on various subjects (units of length in the classics, a formula of spherical trigonometry, comparison between the algebra of the 'jiegenfang' and the Chinese medieval algebra of the 'tianyuan', geometrical construction of the golden ratio, etc.).
    • nnnnnn(a) the length of the circumference given its diameter .
    • nnnnnn(b) the sine of an arc (zhengxian ) .
    • nnnnnn(c) the versed sine of an arc (zhengshi ) .
    • Important work of Mei Wending on mathematics published in this collection included: Bisuan Ⓣ, Chou suan Ⓣ, Du suan shi li Ⓣ , Shao guang shi yi Ⓣ, Fang cheng lun Ⓣ, Gougu ju yu Ⓣ, Jihe tong jie Ⓣ, Ping san jiao ju yao Ⓣ, Fang yuan mi ji Ⓣ, Jihe bu bian Ⓣ, Hu san jiao ju yao Ⓣ, Huan zhong shu chi Ⓣ and Qiandu celiang Ⓣ.
    • List of References (11 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Mei_Juecheng.html .

  6. Sun Zi biography
    • In the 17th century Sun Zi was identified with Sun Wu, a famous military expert of the sixth century BC who wrote Sun Zi's art of war.
    • The Ruan Yuan in his Chouren zhuan or Biographies of astronomers and mathematicians (1799) certainly realised that references in certain problems in the Sunzi suanjing meant that the identification with Sun Wu was incorrect.
    • During the third century Sun Zi, an author of considerable note, published his Sunzi suanjing.
    • Wang Ling [',' L Wang, The date of the Sunzi suanjing and the Chinese remainder theorem, in Proc.
    • History of Science, 1962 (Paris, 1964), 489-492.','10] seems to have the most convincing argument:- .
    • The Sunzi suanjing mentions the mein as an item of taxation, and the hu tiao system.
    • Of course this dating assumes that the text was written as a whole, while it seems more likely that it was compiled, like many of the texts, from older sources.
    • In that case Wang Ling's dating will only establish when part of the text was written, some possibly being earlier, while other parts probably have been written later.
    • Rather strangely it is the first of the three chapters which has a different form to the other two, so perhaps an elementary introduction was added at a later date.
    • What of the modern texts? Lam and Ang in [',' L Y Lam and T S Ang, Fleeting footsteps : Tracing the conception of arithmetic and algebra in ancient China (River Edge, NJ, 1992).','1] suggest it is a '3rd century AD treatise'; Martzloff in [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','2] (also [',' J-C Martzloff, Histoire des mathematiques chinoises (Paris, 1987).','3]) gives 'fifth century very approximately'; Bag and Shen [',' A K Bag and K S Shen, Kuttaka and qiuyishu, Indian J.
    • 19 (4) (1984), 397-405.','4] say 'About 400 AD a Chinese mathematician, Sun Zi, took up the problem ..
    • 13 (1960), 219-230.','5] says it was 'composed in the third or fourth century of our era'; Lam in [',' L Y Lam, A Chinese genesis : rewriting the history of our numeral system, Arch.
    • .'; while Shen [',' K S Shen, Historical development of the Chinese remainder theorem, Arch.
    • 38 (4) (1988), 285-305.','9] gives the very precise date for the Sunzi suanjing of 237 AD.
    • Leaving the question of the date let us look briefly at the content of the treatise, before finally trying to make some guesses about Sun Zi based purely on the text.
    • The Sunzi suanjing consists, as we have already noted, of three chapters.
    • The first chapter describes systems of measuring with considerable detail, and gives instructions on using counting rods to multiply, divide, and compute square roots.
    • It also gives two systems for designating high powers of ten.
    • Attention must be paid to the placing of the digits.
    • So the numbers to be multiplier are placed in the top and bottom of the three rows of the counting board and multiplications by single digits and additions take place in constructing the product in the middle row.
    • The second and third chapters consist of problems (28 problems and 36 problems respectively) concerning fractions, areas, volumes etc.
    • How many are there of each? .
    • One problem, however, is of special interest, this being Problem 26 in Chapter 3:- .
    • Suppose we have an unknown number of objects.
    • This, of course, is important for it is a problem which is solved using the Chinese remainder theorem.
    • It is the earliest known occurrence of this type of problem.
    • Multiply the number of units left over when counting in threes by 70, add to the product of the number of units left over when counting in fives by 21, and then add the product of the number of units left over when counting in sevens by 15.
    • If the answer is 106 or more then subtract multiples of 105.
    • Xu, in [',' X T Xu, Sun Zi suan jing ( Master Sun’s arithmetical manual) was the original source for the ’leap forward and regress method of place determination’ in the extraction of roots (Chinese), J.
    • (1) (1987), 22-27.','11], gives a detailed description of the algorithm used by Sun Zi for the extraction of roots and compares it with the method described in the Nine Chapters on the Mathematical Art.
    • Can we deduce anything of Sun Zi himself? Perhaps the most significant fact is that nothing is known.
    • How can this tell us anything? Well first notice that since the work of modern historians has placed the Sunzi suanjing much later than was thought in ancient times, we can now see that as nothing was known of Sun Zi within say 100 years of his death.
    • For example in the Standard History of the Sui dynasty the treatise is mentioned with no details of the author, and the same occurs in other similar works.
    • Unlike many Chinese mathematicians, Sun Zi cannot have been an important government official, nor to have been from a family of high social standing.
    • The best guess would be that he was a scholar, clearly interested in the social issues of the day, but playing a very minor role in society.
    • One problem suggests that he may have been a Buddhist for he mentions a Buddhist sutra in Problem 4 of Chapter 3.
    • List of References (11 books/articles) .
    • History Topics: Chinese numerals .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: Chinese problems .
    • History Topics: The Ten Mathematical Classics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Sun_Zi.html .

  7. Matteo Ricci biography
    • He then continued his studies in Rome, studying mathematics and astronomy under Clavius.
    • He arrived first in Portugal where he studied at the University of Coimbra for a while.
    • Then, in 1578, he sailed to the Portuguese city of Goa on the west coast of India.
    • Ricci arrived at Macau on the east coast of China in 1582.
    • He settled in Chao-ch'ing, Kwangtung Province and began his study of Chinese.
    • He also worked at acquiring understanding of Chinese culture.
    • While there Ricci produced the first edition of his map of the world Great Map of Ten Thousand Countries which is a remarkable achievement showing China's geographical position in the world.
    • In 1589 Ricci moved to Shao-chou and began to teach Chinese scholars the mathematical ideas that he had learnt from his teacher Clavius.
    • This is perhaps the first time that European mathematics and Chinese mathematics had interacted and it must be seen as an important event.
    • He went instead to Nanking where he lived from 1599, working on mathematics, astronomy and geography.
    • There was at that time a problem with the European's understanding of whether the country which Marco Polo had visited by an overland route, and called Cathay, was the same country as China which had been visited by sea.
    • Ricci's hypothesis was proved by another Jesuit by the name of De Goes, who set out from India in 1602, and although he died in 1607 before reaching Peking, he had by that time made contact by letter with Ricci and proved that Marco Polo's Cathay was China.
    • By the time he was living in Peking, Ricci's skill at Chinese was sufficient to allow him to publish several books in Chinese.
    • He wrote The Secure Treatise on God (1603), The Twenty-five Words (1605), The First Six Books of Euclid (1607), and The Ten Paradoxes (1608).
    • The First Six Books of Euclid was based on Clavius's Latin version of Euclid's Elements which Ricci had studied under Clavius's guidance while in Rome.
    • The Chinese reaction to Ricci's book, which showed them the logical construction in Euclid's Elements for the first time, is discussed in [',' J C Martzloff, De Matteo Ricci a l’histoire des mathematiques en Chine, Bull.
    • Certainly the style of Euclid was far from the style of Chinese mathematics and this mixing of mathematical cultures must have been a cultural shock to both sides.
    • Ricci of course had to dress in the style of a Chinese scholar and be known under a Chinese name, he used 'Li Matou', to become accepted by the Chinese.
    • However he became famous in China for more than his mathematical skills, becoming known for his extraordinary memory and for his knowledge of astronomy.
    • He even became known as a painter and a painting of a landscape around Peking has recently been attributed to him.
    • List of References (10 books/articles) .
    • A Poster of Matteo Ricci .
    • History Topics: Overview of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Ricci_Matteo.html .

  8. Zhu Shijie (about 1260-about 1320)
    • Zhu was, therefore, the last of these four great thirteenth century Chinese mathematicians but it would appear from his writings that he was unaware of the work of his three famous predecessors.
    • Kublai Khan united the whole of China in 1279 and the Yuan dynasty came to power.
    • The capital of the new united China became Dadu (today called Beijing or Peking) which Kublai Khan had built up as a walled city with splendid palaces and government offices.
    • The unification of north and south China would have a significant effect on Zhu's life, for it allowed him to travel throughout the whole of China, and also allowed certain mathematical expertise which was previously known only in northern China to spread through the south.
    • What little we know of Zhu is contained in the following quotation from the preface to the second, and most famous, of his texts.
    • The preface, written by Mo Ruo, tells us that (see [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','4]):- .
    • Zhu Shijie of Yan-shan became famous as a mathematician.
    • He travelled widely for more than twenty years and the number of those who came to be taught by him increased each day.
    • Yan-shan was near the new capital of united China at Dadu (today called Beijing or Peking).
    • We see from this quotation that, as we mentioned above, the stability brought by the Mongol conquest gave Zhu opportunities to travel of which he took full advantage.
    • We know that Zhu wrote two books, the Suanxue qimeng (Introduction to mathematical studies) published in 1299, and the Siyuan yujian (True reflections of the four unknowns) published in 1303.
    • These are remarkable works which led to George Sarton writing that Zhu was (see [',' H Peng-Yoke, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1] where this is quoted):- .
    • one of the greatest mathematicians of his race, of his time, and indeed of all times.
    • It was printed in 1433 in Korea and in 1658 in Japan but a Chinese version only became available again in the nineteen century when a 1660 Korean edition was translated back into Chinese.
    • Zhu's book is based, as so many Chinese mathematics books were, on the Nine Chapters on the Mathematical Art.
    • The book contains examples of computations with fractions and decimals giving results such as 1/16 = 0.0625 and 2/16 = 0.125.
    • Zhu also explains the rule of three, areas and volumes, and the rule of false double position.
    • Some of his discussions extend methods from the Nine Chapters.
    • Some of Zhu's text, however, presents ideas going far beyond Nine Chapters.
    • He treats polynomial algebra, and polynomial equations, by the "coefficient array method" or "method of the celestial unknown" which had been developed in northern China by the earlier thirteenth century Chinese mathematicians, but up till that time had not spread to southern China.
    • In particular, how much was Zhu aware of Yang Hui's work? .
    • Zhu's second book Siyuan yujian (True reflections of the four unknowns) marks the peak of Chinese mathematics and it was a long time before mathematics in China progressed beyond it.
    • It survived in China probably until the second half of the eighteenth century when it appears to have become lost.
    • When Ruan Yuan compiled the Chouren zhuan or Biographies of astronomers and mathematicians in 1799, he failed to find Zhu's text despite his expertise in tracking down old books.
    • However some years later Ruan Yuan found a copy of the Siyuan yujian in Zhejiang (or Chekiang) province when he was governor there.
    • Ruan's hand written copy of the Siyuan yujian was eventually printed.
    • The text of the Siyuan yujian which is available today is therefore not the one which was originally published by Zhu.
    • The problem arises since the version which Ruan Yuan discovered was [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','4]:- .
    • Of the other prefaces, one is written by Ruan Yuan and the others by later commentators.
    • There are various nineteenth century commentaries on the text all of which make it harder to identify Zhu's original work.
    • One of these is Pascal's triangle which gives the coefficients needed to expand sums of unknowns up to the eighth power.
    • the table of the ancient method of powers up to the eighth.
    • There are then four preliminary problems which Zhu uses to explain his methods of using polynomials to solve problems with 1, 2, 3, and 4 unknowns.
    • In fact Zhu uses an extension of the "coefficient array method" or "method of the celestial unknown" to polynomials with several unknowns.
    • that the procedure of the celestial unknown had received previously to him.
    • One of the most interesting aspects of this work is that Zhu, although still using the traditional Chinese approach of presenting mathematics through practical problems, does not in any sense make his examples realistic.
    • Let us illustrate by giving one of Zhu's problems.
    • The sum of the base and height of the triangle is 17 bu.
    • What is the sum of the base and hypotenuse? .
    • Here is another of Zhu's problems.
    • It is phrased in terms of a right angled triangle, but the conditions are so artificial that he is really simply giving a system of equations.
    • The sides of the triangle are x, y, z where z is the hypotenuse.
    • Given the relations 2yz = z2 + xz and 2x + 4y + 4z = x(y2 - z + x) between the sides of a right angled triangle x, y, z where z is the hypotenuse, find d = 2x + 2y.
    • The following problem in the Siyuan yujian is reduced by Zhu to a polynomial equation of degree 5 (see [',' J Hoe, Zhu Shijie and his Jade mirror of the four unknowns, in First Australian Conference on the History of Mathematics (Clayton, 1980) (Clayton, 1981), 103-134.','7] for a detailed solution as given by Zhu):- .
    • Let d be the diameter of the circle inscribed in a right triangle (Zhu uses the relation d = x + y - z where x, y, z are as defined below).
    • Let x, y be the lengths of the two legs and z the length of the hypotenuse of the triangle.
    • The Siyuan yujian also contains a transformation method for the numerical solution of equations which is applied to equations up to degree 14.
    • Similarly he gave the sum of .
    • If the cube law is applied to the rate of recruiting soldiers and it is found that on the first day 3 cubed are recruited, 4 cubed on the second day, and on each succeeding day the cube of a number one greater than the previous day are recruited, how many soldiers in total will have been recruited after 15 days? How many after n days? .
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (14 books/articles) .
    • History Topics: A history of Zero .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: Chinese problems .
    • Index of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Zhu_Shijie.html .

  9. Giovanni Vacca biography
    • Vacca entered the University of Genoa to study mathematics.
    • However in 1894 Crispi dissolved the Party and the leaders of the Party, including Vacca were banished from Genoa.
    • In 1899 Vacca went to Hannover to study the unpublished manuscripts of Leibniz.
    • The following year Vacca attended the First International Congress of Philosophy which was held in Paris in 1900.
    • At the Congress Vacca met Couturat and, the following year, Couturat wrote in the preface of La Logique de Leibniz Ⓣ :- .
    • Our work on the logic of Leibniz was almost completed (at least we thought so) when we had the pleasure, at the International Congress of Philosophy (August, 1900), of making the acquaintance of Mr Giovanni Vacca, at that time mathematical assistant at the University of Turin, who had examined, the year before, the manuscripts of Leibniz preserved in Hannover, and had extracted from them several formulae of logic inserted in the "Formulaire de Mathematiques" of Mr Peano.
    • It was he who revealed to us the importance of the unpublished works of Leibniz, and inspired us with the desire to consult them in turn.
    • In 1903 Vacca published a collection of short works by Leibniz and some of his papers which had not been previously published.
    • Despite the dissolution of the Italian Socialist Party in 1894, it had been revived in the late 1890s and won 32 parliamentary seats in 1900.
    • Vacca, on his return to Genoa in 1902, worked for the Party again becoming a member of the Socialist Council and also a member of the national party administration.
    • However Vacca continued his mathematical work and gave a course at the University of Genoa on mathematical logic.
    • However Vacca had by this time yet a third interest in addition to mathematics and politics.
    • He had become interested in Chinese as early as 1898 when there had been a Chinese exhibition in Turin and he had some lessons in Chinese from two missionaries who had returned from China.
    • In 1905, this interest in Chinese became the road that Vacca decided to follow.
    • He went to Florence to study the Chinese language at the university.
    • They shared mathematical interests and certainly Vacca continued his mathematical research and interest in the history of mathematics.
    • In fact Vacca would be one of the editors of Vailati's collected works, published in 1911, two years after Vailati's death.
    • Vacca spent 1907-08 in western China, spending a year in the city of Cheng-tu.
    • After his return to Florence he was awarded his doctorate for Chinese studies in 1910 and, the following year, he was appointed to a post teaching Chinese literature at the University of Rome.
    • In 1922 Vacca succeeded his old professor of History and Geography of East Asia to the chair at Florence.
    • Vacca taught Chinese language and literature there until he retired in 1947.
    • Despite his change of topic in mid career, Vacca continued his Chinese and mathematical studies in parallel.
    • For example in 1928 Peano presented a paper by Vacca on Fermat's method of descent to the Academy of Sciences of Turin.
    • Throughout his career he published around 130 papers, 47 relating to his Chinese interests, 38 on mathematical research and 45 on the history of mathematics.
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Vacca.html .

  10. Qin Jiushao (1202-1261)
    • Qin Jiushao, also known as Ch'in Chiu-Shao, was born at the time of the Nan (Southern) Sung dynasty.
    • In around 1219, when Qin was about seventeen years old, his father was working as a prefect of Bazhou.
    • Qin's father moved to Hang-chou, the capital of the Nan Sung, in around 1224 and Qin went with his father.
    • He wrote in the preface to his famous work Shushu Jiuzhang (see for example [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','3]):- .
    • In my youth I was living in the capital, so that I was able to study in the Board of Astronomy; subsequently, I was instructed in mathematics by a recluse scholar.
    • We know that Qin was a rebellious youth, famous for his many love affairs, who disliked authority [',' H Peng-Yoke, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • During a banquet given by his father, a commotion was created when a stone suddenly landed among the guests; investigations disclosed that the missile had come from the direction of Qin, who was showing a young girl how to use a bow as a sling to hurl projectiles.
    • Sadly, we do not know which recluse scholar taught Qin mathematics, but we do know that he studied the Nine Chapters on the Mathematical Art.
    • By 1233 Qin was himself the sheriff of a subprefecture in Szechwan province and at this time he was instructed in writing poetry by an official from Chengdu, in central Szechwan province.
    • It is worth noting at this point that as well as being a genius in mathematics and accomplished in poetry, Qin was expert at fencing, archery, riding, music and architecture.
    • He was described by a contemporary in a letter to the Emperor as [',' H Peng-Yoke, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • He wrote in the preface of Shushu Jiuzhang (see for example [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','3]):- .
    • At the time of the troubles with the barbarians, I spent several years on the remote frontier, without care for my safety among the arrows and stone missiles, I endured danger and unhappiness for ten years.
    • Then he was appointed governor of Hui-chou (now She-hsien) in Anhwei province but here he undertook illegal dealings in salt which made him rich.
    • In the middle of 1244 he was posted as a senior administrator to Nanking.
    • During his period of mourning in Hui-chou, Qin wrote his famous mathematical treatise Shushu Jiuzhang (Mathematical Treatise in Nine Sections) which appeared in 1247.
    • This is a remarkable work which led to George Sarton writing that Qin was (see [',' H Peng-Yoke, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1] or [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','3] where this is quoted):- .
    • one of the greatest mathematicians of his race, of his time, and indeed of all times.
    • Before we look at the contents of the Shushu Jiuzhang we continue our description of Qin's life.
    • Appointed governor of Qiongzhou in Hainan in 1259 he was dismissed for corruption and exploitation after a hundred days in office and returned home having again acquired immense wealth illegally.
    • One might expect that by this time Qin would be unemployable, given his record of criminal dealings, but he next managed to gain an appointment as an assistant in the district of Yin (near Ningpo) in Zhekiang where his friend Wu Qian had been appointed as a naval officer.
    • Before looking at his mathematical achievements, let us recount further stories of his character.
    • It is recorded that Qin cheated his friend Wu Qian so that he became the owner of some of his land, and also that Qin punished a female member of his household by confining her without food.
    • Chapter 1 is on indeterminate analysis; it contains remarkable work on the Chinese remainder theorem which occurs right at the beginning of the text.
    • For example one problem asks for the height that rainwater would collect on level ground given that it reaches a certain height h in a vessel with a circular top of diameter a and circular base of diameter b where a > b.
    • Chapter 3 is called Boundaries of fields and looks at surveying.
    • There is a remarkable formula given in this Chapter which expresses the area of a figure as the root of an equation of degree 4.
    • The novelty here is that the coefficients are not numbers but are functions of lengths in the figure which are left as unspecified.
    • Another novelty in this chapter is a formula for the area of a triangle given in terms of its sides, essentially Heron's formula.
    • Again equations of high degree appear, one problem involving the solution of the equation of degree 10.
    • Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points.
    • A tree lies three li north of the northern gate.
    • Find the circumference and the diameter of the city wall.
    • Qin obtains the equation (really an equation of degree 5 in x2, where x2 is the diameter of the city):- .
    • [Solution: x = 3, so diameter of city is x2= 9 li] .
    • Throughout the text, in addition to the tenth degree equation above, Qin also reduces the solution of certain problems to a cubic or quartic equation which he solves by the standard Chinese method (namely that which today is called the Ruffini-Horner method).
    • One further remarkable feature of the text is that Qin uses 0 for zero, so not only does he use a symbol for zero, but that symbol is a little circle.
    • He writes about previous uses of zero:- .
    • As we have mentioned, the most remarkable method in the text is the method for solving simultaneous integer congruences, the Chinese Remainder Theorem.
    • Qin considers problems of the type .
    • Although there is no evidence of progress on such problems in China since the work of Sun Zi which was 800 years earlier, Qin now shows how to handle the case where the mk are not pairwise coprime.
    • He then solves each congruence and finally reassembles the answers to give the solution to the system of simultaneous congruences.
    • Shen discusses Qin's method of solution fully in [',' K S Shen, Number-theoretic propositions in the Da Yan problems of the Shu shu jiu zhang (Chinese), J.
    • 13 (4) (1986), 421-434.','15]; see also [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','3] and [',' J-C Martzloff, Histoire des mathematiques chinoises (Paris, 1987).
    • Libbrecht writes [',' U Libbrecht, Chinese mathematics in the thirteenth century : the Shu-shu chiu-chang of Ch’in Chiu-shao (Cambridge, Mass., 1973).','2]:- .
    • We should not underestimate [Qin's] revolutionary advance, because from [Sun Zi's] single remainder problem, we come at once to the general procedure for solving the remainder problem, even more advanced than Gauss's method, and there is not the slightest indication of gradual evolution.
    • This is such a brilliant piece of work that we are left with asking how Qin could have achieved it.
    • We certainly know that Qin was a rogue who was happy to steal, so could he have stolen his mathematics? Of course if he stole it then who did he steal it from? This does not seem likely, for it raises more questions than it solves.
    • Perhaps he learnt of the method through Indian approaches to such problems.
    • Although Qin's use of the symbol 0 suggests possible Indian knowledge, the Indian approach to such congruence problems is sufficiently different to make this highly unlikely.
    • One is left with no conclusion other than accepting that Qin was one of the great mathematical geniuses of all time.
    • He says that he learn it from the calendar experts when he was studying at the Board of Astronomy in Hang-chou.
    • There must be something in this for, without a doubt, calculating calendars was an important motivation for studying the theory of first-order congruences.
    • It would appear though that Qin must have taken these ideas much further and be showing a modesty in his mathematical work which was certainly lacking in other aspects of his life.
    • How impressive is this work? Well suffice to say that Euler failed to provide a satisfactory solution to these problems and it was left to Gauss, Lebesgue and Stieltjes to rediscovered this method of solving systems of congruences.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (16 books/articles) .
    • Index of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Qin_Jiushao.html .

  11. Shen Kua (1031-1095)
    • Shen Kua's mother came from the Soochow region of China and played an extremely important role in her son's upbringing since all his early education came from his mother.
    • The type of work that officials such as Shen Chou had to undertake was very varied, for he was involved in financial matters, in technical matters such as looking after the waterways and canals and overseeing major building projects, and he would have responsibilities for agriculture in the district to which he was posted.
    • The young boy Shen Kua learnt much from observing his father at work, and having an extremely sharp mind, he took full advantage of the opportunities presented to him.
    • As the son of an official, Shen Kua himself could become an official after the death of his father without taking the formal examinations which others were required to sit.
    • He was appointed to his first minor official post in 1054 and this was followed by a number of other postings.
    • The emperor at the time of Shen Kua's birth was Jen-tsung.
    • However problems were developing in the ranks of his administration.
    • Wang Anshi was one of this new breed of administrator, who progressed through ability rather than through patronage, and followed Confucian ideals.
    • A number of crises, some internal rebellions, some attacks from invaders crossing borders, and falling prosperity, all meant that reforms which the new bureaucrats wanted began to be implemented.
    • By the time that Shen Kua started to progress through the ranks of the administration, reformers had returned now having learnt political skills.
    • He was involved in a number of schemes to control water, always a major problem in China at this time, which turned out to be very successful.
    • Building drains and embankments, he reclaimed large areas of land which then became productive farmland.
    • While subprefect of Ning-kuo in Anhwei province in 1061, Shen undertook a cartographic survey and then undertook another major land reclamation programme.
    • Again it was very successful and after Shen passed national examinations in 1063 and moved to Yang-chou, the governor realised that here was someone of outstanding ability and recommended him for an appointment at court in the capital Kaifeng in Honan province.
    • He now launched a major reform program known as the "New Policies" and Shen was identified as one of eighteen members of Wang's strong reforming group within the court.
    • While Shen had worked as an official in the provinces he had undertaken a large number of highly successful projects but this had not fully occupied him and he had spent his spare time studying astronomy, calendar science, and the mathematics behind these.
    • In 1072 Shen was made Director of the Astronomy Bureau.
    • It was customary for a new calendar to be set up on the succession of a new emperor.
    • After a change of ruler the new Chinese emperor would seek a new official calendar thus establishing a new rule with new celestial influences.
    • He set up a programme which would measure the positions of the moon and planets, plotting exact coordinates three times a night for five years.
    • However, many of the staff at the Astronomy Bureau were incompetent and Shen had to dismiss six who he found were falsifying data.
    • The inefficiency of the staff made the calendar less accurate than it should have been but it came into use in 1075 and lasted about 20 years before being replaced.
    • Shen now undertook a number of projects to support Wang's reform agenda.
    • He created three dimensional maps of areas he visited made from wooden plates on which sawdust mixed with glue and melted wax were used.
    • However Shen was so successful in the tasks he was given that he continued to be asked to undertake pojects of major national importance.
    • In 1075 he undertook the task of revising the defence strategy of the country, and he undertook very successful negotiations with the Khitan tribes in the north who were threatening to invade.
    • Using his knowledge of history to refute the claims of the Khitan tribes to Chinese territory, he brought peace to the area which lasted for a number of years.
    • His writings on the theory of supply and demand, made while holding this office, are quite remarkable.
    • He wrote on methods of forecasting prices, the currency supply, price controls, market intervention and other topics which were not studied again in this depth until modern times.
    • Having reached a position of great power in the government, pushing through effective reforms with great success, he was accused of dishonest practice in the autumn of 1077.
    • These charges were completely false, but brought by political opponents of Wang's reform programme.
    • He had suddenly become exposed since the politically skilful Wang had become frustrated and had retired in the autumn of 1076.
    • There, in 1081, he organised an advance against the Tanguts winning major victories and extending the control of the Sung dynasty over new regions.
    • Although he had been in no way responsible for the defeat he was blamed by Wang's successor and was relieved of his command and banished.
    • He was free to undertake scientific work, however, and he worked on a project he had begun at the Emperor's request in 1076, namely to produce maps of all Chinese territory.
    • He produced 23 maps, all drawn to the scale of 1:900,000.
    • Shen knew that maps tended not to survive for long, whereas books had a better chance of long term survival.
    • He therefore encoded his maps in terms of directional coordinates and distances, writing [',' N Sivin, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • Maps are of great military significance, as well as scientific interest, so his project found favour with the government.
    • After six years of being subjected to what was close to house arrest, Shen was allowed to live in a place of his own choosing.
    • He already had purchased a property when he had been a leading figure ten years earlier but he had never seen his property on the outskirts of Jiangsu (Zhenjiang).
    • In 1086 he was allowed to visit it and found a place of great beauty which he had dreamt of for many years.
    • He spent the last seven years of his life in isolation at Dream Brook.
    • He wrote [',' N Sivin, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • It is a remarkable scientific document which contains his work on mathematics, music, astronomy, calendars, cartography, geology, optics and medicine.
    • He recognised fossils of certain sea creatures in rock far from the sea and understood what this meant.
    • Observing seashells in strata of the T'ai-hang Shan mountains, he deduced that these mountains, though now far from the sea, must once have been a sea shore [',' N Sivin, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • this was once a seashore, although the sea is now hundreds of miles east.
    • What we call our continent is an inundation of silt ..
    • Let us look briefly at some of his achievements.
    • First let us see what he achieved in mathematics.
    • The first thing to note is that Shen took a much more abstract view of mathematics than the Chinese generally did.
    • Most Chinese mathematics was motivated by practical problems, but Shen was happy to look at mathematical problems for their own sake.
    • He considered the problem of the go board which had 19 × 19 lines giving 361 intersections.
    • On each intersection there was one of three possibilities: no counter, a white counter, or a black counter.
    • This, of course, is an exceedingly large number and Shen's problem was to express such a number within the Chinese number system.
    • Here wan = 104 and he is writing a product of, presumably, 43 wan.
    • We say presumably since copyists made errors in reproducing how many wan there were, possibly not understanding the significance of what Shen was doing.
    • The second example that we shall give to illustrate Shen's mathematics is that of a pile of unit sized items.
    • If the top of the pile is a rectangle a × b and the base of the pile is a rectangle c × d, and there are h layers of items in the pile then he showed that the number of items in the pile is ((2b + d)a + (2d + b)c + (c - a))h/6.
    • Shen does not specify what the items are, but leaves the formula general, he takes them to be of unit volume and also neglects (deliberately) volumes left between the items.
    • This requires at least some understanding of spherical geometry and trigonometry.
    • In fact Shen demonstrated a remarkable ability to view spatial arrangements and he gave an approximate formula for the length of a circular arc in terms of the cord subtending the arc.
    • He attempted to explain why the motion of the planets across the sky was as observed with periods of retrograde motion.
    • Other Chinese astronomers had only looked at the positions of the planets without ever trying to explain what was observed.
    • He is also said to have constructed a armillary sphere, a water clock, and a bronze gnomon, a pointer whose shadow gives the time of mid-day.
    • He also wrote a medical work Good prescriptions and Record of longings forgotten which was observations on living in the mountains but it has not survived.
    • His treatise on calendar reform is also lost as is about half of a posthumous compilation of his unpublished writings.
    • Sivin writes in [',' N Sivin, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • Breadth of interest alone does not account for Shen's importance for the study of the Chinese scientific intellect.
    • Above all, one is aware in Shen, as in other great scientific figures, of a special directness.
    • A member of a society in which the weight of the past always lay heavily on work of the mind, he nevertheless often cut past deeply ingrained structures and assumptions.
    • List of References (12 books/articles) .
    • A Poster of Shen Kua .
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Shen_Kua.html .

  12. Jia Xian biography
    • It is recorded that he was a pupil of Chu Yan who was a famous calendarist, astronomer and mathematician.
    • Other evidence would suggest that Chu Yan taught Jia Xian fairly near the beginning of his career.
    • According to Qian [',' B Qian, History of Chinese mathematics (Chinese) (Peking, 1981).','3], Jia Xian was a Palace Eunuch of the Left Duty Group.
    • The Emperor of China would employ eunuchs, castrated men, as guards and servants in his Palace.
    • Although the original role was that of guarding the women's quarters, these men achieved real power and influence.
    • Jia Xian is known to have written two mathematics books: Huangdi Jiuzhang Suanjing Xicao (The Yellow Emperor's detailed solutions to the Nine Chapters on the Mathematical Art), and Suanfa Xuegu Ji (A collection of ancient mathematical rules).
    • Both are lost and we know nothing of the second of the two books other than its title.
    • This is because Yang Hui wrote Xiangjie Jiuzhang Suanfa (A detailed analysis of the mathematical rules in the Nine Chapters) in 1261 with the intention of explaining, and making better known, the work of Jia Xian.
    • A copy of Yang Hui's text has survived and he explicitly states his reasons for writing the work in the preface.
    • What does Yang Hui tell us of Jia Xian's mathematical contribution? The first is an understanding of Pascal's triangle.
    • Here Jia Xian is aware of the expansion of (a + b)n and gives a table of the resulting binomial coefficients in the form of Pascal's triangle.
    • It is clear from Yang Hui's description that Jia Xian understood the method of generating the triangle, namely adding the numbers in the two positions above in order to find the number in the position below.
    • He generalised a method of finding square roots and cube roots to finding nth roots, for n > 3, and then extended the method to solving polynomial equations of arbitrary degree.
    • A fascinating historical account of methods of root extraction used by Chinese and Arabic scholars is given in [',' K Chemla, Similarities between Chinese and Arabic mathematical writings I : Root extraction, Arabic Sci.
    • Chemla defines precisely what constitutes the Ruffini-Horner method so that at each step of the algorithm precisely the same procedure, using multiplication and subtraction, is carried out until the root is obtained.
    • After examining earlier Chinese methods given in the Nine Chapters on the Mathematical Art and those by Zhang Qiujian in the fifth century, she concludes that Jia Xian was the first to use the Ruffini-Horner method.
    • An examination of root extraction methods by Arabic authors leads to the conclusion that al-Samawal in the twelfth century was the first to use the Ruffini-Horner method.
    • It is also shown in [',' K Chemla, Similarities between Chinese and Arabic mathematical writings I : Root extraction, Arabic Sci.
    • 4 (2) (1994), 207-266.','4] that both Jia Xian's method and al-Samawal's method end up with the same form for the approximation of nth roots.
    • If a is the integral portion of the nth root of A, then the approximation is given by .
    • Both Jia Xian and al-Samawal use binomial coefficients, computed with a form of Pascal's triangle, to calculate the denominator of the expression.
    • The intriguing question of whether al-Samawal discovered the method independently, or whether there was transmission of the Chinese method of Jia Xian into Islamic/Arabic mathematics is left unresolved.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (7 books/articles) .
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Jia_Xian.html .

  13. Li Chunfeng biography
    • Education became important and mathematics was taught at the Imperial Academy.
    • It is a positive religion, emphasising the joyful and carefree sides of the Chinese character.
    • It continued the educational development which had already begun, and formalised the teaching of mathematics at the Imperial Academy.
    • Changing the calendar was seen as one of the duties of the office, establishing the emperor's heavenly link on earth.
    • After a change of ruler, and even more significantly after a change of dynasty, the new Chinese emperor would seek a new official calendar thus establishing a new rule with new celestial influences.
    • Although the Chinese calendar had only been in operation for a few years, already predictions of eclipses were getting out of step.
    • Li was promoted to become the deputy director of the Imperial Astronomical Bureau in about 641.
    • We shall describe below some of the contributions Li made to mathematics through his work in astronomy and calendar reform.
    • He assisted in compiling the official histories of the Jin and Sui dynasties.
    • In the Jinshu (History of the Jin Dynasty) and Suishu (History of the Sui Dynasty) Li wrote the chapters on the developments in Chinese astronomy, astrology, metrology, and the mathematics of music through the relevant periods.
    • In 648 Li was appointed as director of the Imperial Astronomical Bureau.
    • He also became editor-in-chief for a collection of mathematical treatises now called The Ten Classics a name given to them in 1084.
    • The History of the T'ang records (see [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','5]):- .
    • As a consequence Li Chunfeng together with Liang Shu, an expert in mathematics from the ministry of education, and Wang Zhenru, a teacher from the national university and others were ordered by imperial decree to annotate the ten mathematical texts such as the Wucao suanjing or the Sunzi suanjing.
    • This calendar gave better results than the calendar it replaced in predicting the positions of the planets.
    • There were long months of 30 days, short months of 29 days and, in addition, intercalary months.
    • An intercalary month was added every three years to allow for the fact that a solar year has 365.2422 days while 12 lunar months of 29.5306 days contains 354.3672 days.
    • The quotation we gave above suggests that Li was given the task of correcting and annotating mathematical texts.
    • Do we know of errors that he corrected? Yes, we do.
    • To give an example, he corrected a mistake in Liu Hui's comment on the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 in Problem 11 of Chapter 4 of the Nine Chapters on the Mathematical Art, giving the correct answer 27720.
    • In all he made 110 comments in the text of this work providing useful historical information as well as corrections.
    • For example he gave Zu Geng's derivation of the volume of a sphere, then improved it by using π = 22/7 instead of π = 3.
    • He begins each comment with the words (see for example [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','5]):- .
    • Li did make some original contributions to mathematics although he is not particularly famous in this respect.
    • His contributions arose through his astronomical work, in particular in computing the angular speed of the sun's apparent motion.
    • He developed a method of finite differences in his computations which he used in his work on the Linde calendar.
    • We mentioned near the beginning of this article the influence of Taoist beliefs on Li.
    • In the area of astrology he wrote Yisizhan in 645 which is a classic work of astrology which was important in the development of Chinese culture.
    • This is a collection of sixty attempts to predict the future using numerology.
    • thousands of years need endless telling, so we'd better stop and enjoy a conformable massage.
    • List of References (8 books/articles) .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: The Ten Mathematical Classics .
    • History Topics: Nine Chapters on the Mathematical Art .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Li_Chunfeng.html .

  14. Wen-Tsun Wu biography
    • He studied mathematics at Chiao-Tung University, Shanghai during a particularly difficult period since war broke out between China and Japan in 1937.
    • In August 1937 the Japanese began an attack on Shanghai which was strongly defended by the Chinese.
    • For over three months Shanghai was the centre of a fierce battle during which time it faced air and sea attacks from the Japanese who made amphibious landings on the beaches of the Jiangsu coast.
    • The University was temporarily relocated to the French Concession in Shanghai and Wu continued his studies of mathematics, graduating with his first degree in 1940.
    • He took a position as a mathematics teacher in a school.
    • The war continued but from 1941 it became part of the wider conflict of World War II.
    • In 1946 he met Shiing-Shen Chern who was at that time working on setting up an Institute of Mathematics as part of the Academia Sinica.
    • Wu writes [',' Autobiography of Professor Wu Wen-Tsun.','1]:- .
    • This meeting with Chern was decisive for the future of my career in mathematics.
    • Wu was admitted as a research student at the Institute of Mathematics and, also in 1946, took the examinations to compete for a scholarship to study abroad.
    • He was successful and in 1947 he went to France as part of the Sino-France Exchange Program.
    • He went to the University of Strasbourg where he undertook research with Charles Ehresmann as his advisor.
    • The following year saw a wealth of papers from Wu: On the product of sphere bundles and the duality theorem modulo two; Sur l'existence d'un champ d'elements de contact ou d'une structure complexe sur une sphere Ⓣ; Sur les classes caracteristiques d'un espace fibre en spheres Ⓣ; Sur le second obstacle d'un champ d'elements de contact dans une structure fibree spherique Ⓣ; and Sur la structure presque complexe d'une variete differentiable reelle de dimension 4 Ⓣ.
    • Wu was awarded his doctorate in 1949 for his thesis Sur les classes caracteristiques des structures fibrees spheriques Ⓣ in which he made a detailed study of characteristic classes via Grassmannian varieties.
    • Following the award of his doctorate, Wu went to Paris where he studied with Henri Cartan.
    • Thom, a student of Cartan's, had held a CNRS research post at Strasbourg while Wu was studying there and they had got to know each other at this time and exchanged mathematical ideas, beginning a good collaboration.
    • While working in Paris during the early months of 1950, Thom discovered the topological invariance of Stiefel-Whitney classes, while Wu discovered a set of invariants and formulas, now called the Wu classes and Wu formulas, which have also proved important.
    • From 1953 onwards I made a somewhat systematic investigation of classical topological but non-homotopic problems which were being ignored at that time owing to the rapid development of homotopy theory.
    • I introduced the notion of imbedding classes, and established a theory of imbedding, immersion, and isotopy of polyhedra in Euclidean spaces which was published in book form later in 1965.
    • The cultural revolution was launched by Mao Zedong, Chairman of the Communist Party of China, in May 1966.
    • One of the aims was to send intellectuals to undertake manual jobs in the countryside and factories.
    • I was initially struck by the power of the computer.
    • I was also devoted to the study of Chinese ancient mathematics and began to understand what Chinese ancient mathematics really was.
    • I was greatly struck by the depth and powerfulness of its thought and its methods.
    • It was under such influence that I investigated the possibility of proving geometry theorems in a mechanical way.
    • Although at first sight the two ideas of computer proof and ancient Chinese mathematics appear to be at the opposite ends of the spectrum, Wu saw that the philosophy behind ancient Chinese mathematics was the development of algorithms rather than the axiomatic abstract approach begun by the ancient Greeks and developed in the West.
    • In 1977 Wu introduced a new way of studying geometry on a computer.
    • He based his method on the idea of a characteristic set which had been introduced by Joseph Ritt in his algebraic and algorithmic approach to differential equations.
    • (It is worth noting that Ritt's approach was based on earlier ideas of van der Waerden.) With his new ideas Wu could take a problem in elementary geometry and transform it into an algebraic question about polynomials.
    • Computers were able to answer questions about polynomials so Wu had a powerful method of proving geometric theorems on a computer.
    • He wrote the important book Mechanical theorem proving in geometries (1984) in Chinese which was translated into English and published ten years later.
    • Mechanization of theorem proving in geometry and Hilbert's mechanization theorem.
    • The mechanization theorem of (ordinary) unordered geometry.
    • Mechanization theorems of (ordinary) ordered geometries.
    • Mechanization theorems of various geometries.
    • In 2000 Wu published Mathematics mechanization : Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equations-solving.
    • Wu and his followers have made great progress and this book should be of interest to Western readers in particular, not least for the diversity of topics addressed by the methods.
    • He became a member of the Chinese Academy of Sciences in 1956, receiving one of the three national prizes for natural sciences.
    • He was an invited speaker at the International Congress of Mathematicians in Edinburgh, Scotland, in 1958 but was unable to attend.
    • He was invited for the second time to give an invited lecture, this time on the history of ancient Chinese mathematics, to an International Congress of Mathematicians, this time in Berkeley in 1986.
    • He was elected to the Academy of Sciences for the Developing World in 1991, receiving their mathematics award.
    • He also received the first State Supreme Science and Technology Award of the Chinese government in 2001:- .
    • in recognition of achievements in mathematics research, both in pure mathematics and in mathematics mechanization.
    • For his contributions to the new interdisciplinary field of mathematics mechanization.
    • In 2008 Selected works of Wen-Tsun Wu was published.
    • The present 'Selected papers' may be considered as a brief survey of my scientific career in mathematical sciences.
    • My researches in mathematical sciences consist of two stages.
    • The researches in the first stage, started in 1947, are in pure mathematics, mainly in algebraic topology, occasionally also in algebraic geometry.
    • This ended actually in 1965, the beginning of the Cultural Revolution.
    • During the Cultural Revolution there were however some sporadic research works in pure mathematics, with papers published a little later.
    • Such researches stopped completely at the end of the Cultural Revolution, viz.
    • The second stage of my mathematical researches took place during the Cultural Revolution.
    • It took place owing to my learning of the history of our proper mathematics in ancient times.
    • Being struck by the powerfulness of computers I began to consider applying computers to the study of mathematics.
    • This resulted in a method of proving geometry theorems by means of computers.
    • Extending further the method gave rise to the subject that I called mathematics mechanization, which had an immense variety of applications in science and technology, besides mathematics itself.
    • List of References (2 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Wu_Wen-Tsun.html .

  15. Wang Xiaotong biography
    • He lived the early part of his life under the Sui dynasty (581-618).
    • Although it was only a short-lived dynasty, during these years the north and south of China were unified.
    • Wang is famed as the author of the Jigu suanjing (Continuation of Ancient Mathematics) one of The Ten Classics.
    • He presented his treatise to the Emperor of China, Li Yuan the first emperor of the T'ang dynasty, who came to power in 618.
    • He had full control of the eastern part by 621 and of the north and south as well by 624.
    • Wang Xiaotong's biography tells us that he became interested in mathematics at a young age.
    • He made a careful study of the Nine Chapters on the Mathematical Art and was particularly impressed with the commentary on that text written by Liu Hui.
    • Wang went on to become a teacher of mathematics, and later he was appointed as deputy director of the Astronomical Bureau.
    • It was known that the Chinese calendar at that time was in need of reform since, although only in operation for a few years, already predictions of eclipses were getting out of step.
    • In 623 Wang and Zu Xiaosun, a Civil Servant, were given the task of reporting on the problems with the calendar and making recommendations.
    • There was disagreement between Wang and another calendar expert Fu Renjun about certain aspects of the calendar and in fact Wang's ideas were not particularly good since he wished to ignore the irregularity of the sun's motion and he also wanted to ignore the precession of the equinoxes which had first been incorporated in calendar calculations by Zu Chongzhi in the fifth century.
    • Well, he has not been included for this work but rather for the remarkable contribution he made to Chinese mathematics in the Jigu suanjing (Continuation of Ancient Mathematics).
    • The first is a pursuit problem of a dog chasing a hare, but Wang tells us that this is really a problem about movements of astronomical bodies.
    • The next 13 problems concern engineering constructions and the volume of granaries.
    • The important innovation which is incorporated in most of these problems is that they reduce to a cubic equation which Wang solves numerically.
    • We do not know of any earlier Chinese work on cubic equations.
    • Of course one has to understand that when we say that the text is concerned with cubic equations, we do not see expressions with x, x2 and x3 in them.
    • Let the height be the side of a cube.
    • Let the depth be the side of a cube .
    • It is interesting to compare this geometrical way of thinking about cubics with that of Cardan 900 years later.
    • Many see this work by Wang as the first steps towards the "tian yuan" or "coefficient array method" or "method of the celestial unknown" of Li Zhi.
    • For example Ruan Yuan writes in his Biographies of astronomers and mathematicians (1799) (see [',' Y Ruan, Biographies of Mathematicians and Astronomers (Chinese) 1 (Shanghai, 1955).','5]):- .
    • Really his work is the true source of the later "tian yuan" method.
    • Mikami writes [',' Y Mikami, The Development of Mathematics in China and Japan (New York, 1974).','3]:- .
    • Thus the rule is new in form, but, in fact, it is the continuation of the ancient style handed down from earlier generations.
    • The problems in the Jigu suanjing (Continuation of Ancient Mathematics) are stated in a very complicated form.
    • Here is a flavour of the style with a translation of Problem 3 as given in [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','1]:- .
    • The difference between the lower and upper widths of the west face is 6 zhang 8 chi 2 cun ..
    • ., the height of the east face is 3 zhang 1 chi less than the height of the west face ..
    • Each man of the four sub-prefectures excavates 9 dan 9 dou 2 sheng of earth per day and packs down on average 11 chi 4 cun and 6/12 cun per day ..
    • Previously a man was able to cover a horizontal road of length 192 bu 62 times per day carrying 2 dou 4 sheng and 8 he of earth on his back.
    • A climb of 3 bu is equivalent to 4 bu of a flat road, 1 bu for a ford is equivalent to 2 bu for a flat road, ..
    • What is the daily task of a man who digs, transports and constructs; what are the heights and the upper and lower widths of the dyke ..
    • If we strip away some of the complications then we are left with workers from four countries, A, B, C, and D who cooperate to build a dyke.
    • The east and west ends of the dyke are trapezoidal.
    • The length, upper width, lower widths of the east and west ends and the height of the west end are all known as functions of the height x of the east end.
    • This allows the volume to be calculated as a function of x.
    • To be able to solve this problem Wang has not only to be able to set up a cubic equation and solve it, but he also needs to know a formula for the volume of his dyke with trapezoidal ends and varying cross-section.
    • One of Wang's achievements was calculating this formula and in the preface to the treatise he gives a reference to ideas which had led him to the volume formula (see for example [',' K Shen, J N Crossley and A W-C Lun, The nine chapters on the mathematical art : Companion and commentary (Beijing, 1999).','6]):- .
    • We use letters to name the sides of the right angled triangle where Wang uses names:- .
    • What are the values of the three sides.
    • We know ab, c - a and of course that a2 + b2 = c2 by the Gougu rule (Pythagoras).
    • Wang calls a the unknown and finds a cubic equation in terms of a.
    • Other problems here are of the same type.
    • In 656, after editing by Li Chunfeng, the treatise Jigu suanjing (Continuation of Ancient Mathematics) became a text for the Imperial examinations and it became one of The Ten Classics when reprinted in 1084.
    • Not only did Wang's work influence later Chinese mathematicians, but it is said that it was his ideas on cubic equations which Fibonacci learnt, probably first transmitted into the Islamic/Arabic world, and then brought to Europe.
    • List of References (7 books/articles) .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: Chinese problems .
    • History Topics: The Ten Mathematical Classics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Wang_Xiaotong.html .

  16. Ruan Yuan biography
    • He came from a distinguished Yangzhou family, with his grandfather having been Major General of Hunan province.
    • The city of Yangzhou where Ruan was born is about halfway between Nanjing and Shanghai.
    • We should note that this put Ruan in a position of authority since scholars there acted as advisors to the emperor and as his confidential secretaries.
    • The Academies basically trained students for examinations and the most advanced of these, the civil service examinations, were very much based on the ancient Chinese classics.
    • The underlying philosophy was that of Confucianism as interpreted in commentaries by Cheng Yi and Zhu Xi in the 11th and 12th centuries.
    • To achieve the level which Ruan Yuan achieved required many years of study and he would have been a very elite scholar of Confucianism.
    • He was then in a position to carry out duties at the Hanlin Academy where scholars worked on new interpretations of the Confucian Classics as well as keeping historical records, and preparing encyclopaedias recording as complete an account of knowledge as they could.
    • Starting in 1772, when Ruan Yuan was a young boy, the Chinese Emperor Qianlong organised major searches for lost texts.
    • In 1791 he wrote the Verse of astronomer Zhang Heng.
    • This work was on the calendarist and constructor of armillary sphere who lived from 78 to 139 AD.
    • Ruan's career as a government official led to him holding a variety of offices such as that of a minister in the Education Department of Shandong province from 1793 to 1795 when he took up a similar post in Hangzhou in the Education Department of Zhejiang province.
    • It was while he held this post that Ruan began his most famous work, namely the Chouren zhuan or Biographies of astronomers and mathematicians.
    • In this work Ruan put forward the idea, which had been gaining credence in China for some time, that all Western sciences were of Chinese origin.
    • The work consists of 46 chapters containing biographies of 275 Chinese and 41 Western "mathematicians".
    • More interesting are the numerous quotations from the works of the scientists whose biographies are given.
    • Martzloff notes that a studies of the European mathematicians included in the work [',' J-C Martzloff, Histoire des mathematiques chinoises (Paris, 1987).
    • confirm the importance of Jesuit and Protestant missionaries.
    • They also reveal, at various levels, some limitations and obstacles faced by astronomy and mathematics in the process of transmission from Western countries to China.
    • For example, European names of persons were not always rendered consistently ..
    • Examples of these difficulties are illustrated by the fact that Copernicus is given two different names in Chinese and given two separate biographies without it being recognised that they are of the same person.
    • Again it is not surprising, if the length of the entry in any way indicates importance, that certain biographies look strange.
    • Despite obvious limitations and misrepresentations, Western astronomy and mathematics made accessible to Chinese readers, was none the less, for the most part, based on significant excerpts from texts of primary importance such as Ptolemy's Almagest Ⓣ, Copernicus's De revolutionibus or Euclid's Elements and never on mythical accounts.
    • That was all that Ruan Yuan needed to bring to the fore "the strong and weak points" of both sciences in order to prepare a future synthesis.
    • While still working on this major compilation of biographies of mathematicians, he edited the Dictionary of Old Literature (1797).
    • The Biographies of astronomers and mathematicians was published two years later in 1799.
    • In this year Ruan Yuan was appointed a professor of mathematics in the Imperial Academy as well as holding a number of posts in government departments.
    • In 1800 he was appointed governor of the province of Zhejiang (or Chekiang), a coastal province bounded by the East China Sea.
    • He wrote the foreword to the Ti Chhiu Thu Shuo (An Illustrated Account of the Earth) by Chhien Ta-hsin in about 1800.
    • In it Ruan advised the readers of the book not to follow the Copernican theory just because of its "newness".
    • He also combined his administrative and academic interests when he founded the Gujing jingshe (School of Classical Literature).
    • The school was set up with an innovative curriculum, and literature, astronomy, geometry and mathematics were taught there.
    • He continued to edit, catalogue and collect books and his career moved forward when he was appointed Chancellor of Henan province in 1806.
    • Sent to the Hanlin Imperial Academy as a rather lowly scholar, he continued his research into ancient texts, being promoted to a researcher in the Historical Library, and then in 1810 he became director of the Library.
    • Slowly he made his way back up to high office, becoming Governor of Jiangxi province in 1814.
    • Zhu Shijie wrote the Ssu-yuan yu-chien (Precious mirror of the four elements) but the work was lost during the latter part of the 18th century and was not found by Ruan Yuan when he compiled his Biographies of astronomers and mathematicians in 1799.
    • In 1817 Ruan Yuan reached his highest position of importance when he was appointed Grand Governor of the two Guangzhou, so becoming governor of the two provinces of Guangdong and Guangxi in southern China.
    • It was in the city of Guangzhou that Ruan Yuan founded his most famous academy the Xuehaitang, meaning the Sea of Learning Hall.
    • Then in 1824 he selected a site for new academy buildings on a hill just inside the northern city wall of Guangdong.
    • That the Xuehaitang should be on a hill was significant for he intended that the people should look up to the place of learning.
    • The Xuehaitang Academy, designed to promote evidential studies and Han Learning, went on to achieve the high aims of its founder.
    • The Xuehaitang Academy had only been operating from its new buildings for about a year when Ruan moved again, for in 1826 he was appointed Grand Governor of Yunnan and Guizhou.
    • We should mention another of Ruan's scholarly interests, namely that he was an avid collector of bronze ritual vessels and is said to have owned over 460 of these which had been made between 1600 BC and 220 AD.
    • He was not simply collecting these historic vessels for their own sake, for he studied the inscriptions on them and, in 1804, published a book which reproduced many of them.
    • This study made him a leading authority on calligraphy and his book on the inscriptions was a major influence on the study of calligraphy throughout the 19th century.
    • Finally let us note the importance of Ruan's Biographies of astronomers and mathematicians in Chinese scholarship.
    • It represented the integration of the mathematical sciences with evidential studies and, after its publication, mathematical study was no longer independent of classical studies.
    • List of References (3 books/articles) .
    • A Poster of Ruan Yuan .
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Ruan_Yuan.html .

  17. Yang Hui (about 1238-about 1298)
    • He was a contemporary of both Qin Jiushao and Li Zhi, which we know from the dates on which his texts appeared, showing that he lived towards the end of the Nan (Southern) Sung dynasty.
    • However, both Qin and Li's major works appeared about fifteen years before the first work of Yang.
    • Zhu Shijie was only born about the time Yang Hui's first texts were appearing so his life also overlapped that of Yang.
    • There is a small amount of information about Yang Hui which he relates in his books.
    • He tells us that he was taught mathematics by Liu I who was a native of Chung-shan, in Kwangtung province, which is south of Chekiang province where Yang Hui was born.
    • Nothing at all is known of Liu I, so this information is less helpful in giving us details of Yang Hui than it might be.
    • Again we know the names of four of Yang's friends who were also interested in mathematics, but again as these men are unknown except for Yang's reference to them.
    • The best guess that historians make is that Yang was a minor Chinese official.
    • Most Chinese scholars of the period were officials, for there were no professional mathematicians, but he could not have held an important post since were he a major official he would appear in records of the dynasty.
    • I base my argument on the style and content of Yang's books, for it is clear from these that he was an experienced teacher.
    • Any teacher of mathematics today can identify with what Yang is trying to do here.
    • Of course, this in no way proves that the view of Yang as a minor official is false, indeed he could be an official with responsibility for teaching mathematics, but I suggest that it is more likely that he was an active teacher of mathematics who would have had a group of young students around him.
    • In 1261 Yang wrote the Xiangjie jiuzhang suanfa (Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications).
    • He tells us that he had obtained a fine edition of the Nine Chapters on the Mathematical Art which contained notes by Jia Xian on the edition commented on by Liu Hui and later by Li Chunfeng.
    • The notes by Jia Xian have not survived so we know of them only through the references from Yang.
    • What Yang produced was not intended to be a further commentary on the ancient classic but instead he selected 80 of the 246 problems for his discussion.
    • He chose these 80 since he felt that they were representatives of the different techniques which were presented in the Nine Chapters.
    • Nine of the twelve correspond to those of the Nine Chapters but there are three further chapters: one containing geometrical figures, one containing the fundamental methods, and one in which Yang presents a new classification of the problems.
    • For example, if the problem reduced to the solution of a quadratic equation, then Yang would solve it numerically, then show how to solve a general quadratic equation numerically.
    • Problem 16 in Chapter 7 of the Nine Chapters is as follows:- .
    • Now 1 cubic cun of jade weighs 7 liang, and 1 cubic cun of rock weighs 6 liang.
    • Now there is a cube of side 3 cun consisting of a mixture of jade and rock which weighs 11 jin.
    • Tell: what are the weights of jade and rock in the cube.
    • If there are x cubic cun of jade and y cubic cun of rock in the cube then .
    • Although Yang has presented a problem straight from the Nine Chapters his method of solution is quite different.
    • What Yang's method essentially reduces to is finding the determinant of the matrix of coefficients of the system of equations.
    • Of course he gets the same answer as the earlier authors and commentators, namely that the cube contains 14 cubic cun of jade weighing 6 jin 2 liang, and 13 cubic cun of rock weighing 4 jin 14 liang.
    • Yang also gave formulae for the sum of certain series, for example he found the sum of the squares of the natural numbers from m2 to (m+n)2 and showed that .
    • See [',' M K Siu, Pyramid, pile, and sum of squares, Historia Math.
    • 8 (1) (1981), 61-66.','14] for a discussion of the geometrical ideas which lie behind Yang's approach to summing series.
    • A year after producing Detailed analysis Yang wrote Riyong suanfa (Mathematics for everyday use).
    • Although the text of this has been lost, we know enough about it from quotes in other works to know that it was an elementary text.
    • to assist the reader with the numerous matters of daily use and also to instruct the young in observation and practice.
    • In [',' L Y Yong, The Jih yung suan fa: an elementary arithmetic textbook of the thirteenth century, Isis 63 (218) (1972), 370-383.','19] some of the quotes which allow a partial reconstruction of this work are translated into English.
    • the additive method of multiplication and the subtractive method of division [relative to the] ten problems and their solutions.
    • Over the next years Yang must have continued to produce material for mathematics texts but he published nothing more until 1274 when Cheng Chu Tong Bian Ben Mo which means Alpha and omega of variations on multiplication and division appeared.
    • The first chapter is Fundamental changes in calculation, the second is Computational treasure for variations in multiplications and divisions, and the third, written in collaboration with Shih Chung-yung who was one of his friends, is Fundamentals of the applications of mathematics.
    • In 1275 two further works by Yang appeared; the Practical mathematical rules for surveying and Continuation of ancient mathematical methods for elucidating strange properties of numbers, both being works of two chapters.
    • All Yang's volumes of 1274 and 1275 were assembled into what were essentially his collected works called Yang Hui suanfa (Yang Hui's methods of computation).
    • An English translation of the Yang Hui suanfa appears in [',' Lay Yong Lam, A Critical Study of the Yang Hui Suan Fa: A Thirteenth-Century Chinese Mathematical Treatise, translated from Chinese (1977).','3].
    • The topics covered by Yang include multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
    • He also gives a wonderful account of magic squares and magic circles which we give more information about below.
    • One of the more remarkable aspects of this work is the document on mathematics education Xi Suan Gang Mu (A syllabus of mathematics) which prefaced the first chapter of Cheng Chu Tong Bian Ben Mo.
    • Man Keung Siu, reviewing [',' G Abe, Magic squares that occur in Yang-Hui’s mathematics (Japanese), Sugakushi Kenkyu 70 (1976), 11-32.','10], writes that the syllabus:- .
    • is an important and unusual extant document in mathematics education in ancient China.
    • Not only does it specify the content and the time-table of a comprehensive study program in mathematics, it also explains the rationale behind the design of such a curriculum.
    • This program is a marked improvement on the traditional way of learning mathematics by which a student is assigned certain classical texts, to be studied one followed by the other, each for a period of one to two years! .
    • The syllabus is a fascinating document for it shows Yang's concern that mathematics is properly taught to those meeting the subject for the first time.
    • This was not the first time Yang had shown such concerns, for his elementary text of 1262 was also clearly designed to help beginners.
    • Here is a problem taken from Chapter 2 of Continuation of ancient mathematical methods for elucidating strange properties of numbers.
    • If a Wenzhou orange costs 7 coins, a green orange 3 coins, and 3 golden oranges cost 1 coin, how many oranges of the three kinds will be bought? .
    • From 3 times 100 coins subtract 100 coins; from 3 times the cost of a Wenzhou orange i.e.
    • From 3 times the cost of a green orange, i.e.
    • The sum of the remainder is 28.
    • And then (200 - 6 × 28) ÷ 8 = 4, this is the difference of the number of Wenzhou oranges and green oranges.
    • Hence the sum of them is 16, whereas the number of gold oranges to be found is 84.
    • Then let d, say, be the difference of the number of Wenzhou oranges and green oranges, so y = x - d.
    • Hence d = 4, x = 6, y = 10 and 100 - (6 + 10) = 84 which is the number of golden oranges.
    • If you want to try one of Yang's problems, here is another of the same type, being the first problem in Chapter 2:- .
    • A number of pheasants and rabbits are placed together in the same cage.
    • Find the number of pheasants and rabbits.
    • He gives a magic square of order 3, two squares of order 4, two squares of order 5, two squares of order six, two squares of order 7, two of order 8, one of order nine, and one of order 10.
    • Yang's 3 × 3 squarennnnOne of Yang's 4 × 4 squaresnn .
    • One of Yang's 5 × 5 squaresnnnnnOne of Yang's 6 × 6 squaresnn .
    • One of Yangs 7 x 7 squares is at THIS LINK.
    • One of Yangs 8 x 8 squares is at THIS LINK.
    • This said, no record of the higher order magic squares now exist in the writings of earlier Chinese mathematicians.
    • Adding the central number and the four numbers on the circumference gives 65 for each of the seven circles.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (23 books/articles) .
    • A Poster of Yang Hui .
    • One of Yangs 7 x 7 squares .
    • One of Yangs 8 x 8 squares .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: Chinese problems .
    • Index of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Yang_Hui.html .

  18. Li Zhi (1192-1279)
    • Usually Chinese names have a number of different spellings, each trying in a different way to match the pronunciation of the original.
    • Li Zhi's father was named Li Yu and he was the secretary to a Jurchen officer by the name of Hu Shahu.
    • The Jurchen empire, formed by the Jurchen tribes of Manchuria, covered much of Inner Asia and all of North China.
    • The capital of the empire became Ta-hsing (sometimes written Daxing and now called Peking) in 1192 and it was in this capital city that Li Yu's son Li Zhi was born.
    • Between 1207 and 1215 the armies of the Mongol leader Genghis Khan pushed into North China and in 1215 they sacked the capital Ta-hsing (now Peking) of the Jurchen empire.
    • Li Yu's home was in Luan-ch'eng, in the Hopeh province which included Ta-hsing, and he sent his family back to his home but not Li Zhi who went alone to the Yuan-shih district of Hopeh province for his education.
    • After the death of Genghis Khan, one of his sons Ogodei became the Great Khan in 1229.
    • He expanded the Mongol empire sending armies to complete the defeat of the Jurchens.
    • After successfully completing his examinations, Li Zhi was appointed registrar of the district of Kaoling but the advancing Mongol armies prevented him taking up the appointment.
    • Instead he became governor of the Jun prefecture of Honan province.
    • Li Zhi only escaped being massacred himself thanks to the intervention of one of the Jurchen officials who had gone over to the side of the Mongols.
    • By 1234 the Mongols had completed the destruction of the Jurchen empire and turned their attention to the south.
    • It was here, in 1248, that he completed his most famous work the Ce yuan hai jing (Sea mirror of circle measurements).
    • We look at the contents of this remarkable treatise below.
    • He lived here until 1257 when Kublai, a grandson of the Mongol leader Genghis Khan who was leading further Mongol advances, sent for Li Zhi to ask his advice on governing the state and on civil service examinations.
    • Li Zhi continued to work on mathematics and completed another important text Yi gu yan duan (New steps in computation) in 1259.
    • In 1261 Kublai Khan offered Li Zhi a government position but by now he was 69 years old and [',' H Peng-Yoke, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • politely declined with the plea of ill health and old age.
    • We are lucky to have any works by Li Zhi other than the Sea mirror of circle measurements for he told his son to burn all his books except the Sea mirror of circle measurements on his death since this was the only work of which he was proud.
    • We now look at some of the very remarkable contributions which Li Zhi made to mathematics.
    • First let us look briefly at the "tian yuan" or "coefficient array method" or "method of the celestial unknown".
    • This was a notation for an equation and, although the work of Li Zhi is the earliest source of the method, it must have been invented before his time.
    • We have given the example using numerals which are natural with the language that we write this archive but, of course, Li Zhi would have used Chinese characters.
    • Here the numbers which in our notation correspond to the coefficients of the equation are placed above each other so that the coefficient of x is placed above the constant, the coefficient of x2 is placed above the coefficient of x etc.
    • Perhaps even more surprisingly, negative powers of x were placed in descending order below the constant term.
    • But he does not limit his reflections to equations of degree two or three; for him, the fact that polynomial equations of arbitrarily high degree are involved is of little importance.
    • Moreover, he never explains what he understands by an equation, an unknown, a negative number, etc., but only describes the manipulations which should be carried out in specific problems, without worrying about arranging his text in terms of definitions, rules and theorems.
    • In other words, like many other algebraists, Chinese or not, he demonstrates algebra by using it ..
    • The type of problem which worried mathematicians in Islamic countries, and in Europe, concerning the solution of cubic, quartic, and higher order equations did not seem to arise in China.
    • Li Zhi seems happy with equations of any degree and, although methods to solve equations do not appear explicitly, one has to assume that he used methods similar to those Ruffini and Horner discovered over 600 years later.
    • By any standards the Sea mirror of circle measurements is a most remarkable work.
    • In the diagram O is the centre of the circle inscribed in the square EDCF.
    • AB is the hypotenuse of the triangle ACB which meets the square at G and H.
    • The figure represents a circular town and follows the Chinese convention of having north at the bottom, south at the top, east on the left, and west on the right.
    • This is the only figure in the book and every one of the 170 problems which make up Chapters 2 to 12 concern this figure.
    • Chapter 1 contains three sections, the first giving the names of the constituents, the second section lists all the values of the lengths of the segments, so in essence contains all the answers to the problems, while the third section comprises of 692 formulae for areas of triangles and lengths of segments.
    • As one sees this is nothing like any mathematics book of today! .
    • Here are some sample problems taken from [',' H Peng-Yoke, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1].
    • B walks a distance of 256 pu eastwards.
    • Then A walks a distance of 480 pu south before he can see B.
    • Find the diameter of the town.
    • B leaves the east gate and walks straight ahead a distance of 16 pu, when he just sees A.
    • Find the diameter of the town.
    • If we examine Li Zhi's solution closely we see a remarkable depth of understanding of equations.
    • Try some of these problems, they are fun.
    • 135 pu directly out of the south gate is a tree.
    • If one walks 15 pu out of the north gate and then turns east for a distance of 208 pu, the tree comes into sight.
    • Find the diameter of the town.
    • It is thought by many historians to have been written because people found understanding the Sea mirror of circle measurements was beyond them.
    • The New steps in computation is based on an earlier book which it is said was written by Chaing Chou of P'ing-yang (although nothing else is known of the author, nor is there any knowledge of the date of this earlier work).
    • Li Zhi's book contains 64 problems, of which he says that 21 are from the earlier text.
    • The central theme is the construction and formulation of quadratic equations.
    • Some of these equations are solved by the "coefficient array method" described above, but some are formulated using the tiao duan or "method of sections".
    • This older geometric style method of solving equations was used by Chinese mathematicians before Li Zhi and so the New steps in computation gives historians a unique opportunity to see the new coefficient array method beside the older method of sections.
    • A fascinating comparison of the methods is described in [',' L Y Lam and T S Ang, Li Ye and his Yi gu yan duan (old mathematics in expanded sections), Arch.
    • Let us finish this biography by giving the first problem of New steps in computation.
    • The land area is 13 mou and 7 1/2 tenths of a mou.
    • Find the length of the side of the farm and the diameter of the pond.
    • Area of square is (x + 40)2.
    • Area of pond 0.75x2.
    • He gives the solution 20 pu which is the diameter of the pond.
    • The side of the square farm is then 60 pu.
    • One final comment on Li Zhi's use of π = 3 in this problem.
    • When he takes π = 3 it is not because he is obtaining the best approximate answer that he can, rather it is the method of solving the problem which is important and he is better able to illustrate this with "nice" numbers.
    • That dictates his choice of π = 3 in this problem.
    • List of References (7 books/articles) .
    • History Topics: Chinese numerals .
    • History Topics: Chinese problems .
    • Index of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Li_Zhi.html .

  19. Cheng Dawei biography
    • He published the Suanfa tong zong (General source of computational methods) in 1592 and almost all that is known about his life is contained in a passage written in the Preface of the book by one of his descendents when the book was being reprinted.
    • We reproduce it here (see [',' K Takeda, The characteristics of Chinese mathematics in the Ming dynasty (Japanese), J.
    • Tokyo 29 (1954), 8-18.','7] and also [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','1] and [',' J-C Martzloff, Histoire des mathematiques chinoises (Paris, 1987).
    • In the prime of his life he visited the fairs of Wu and Chu.
    • He questioned respectable old men who were experienced in the practice of arithmetic and gradually and indefatigably formed his own collection of difficult problems.
    • What can we deduce from this description? Firstly we know Cheng Da Wei lived in the latter half of the Ming dynasty which was a period of prosperity with increasing trade and commerce.
    • It was also a period of relatively good stable government.
    • A complex system of land tax led to a farmer's tax bill involving complicated reckoning of many different tax items.
    • The need for arithmetical skills led to the invention of the abacus and Cheng Da Wei's book General source of computational methods was an arithmetic book for the abacus.
    • It is not an academic work on mathematics, rather it is a practical book aimed at assisting those who need to calculate.
    • That Cheng Da Wei was not a professional mathematician is typical of what one would expect from this period in China.
    • His occupation in local government is also typical of the type of profession which contained highly skilled mathematicians.
    • Although mathematics did not rate highly as an academic discipline, as we indicated above it was essential for many people to posses arithmetical skills.
    • Again we see that he was an avid collector of books on mathematics and this is borne out by the General source of computational methods which is not particularly original, but is important for the compilation of problems from earlier works which it contains.
    • Cheng Da Wei wrote the General source of computational methods in 1592.
    • By this time he was quite old and making use of the large collection of works which he had collected throughout his younger days.
    • It is written in the style of the Nine Chapters on the Mathematical Art and contains 595 problems in 12 chapters.
    • unlike the authors of the venerable classic, Cheng Dawei was not afraid of superfluity or verbosity.
    • His book is an encyclopaedic hotch-potch of ideas which contains everything from A to Z relating to the Chinese mystique of numbers (magic squares, ..
    • generation of the eight trigrams, musical tubes), how computation should be taught and studied, the meaning of technical arithmetical terms, computation on the abacus with its tables which must be learnt by heart, the history of Chinese mathematics, mathematical recreations and mathematical curiosities of all types.
    • Let us give examples of the problems.
    • Let x be the number of sheep in shepherd A's flock.
    • In Chapter 2 of Cheng Da Wei's text there is the following problem.
    • Now a pile of rice is against the wall with a base circumference 60 chi and an altitude of 12 chi.
    • What is the volume? Another pile is at an inner corner, with a base circumference of 30 chi and an altitude of 12 chi.
    • What is the volume? Another pile is at an outer corner, with base circumference of 90 chi and an altitude of 12 chi.
    • Cheng Da Wei goes on to explain what one expects the altitude of grain for a given base circumference to be.
    • Of course in practice it will depend on how coarse the grain is, but Cheng Da Wei's values are quite close to what experimental evidence suggests.
    • In problems of piles on the ground, against a wall, at an inner corner or an outer corner, the ancients always measured their altitude and then calculated.
    • Instead of measuring the altitude we now take 1/10 the base circumference as the altitude for a pile on the ground; take 1/5 base circumference as the altitude for a pile against a wall, for it is half a cone; take 10/25 base circumference as the altitude for a pile at an inner corner, for it is quarter of a cone; take 10/75 base circumference as the altitude for a pile at an outer corner, for it is three quarters of a cone.
    • Here is another two of the problems of General source of computational methods:- .
    • Given the diameter of the field and the breadth of the river find the area of the non-flooded part of the field.
    • In the right-angled triangle with sides of length a, b and c with a > b > c, we know that a + b = 81 ken and a + c = 72 ken.
    • A descendant of Cheng Da Wei wrote in 1716 about the reputation of General source of computational methods:- .
    • A century and several decades have passed since the first edition of "Suanfa tong zong" during which period this work has remained in vogue.
    • Practically all those involved in mathematics have a copy and consider it a classic ..
    • Even in 1964 two authors of a book on the history of Chinese mathematics wrote:- .
    • Nowadays, various editions of the "Suanfa tong zong" can still be found throughout China and some old people still recite the versified formulas and talk to each other about its difficult problems.
    • List of References (7 books/articles) .
    • A Poster of Cheng Dawei .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: Chinese problems .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Cheng_Dawei.html .

  20. Zhang Heng biography
    • Zhang Heng was born at the time of the Eastern Han (sometimes called Later Han) dynasty, the second half of the longest lasting Chinese dynasty.
    • The Eastern Han was established in 25 AD after the brief 15 year reign of Wang Mang's Hsin dynasty had replaced the Western Han dynasty.
    • At the time of Zhang Heng's birth the Emperor was Chang-ti, the third of the Eastern Han emperors.
    • The capital of the country had moved to Lo-yang where a large ornate palace had been built.
    • In Encyclopaedia Britannica the aims and achievements of the Han rulers are described as follows:- .
    • the Han came to require cultural accomplishment from their public servants, making mastery of classical texts a condition of employment.
    • The title list of the enormous imperial library is China's first bibliography.
    • Its text included works on practical matters such as mathematics and medicine, as well as treatises on philosophy and religion and the arts.
    • Over the years that Zhang Heng grew up, Chinese influence and prestige were growing rapidly and reached their peak in around 90 when he was about 12 years old.
    • Zhang, who had been born into an important family, was educated in the moral and political philosophy of Confucianism.
    • He published a number of literary works which gained him considerable fame.
    • We shall give more information below on these aspects of Zhang's achievements as well as examples of his poetry.
    • The court, however, was beginning to provide a less efficient government due to the weakness of successive emperors who were manipulated by those around them seeking advantage for themselves.
    • His biography in The History of the Eastern Han Dynasty (see [',' Ngo van Xuyet, Divination, Magie et Politique dans la Chine Ancienne (Paris, 1976).','5]) suggests that he was not as successful an official as he might have been precisely because of a lack of ambition.
    • He refused advancement in his career on several occasions when he turned down posts that were offered to him, and he also spent periods away from the capital when he lived in isolation and thought about the nature of the universe and about a wide variety of scientific topics.
    • He held the position of chief astrologer on a number of occasions.
    • We will describe below some of Zhang's outstanding scientific achievements.
    • However, as we indicated above, he first achieved fame as a poet and writer of over twenty works, and in this capacity he had a lasting influence on Chinese culture.
    • His works Si Chou Shi (Four Chapters of Distressed Poems) and Gui Tian Fu (To Live in Seclusion) are considered literary masterpieces.
    • Zhang's poem, in his highly influential style of prose poetry, which we now quote comes from [',' D R Knechtges (trns.), Wen Xuan, or Selections of Refined Literature 1 (Princeton, New Jersey, 1982).','1].
    • It is a telling criticism of the last rulers of the Western Han dynasty:- .
    • Zhang wrote the Four Stanzas of Sorrow which is the first seven-syllabic poem which we know of in China.
    • We quote (in translation of course) only the first of its four stanzas:- .
    • I'm at a loss as she is out of sight; .
    • Changing the calendar was seen as one of the duties of the office, establishing the emperor's heavenly link on earth.
    • After a change of ruler, and even more significantly after a change of dynasty, the new Chinese emperor would seek a new official calendar thus establishing a new rule with new celestial influences.
    • One has to understand how significant earthquakes were in China at this time, not only for the destructive power which they unleashed but also because they were seen as punishment from the gods for poor governance of the country.
    • In his role as chief astrologer he was responsible for detecting signs of bad government which were indicated by earthquakes.
    • Zhang's device, which he called Hou Feng Di Dong Yi, was made of copper.
    • It was in the shape of an egg with eight dragon heads around the top, each with a copper ball in its mouth, and a pendulum in the centre.
    • When an earthquake occurred, a ball fell out of a dragon's mouth into a frog's mouth, making a noise.
    • In fact the seismograph detected an earthquake in February of 138 and Zhang reported this fact to the Emperor despite no other evidence of the earthquake being felt in the capital Lo-yang.
    • He was even able to indicate that the earthquake was to the west of the capital.
    • He achieved fame when reports of an earthquake more than a thousand kilometres north west reached the capital several days later.
    • It consisted of a system of rings corresponding to the great circles of the celestial sphere with a central tube which was used to line up stars and planets.
    • With this instrument Zhang was able to make more accurate star maps than earlier Chinese astronomers.
    • He wrote about his instrument in the work Hun-i chu where he described his version of the universe as follows:- .
    • The sky is like a hen's egg, and is as round as a crossbow pellet, the Earth is like the yolk of the egg, lying alone at the centre.
    • North and south of the equator there are 124 groups which are always brightly shining.
    • Of the very small stars there are 11520.
    • Only the first part of this text by Zhang has survived.
    • In mathematics Zhang studied 3 by 3 magic squares.
    • He also proposed, in a treatise on inscribed and circumscribed circles of a square, that π = √10 or approximately 3.162.
    • Although this is not particularly accurate the significance of his work is pointed out by S K Mo in [',' S K Mo, The ratio of circumference to diameter according to Heng Zhang (Chinese), J.
    • Zhang also gave formulae for the volume of a sphere in terms of the volume of the circumscribing cube.
    • In an attempt to make his statement consistent and harmonise his philosophy of yin and yang, and the doctrine of odd and even, [Zhang] neglected the precision of the data.
    • One interesting point to note in some of Zhang's mathematical work is that he leaves square roots as unevaluated.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (8 books/articles) .
    • A Poster of Zhang Heng .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Zhang_Heng.html .

  21. Shiing-shen Chern biography
    • Shiing-shen Chern, whose name can also be written as Chen Xingshen, was educated at home as well as occasionally attending Chia-hsing elementary school from the winter of 1917 until 1920.
    • [Schooling in China at this time was just beginning to get organised following the revolution in the country around the time Chern was born.] He was taught Chinese by his aunt and mathematics by his father.
    • The daily newsletter of ICM 2010 (21 August 2010).','32]:- .
    • He did things on his own and spent a lot of time in his younger days with his grandmother who probably spoiled him as much as she could.
    • There Chern came to love mathematics and avidly solved the problems in Higher Algebra by H S Hall and S R Knight, and in geometry and trigonometry books by George Albert Wentworth and David Eugene Smith.
    • This small university had about 300 students in total and Chern was one of a mathematics class of four students.
    • He was particularly inspired by a geometry course given by Lifu Jiang, the only professor of mathematics at the university, who had studied at Harvard under Julian Coolidge.
    • Lifu Jiang [',' Z Wang, Shiing-Shen Chern, 1911-2004, in X Zhao and E J W Park (eds.), Asian Americans: An Encyclopedia of Social, Cultural, Economic, and Political History (ABC-CLIO, 2013), 206-207.','65]:- .
    • After four years of study at Nankai University Chern was awarded a diploma and a B.Sc.
    • in mathematics in 1930.
    • Dan Sun had obtained a doctorate from the University of Chicago in 1928, advised by Ernest Preston Lane, with his thesis Projective Differential Geometry of Quadruples of Surfaces with Points in Correspondence.
    • After taking the entrance examination for the Graduate School at Tsing Hua University, Peking, Chern was appointed as an assistant in the Department of Mathematics at that university in August 1930.
    • In August 1931 he continued to undertake research in the Graduate School of Tsing Hua University.
    • He was the only graduate student in mathematics to enter the university in 1930 but during his four years there he not only studied widely in projective differential geometry but he also began to publish his own papers on the topic.
    • Chern wrote [',' S S Chern, May mathematical education, in Shing-Tung Yau (ed.), Chern - a great geometer of the twentieth century (Int.
    • In the spring of 1932 Blaschke visited Peking and gave a series on topological questions in differential geometry.
    • It was really local differential geometry where he took, instead of a Lie group as in the case of classical differential geometries, the pseudo-group of all diffeomorphisms and studied the local invariants.
    • It was my first introduction to modern mathematics and it opened my eyes ..
    • He received a scholarship from Tsing Hua University in 1934 to study in the United States, but he made a special request that he be allowed to go to the University of Hamburg.
    • His reason was that he believed the mathematics he was interested in was being done in Europe and not, at that time, in the United States.
    • His meeting with Wilhelm Blaschke when he visited Peking had convinced him that Hamburg would be better for him than the other big European mathematics centres such as Paris, Gottingen or Berlin.
    • In 1932 he visited Peking as part of his world tour.
    • I was immediately impressed by his fresh ideas and his insistence on mathematics being a lively and intelligible subject.
    • When Chern arrived in Hamburg he was told that Erich Kahler, a Privatdozent at Hamburg, had just written a book describing Elie Cartan's mathematics and was about to run a seminar on the topic.
    • Kahler came in with a pile of the books and gave everybody a copy.
    • But the subject was difficult, so after a number of times, people didn't come anymore.
    • Not only that, I was writing a thesis applying the methods to another problem, so the seminar was of great importance to me.
    • His scholarship was for three years so he had still another year of financial support.
    • His time in Paris was a very productive one and he learnt to approach mathematics, in the same way that Cartan did, see [',' W G Chinn and J B Lewis, Shiing-Shen Chern: A man and his times, The Two-Year College Mathematics Journal 14 (5) (1983), 370-376.','28]:- .
    • Cartan's writings were generally regarded as very difficult, but Chern quickly accustomed himself to Cartan's way of thinking.
    • There is a tendency in mathematics to be abstract and have everything defined, whereas Cartan approached mathematics more intuitively.
    • That is, he approached mathematics from evidence and the phenomena which arise from special cases rather than from a general and abstract viewpoint.
    • Speaking of Cartan's ideas, Chern said in the interview [',' A Jackson, Interview with Shiing Shen Chern, Notices Amer.
    • Without the notation and terminology of fibre bundles, it was difficult to explain these concepts in a satisfactory way.
    • In 1937 Chern left Paris to become professor of mathematics at Tsing Hua University.
    • However the Chinese-Japanese war began in July 1937 while he was on the journey and the university moved twice to avoid the war.
    • He worked at what was then named Southwest Associated University (consisting of the former Tsing Hua University, Peking University and Nankai University) from 1938 until 1943.
    • This university operated from the city of Kunming in south west China.
    • So the Chinese government arranged for me a seat on an US Air Force plane from Calcutta, India to Miami, US.
    • He became friendly with Lefschetz who persuaded him to become an editor of the Annals of Mathematics.
    • In [',' A Weil, S S Chern as geometer and friend, in Shing-Tung Yau (ed.), Chern - a great geometer of the twentieth century (Int.
    • Press, Hong Kong, 1992), 72-78.','67], Weil wrote about talking about Cartan's mathematics to Chern at this time:- .
    • we seemed to share a common attitude towards such subjects, or towards mathematics in general; we were both striving to strike at the root of each question while freeing our minds from preconceived notions about what others might have regarded as the right or the wrong way of dealing with it.
    • These talks between Weil and Chern were very influential for Chern and led to some of his most important work on characteristic classes.
    • At the end of World War II, Chern returned to China reaching Shanghai in March 1946.
    • He was asked to set up the Institute of Mathematics of the Academia Sinica in Nanking which he did very successfully.
    • Andre Weil, who by this time was at the University of Chicago, arranged for Chern to be offered a full professorship at University of Chicago.
    • From 1949 Chern worked in the United States accepting the chair of geometry at the University of Chicago after first making a short visit to Princeton.
    • He was an invited one-hour plenary speaker at the International Congress of Mathematicians held in Cambridge, Massachusetts, from 30 August to 6 September 1950.
    • He gave the address Differential Geometry of Fibre Bundles.
    • The starting point of this lecture is the definition of a connection in a principal fibre bundle (all spaces are differentiable manifolds, the structure group is a Lie group ..
    • Geometrically the connection is a field of contact elements in the bundle, transversal to the fibres, and invariant under the action of the group.
    • Chern remained at Chicago until 1960 when he went to the University of California, Berkeley.
    • My election to the US National Academy of Sciences was a prime factor for my US citizenship.
    • In 1960 I was tipped about the possibility of an academy membership.
    • The process was slowed because of my association to Oppenheimer.
    • In 1970 he was an invited one-hour plenary speaker at the International Congress of Mathematicians held in Nice, France, from 1 September to 10 September 1970.
    • This was a great honour since very few mathematicians have been asked to be one-hour plenary speaker at two International Congresses of Mathematicians.
    • He continued working at Berkeley, retiring officially in 1979 but remaining highly mathematically active there for six of seven more years.
    • He continued to live in Berkeley until 1999 when, at the age of 88, he returned to China where he made his home in Tianjin, where the Chern Institute of Mathematics of Nankai University had been set up in 1985.
    • In the paper A summary of my scientific life and works which Chern wrote in 1978 (and is included in the volumes of his selected papers) Chern wrote about the contribution of his wife:- .
    • He died at his home in Tianjin at the age of 93 from heart failure following a heart attack.
    • As we have already seen, his area of research was differential geometry where he studied the (now named) Chern characteristic classes in fibre spaces.
    • These are important not only in mathematics but also in mathematical physics.
    • He worked on characteristic classes during his 1943-45 visit to Princeton and, also at this time, he gave a now famous proof of the Gauss-Bonnet formula.
    • His work is summed up in [',' C N Yang, S S Chern and I, in Shing-Tung Yau (ed.), Chern - a great geometer of the twentieth century (Int.
    • When Chern was working on differential geometry in the 1940s, this area of mathematics was at a low point.
    • Global differential geometry was only beginning, even Morse theory was understood and used by a very small number of people.
    • Today, differential geometry is a major subject in mathematics and a large share of the credit for this transformation goes to Professor Chern.
    • Richard Palais and Chuu-Lian Terng give an excellent overview of Chern's mathematics in [',' R S Palais and C-L Terng, The life and mathematics of Shiing Shen Chern, in Shing-Tung Yau (ed.), Chern - a great geometer of the twentieth century (Int.
    • Chern's mathematical interests have been unusually wide and far-ranging and he has made significant contributions to many areas of geometry, both classical and modern.
    • we would like to point out a unifying theme that runs through all of it: his absolute mastery of the techniques of differential forms and his artful application of these techniques in solving geometric problems.
    • They admit, on the one hand, the local operation of exterior differentiation, and on the other the global operation of integration over cochains, and these are related via Stokes's Theorem.
    • He was awarded the Chauvenet Prize from the Mathematical Association of American 1970, the National Medal of Science in 1975, the Humboldt Prize in 1982, the Leroy F Steele Prize from the American Mathematical Society in 1983, the Wolf Prize in 1984, the Lobachevsky Medal in 2002 and the first Shaw Prize in Mathematics from Hong Kong in 2004:- .
    • for his initiation of the field of global differential geometry and his continued leadership of the field, resulting in beautiful developments that are at the centre of contemporary mathematics, with deep connections to topology, algebra and analysis, in short, to all major branches of mathematics of the last sixty years.
    • In 1985 he was elected a Fellow of the Royal Society of London and the following year he was made an honorary member of the London Mathematical Society.
    • He has also been made an honorary member of the Indian Mathematical Society (1950), the New York Academy of Sciences (1987).
    • He was elected to the Academia Sinica (1948), the United States National Academy of Sciences (1961), the American Academy of Arts and Sciences (1963), the Brazilian Academy of Sciences (1971), the Academia Peloritana, Messina, Sicily (1986), the Accademia dei Lincei (1989), the Academie des Sciences, Paris (1989), the American Philosophical Society (1989) the Chinese Academy of Sciences (1994), and the Russian Academy of Sciences (2001).
    • He was awarded honorary degrees by the University of Chicago (1969), the Chinese University of Hong Kong (1969), Eidgenossische Technische Hochschule Zurich (1982), the State University of New York Stony Brook (1985), University of Hamburg (1971), Nankai University (1985), University of Notre Dame (1994), Technische Universitat Berlin (2001), and Hong Kong University of Science and Technology (2003).
    • Hail to Chern! Mathematics Greatest! .
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (81 books/articles) .
    • 6.nFellow of the Royal Societyn1985 .
    • Index of Chinese mathematics .
    • Chinese Academy .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Chern.html .

  22. Shing-Tung Yau biography
    • Shing-Tung Yau was the fifth of the eight children of his parents Chen Ying Chiou and Yeuk-Lam Leung Chiou.
    • However, by late 1949 the Communists were in control of almost all of China and Yau's family fled to Hong Kong where his father obtained a position teaching at a College.
    • (The College later became a part of the Chinese University of Hong Kong.) Times were difficult for the family, however, and Yau's mother knitted goods to sell in order to supplement their low income.
    • Life was tough for Yau living in a village outside Hong Kong city in a house which had no electricity or running water and at this stage of his life he often played truant from school preferring his role as leader of a street gang.
    • Yau's father was a major influence on him, encouraging his interest in philosophy and mathematics.
    • By good fortune, one of his lecturers at the College had studied at the University of California, and, seeing Yau's enormous potential, suggested that he go there to study for a doctorate.
    • Yau studied for his doctorate at the University of California at Berkeley under Chern's supervision.
    • in 1971 and, during session 1971-72, Yau was a member of the Institute for Advanced Study at Princeton.
    • Yau was appointed assistant professor at the State University of New York at Stony Brook in 1972.
    • In 1980 he was made a professor at the Institute for Advanced Study at Princeton, a position he held until 1984 when he moved to a chair at the University of California at San Diego.
    • Yau was awarded a Fields Medal in 1982 for his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampere equations.
    • In fact the 1982 Fields Medals were announced at a meeting of the General Assembly of the International Mathematical Union in Warsaw in early August 1982.
    • Writing in [',' L Nirenberg, The work of Shing-Tung Yau, Notices Amer.
    • Nirenberg describes briefly the areas of Yau's work.
    • comes from algebraic geometry and involves proving the existence of a Kahler metric, on a compact Kahler manifold, having a prescribed volume form.
    • The analytic problem is that of proving the existence of a solution of a highly nonlinear (complex Monge-Ampere ) differential equation.
    • His derivation of the estimates is a tour de force and the applications in algebraic geometry are beautiful.
    • Yau, in joint work, constructed minimal surfaces, studied their stability and made a deep analysis of how they behave in space-time.
    • His work here has applications to the formation of black holes.
    • .for his work in nonlinear partial differential equations, his contributions to the topology of differentiable manifolds, and for his work on the complex Monge-Ampere equation on compact complex manifolds.
    • In joint work of Yau with Karen Uhlenbeck On the existence of Hermitian Yang-Mills connections in stable bundles (1986), they solved higher dimensional versions of the Hitchin-Kobayashi conjecture.
    • Their work extended that of Donaldson on this topic in 1985.
    • The Crafoord Prize of the Royal Swedish Academy of Sciences was awarded to Yau in 1994:- .
    • for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.
    • As a result of Yau's work over the past twenty years, the role and understanding of basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics.
    • His work has had, and will continue to have, a great impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory, and general relativity as well as differential geometry and partial differential equations.
    • Yau was elected to the National Academy of Sciences in 1993.
    • He was awarded the National Medal of Science in 1997.
    • He put a great deal of effort into building Chinese mathematics, visiting China during the Harvard summer vacation, helping top Chinese students go to the United States for doctoral studies, and working hard for the founding of mathematical institutes in Hong Kong, Beijing and Hangzhou.
    • In 2004 he was honoured in the Great Hall of the People, located on the western side of Tiananmen Square in Beijing, for his contributions to Chinese mathematics.
    • However, recently he has been involved in an unfortunate dispute regarding the proof of the Poincare conjecture.
    • The Russian mathematician Grigory Perelman sketched a proof of the conjecture in 2003 and several teams began work on giving a full comprehensive proof.
    • Yau's team is one of these and he has been criticised by some for comments which people felt did not give Perelman full credit.
    • However Yau has clearly stated that it had only been his intention to say that his team made Perelman's proof understandable to a much wider range of mathematicians.
    • Let us end this biography by quoting Bun Wong and Yat Sun Poon, Professors of Mathematics at the University of California at Riverside:- .
    • Yau's achievement in Mathematics is well known within mathematics community.
    • It is equally well known that he has successfully produced nearly 50 PhD students in mathematics and has many collaborators across the globe.
    • Perhaps, it is less well known that he has donated personal fund to establish scholarships for mathematics students, has donated tens of thousands of books to educational institutions, has helped raise tens of millions of dollars to promote mathematics education and research, and has raised fund to promote interaction among scientists across subject boundaries and national borders.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (6 books/articles) .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Yau.html .

  23. Guo Shoujing biography
    • We do not know the names of his parents, but his paternal grandfather, who was clearly more famous than his parents, was Guo Yong who was famed as an expert in a wide range of topics from classical studies to mathematics and hydraulics.
    • By the age of sixteen Guo was studying mathematics.
    • Before we continue to describe events of Guo's life we should look briefly at the political situation, for it was a troubled time with many wars.
    • After the death of Genghis Khan, one of his sons Ogodei had become the Great Khan in 1229.
    • He had expanded the Mongol empire sending armies to complete the defeat of the Jurchens.
    • By 1234 the Mongols had completed the destruction of the Jurchen empire and turned their attention to the south.
    • This then was the situation in the northern part of China where Guo was growing up.
    • By the age of twenty Guo was working as an hydraulic engineer.
    • This was a river in Guo's home province of Hebei and the bridge he renovated was a little way north of the town of Xinzhou.
    • Kublai, a grandson of the Mongol leader Genghis Khan began leading further Mongol advances in the latter years of the 1250s.
    • Zhang Wenqian, who was a friend of Guo and like him was a central government official, was sent by Kublai Khan in 1260 to Daming where unrest had been reported in the local population.
    • Kublai knew the importance of water management, for irrigation, transport of grain, and flood control, and he asked Guo to look at these aspects in the area between Dadu (now Beijing or Peking) and the Yellow River.
    • To provide Dadu with a new supply of water, Guo found the Baifu spring in the Shenshan Mountain and had a 30 km channel built to bring the water to Dadu.
    • This pleased Kublai Khan and led to Guo being asked to undertake similar projects in other parts of the country.
    • In 1264 he was asked to go to Gansu province to repair the damage that had been caused to the irrigation systems by the years of war during the Mongul advance through the region.
    • Guo travelled extensively along with his friend Zhang taking notes of the work which needed to be done to unblock damaged parts of the system and to make improvements to its efficiency.
    • The advance of the Monguls was continuing under Kublai Khan and in 1276 he captured the city today named Hangzhou south of Shanghai.
    • the fundamental method is to carry out observation and tests in the area of astronomical changes.
    • An accurate calendar depends on the quality of the astronomical data used to support it and Guo's first move was to built seventeen new astronomical instruments so that accurate data could be collected.
    • Of these seventeen instruments, thirteen were to be set up in an observatory in Kublai's capital Dadu (today called Beijing or Peking) while the other four were portable instruments which could make observations from different locations.
    • The simplest astronomical instruments was the gnomon, nothing other than a stick which was erected and the length of its shadow measured.
    • The minimum length of shadow during a day is less in summer than in winter and at the solstices it changes from lengthening to shortening or visa versa.
    • To make it easier to determine these points Guo fixed a crossbar to the top of a gnomon and used the principle of the pinhole camera to cast its shadow onto a measuring scale.
    • Building began in March of that year and, following a design proposed by Guo, the work was completed in two months.
    • Making sense of the data gathered from the instruments required a knowledge of spherical trigonometry and Guo devised some remarkable formulae.
    • We look below at the clever mathematics which he introduced in undertaking his project on the new calendar.
    • The work was completed by 1280, Guo having calculated the length of the year correct to within 26 seconds, and in the following year Kublai Khan introduced the use of this extremely accurate calendar.
    • Zhang Wenqian died in 1283 and Guo was promoted to be director of the Observatory in Beijing.
    • In 1292, in addition to his role of director of the Observatory, he was made head of the Water Works Bureau.
    • As always he met with success and even after the death of Kublai Khan, although Guo was by this time an old man, his advice continued to be sought by Kublai's successor.
    • He produced a number of formulae for triangles, two sides of which were straight lines and the third was the arc of a circle.
    • These formulae are approximate ones, but Guo was well aware of this.
    • In a sense approximation was not regarded as important by the Chinese and they never became obsessed by the "squaring the circle" type of question like the ancient Greeks, since the Chinese approach was more practical and never axiomatic.
    • In the diagram d is the diameter of the circle, a is the length of the arc AB and x is the length of NB which Guo wanted to calculate.
    • The equation has two real roots, the smaller being the solution to the problem while the other, being numerically larger than the length of the arc, was rightly discarded by Guo.
    • Two of the coefficients of the equation, namely the constant term and the coefficient of x2, involve the length a of the arc, so require a value to be chosen for π.
    • Why did Guo choose π = 3? Surprisingly this gives a better answer than the more accurate values of π, for remember the formula is itself based on approximations.
    • One has to believe that Guo chose π = 3 because he knew that the answers that he then found for different sizes of the angle at O more closely approximated values he found by direct measurement.
    • The values are computed for increasing angles up to 90° for a circle of radius 100.
    • The first column is the value of x using Guo's formula taking an accurate modern approximation to π, the second column is the result given by the formula with π = 3, while the third column is the correct answer calculated using trigonometry (in fact the cosine).
    • The reason that interpolation was required was that the motion of the sun through the stars throughout the year is irregular.
    • This was discovered by Chinese astronomers in the sixth century.
    • He then tabulated first, second, and third differences of the accumulated difference as in Newton's forward difference interpolation method.
    • List of References (6 books/articles) .
    • A Poster of Guo Shoujing .
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Guo_Shoujing.html .

  24. Zu Chongzhi biography
    • His great grandfather was an official at the court of the Eastern Chin dynasty which had been established at Jiankang (now Nanking).
    • Zu Chongzhi's grandfather and father both served as officials of the Liu-Sung dynasty which also had its court at Jiankang (now Nanking).
    • The Zu family handed their mathematical and astronomical skills down from father to son and, indeed, this was one of the main ways that such skills were transmitted.
    • Zu Chongzhi, in the family tradition, was taught a variety of skills as he grew up.
    • In particular he was taught mathematics, astronomy and the science of the calendar from his talented father.
    • He learnt mathematics from a number of sources, but mainly from Liu Hui's commentary on the Nine Chapters on the Mathematical Art.
    • Zu Chongzhi followed in the family tradition of serving the emperors.
    • During this time Zu worked on mathematics and astronomy.
    • The calendar which had been in use was based on a 19 year cycle with years consisting of 12 months of 29 or 30 days.
    • In seven of the 19 years an extra month was inserted making it a calendar based both on the sun and the moon with 235 months in 19 years.
    • This had been changed in 412 to a calendar based on a 600 year cycle with an extra month inserted in 221 of the years.
    • In 462 Zu proposed a new calendar, the Tam-ing Calendar (Calendar of Great Brightness), to the Emperor which was based on a cycle of 391 years.
    • In 144 of the 391 years an extra month was inserted, so there were 4836 months in 391 years.
    • He was able to make a calendar with this degree of accuracy since he had calculated the length of the tropical year (time between two successive occurrences of the vernal equinox) as 365.24281481 days (an error of only 50 seconds from its true value of 365 days 5 hours 48 minutes 46 seconds), and a nodal month for the moon of 27.21233 days (compare the modern value of 27.21222 days).
    • This was Tai Faxin, one of the Emperor's ministers, who declared that Zu was:- .
    • distorting the truth about heaven and violating the teaching of the classics.
    • However, Xiao-wu died in 464 before the calendar was introduced, and his successor was persuaded by Tai Faxin to cancel the introduction of the new calendar.
    • Zu left the imperial service on the death of Emperor Xiao-wu and devoted himself entirely to his scientific studies.
    • Of course, it is not unreasonable to ask where the numbers 144 and 391 came from.
    • Having accurate knowledge of the lengths of the year and the month were necessary, but it is still not clear how Zu translated this into a cycle of 391 years.
    • and hence the extra month in 144 out of 391 years.
    • Before we leave our discussion of Zu's astronomical work we give further details of his work in this area.
    • He was not the first Chinese astronomer to discover the precession of the equinoxes (Yu Xi did so in the fourth century) but he was the first to take this into account in calendar calculations.
    • Because of the precession of the equinoxes the tropical year is shorter by about 21 minutes than the sidereal year (the time taken by the Sun to return to the same place against the background stars).
    • Zu's calculations of the length of the year were well within the range that allowed him to differentiate between the tropical and sidereal year.
    • He discovered that in 7 cycles of 12 years, Jupiter had completed seven and one twelfth orbits, giving its sidereal period as 11.859 years (accurate to within one part in 4000).
    • He gave the rational approximation 355/113 to in his text Zhui shu (Method of Interpolation), which is correct to 6 decimal places.
    • It is reported in the History of the Sui dynasty, compiled in the 7th century by Li Chunfeng and others, that (see [',' A Kobori, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1] or [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','3] for a different translation):- .
    • Zu Chongzhi further devised a precise method [of calculating ].
    • Taking a circle of diameter 10,000,000 chang, he found the circumference of this circle to be less than 31,415,927 chang and greater than 31,415,926 chang.
    • He deduced from these results that the accurate value of the circumference must lie between these two values.
    • Therefore the precise value of the ratio of the circumference of a circle to its diameter is as 355 to 113, and the approximate value is as 22 to 7.
    • To compute this accuracy for π, Zu must have used an inscribed regular 24,576-gon and undertaken the extremely lengthy calculations, involving hundereds of square roots, all to 9 decimal place accuracy.
    • Martzloff, in [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','3] or [',' J-C Martzloff, Histoire des mathematiques chinoises (Paris, 1987).','4], presents another possible way that Zu might have found 355/113 by luck rather than mathematical skill.
    • In 656, after editing by Li Chunfeng, the treatise Zhui shu (Method of Interpolation) became a text for the Imperial examinations and it became one of The Ten Classics when reprinted in 1084.
    • In the latter part of his life Zu Chongzhi collaborated with his son, Zu Geng (or Zu Xuan), who was also an outstanding mathematician.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (10 books/articles) .
    • A Poster of Zu Chongzhi .
    • Astronomy: A Brief History of Time and Calendars .
    • History Topics: Pi through the ages .
    • History Topics: A chronology of pi .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: The Ten Mathematical Classics .
    • Index of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Zu_Chongzhi.html .

  25. Liu Hui (about 220-about 280)
    • Liu Hui lived in the Kingdom of Wei so it is likely that he worked in what is now the Shansi province in north-central China.
    • The Kingdom of Wei had come about after the Han Empire, which lasted from around 200 BC to 220 AD, collapsed.
    • However, the collapse of the Han Empire led to three Kingdoms coming into existence for, in addition to the Kingdom of Wei, two former Han generals set up Kingdoms, one to the south of the Yangtze and one in the west of China in the present Szechwan Province.
    • This situation lasted for about sixty years, from 220 to 280, which must have been almost exactly the period of Liu Hui's life.
    • The period of the Three Kingdoms was one of almost constant warfare and political intrigue.
    • However this fascinating period is now thought of as the most romantic in all of Chinese history.
    • What influence the events of the period had on Liu Hui is unknown, for nothing is known of his life except that he wrote two works.
    • That no record of Liu Hui's life was written, or at least if it was it was not considered worth preserving, does not mean that he was particularly obscure during his lifetime.
    • Although mathematics was an important topic in China, nevertheless being a mathematician seems to have been considered an occupation of minor importance.
    • As a consequence many Chinese mathematical works are anonymous.
    • The only precise information about Liu Hui comes from a later work which states that he wrote his commentary on the Nine Chapters on the Mathematical Art in the fourth year of the era of the Jingyuan reign of Prince Chenliu of the Wei, which gives a date of 263 AD.
    • One piece of information he gives us about his life in the Preface of the book is:- .
    • What exactly was is the text that Liu Hui is commenting on? It was a practical handbook of mathematics meant to provide methods to be used to solve everyday problems of engineering, surveying, trade, and taxation.
    • Liu Hui himself believed that the text which he was commentating on was originally written around 1000 BC but incorporating much material of later eras.
    • In the past, the tyrant Qin burnt written documents, which led to the destruction of classical knowledge.
    • Later, Zhang Cang, Marquis of Peiping and Geng Shouchang, Vice-President of the Ministry of Agriculture, both became famous through their talent for calculation.
    • Most historians, however, would not believe that the original text of the Nine Chapters was nearly as old as Liu Hui believed.
    • In fact most historians think that Liu Hui was quite wrong in what he wrote, for it is now thought that the text originated around 200 BC after the burning of the books by Shih Huang Ti.
    • First we should note that he introduced a different approach to mathematics from that of the text on which he was commentating.
    • His methods are not exactly "proofs" in our understanding of a mathematical proof today.
    • They are more the type of brief explanation that a mathematician will give to convince you that if you wanted to you could construct a proof.
    • Liu Hui also shows that he understands that some of the methods of the original text are approximations, and he investigates the accuracy of the approximations.
    • This appears in the first chapter of the Nine Chapters.
    • He found a recurrence relation to express the length of the side of a regular polygon with 3 × 2n sides in terms of the length of the side of a regular polygon with 3 × 2n-1 sides.
    • This is achieved with an application of Pythagoras's theorem, which Liu Hui knew as the Gougu theorem.
    • In the diagram we have a circle of radius r with centre O.
    • We know AB, it is pn-1 , the length of the side of a regular polygon with 3×2n-1 sides, so AY has length pn-1/2.
    • √{r[',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','2r - √(4r - pn-12)]}.
    • Then pn= AX is the length of a side of a regular polygon with N = 3 × 2n sides.
    • Putting r = 1 and taking n = 1 gives a regular hexagon of side p6= 1.
    • Then the perimeter of the hexagon is 6p1= 6 giving an approximate value of π as 6p1/2 = 3 (assuming the circumference of the circle is approximately the perimeter of the hexagon and using π = circumference/diameter).
    • In general we obtain an approximate value of π as N pn/2.
    • Larger values of n give more accurate values of π.
    • Liu Hui used the approximation 3.14 which he obtained from taking n = 5, in other words using a regular polygon of 96 sides.
    • In fact Liu Hui stopped one step short of our computer calculation, for he also obtained the better approximation from n = 3072, namely 3.14159.
    • We must emphasise that, of course, Liu Hui did not use algebraic notation as we have done above, nor did he use the number system that we have used.
    • He also understood the notion of a limit.
    • Other interesting examples of Liu Hui's contributions to the Nine Chapters on the Mathematical Art is in Chapter 5 on engineering works, where he computes the volume of various solids such as a prism, pyramid, tetrahedron, wedge, cylinder, cone and frustum of a cone.
    • He fails, however, to find the volume of a sphere which he says he leaves to a future mathematician to compute.
    • This is a small work consisting of nine problems and it was originally written as part of his commentary on Chapter Nine of the Nine Chapters but later removed and made into a separate work by later editors.
    • It shows how to use the Gougu theorem (Pythagoras theorem) to calculate heights of objects and distances to objects which cannot be measured directly.
    • When viewed from X at ground level, 123 pu behind P1, the summit S of the island is in line with the top of P1.
    • Similarly when viewed from Y at ground level, 127 pu behind P2, the top of the island is in line with the top of P2.
    • Calculate the height of the island and its distance from P1.
    • Suppose the poles are of height h and the distance between the poles is d.
    • Liu Hui gives the height of the island as h × d/(P2Y-P1X) + h and the distance to it to be P1X × d/(P2Y-P1X).
    • He then gives: height of the island: 1255 pu; distance from P1 to the island: 30750 pu.
    • Other problems in this work are the height of a tree on the side of a mountain, the distance to a square town, the depth of a gorge, the height of a tower on a hill, the width of a river, the depth of a valley with a lake at the bottom, the width of a ford viewed from a hill, and the size of a town seen from a mountain.
    • Since we have no information about Liu Hui's life, can we at least deduce some information about him from his work? Firstly we can see that he is an outstanding mathematician with a deep understanding of difficult concepts.
    • He is also highly original, coming up with ideas which rank him among the leading mathematicians of all time.
    • But we can deduce more: as the authors of [',' K Shen, J N Crossley and A W-C Lun, The nine chapters on the mathematical art : Companion and commentary (Beijing, 1999).','4] write:- .
    • The techniques Liu employed are typical of a teacher of skill, patience and tireless zeal.
    • Liu Hui was a learned man, not only having great expertise in mathematics but also being familiar with the literary and historical classics of China.
    • He could write with clarity and also with style, quoting from a wide variety of sources.
    • We can also see that he was a modest man who never claimed results of which he was not fully confident, preferring to write:- .
    • He also shows himself to be someone who cared about the conditions of people and also about the economy of the country.
    • This suggests that he may have held high office in the administration of his country, and if he did so then his comments would have us believe that he was very fair in his policies.
    • List of References (28 books/articles) .
    • A Poster of Liu Hui .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: The Ten Mathematical Classics .
    • History Topics: Nine Chapters on the Mathematical Art .
    • Index of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • D B Wagner (Volume of a sphere) .
    • D B Wagner (Volume of a pyramid) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Liu_Hui.html .

  26. Zhang Qiujian biography
    • At least the reader should be aware of this lack of knowledge.
    • The book comprises of three chapters, with 32 problems in the first chapter, 22 problems in the second, and 38 problems in the third chapter.
    • Each problem is followed by an answer and many are followed by a description of the method of determining the answer.
    • However, no reasons are given for the method of solution.
    • There are problems on extracting square and cube roots, problems on finding the solution to quadratic equations, problems on finding the sum of an arithmetic progression, and on solving systems of linear equations.
    • The book certainly represents progress in Chinese mathematics beyond the Nine Chapters on the Mathematical Art.
    • Much of the book is designed to give the reader practice at manipulating fractions.
    • Anyone who studies mathematics should not be afraid of the difficulty of multiplication and division, but should be afraid of the mysteries of manipulating fractions.
    • Zhang asks the reader to divide 12380/7 by 138/5 in Problem 5 of Chapter 1 and to calculate (6587 + 2/3+ 3/4) divided by 58 1/2(= 117/2) in the next problem.
    • He gives the reader the same method of dividing fractions as taught in schools today, namely invert the divisor and multiply.
    • In particular Zhang sees the reduction of fractions to a common denominator as hard.
    • The method Zhang gives is to find the greatest common divisor of 150, 120 and 90.
    • Then divide the length of the road by the greatest common divisor to get 325/30= 10 5/6 days.
    • In Problem 22 of Chapter 2 segments of a circle are considered.
    • The chord of the segment is given, as is its area, and the student is asked to compute its height (the length of the perpendicular bisector of the chord to the circle).
    • Martzloff [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','2] points out that the length as calculated by Zhang in this problem is in error by about 14%.
    • In Chapter 3 problems which involve solving systems of equations occur.
    • For example Problem 4 of Chapter 3:- .
    • There are three persons, A, B, and C each with a number of coins.
    • A says "If I take 2/3 of B's coins and 1/3 of C's coins then I hold 100".
    • B says If I take 2/3 of A's coins and 1/2 of C's coins then I hold 100 coins".
    • C says "If I take 2/3 of A's coins and 2/3 of B's coins, then I hold 100 coins".
    • The method Zhang uses is essentially Gaussian elimination on the matrix of coefficients.
    • Problem 38 of Chapter 3 is perhaps the most famous in the whole of Zhang's treatise:- .
    • 4 cockerels costing a total of 20 qian, 18 hens costing a total of 54 qian and 78 chickens costing a total of 26 qian.
    • 8 cockerels costing a total of 40 qian, 11 hens costing a total of 33 qian and 81 chickens costing a total of 27 qian.
    • 12 cockerels costing a total of 60 qian, 4 hens costing a total of 12 qian and 84 chickens costing a total of 28 qian.
    • We do not know if Zhang had a systematic method to solve problems of this type or whether he solved them by trial and error.
    • In 656, after editing by Li Chunfeng, the treatise Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual) became a text for the Imperial examinations and it became one of The Ten Classics when reprinted in 1084.
    • We can certainly deduce that Zhang was a fine teacher and may well have had experience in teaching, to be able to write with so much knowledge of how students learn.
    • List of References (8 books/articles) .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: Chinese problems .
    • History Topics: The Ten Mathematical Classics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Zhang_Qiujian.html .

  27. Li Rui biography
    • When Li Rui was a young boy, about sixteen years old, he studied Cheng Dawei's Suanfa tong zong (General source of computational methods) written in 1592.
    • In 1788 he graduated from the Yuanhe province school and went on to study mathematics at the Ziyang Academy in Suzhou.
    • At the Ziyang Academy Li Rui was taught mathematics by the director of the Academy, Qian Daxin (1728-1804).
    • It was the first of many postscripts which he would write.
    • In 1795 Ruan Yuan became a minister in the Education Department of Zhejiang province.
    • He was based in Hangzhou and had an ambitious programme to compile a massive work of biography on mathematicians and astronomers.
    • He gathered a team round him to undertake this task and two members of the team were Qian Daxin and Li Rui.
    • Li Rui now undertook a number of important tasks.
    • Li Rui also wrote a commentary on another of Li Zhi's works, namely Yi gu yan duan (New steps in computation) which was originally written in 1259.
    • This was a work by Qin Jiushao which presents a general algorithm for the solving simultaneous congruences, that is the Chinese remainder theorem.
    • From 1795 when he went to assist Ruan Yuan in Hangzhou until the Chouren zhuan or Biographies of astronomers and mathematicians was published in 1799 Li Rui contributed to the work and to several other works.
    • In addition to those listed above, he wrote the Hu Shi Suan Shu Xi Cao (Commentary of Calculations of Arcs and Segments), the Zhong Ke Ce Yuan Hai Jing Xi Cao, and the Ri Fa Shuo Yu Qiang Ruo Kao (Studies of Denominator of Tropical Year).
    • After the compilation was completed Li Rui and Jiao Xun studied mathematics together around 1800.
    • Li Rui continued to contribute to commentaries produced by others, and in 1802 he wrote a postscript to Commentary of Qi Gu Suan Jing by Zhang Dunren (1754-1834).
    • Although the name of Li Rui's wife is not known, we do know that she died in the year he wrote this postscript.
    • Li Huang was one of the commentators on the Nine Chapters on the Mathematical Art.
    • Li Rui failed his examinations, but he made an important discovery in Li Huang's home for there he found a copy of the Yang Hui Suan Fa ("Yang Hui's Method of Computation").
    • This was a copy of commentary of a book originally written in 1275 which had been lost.
    • In 1813 while working on his edition of the lost work by Yang Hui, Li Rui wrote the Kai Fang Shuo (Theory of Equations of Higher Degree).
    • He was a highly productive mathematician at the height of his abilities, making very useful contributions to Chinese mathematics, when sadly he died in 1817.
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Li_Rui.html .

  28. Xiahou Yang biography
    • Nothing is known of Xiahou Yang except as the supposed author of the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual).
    • However, according to [',' B Qian (ed.), Ten Mathematical Classics (Chinese) (Beijing, 1963).','3], there is no doubt that the Xiahou Yang suanjing was not written by Xiahou Yang.
    • The dating of the work is fairly well pinned down.
    • We have comments by Zhang Qiujian which criticise the accuracy of one of the solutions given in the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual).
    • On the other hand a change of volume standard which took place in 425 in mentioned in the text so it must have been written after that date.
    • The dates we give cannot therefore be more than 40 years in error for the author of the work.
    • The treatise contains three chapters in the usual style of problems and solutions.
    • One significant idea which appears in the text concerns representation of numbers in the decimal notation.
    • What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10.
    • Although Xiahou Yang has no symbol for 0 in an empty place, there is good evidence from his description of moving numbers to the right and left that he at least has a virtual zero in mind despite the lack of a symbol.
    • Certainly when Li Chunfeng came to edit this text to make it a suitable text for the Imperial examinations, which it became in 656, he had to correct some of the problems.
    • In particular some of the problems dealing with areas in the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) use incorrect formulae.
    • List of References (6 books/articles) .
    • History Topics: Chinese numerals .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: The Ten Mathematical Classics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Xiahou_Yang.html .

  29. Wang Yuan biography
    • Wang Yuan spent the first seven years of his childhood in Lanhsi which at that time was in an area benefiting from modernization programs, but things changed in 1937 when war broke out between China and Japan.
    • This is where Wang Yuan's family moved in 1937, leaving Chekiang province before the Japanese occupied much of it which they did by 1938.
    • It lies close to Chungking (also called Ch'ung-ch'ing, or Chongqing) the largest city of Szechwan province.
    • In 1942 Wang Yuan's father became the chief secretary in the Academia Sinica, the Chinese national research organization.
    • Wang Yuan enrolled in the National Second Middle School in Ho-ch'uan (also called Ho-yang, Hechuan, or Heyang), still in Szechwan province, However, in 1946 his family moved to Nanking (also called Nan-ching or Nanjing) the capital of Kiangsu province where he attended the middle school attached to the Social Education College.
    • He graduated in 1948 and entered Yingshi University to study mathematics.
    • After one year of study Yingshi University became part of Chekiang University (founded in 1897) in Hangchow.
    • This was fortunate since Chekiang University had a strong mathematics department with an excellent seminar organised for graduate students.
    • Wang Yuan fell in love with analytic number theory and gave a series of lectures to the graduate seminar based on Ingham's book The distribution of prime numbers.
    • Wang Yuan graduated in 1952 and was assigned a position by the government in the Institute of Mathematics at the Academia Sinica in Nanking.
    • Wang Yuan was assigned to the number theory section where he worked under Professor Hua Loo Keng, the director of the Institute.
    • Most of Wang Yuan's research has been in the area of number theory.
    • In 1956 he published (in Chinese) On the representation of large even integer as a sum of a prime and a product of at most 4 primes in which he assumed the truth of the Riemann hypothesis and with that assumption proved that every large even integer is the sum of a prime and of a product of at most 4 primes.
    • He also proved that there are infinitely many primes p such that p + 2 is a product of at most 4 primes.
    • In 1957 Wang Yuan published four papers: On sieve methods and some of their applications; On some properties of integral valued polynomials; On the representation of large even number as a sum of two almost-primes; and On sieve methods and some of the related problems.
    • In the first of these he proved, among other results, that for infinitely many integers n, n3 + 2 has at most 4 prime factors.
    • In the third paper he proved that every even integer is the sum of two integers each of which has at most 5 prime factors.
    • The last of the four 1957 papers proved, this time assuming the Riemann hypothesis, that every sufficiently large even integer is the sum of a prime and a product of at most 3 primes, and there are infinitely many primes p such that p + 2 is the product of at most 3 primes.
    • In 1958 On sieve methods and some of their applications.
    • I showed that every even integer is the sum of two integers, one of which has at most 2 prime factors, the other having at most 3 prime factors.
    • He also attacked other questions in number theory in papers such as On the least primitive root of a prime (1959) and On Diophantine approximations and numerical integrations.
    • Also in 1964 he published two papers on orthogonal Latin squares: A note on the maximal number of pairwise orthogonal Latin squares of a given order; and On the maximal number of pairwise orthogonal latin squares of order s, an application of the sieve method.
    • The Chinese Communist Party chairman Mao Zedong launched the Cultural Revolution in August 1966.
    • From 1966 to 1972 he did no mathematical research but after 1972 he resumed his research activities, although never as vigorously as before the start of the Cultural Revolution.
    • However he did write a number of books such as: (with Hua Loo Keng) Applications of number theory to numerical analysis (1978); Goldbach Conjecture (1984); (with Hua Loo Keng) Popularising mathematical methods in the People's Republic of China (1989); Diophantine equations and inequalities in algebraic number fields (1991); (with Fang Kai-Tai) Number theoretic methods in statistics (1994); Hua Loo Keng (1995); and (with Fong Yuen) Calculus (1997).
    • In 1978 Wang Yuan was promoted to professor at the Institute of Mathematics at the Academia Sinica and elected to membership of the Academia Sinica in 1980.
    • In 1984 he became director of the Institute of Mathematics at the Academia Sinica.
    • He was elected as president of the Chinese Mathematical Society during 1988-92.
    • Finally let us note that he became interested in the history of mathematics and published Analytic number theory in China in 2001.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (3 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Wang_Yuan.html .

  30. Frank Jackson biography
    • Frank Jackson was the son of William and Alice Jackson.
    • He was an outstanding pupil and, since his parents were not in a good position financially to support their son, he was encouraged to try for the award that Peterhouse, University of Cambridge, was offering to allow early entry of talented young students.
    • Jackson was one of the first victims of this idea.
    • We have noted above that Jackson wanted to read classics at Cambridge, but his father thought that mathematics would lead to his son gaining better paid employment so Jackson began to study the Mathematical tripos.
    • Being a Wrangler meant that his performance had been first class but Jackson was ranked lowest in the list of Wranglers and so had no real opportunity to continue his study of mathematics at Cambridge.
    • Jackson now decided that he would enter the Church and he was ordained, becoming a Curate of Bemerton, Salisbury in 1896.
    • He had, however, already published his first mathematical paper with Theorems in the products of related quantities appearing in the Proceedings of the Edinburgh Mathematical Society in 1895.
    • Over the following couple of years he published two further papers in the same Proceedings, namely A certain linear differential equation (1896) and Certain expansions of xn in hypergeometric series (1897).
    • On 20 June 1900, when the vessel was in Castellammare di Stabia, in the Bay of Naples, Italy, telegraphic orders were received ordering the vessel to sail for Chinese waters.
    • The Boxer movement was a Chinese nationalist militia who were opposed to foreign rule and opposed to Christian missionaries in China.
    • Three weeks before HMS Dido left Italy, troops from eight countries had already begun to deploy to confront the Boxers who were attacking Christian churches, Chinese Christians and Chinese officials.
    • The Boxer Rebellion was put down by 1901 and Jackson received the China Medal (1900) for his part, as did many other members of the crew of HMS Dido.
    • He left the navy in 1907 and, in the following year, was appointed as a Curate at Christ Church, Isle of Dogs.
    • He served as Curate at Featherstone in Yorkshire from 1910 to 1912 when he was appointed Vicar of Thornton-le-Street and North Otterington, Yorkshire.
    • His appointment as Vicar in 1912 allowed him to marry Elizabeth Lucy Bernarda Mulhern, the daughter of Edward Bernard Mulhern of Tunbridge Wells.
    • He wrote much on basic hypergeometric functions, including the basic functions of Legendre and Bessel.
    • Bruce Berndt writes [',' B C Berndt, What is a q-series, University of Illinois at Urbana-Champaign.','1]:- .
    • Two English mathematicians, Frank H Jackson and L James Rogers, at the end of the 19th and beginning of the 20th centuries devoted most of their mathematical careers to further developing the theory of q-series, but their efforts were not appreciated by their contemporary researchers.
    • The Bible in the theory of basic hypergeometric series is the text 'Basic Hypergeometric Series' by G Gasper and M Rahman.
    • Readers of this book will find many results due to Jackson and rightly conclude that he, indeed, is one of the founders of the subject.
    • In the first of these three papers what is now called the 'Jackson integral' appears.
    • It is the operation in the theory of special functions that expresses the inverse to q-differentiation.
    • The first appearance of this integral was in Jackson's earlier paper A generalization of the functions G(n) and xn (1904).
    • On 21 May 1919, Jackson was inducted as rector of Chester-le-Street, St Mary Parish.
    • In 1929 he published the book The Collegiate Church of the Blessed Virgin Mary & S.
    • He continued as rector of Chester-le-Street until 1935 but, from 1925, he was also Rural Dean.
    • He only published a couple of mathematics papers between 1917 and 1941 although, as we have just seen, he wrote a book during this time.
    • However, although over the age of seventy, he published four papers in 1941-42 all in the Oxford Quarterly Journal of Mathematics.
    • Several of Jackson's early papers were published by the London Mathematical Society but after 1905 he never published a paper with that Society.
    • Once (with a whimsical smile, one imagines) he recounted the occasion of his quarrel with our Society [the London Mathematical Society]: he had read a paper, when someone remarked: "Surely, Mr President, we have heard all this before." He strode from the room and never darkened our pages again.
    • He had a ceaseless and vivid imagination which (to this writer) seemed often to tax his powers of exposition.
    • We enjoyed a long and wide-ranging correspondence from the days when he first began to write for the 'Quarterly Journal of Mathematics', in which he showed himself a very human and humorous personality.
    • When one of his parishioners protested, "But surely, rector, there can be two opinions on this matter", he was met with the retort, "Not in this parish".
    • Yet to those who met him he was, despite his extreme deafness, a charming companion, versatile in conversation out of his rich store of experience.
    • In later life his interests included pottery and Greek coins, of which he was both knowledgeable and appreciative: his interest in classical remains was stimulated while he held the living of Chester-le-Street.
    • To the end he continued to meditate on mathematics: in some of his last letters he proposed to explore infinity through a use of basic numbers, but he left some of his readers doubting whether this could be mathematics or mythology.
    • On our bookshelves we have a first edition of Charles Babbage's 1864 book Passages from the life of a philosopher.
    • The book has had a number of owners, but one of them was "F H Jackson" so we add to this biography a photo of his signature on this book .
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (5 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Jackson_Frank.html .

  31. Takebe Katahiro biography
    • He had an older brother Takebe Kataaki (1661-1716) who was also a mathematician and the two brothers were both pupils of Takakazu Seki.
    • Let us note now that when we speak of Takebe in this article, we are referring to Takebe Katahiro and if we refer to his brother we shall use his full name Takebe Kataaki.
    • Takebe was only thirteen years of age when he became Seki's pupil and both brothers remained with their teacher until his death in 1708.
    • Not only did the brothers gain much from Seki's teaching but they also had access to his extensive library of Japanese and Chinese books on mathematics.
    • They were the main craftsmen of Seki's project (launched 1683) to record mathematical knowledge in an encyclopaedia.
    • The 'Taisei sankei' (Comprehensive Classic of Mathematics), in 20 volumes, was finally completed by Takebe Kataaki in 1710.
    • It gives a good picture of Seki's skill at reformulating problems, as well as Takebe Katahiro's ability to correct, perfect, and extend his master's intuitions.
    • He held the position of examiner of accounts to Tokugawa Ienobu, the Lord of Kofu.
    • When his lord became heir to the Shogun, the hereditary military dictator of Japan, Seki became Shogunate samurai and in 1704 was given a position of honour as master of ceremonies in the Shogun's household.
    • This brought Takebe into contact with the leading members of the Tokugawa family.
    • He was an officer to Tokugawa Bakuhu, and then he was close to Tokugawa Ienobu, the Lord of Kofu, as he rose to become Shogun in 1709.
    • From 1695 when he began his close association with members of the Tokugawa family, Takebe spent less time on his study of mathematics.
    • Takebe's enthusiasm for the study of mathematics and astronomy was invigorated again from 1716.
    • Encouraged by Takebe, Yoshimune relaxed the edict forbidding the introduction of foreign books, including scientific books, which led to the growth of interest in Western science in Japan.
    • Tatebe was only nineteen years old when he published his first mathematics book the Kenki Sanpo Ⓣ (1683).
    • Tatebe improved and extended the methods of his teacher and applied them to a wide range of problems in these books.
    • Takebe had made a careful study of Zhu Shijie's Chinese text Suanxue qimeng Ⓣ published in 1299 and the work had been a great help to him in developing his theory of polynomials.
    • In 1690 Takebe published an annotated Japanese translation of the Suanxue qimeng Ⓣ which he intended as a text for students of mathematics.
    • In 1683 Seki started a project to compile an encyclopaedia of mathematics.
    • This 20 volume work appeared in 1710 with the title Taisei sankei (Comprehensive Classic of Mathematics).
    • None of the work was written by Seki himself and it is clear that the first twelve volumes were written by Takebe.
    • One of the most significant ideas introduced in Takebe's part of the text is the method of Enri, which is definite integration.
    • However, modern research leads most historians to claim that the method of Enri was in fact due to Takebe.
    • The most important of Takebe's work is Tetsujutsu Sankei Ⓣ.
    • Morimoto writes in [',' M Morimoto, Differentiation and Integration in Takebe Katahiro’s Mathematics, Sixth International Symposium on the History of Mathematics and Mathematical Education Using Chinese Characters (University of Tokyo, 2005), 131-143.','10]:- .
    • In 1722, Takebe wrote the 'Tetsujutsu Sankei' (Mathematical Treatise on the Technique of Linkage) to explain how mathematical research could be done in accordance of one's inclination, based on 12 examples of mathematical investigation.
    • In Chapter 2 of this work, Takebe explains the "method of celestial element" which Zhu Shijie had introduced in Suanxue qimeng Ⓣ as a method of representing a polynomial in one variable on a counting board.
    • The extension of this method to polynomials with variable coefficients, due to Seki and Takebe, was presented in Chapter 6 of the Tetsujutsu Sankei Ⓣ.
    • Although Takebe did not explicitly have the operation of differentiation, nevertheless, he stated a result in Chapter 6 which is equivalent to the statement that if a cubic polynomial takes an extreme value at a point the derivative vanishes at that point.
    • We should note, however, that Ogawa writes in [',' T Ogawa, On a Calculation of an Extremum by Takebe Katahiro (Japanese), Yokkaichi University Journal of Environmental and Information Sciences 2 (2) (1999), 247-267.','22]:- .
    • The purpose of this paper is to consider the essence of mathematics of Takebe Katahiro (1664-1739) by investigating his method of finding the maximum in 'Tetsujutsu Sankei' (1722).
    • It has been said that he had first done a calculation of a derivation for finding the maximum in Chapter 6 of the treatise, but that is not strictly true.
    • A close look at the chapter will reveal that his method has nothing to do with the theory of differential.
    • Perhaps Takebe's greatest achievement was to devise a method to calculate a series expansion of a function.
    • Ancient Greek mathematicians had been perplexed by the problem of squaring the circle and in the 17th century Japanese mathematicians looked at a similar problem, namely the problem of finding a polynomial which expressed the length s of an arc of a circle subtended by a chord with sagitta k.
    • The sagitta is the line from the midpoint of the chord to the midpoint of the arc of the circle it subtends.
    • Now we know that no such polynomial exists but Takebe found an infinite series expressing s in terms of k, namely .
    • In fact, looked at in modern terms, what Takebe was calculating was the Taylor series expansion of arcsin(√k)2 about k = 0.
    • We have not yet mentioned the result for which many know Takebe's name, namely his calculation of π.
    • He describes in Chapter 9 of the Tetsujutsu Sankei Ⓣ the right way to proceed if one wants to calculate the circumference of a circle of a given diameter.
    • He writes (in Morimoto's translation from [',' M Morimoto, Differentiation and Integration in Takebe Katahiro’s Mathematics, Sixth International Symposium on the History of Mathematics and Mathematical Education Using Chinese Characters (University of Tokyo, 2005), 131-143.','10]):- .
    • If he who decomposes the circumference of a circle cuts the diameter equally and horizontally into thin slices, seeks the [length of the] right and left oblique chords cut by the horizontal lines and adds the oblique chords to seek the [approximate] circular circumference, then the parts of circumference are not equal even if he cuts the diameter equally.
    • Therefore, if he seeks the circumference doubling the sections of the diameter, these numbers being disobedient to the attribute, he stagnates in determining the extreme number and never obtain a basis to understand the attribute of circle.
    • On the other hand, when he cuts the circumference into the four angular forms [i.e., by an inscribed square] and further doubling angles [i.e., forming an inscribed octagon, etc.], the circumference is cut into equal length and the numbers are obedient to the attribute of circumference.
    • Therefore, doubling the number of angles and seeking the angular circumferences at each step, by the repeated application of the procedure of incremental divisor he can determine the extreme number rapidly and obtain a basis to understand the attribute of a circle.
    • Now of course this gives a method to approximate π but the method converges very slowly.
    • Takebe invented a method for the acceleration of convergence using a technique for the successive removal of various powers of the argument, in this case the error term.
    • Let us give two quotations dealing with this part of his thought; first the description by Annick Horiuchi in [',' Biography in Encyclopaedia Britannica.','1]:- .
    • He distinguished two ways of solving a mathematical problem (and two corresponding types of mathematicians): an "investigation based on numbers," an inductive approach that involves scrutinizing and manipulating data until one finds a general law; and an "investigation based on principle," a reasoned approach that involves directly utilizing rules and procedures, as in algebra.
    • His procedure for calculating the infinite series played a key role in the development of analysis in Japan in the following decades.
    • In view of mathematical methodology and the Neo-Confucianism of the Song and Yuan dynasties, this paper discusses the essential character of 'Tetsujutsu' which runs through 'Tetsujutsu Sankei', and Takebe's mathematical thought and methodology reflected from the 'Jishitsu Setsu' at the end of the book.
    • The paper comes to the conclusion that Takebe's 'Tetsujutsu' is, actually, the colligation of the methods of deduction and induction, in which Takebe pays more attention to the method of induction, but he ignores the proof of precisely relying on the method of deduction.
    • Based on our analysis, we conclude that the mathematical thought behind the author's preface and the 'Jishitsu Setsu' of the 'Tetsujutsu Sankei' is connected to the mainstream philosophy of Zhu Xi (1130-1200) and Wang Shouren (1472-1528) of the time, rather than having nothing to do with the philosophical thought, which is the view of some scholars.
    • Furthermore, the origin of Takebe's mathematical thought may be traced back to the Neo-Confucianism of the Song and Yuan dynasties in China.
    • presents the contribution in astronomy and calendar science of Japanese mathematician Takebe, who worked around 1720-1730 on those subjects.
    • The author explains in particular the work of Takebe on two topics: (i) the variation of the tropic year, i.e.
    • the time between two consecutive winter solstices; (ii) the way of calculating the position of the polar star in the sky.
    • By this study, the author draws the conclusion that Takebe (whose work is still unpublished, and partly lost) had a role in the history of scientific thought, as his aim was to improve, by the assistance of geometry and mathematics, calculation techniques so that astronomy and calendar science could become, in Japan too, exact sciences.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (32 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Takebe.html .

  32. Hsien C Wang (1918-1978)
    • On 7 July 1937 Japanese and Chinese troops clashed near Peking.
    • Wang had to journey to the new site of his university and begin his studies again.
    • Perhaps the political events had a positive effect as far as mathematics was concerned since Wang changed his studies to mathematics when he took them up again at the re-established university.
    • On his return to China, Wang took up a research post at the Chinese National Academy of Sciences.
    • The Chinese National Academy of Sciences was set up on Taiwan and Wang followed the Academy there.
    • This was not an easy time to obtain a mathematics post in the United States and Wang, although he had an impressive reputation as a mathematician by this time, could only manage a succession of temporary posts.
    • Again he held temporary posts, this time for two years at Alabama Polytechnic, then 1954-55 at Princeton again, 1955-57 at the University of Washington in Seattle followed by a time at Columbia in New York.
    • The year 1957 saw Wang receive an offer of a permanent post for the first time.
    • Wang also solved, at that time, an important open problem in determining the closed subgroups of maximal rank in a compact Lie group.
    • Wang's most important work was on discrete subgroups of Lie groups, a topic on which he continued to work.
    • He published Two-point homogeneous spaces in 1952 which dealt with a homogeneous space of a compact Lie group.
    • In 1960 he studied transformation groups of n-spheres and wrote the highly original paper Compact transformation groups of Sn with an (n-1)-dimensional orbit.
    • The latter part of Wang's life is described in [',' W M Boothby, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1] as follows:- .
    • Wang's last paper was published in 1973, after which his research was much curtailed because of anxiety for his wife, who had developed cancer.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (5 books/articles) .
    • Index of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Wang.html .

  33. Luoxia Hong biography
    • Luoxia Hong lived in the time of the Former Han dynasty.
    • In particular, as part of his efforts to improve the administration, he decided to introduce a new calendar.
    • We should say a little about why calendar reform was such a frequent event in China while in Europe there has only been one major reform of the calendar.
    • Changing the calendar was seen as one of the duties of the office, establishing the emperor's heavenly link on earth.
    • After a change of ruler, and even more significantly after a change of dynasty, the new Chinese emperor would seek a new official calendar thus establishing a new rule with new celestial influences.
    • Luoxia Hong was one of the astronomers who went to Chang'an in answer to Emperor Wu-ti's request.
    • Some of the astronomers were already part of Wu-ti's civil service, being imperial astronomers.
    • Luoxia, however, lived in the southwest of China.
    • Emperor Wu-ti received eighteen proposals for a new calendar, and he judged the best one was that received from Luoxia Hong and one of his colleagues Deng Ping.
    • There were 12 months of 29 or 30 days and the calendar was based on a cycle of 19 years.
    • In seven of the 19 years an extra month was inserted making it a calendar based both on the sun and on the moon.
    • Luoxia Hong's calendar was much more than simply an attempt to bring the sun and moon into a common system for he also gave predictions for the positions of the planets and predictions of eclipses.
    • List of References (2 books/articles) .
    • Astronomy: A Brief History of Time and Calendars .
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Luoxia_Hong.html .

  34. Ivor Etherington biography
    • Baptists are much interested in the departure of one of their members Mr Bruce Etheringyon, B.A.
    • as a missionary to Ceylon, the son of the Rev W D Etherington, M.A.
    • of Babbicombe, for 17 years missionary in India.
    • He carries with him a Kodak camera, a memento of his friends, and the special wishes of the Sunday School and Y.P.S.C.E.
    • with both of which he was identified.
    • Bruce Etherington married Annie Margaret Ferguson who born about 1873 in Ceylon (now Sri Lanka) but, at the age of seven, on 3 April 1881, she was living in Hougham, Kent, England.
    • Bruce Etherington returned to England from Ceylon on the 4th April 1905 [',' Bruce Etherington, Western Times Newspaper (20 April 1905).','2] but the family, consisting of Bruce, Annie Margaret, and their two children E Margaret Etherington (born about 1904) and Bruce Etherington Jr, were back in Ceylon when Bruce Etherington Sr.
    • At this time Annie Margaret was carrying Ivor, the subject of this biography, and she returned to England with her two children.
    • The unveiling of a commemorative plaque to Bruce Etherington was reported in the Western Times Newspaper on 12 October 1908 [',' Bruce Etherington, Western Times Newspaper (12 October 1908).','3]:- .
    • Mention was made of the zeal and enthusiasm of the late Mr Etherington, who was himself the son of a former Indian missionary.
    • By the end of 1916, Ivor was the sixth of a family of eight.
    • Life in a large family of lively intelligent people formed the grounding of Ivor's education.
    • Their house was full of gadgets which stimulated infectious curiosity.
    • Unfortunately it was not allowed to be used until after lunch on Sundays! He showed early signs of his lifelong interest in mathematics when he entertained himself during his stepfather's sermons by factorising the hymn numbers.
    • Edwin made use of his engineering expertise during the years of World War I, working for the Ministry of Munitions in addition to his work as a Baptist Church minister.
    • In 1922, Ivor was sent to Mill Hill School, North London, where he quickly developed a love for mathematics under the expert teaching of the Senior Mathematics Master, Herbert Coates.
    • In 1927 Etherington matriculated at Hertford College, Oxford where he studied mathematics.
    • His tutor was Bill Ferrar who had been appointed to Oxford in 1925 after spending two years as a Senior Lecturer at the University of Edinburgh working under Edmund Whittaker.
    • Etherington was a first class student in the examinations at the end of his first year but after this his time to study became restricted because he became Secretary of the Hertford College branch of the Student Christian Movement.
    • His upbringing had seen him surrounded by a deeply religious family and Etherington felt that he had to undertake Christian duties out of respect to his family, although his own attitude towards religion was that of a sceptic.
    • in the general theory of relativity advised by Whittaker.
    • by the University of Edinburgh in 1932 for his thesis On Relativistic Cosmology, and the Definition of Distance in General Relativity.
    • Although he was working on general relativity, he published the following two papers in 1932: On errors in determinants and A simple method of finding sums of powers of the natural numbers.
    • He published a paper, On the definition of distance in general relativity, based on his doctoral thesis, in the Philosophical Magazine in 1933.
    • In recent papers Professor E T Whittaker and H S Ruse have discussed the problem of defining, in a general Riemannian space-time, the concept of distance between two particles, as distinct from that of interval (or integrated line element) between two events.
    • Ruse's procedure is purely mathematical, being a natural extension of the concept of spatial distance in Special Relativity.
    • Whittaker and Tolman, on the other hand, related their definitions to the astronomical methods of calculating great distances, such as those of the extragalactic nebulae.
    • These methods depend ultimately on a comparison of absolute and apparent brightness, it being assumed that brightness decreases with the square of the distance; or, alternatively, on a similar comparison of absolute and apparent size.
    • It is the purpose of the present paper to investigate definitions which translate that procedure more exactly.
    • Perhaps the most remarkable fact about this paper is that 74 years later, it was thought worthy of republication.
    • This is a reprint of a beautiful paper by I M H Etherington, written in the early times (1933) of the theory of general relativity.
    • It deals with the very important issue of defining, in the general relativistic context (i.e., in an arbitrary space-time), the notion of (spatial) distance between two particles, this being relevant in the areas of observational astrophysics and cosmology.
    • The paper emerged in the context of an on-going discussion amongst Whittaker, Ruse and Tolman on this central concept.
    • Using very unsophisticated mathematical tools and concepts (normal coordinates and the very notion of tensorial invariance), Etherington managed to prove what is nowadays called the 'reciprocity theorem for null geodesics', which states, roughly speaking, that many geometric properties are invariant when the roles of the observer and the source are exchanged in astronomical observations, this holding in any general space-time, regardless of its geometry.
    • After the award of his doctorate, Etherington was appointed to Chelsea Polytechnic where he taught for the year 1932-33.
    • After only one year, he was appointed to a lectureship at the University of Edinburgh and taught there for 41 years until he retired at the age of 66 in 1974.
    • To emphasise his achievements we note that, at the age of 26, he was elected to the Royal Society of Edinburgh on 5 March 1934.
    • Ivor and Betty Etherington [',' T A Gillespie, Ivor Malcolm Haddon Etherington BA(Oxon), PhD, DSc(Edin), Royal Society of Edinburgh Yearbook 1995 (1995), 101-.','8]:- .
    • As the situation in Europe deteriorated, their home became a sanctuary to a large number of refugees seeking to start new lives beyond the reach of Nazi persecution.
    • The Etheringtons, with the help of anyone else they could involve, managed to effect the escape of some 32 people from Germany.
    • Betty even undertook a journey to Germany herself just before the outbreak of war to smuggle back the belongings of escaping refugees.
    • Etherington's friendship with Ffoulkes Edwards was to set the path for his research for the rest of his career.
    • They begin the first of these papers writing:- .
    • Several theories of the inheritance of human blood groups have been proposed, but none has been completely satisfactory.
    • The importance of this work for Etherington's future research was that it led him to study non-associative algebras which he called genetic algebras.
    • For a list of Etherington's papers see THIS LINK.
    • To see the motivation for Etherington's research in this area we quote from the introduction to his paper Non-associative algebra and the symbolism of genetics published by the Royal Society of Edinburgh in 1941:- .
    • The statistical material of genetics usually consists of frequency distributions - of genes, zygotes and mating couples - from which new distributions referring to their progeny arise.
    • Combination of distributions by random mating is usually symbolised by the mathematical sign for multiplication; but this sign is not taken literally for the simple reason that the genetical laws connecting the distributions of progenitors and progeny are inconsistent with the laws governing multiplication in ordinary algebra.
    • However, there is no insuperable reason why the genetical sign of multiplication should not be taken literally; for it is possible with any particular type of inheritance to construct an "algebra" - distinct from ordinary algebra but of a type well known to mathematicians - such that the laws governing multiplication shall represent exactly the underlying genetical situation.
    • These "genetic algebras" are of a kind known as "linear algebras," ..
    • It is not suggested that the use of ordinary algebraic methods in conjunction with the specific principles of genetics will not lead to correct results.
    • It seems, however, that the systematic use of genetic algebras would simplify and shorten the way to their attainment, and perhaps enable much more difficult problems to be tackled with equal ease.
    • The construction of genetic algebras has been described in a somewhat abstract way in a previous paper (Etherington, 1939), ..
    • Here I propose to consider the symbolism more from the geneticist's point of view, applying it to some simple population problems, without going into the details of the mathematical background.
    • It will be recognised that the current treatment of such problems does in reality make use of genetic algebras without noticing them explicitly.
    • I wish that this thesis may not be judged as a finished achievement in biological investigations but may be judged primarily as a contribution to algebra, suggested by biological problems, and perhaps having possibilities of applications beyond the simple ones so far demonstrated.
    • He explained what train algebras were in the introduction to Non-commutative train algebras of ranks 2 and 3 (1951):- .
    • They are not necessarily of finite order.
    • Train algebras arose in the study of genetic algebras, which provide simplified mathematical models of the transmission of genes in sexual reproduction.
    • In 1981, in the review [',' I M H Etherington, Non-associative algebra and the symbolism of genetics, Proc.
    • I initiated the study of genetic algebras in several papers between 1939 and 1951 ..
    • I was intrigued by the unusual properties of these algebras, which were quite unlike anything in the literature.
    • I also thought that, from the point of view of geneticists, the subject had a future.
    • My belief was fortified by kindly letters from J B S Haldane and Lancelot Hogben, and still more by the appearance of R.
    • Schafer's seminal paper in 1962 in 'Mathematica Scandinavica'; also by Olav Reiersol's paper in 1949 in the 'American Journal of Mathematics'.
    • Our colleague, Colin Campbell, was a student at the University of Edinburgh in the early 1960s and was taught by Etherington.
    • After he retired in 1974 the University gave him the title of the title Professor Emeritus.
    • He served the Edinburgh Mathematical Society during his career, as secretary for ten years between 1933 and 1944, as editor of the Edinburgh Mathematical Notes and as President of the Society in 1947-48.
    • He was awarded the Keith Prize by the Royal Society of Edinburgh in 1958.
    • He was a faithful attender of the EMS Colloquia in St Andrews as the set of portraits above shows.
    • Let us end this biography by giving a quote from [',' T A Gillespie, Ivor Malcolm Haddon Etherington BA(Oxon), PhD, DSc(Edin), Royal Society of Edinburgh Yearbook 1995 (1995), 101-.','8] concerning Etherington's interests outside mathematics:- .
    • He had a wide range of interests in all manner of things, a sharp intellect and an impish sense of humour.
    • He had a reading knowledge of seven foreign languages, not to mention Esperanto, and harboured ambitions to read (though not to speak) Chinese.
    • He obtained some books on Chinese and a Chinese mathematical dictionary, confessing in typically modest fashion to making just a little progress with the language.
    • He also had a lifelong love of music and particularly enjoyed playing the piano, especially Beethoven sonatas.
    • He had learnt the rudiments of piano-playing from his friend John Ffoulkes Edwards and started to teach himself, but later received more formal instruction on the insistence of his step-grandmother.
    • List of References (10 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Etherington.html .

  35. Jacob Gool biography
    • Jacob Gool is better known by the Latin version of his name, Jacobus Golius.
    • Dirck Gool had been a resident of the city of Leiden and had lived through the siege of that city by the Spanish from May until October 1574.
    • Dirck Gool held various important administrative positions in the Court of Holland in The Hague and was first clerk in the State Council.
    • Later in his life he was an amanuensis to Constantin Huygens, the father of Christiaan Huygens.
    • Constantin Huygens was, in fact, born in the same year as Jacob, the subject of this biography.
    • Little is known of Jacob's mother, Anna, but it is recorded that Jacob was named after the pastor Jacob Starck who was a great uncle of his mother.
    • Both Willebrord Snell and his father Rudolph Snell (1546-1613), the professor of mathematics at Leiden, were teaching mathematics at Leiden when Golius began his studies.
    • Also teaching mathematics at the university was Frans van Schooten The Elder, father of the better known Frans van Schooten.
    • Golius was also taught by Aelius Everhardus Vorstius (1565-1624) who had become an extraordinary professor of Natural Philosophy at Leiden in 1598, and a full professor of Natural History and Medicine in the following year.
    • Golius initially focused on the study of medicine, mathematics and astronomy.
    • However, during his studies he became fascinated reading the texts of the ancient Greek mathematicians, particularly the Conics by Apollonius of Perga.
    • In Arabic, however, the first seven of the eight books of Conics survive.
    • Thomas van Erpe (1584-1624), who is also known as Thomas Erpenius, was appointed professor of Arabic and other Oriental languages in Leiden in February 1613.
    • From 1618 he began teaching Arabic to Golius who became extremely enthusiastic about the language and also became a close friend of van Erpe.
    • Golius, who eventually became an expert on Persian, Turkish, Armenian, and Chinese, wanted to extend his knowledge of Oriental languages by travelling to Oriental countries.
    • In 1622-24 he was sent by the Dutch States General to Morocco to work as an engineer, being a member of a mission headed by Albert Ruyl.
    • Golius was responsible for an investigation into the condition of the Bay of Agadir and asked to study the feasibility of building a harbour there.
    • He visited Muley Zidan, Sultan of Morocco, who had become Sultan in 1594 and spent much of his time in the city of Safi, on the coast about 250 km north of the Bay of Agadir, where he worked with Muslim scholars.
    • (For information on this scholar, see [',' G Wiegers, A life between Europe and the Maghrib: the writings and travels of Ahmad b.
    • Qasim al-Hajari al-Andalusi, in G J H van Gelder and Ed de Moor, The Middle East and Europe: Encounters and Exchanges (Rodopi, Amsterdam-Atlanta, GA, 1992), 87-115.','8].) Ahmad ben Qasim had made copies of some manuscripts in his own hand and at least one of these is now in the Leiden University Library.
    • Van Erpe died in 1624 and Golius was appointed to succeed him in 1625 as the professor of Arabic and other Oriental languages at Leiden.
    • He was particularly interested in visiting Mesopotamia with its strong reputation for ancient studies in mathematics, astronomy and medicine.
    • His learning, especially as a physician and astronomer, impressed the scholars of Constantinople.
    • During his travels Golius collected a large number of Oriental manuscripts which he brought back to Leiden.
    • This gave Leiden the largest collection of Oriental manuscripts anywhere in Europe and aroused a great deal of interest from scholars all over Europe.
    • A list of about 300 titles of these Arabic, Turkish and Persian works were listed in 1630 in Paris in a catalogue compiled by Pierre Gassendi.
    • His extensive personal collection of manuscripts is now held in the Bodleian Library in Oxford, England.
    • The manuscripts arrived there after they were purchased by Narcissus Marsh, Archbishop of Armagh, in 1696 who then left them to the Bodleian.
    • Willebrord Snell died in 1626 and, in 1629 Golius was appointed as professor of mathematics at Leiden in addition to his position as professor of Arabic and other Oriental languages.
    • Cornelis Schoneveld writes [',' C W Schoneveld, Sea-changes: Studies in Three Centuries of Anglo-Dutch Cultural Transmission (Rodopi, 1996).','3]:- .
    • In the autumn of 1633, when walking to the academy building, to the botanical garden adjacent to it, or to the library ..
    • housed in the church-building across the Rapenburg canal, [one] must have heard the noise made by the carpenters on the roof of the academy erecting a small wooden shed-like structure for the use of Jacobus Golius, Professor not only of Arabic but of Mathematics.
    • He needed it as a kind of observatory "to demonstrate to the students the course of the heavens and the stars".
    • Golius used this observatory to make good quality observations of lunar eclipses, comets and planets.
    • He also lectured on mathematics, teaching his students about the advances made by the Arabs, particularly the decimal place-value number system.
    • He also taught his students about the ancient Greek mathematics, usually using the Arabic versions of the Greek texts.
    • One of his contributions is related in [',' H T Zurndorfer, Sociology, Social Science, and Sinology in the Netherlands before World War II: with special reference to the work of Frederik van Heek, in Sociologie de la Chine et Sociologie chinoise, Revue Europeenne des Sciences Sociales XXVII (84) (1898), 19-32.','9]:- .
    • Golius's interest in chronology led him to investigate the "Catayan" system of twelve branches or duodecimal cycles from a Persian work of the 15th century.
    • With the aid of Heurnius's dictionary [which Golius had in his library] and Martini's advice, he was to publish the cycle of twelve in Chinese characters.
    • It was the first instance of Chinese letters printed (from wood) in Europe.
    • Let us give some details of two of the people mentioned in this quote.
    • In around 1630 Golius married Rensburg van der Goes, the daughter of Matthias van der Goes and Aleyt van Beveren.
    • Golius became known as the "father of Arabic literature".
    • Through Golius's work the scope of Persian studies, as they had been pursued by Dutch Arabists since the end of the 16th century, was widened.
    • In the course of the preparation of his famous "Lexicon Arabico-Latinum' (Leiden, 1653), which is partly based on Persian and Turkish lexicographical sources, he also assembled dictionaries for Persian and Turkish.
    • Further details of his contributions in this area are given in [',' Review: Georgii Wilhelmi Freytagii, Lexicon Arabico-Latinum, praesertim ex Djeuharii Firuzabadiique et aliorum Arabum operibus, adhibitis Golii quoque et aliorum libris, confectum, The North American Review 48 (103) (1839), 461-478.','7].
    • The "Arabic Grammar" of Erpenius, first published in 1636 ..
    • had given a new impulse to the study of the Arabic, and prepared the philologists of Holland and Germany to demand a work like that of Golius.
    • He, instead of republishing and enlarging Anton Giggaeus 'Thesaurus Linguae Arabicae (1632), wisely chose to translate the Arabic lexicon of Jouhari, partly, that the public might thus have both the great native Arabic lexicons in a European dress, and partly for the sake of the passages from various authors which are cited in that work.
    • Jouhari, also, being by birth a Turk, gives his definitions and difficult forms in a way to suit the wants of a foreigner more completely than any native could have done.
    • We note that Jouhari's lexicon was one which gave definitions of Arabic words in Arabic, in the same way that the OED gives the meaning of English words with an English description.
    • However, what Golius produced with his translation of Jouhari was a lexicon in which the definition of an Arabic word was given in Latin.
    • We have already explained above how many of Golius's manuscripts ended up in the Bodleian Library in Oxford, England.
    • The Leiden University would have liked to have been able to make a good offer for Golius's books and manuscripts but they could not make any realistic offer due to a lack of funds.
    • However, one of Golius's students, Levinus Warner (1618-1665), who had studied Middle Eastern languages with Golius, was inspired by him to become a collector of ancient Middle Eastern books and manuscripts.
    • When he died in 1665 he left his private collection of over 900 manuscripts to Leiden University.
    • List of References (9 books/articles) .
    • A Poster of Jacob Gool .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Golius.html .

  36. Wadysaw Orlicz (1903-1990)
    • In 1919 Orlicz's family moved to Lvov (Lwow in Polish which is now Lviv in Ukraine), where he completed his secondary education and then studied mathematics at the Jan Kazimierz University in Lvov having Stefan Banach, Hugo Steinhaus and Antoni Lomnicki as teachers.
    • From 1922 to 1929 he worked as a teaching assistent at the Department of Mathematics of Jan Kazimierz University in Lvov.
    • In 1928 he wrote his doctoral thesis Some problems in the theory of orthogonal series under the supervision of Eustachy Żyliński.
    • Orlicz spent the academic year 1929/30 at Gottingen University on a scholarship in theoretical physics, not in mathematics.
    • It should be emphasized that from the functional analysis point of view (that is, as function spaces) Orlicz spaces appeared for the first time in 1932 in Orlicz's paper: Uber eine gewisse Klasse von Raumen vom Typus B Ⓣ in Bull.
    • In 1934 he was granted the habilitation (venia legendi) for a thesis entitled Investigations of orthogonal systems.
    • The group gained international recognition and was later described as the Lvov School of Mathematics.
    • A collection of 193 mathematical problems from meetings at the Scottish Cafe appeared later on as the Scottish Book.
    • Orlicz is the author or co-author of 14 problems there.
    • (R D Mauldin edited The Scottish Book, Mathematics from the Scottish Cafe (Birkhauser 1981) which contains problems and also commentaries on them by specialists).
    • You can see a picture of the Scottish Cafe at THIS LINK.
    • In 1968 when presenting the mathematical output of Steinhaus (in an article published in Wiadom.
    • In Lvov under the leadership of our dear masters Banach and Steinhaus we were practising intricacies of mathematics.
    • He was professor at the State University of Iwan Franko from January 1940 to June 1941 and from August 1944 to February 1945 he also taught at the school of commerce and handicrafts and lectured at forestry courses.
    • In March 1945 Orlicz went back to Poland and in May 1945 he returned to University of Poznań.
    • Until his retirement in 1974 he worked both at the University of Poznań and the Mathematical Institute of the Polish Academy of Sciences, Poznań Branch.
    • Orlicz continued his seminar Selected Problems of Functional Analysis until 1989.
    • He was interested in works of other mathematicians and in branches far removed from functional analysis.
    • 4 (1961), 257 or [',' H Steinhaus, Between spirit and matter mediate mathematics (Polish) (Warsaw-Wrocław, 2000).','2], 242): .
    • Mazur and Orlicz are direct pupils of Banach; they represent the theory of operations today in Poland and their names on the cover of "Studia Mathematica" indicate direct continuation of Banach's scientific program.
    • Altogether Orlicz published 171 mathematical papers, about half of them in cooperation with several authors.
    • He was the supervisor of 39 doctoral dissertations and over 500 master's theses.
    • Orlicz participated in congresses of mathematics in Oslo (1936), Edinburgh (1958), Stockholm (1962) and Warsaw (1983), and in many scientific conferences.
    • His book Linear Functional Analysis, (Peking 1963, 138 pp - in Chinese), based on a series of lectures delivered in German on selected topics of functional analysis at the Institute of Mathematics of the Academia Sinica in Beijing in 1958, was translated into English and published in 1992 by World Scientific, Singapore.
    • Orlicz is also a co-author of two school textbooks.
    • Orlicz was the editor of Commentationes Mathematicae (1955 - 1990), and of Studia Mathematica (1962 - 1990), and President of the Polish Mathematical Society (1977 - 1979).
    • In 1956 Orlicz was elected a corresponding-member of the Polish Academy of Sciences and in 1961 its full member.
    • Three universities (York University in Canada, Poznań Technical University and Adam Mickiewicz University in Poznań) conferred upon him the title of doctor honoris causa, in 1974, 1978 and 1983, respectively.
    • Orlicz was awarded many high state decorations, prizes as well as medals of scientific institutions and societies, including the Stefan Banach Prize of the Polish Mathematical Society (1948), the Golden Cross of Merit (1954), the Commander's Cross of Polonia Restituta Order (1958), Honorary Membership of the Polish Mathematical Society (1973), the Alfred Jurzykowski Foundation Award (1973), Copernicus Medal of the Polish Academy of Sciences (1973), Order of Distinguished Teacher (1977), Wacław Sierpiński Medal of the Warsaw University (1979), Medal of the Commission for National Education (1983) and the Individual State Prizes (second degree in 1952, first degree in 1966).
    • Orlicz's contribution is important in the following areas in mathematics: function spaces (mainly Orlicz spaces), orthogonal series, unconditional convergence in Banach spaces, summability, vector-valued functions, metric locally convex spaces, Saks spaces, real functions, measure theory and integration, polynomial operators and modular spaces.
    • Orlicz spaces Lφ = Lφ (Ω, Σ, μ) are Banach spaces consisting of all x ∈ L0(Ω, Σ, μ) such that ∫Ω φ(λ|x(t)|)dμ(t) < ∞ for some λ = λ(x) > 0 with the Orlicz norm: .
    • Orlicz spaces Lφ are a natural generalization of Lp spaces.
    • Orlicz's ideas have inspired the research of many mathematicians.
    • In recent decades those spaces have been used in analysis, constructive theory of functions, differential equations, integral equations, probability, mathematical statistics, etc.
    • monographs on Orlicz spaces: M A Krasnoselskii and Ya B Rutickii, Convex Functions and Orlicz Spaces (Groningen 1961), J Lindenstrauss and L Tzafriri, Classical Banach Spaces I, II (Springer 1977, 1979), C Wu and T Wang, Orlicz Spaces and their Applications, (Harbin 1983 - Chinese), A C Zaanen, Riesz Spaces II, (North-Holland 1983), C Wu, T Wang, S Chen and Y Wang, Theory of Geometry of Orlicz Spaces (Harbin 1986 - Chinese), L Maligranda, Orlicz Spaces and Interpolation, (Campinas 1989), M M Rao and Z D Ren, Theory of Orlicz Spaces (Marcel Dekker 1991) and S Chen, Geometry of Orlicz Spaces (Dissertationes Math.
    • The term Orlicz spaces appeared in the sixties in the Mathematics Subject Classification index of the American Mathematical Society in Section 4635, which is now 46E30, Spaces of measurable functions (Lp-spaces, Orlicz spaces, etc.
    • To emphasize the importance of Orlicz spaces in a jocular way, Professor Orlicz used to say that when he was occasionally asked: .
    • The answer of an employee was: .
    • Orlicz's name is associated not only with the Orlicz spaces but also with the Orlicz-Pettis theorem, Orlicz property, Orlicz theorem on unconditional convergence in Lp, Mazur-Orlicz bounded consistency theorem, Mazur-Orlicz theorem on inequalities, Mazur-Orlicz theorem on uniform boundedness in F-spaces, Orlicz category theorem, Orlicz interpolation theorem, Orlicz norm, Orlicz function, convexity in the sense of Orlicz, F-norm of Mazur-Orlicz, Drewnowski-Orlicz theorem on representation of orthogonal additive functionals and modulars, Orlicz theorem on Weyl multipliers, Matuszewska-Orlicz indices, Hardy-Orlicz spaces, Marcinkiewicz-Orlicz spaces, Musielak-Orlicz spaces, Orlicz-Sobolev spaces and Orlicz-Bochner spaces.
    • For example, the Orlicz-Pettis theorem says that in Banach spaces the classes of weakly subseries convergent and norm unconditionally convergent series coincide.
    • In 1988, on the occasion of Orlicz's 85-th birthday, Polish Scientific Publishers (PWN) published his Collected Papers [',' Wladyslaw Orlicz Collected Papers I, II (Warsaw, 1988).','3] in two volumes with a total of 1754 pages, reproducing 141 his articles from 1926-1985.
    • Kuratowski [',' K Kuratowski, A half century of Polish mathematics (Warsaw, 1980).','1, p.
    • 40] has written about creation of the Polish School of Mathematics stating that: .
    • Orlicz's scientific achievements are presented in detail in the papers by Maligranda-Matuszewska [',' L Maligranda and W Matuszewska, A survey of Wladyslaw Orlicz’s scientific work, In: Wladyslaw Orlicz Collected Papers, (Warsaw 1988) xv-liv.','7], Maligranda-Wnuk [','L Maligranda and W Wnuk, Wladyslaw Orlicz (1903-1990) (Polish), Wiadom.
    • 36 (2000), 85-147.','11] and Maligranda [',' L Maligranda, Wladyslaw Orlicz (1903-1990) - his life and contribution to mathematics (Polish), In: Wladyslaw Orlicz (1903-1990) - Founder of the Poznan School of Mathematics, Poznan 2001, 33-79.','6].
    • 36 (2000), 85-147.','11] contains a complete list of Orlicz's publications (171 papers and 3 books).
    • In the late seventies Orlicz started to collect information about mathematicians from Lvov and he was planning to write a book on the History of the Lvov School of Mathematics (he published only two articles: The Lvov School of Mathematics between the Wars, Wiadom.
    • 23(1981), 222-231 and Achievements of Polish Mathematicians in the Domain of Functional Analysis in the Years 1919 - 1951, and biographies of S Banach, S Kaczmarz, A Lomnicki, S Mazur, J P Schauder).
    • Orlicz died on 9 August, 1990 in Poznań when correcting the galley proofs of his last paper accepted for publication in Mathematica Japonica.
    • Three conferences were organized in the memory of Władysław Orlicz: .
    • Orlicz Memorial Conference (March 21 - 23, 1991) by the University of Mississippi in Oxford, USA; .
    • Function Spaces V (28 August to 2 September, 1998) by the University of A.
    • Mickiewicz in Poznań, Poland (paper [','L Maligranda and W Wnuk, Wladyslaw Orlicz: his life and contributions to mathematics, In: Function Spaces (New York-Basel 2000), 23-29.','10] appeared in the proceedings of this conference); .
    • Scientific Session in the Memory of Professor Władysław Orlicz (September 27-29, 2000) by the University of A.
    • Mickiewicz and the Institute of Mathematics of the Polish Academy of Sciences in Bedlewo, Poland (the proceedings include paper [',' L Maligranda, Wladyslaw Orlicz (1903-1990) - his life and contribution to mathematics (Polish), In: Wladyslaw Orlicz (1903-1990) - Founder of the Poznan School of Mathematics, Poznan 2001, 33-79.','6] with over forty photos of Orlicz).
    • Article by: Lech Maligranda, Lulea University of Technology, Sweden.Click on this link to see a list of the Glossary entries for this page .
    • List of References (15 books/articles) .
    • Picture of the Scottish Cafe .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Orlicz.html .

  37. Wei-Liang Chow biography
    • He did, however, receive private tutoring in the Chinese language and in Chinese history before going to the United States.
    • He attended Asbury College in Wilmore, Kentucky and then studied at the University of Kentucky.
    • He next went to the University of Chicago where he was awarded his B.S.
    • It was only during his years in Chicago that he made the decision to concentrate his efforts on the study of mathematics.
    • Having decided to study mathematics there was, without doubt, no better place to go than Gottingen in Germany.
    • However, arriving there in the autumn of 1932 he was soon affected by the rise to power of Hitler and the serious consequences for the mathematics department there due to the racialist policies of the National Socialist party.
    • It was van der Waerden who introduced Chow to algebraic geometry at this time, pointing him towards the work of Severi, Bertini and Enriques.
    • Chow went to Hamburg for a vacation in the summer of 1934 and there he met Margot Victor.
    • Also in July 1936 he married Margot Victor in Hamburg and the newly married couple went to China where Chow began teaching at the National Central University in Nanking in September of that year.
    • In the same volume of Mathematische Annalen he published a paper with the same name as his doctoral thesis.
    • If Chow and his family had been in a difficult position in Germany, forced to leave by the racial policy of the Nazi regime, but he was soon in an equally bad position in Nanking.
    • The Sino-Japanese War began with a minor clash between the troops of the two countries near Peking in July 1937.
    • In August there was savage fighting in Shanghai, but Chow decided that Shanghai was safer than Nanking so in September he escaped from Nanking to Shanghai (which was of course the city of his birth).
    • By the end of December the Japanese had captured both Nanking and Shanghai.
    • Chow had published three further papers in 1939 one of which was Uber Systeme von linearen partiellen Differentialgleichungen erster Ordnung in which he extended results by Caratheodory on Pfaffian systems.
    • In 1940 he published On electric networks in the Journal of the Chinese Mathematical Society but at this stage he found conditions in Shanghai so difficult that it became impossible for him to continue to study mathematics.
    • 43 (10) (1996), 1117-1118.','3] which is reprinted in [',' S S Chern and V V Shokurov (eds.), The collected papers of Wei-Liang Chow (World Scientific Publishing Co., Inc., River Edge, NJ, 2002).
    • In a decade of war years Wei-Liang had practically stopped his mathematical activities, and the question was whether it was advisable or even possible for him to come back to mathematics.
    • His return to mathematics was most successful; I would consider it a miracle.
    • The 'miracle', of course, was partly due to Chern for without his help and encouragement it must rate as extremely unlikely that Chow would ever have even attempted to return to mathematics.
    • He remained there until September 1948 when he joined the Mathematics Faculty of Johns Hopkins University in Baltimore, Maryland, having been recommended by van der Waerden.
    • He served as chairman of the Mathematics Department from 1955 to 1965.
    • Luca Barbieri Viale gives a beautiful survey of Chow's later mathematical achievements in a review of [',' S S Chern and V V Shokurov (eds.), The collected papers of Wei-Liang Chow (World Scientific Publishing Co., Inc., River Edge, NJ, 2002).
    • In 1955 Chow proved the so-called "Chow's moving lemma" in algebraic geometry, providing an intersection theory for algebraic cycles based on ideas and results of Severi, later also developed by van der Waerden, Hodge and Pedoe.
    • The original 1956 Annals of Mathematics paper follows the general setting of Weil's Foundations.
    • In this paper, Severi's idea of intersection by moving cycles in a suitable equivalence class is proven to be appropriate for rational equivalence classes of cycles.
    • Such equivalence classes make up the "Chow ring" of a nonsingular projective variety and provide the algebraic counterpart of the topological singular cohomology ring.
    • Actually, wonderful developments of this analogy were included in Grothendieck's theory of motives, where algebraic cycles provide correspondences between algebraic varieties and their intersections provide their composition, yielding the category of "Chow motives".
    • Notably, "Chow motives" are naturally included in Voevodsky's triangulated category of motives, showing the deepest roots of this analogy.
    • Igusa, who was at Johns Hopkins University from 1955, writes about Chow's leadership in [',' J Igusa, In memory of Prosessor Wei-Liang Chow, Notices Amer.
    • It is a historical fact that this school of algebraic geometry [at Johns Hopkins University] was 'created' by Chow.
    • We often saw how many of those ideas evolved into beautiful theorems.
    • We became almost like relatives - some of us even spent summers together with the Chows at China Lake, Maine.
    • He was fascinating to listen to about his personal history.
    • In addition to his research and leadership of the algebraic geometry group, Chow played an important role as editor-in-chief of the American Journal of Mathematics from 1953 to 1977.
    • It was typical of Chow that every thing he did was at the highest level of expertise.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (8 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Chow.html .

  38. Wilhelm Klingenberg biography
    • Wilhelm was the oldest of his parents' six children.
    • Wilhelm attended schools in Berlin where he learnt Latin, Greek and French but he had to study mathematics on his own.
    • He applied to enter the University of Berlin but this was not permitted and he had to serve in the army.
    • He writes in [',' Wilhelm P A Klingenberg, Selected papers of Wilhelm P A Klingenberg (Singapore, 1991).','1]:- .
    • When the end of the war finally gave me my freedom, I changed my handwriting and started looking for a place to study.
    • The devastated and Soviet occupied city of Berlin was out of the question, Gottingen and Hamburg were filled up, so I went to Kiel University.
    • from 1950 to 1952 he was a research assistant at Kiel where F Bachmann interested him in the foundations of geometry.
    • At this time he solved a problem on equivalences of configurations in an affine plane which Ruth Moufang had worked on.
    • Blaschke advised him regarding trips to Italy and he spent time in 1952/53 at the University of Rome where he was strongly influenced by F Severi, E Bompiani and Beniamino Segre.
    • He wrote in [',' Wilhelm P A Klingenberg, Selected papers of Wilhelm P A Klingenberg (Singapore, 1991).','1]:- .
    • I have fond memories of our years there - Reidemeister had a brilliant mind and a wide range of interests, his wife Elisabeth was a renowned photographer.
    • He also spent 1962 at the University of California at Berkeley at the invitation of S S Chern.
    • Klingenberg wrote in [',' Wilhelm P A Klingenberg, Selected papers of Wilhelm P A Klingenberg (Singapore, 1991).','1]:- .
    • While at Berkeley, Klingenberg received offers of chairs at Wurzburg and Mainz - he chose Mainz.
    • and some of the intimate charm of a close-knit group thereby went down the drain.
    • Of this last work Klingenberg comments:- .
    • It was the first book on this subject since the monograph of L P Eisenhart in 1926.
    • Among the honours he has received, we mention his election to the Academy of Sciences and Literature in Mainz, and the award of an honorary doctorate by the University of Leipzig.
    • He gives his hobbies as: piano, horse-back riding, Chinese art and the art of Albrecht Durer.
    • His interest in Chinese art led to him making trips to China and Tibet.
    • This in turn led to his publication of the book Tibet - Erfahrungen auf dem Dach der Welt Ⓣ published first in 1997 and in a paperback version in 2001:- .
    • He goes by the old pilgrim routes and knows the life of the yak herders.
    • With each trip he became more and more under the spell of this unique civilization where the great edifice of Lamaism was built.
    • [The book provides] an unusual blend of practical information and personal travel log ..
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Klingenberg.html .

  39. Pao Lu Hsu biography
    • His family came from the lake city of Hangzhou in Zhejiang Province but he was born and brought up in Peking.
    • Perhaps because of this background he spoke the "common" or "official" dialect with an interesting soft overtone.
    • Hsu's school education was in Peking and he did not choose mathematics as a career at this stage but rather it was chemistry which he decided to study at university.
    • Two years later he decided to change subject, to change universities, and so he went to Tsing Hua University to read for a degree in mathematics.
    • He obtained his Bachelor of Science degree from Tsing Hua University in 1933 and then moved to the Mathematics Department of Peking University where he was employed as an assistant.
    • Egon Pearson, following the retiral of his father Karl Pearson as Galton Professor of Statistics, had been made Reader and became Head of the Department of Applied Statistics three years before Hsu arrived there.
    • Jerzy Neyman had been appointed in 1934 while R A Fisher held Karl Pearson's Galton Chair of Statistics and was Head of the Department of Eugenics at University College.
    • During this period [at University College, London] Hsu wrote a remarkable series of papers on statistical inference which show the strong influence of the Neyman-Pearson point of view.
    • One concerned what is now known as the Behrens-Fisher problem, while the second Hsu examined the problem of optimal estimators of the variance in the Gauss-Markov model.
    • He was awarded the degree of Ph.D.
    • and then that of D.Sc.
    • [Hsu's] British education formed his taste in mathematics; he preferred the hard and concrete to the general and abstract.
    • This is a very fair comment on the style of British statistics during this period, in contrast to the style in Continental Europe.
    • Hsu chose to leave Britain to return to his homeland of China where he was appointed as Professor at Peking University.
    • It was a period of great difficulty and hardship for Hsu.
    • Many of his publications on multivariate analysis from this period show that he had been strongly influenced by R A Fisher while at University College.
    • His role in promoting the use of matrix theory in statistics should also be emphasised.
    • the forefront of the development of the mathematical theory of multivariate analysis.
    • During the next two years he taught at the University of California, Columbia University, and the University of North Carolina where he was offered an associate professorship.
    • After spending 1946-47 at the University of North Carolina at Chapel Hill, in 1947 Hsu returned to his professorship at Peking University.
    • One of his students at Chapel Hill wrote:- .
    • In was Hsu's insistence on simplicity combined with depth of understanding, clarity without avoidance of difficulties, and above all a deep and obvious but unspoken commitment to the highest goals and standards of scholarship which attracted us to him.
    • 7 (3) (1979), 467-470.','5] (see also [',' T W Anderson, K L Chung and E L Lehmann, Pao Lu Hsu : 1910-1970 (Chinese), Knowledge Practice Math.
    • A particular hobby was chanting the musical drama of the Yuan dynasty with a small group of connoisseurs accompanied by ancient instruments.
    • Hsu had turned down many offers of position, one particularly attractive one from Wald, from universities in the United States but he felt that there he could be [',' T W Anderson, K L Chung and E L Lehmann, Pao Lu Hsu : 1909-1970, Ann.
    • part of the emerging new society in his homeland.
    • He recovered but the extremely hard work which he undertook brought about a recurrence in the summer of 1951 and he spent some time in hospital.
    • with a blackboard hanging on the wall of his room, he gave lectures to upper-class students, graduate students, and young teachers; he was in charge of seminars and other academic activities.
    • By the early 1960s his health had deteriorated to such an extent that he could stand in front of the blackboard for only a few minutes before he had to sit down and rest.
    • Found beside his bed the day after his death were piles of manuscripts which serve as a testimony to the superhuman fortitude with which he exerted himself over a period of more than 20 years ..
    • Hsu died in his home on the campus of Peking University in 1970.
    • He had published a total of 40 mathematical papers.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (11 books/articles) .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Hsu.html .

  40. Zu Geng biography
    • He was the son of Zu Chongzhi and so came from a famous family who had for many generations produced outstanding mathematicians.
    • Zu Geng's great great grandfather was an official at the court of the Eastern Chin dynasty which had been established at Jiankang (now Nanking).
    • Zu Geng's great grandfather, grandfather, and father Zu Chongzhi, all served as officials of the Liu-Sung dynasty which also had its court at Jiankang (now Nanking).
    • They handed their mathematical and astronomical skills down from father to son and, indeed, this was one of the main ways that such skills were transmitted in China at this time.
    • Zu Geng, in the family tradition, was taught a variety of skills as he grew up.
    • In particular he was taught mathematics by his talented father Zu Chongzhi.
    • Zu Geng's greatest achievement was to compute the diameter of a sphere of a given volume.
    • We know of this work through the commentary of Li Chunfeng on the Nine Chapters on the Mathematical Art .
    • The final two problems in Chapter 4: Short Width ask for the diameter of a sphere of given volume.
    • Given a volume of 1644866437500 cubic chi, tell what is the diameter of the sphere.
    • His Rule states: "Double the given volume and extract its cube root, and we have the diameter of the sphere." But why should this be? .
    • (Notice of course, that the formula presented assumes that π = 3.) The proof is based on what is now called the principle of Liu Hui and Zu Geng, namely:- .
    • The volumes of two solids of the same height bear a constant ratio if the areas of the plane sections at equal heights have the same ratio.
    • This is a generalisation of what is often called Cavalieri's principle.
    • He now only has to consider one of the small cubes.
    • He makes two dissections of the small cubes by cylindrical cuts, obtaining 4 smaller pieces.
    • He then applies his principle to find the volumes of his pieces.
    • After showing how Zu Geng used the principle to justify the formula for the volume of a sphere, Li Chunfeng goes on to give Zu Geng's more precise form:- .
    • According to the precise rate, the volume of the sphere is the cube of the diameter multiplied by 11 and divided by 21.
    • This is significant, for Zu Geng has seen the link between squaring the circle and cubing the sphere, and has been able to translate progress on the first of these two problems into progress on the second.
    • As a final comment let us note that Zu Geng was certainly not the last member of the famous Zu family to make important contributions.
    • Zu Geng's son, Zu Hao, was also well known as a mathematical astronomer making contributions to the science of the calendar.
    • List of References (9 books/articles) .
    • History Topics: Overview of Chinese mathematics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Zu_Geng.html .

  41. Kurt Mahler biography
    • Hermann Mahler had become an apprentice bookbinder, working his way up to become the owner of a small printing and bookbinding firm.
    • In this respect he was following the Mahler family tradition of working in the printing and bookbinding trade.
    • Of the six older children, Lydia married a printer, Josef took over Hermann's printing firm but died in a Nazi concentration camp during World War II, while the other four children died young.
    • Mahler recalled that when he was a child, their home was [',' J W S Cassels, Obituary of Kurt Mahler, Acta Arithmetica 58 (3) (1991), 215-228.','1]:- .
    • Although there was no academic tradition in the family, the [',' J W S Cassels, Obituary of Kurt Mahler, Acta Arithmetica 58 (3) (1991), 215-228.','1]:- .
    • children acquired a love of reading from their father.
    • In particular, there was an elementary book on geometry which he would not then understand, but of which he liked to copy the figures.
    • Kurt contracted tuberculosis at the age of five years and, as a result, had severe problems with his right knee.
    • Because of these health problems he attended school for only four years leaving at Easter 1917 at the age of 13.
    • Of course, this led to him being introduced to a certain amount of mathematics which he loved [',' J H Coates and A J Van der Porten, Kurt Mahler 1903-1988, Historical Records of Australian Science 9 (1993), 369-385.','3]:- .
    • He very quickly decided that mathematics was what he really liked doing.
    • Already, from the summer vacation of 1917, he began teaching himself logarithms (the arithmetic properties of which turned out to be one of his abiding interests in transcendental number theory) plane and spherical trigonometry, analytic geometry and calculus.
    • In 1918, at the age of fifteen, he took a job as an apprentice in a machine factory in Krefeld.
    • He worked there for almost three years, the first of which he spent in the drawing office, the rest of the time being spent working in the factory itself.
    • He was self-taught in mathematics teaching himself while working in the factory.
    • His father sent small articles his son had written to Josef Junker, the head of the local high school.
    • He received some help from the local high school teachers for the papers on German, French, and English which he was required to take, continuing to study mathematics on his own.
    • 85 (1) (1983), 50-53.','10] is an essay Why I have a special liking for mathematics written in 1923.
    • is an enthusiastic and moving confession of the author's interest in mathematics.
    • It is followed by a brief account of his (unusual and remarkable) educational experience up to the beginning of his university studies.
    • By the time that Mahler had passed his Abitur, Carl Siegel had moved to the University of Frankfurt and he arranged for him to study there.
    • At Frankfurt, supported financially by his parents and several members of the Krefeld Jewish community, he attended lectures by Max Dehn on topology, Ernst Hellinger on elliptic functions, Carl Siegel on calculus and Otto Szasz [',' J H Coates and A J van der Poorten, Kurt Mahler, Biographical Memoirs of Fellows of the Royal Society of London 39 (1994), 265-279.','2]:- .
    • In 1925 Siegel left Frankfurt for a period of overseas visits, and Mahler moved to Gottingen where he attended lectures by Emmy Noether, Richard Courant, Edmund Landau, Max Born, Werner Heisenberg, David Hilbert and Alexander Ostrowski, and acted as an unpaid assistant to Norbert Wiener.
    • It was through lectures by Emmy Noether that he learnt about p-adic numbers which were to be one of the major topics of his research throughout his life.
    • In 1927 he submitted his doctoral dissertation Uber die Nullstellen der unvollstandigen Gammafunktion, gamma function ]',4608)">Ⓣ, to the University of Frankfurt.
    • During his tenure of the fellowship [',' J W S Cassels, Obituary of Kurt Mahler, Acta Arithmetica 58 (3) (1991), 215-228.','1]:- .
    • he developed a new method in transcendental theory, found his celebrated classification of transcendental numbers and pioneered diophantine approximation in p-adic fields.
    • In 1933 Mahler was appointed to his first post at the University of Konigsberg but, before he could take up the post, Hitler came to power.
    • After visiting van der Corput, and his two pupils Koksma and Popken, in Amsterdam during the summer of 1933, he accepted an invitation from Louis Mordell to go to Manchester where he spent 1933-34 supported by a Bishop Harvey Goodwin Fellowship.
    • He attended the International Congresses of Mathematicians at Oslo in July 1936 where he met Paul Erdős for the first time.
    • Erdős writes [',' P Erdős, Some personal and mathematical reminiscences of Kurt Mahler, Austral.
    • I knew of his work many years earlier and was very glad to meet him.
    • I was a bit crestfallen since I felt that I should have thought of this myself.
    • He underwent several operations on his knee back home in Krefeld, where eventually his kneecap was removed, and also spent some time in Switzerland during the summers of 1937 and 1938 where he was finally cured.
    • However, despite being cured, he walked with a limp for the rest of his life.
    • Mahler returned to Manchester in 1937, where again he interacted with Erdős [',' P Erdős, Some personal and mathematical reminiscences of Kurt Mahler, Austral.
    • However, during 1940 he was interned as "an enemy alien" for three months and spent some time in the same camp on the Isle of Man as Kurt Hirsch.
    • He had learnt Chinese in 1939 in preparation for getting a post in China, but the onset of the war prevented this being realised.
    • Returning to Manchester after his period of internment, he was appointed as an assistant lecturer in 1941 (his first permanent post) and remained there until 1962.
    • Cohn writes [',' J W S Cassels, Obituary of Kurt Mahler, Acta Arithmetica 58 (3) (1991), 215-228.','1]:- .
    • The only other mathematician I found there was Professor Mahler, so we saw a good deal of each other for the next six years.
    • He was without any pretensions and one could discuss anything under the sun with him, though preferably mathematics, photography or Chinese (in that order).
    • For his holidays he always went to the island of Herm, saying that it suited him because he could not walk far.
    • His attitude to mathematics was like his attitude to life: he liked things as simple as possible and usually eschewed abstraction, but with his direct methods was often able to go surprisingly far.
    • In 1962 he went to Canberra for the last 6 years of his career.
    • One of the undergraduate students who took this course was John Coates.
    • After officially retiring from Canberra in 1968, Mahler accepted an invitation from Ohio State University where he worked until 1972 when he returned to Canberra [',' J W S Cassels, Obituary of Kurt Mahler, Acta Arithmetica 58 (3) (1991), 215-228.','1]:- .
    • Erdős writes [',' P Erdős, Some personal and mathematical reminiscences of Kurt Mahler, Austral.
    • I visited Australia fairly often and of course always visited Mahler in Canberra.
    • I had dinner and lunch with Mahler at University House and met him in the Department of Mathematics of the Institute for Advanced Studies.
    • He worked on transcendence of numbers, showing in 1946 that .
    • In [',' A J van der Poorten, Obituary of Kurt Mahler, J.
    • Mahler regretted that, apart from his own work, little interest had been shown by 20th century mathematicians in the study of arithmetical properties of decimal expansions.
    • Other major themes of his work were rational approximations of algebraic numbers, p-adic numbers, p-adic Diophantine approximation, geometry of numbers (a term coined by Minkowski to describe the mathematics of packings and coverings) and measure on polynomials.
    • He published a number of excellent books, but these were all based on lecture courses he had given, often using notes taken by someone who attended the course.
    • For example Lectures on diophantine approximations : g-adic numbers and Roth's theorem (1961) was prepared from notes by R P Bambah of lectures given by Mahler at the University of Notre Dame in autumn 1957 and was described as an "extremely valuable contribution".
    • This set of notes contains an elementary introduction to the theory of p-adic numbers and their analysis.
    • These numbers were introduced by K Hensel some eighty years ago and have slowly become of importance in more and more parts of mathematics.
    • Nevertheless, while many recent books on algebra have short chapters or paragraphs on the subject, a really good introduction to p-adic numbers from the standpoint of elementary analysis does not seem to exist.
    • We shall begin by studying the g-adic rings and p-adic fields, and then finally investigate continuous and differentiable functions of a p-adic variable.
    • A second, improved and expanded, edition of the book was published in 1981:- .
    • The text is written in complete detail, sometimes several proofs of a result are given and lots of concrete examples and special cases are calculated (the results of which are also useful for the specialist).
    • He was elected a Fellow of the Royal Society in 1948.
    • He was elected a fellow of the Australian Academy of Sciences in 1965 and awarded its Lyle Medal in 1977.
    • He was elected an honorary member of the Dutch Mathematical Society in 1957 and the Australian Mathematical Society elected him an honorary member in 1986.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (13 books/articles) .
    • An entry in The Mathematical Gazetteer of the British Isles .
    • 1.nFellow of the Royal Societyn1948 .
    • History Topics: The real numbers: Attempts to understand .
    • Dictionary of National Biography .
    • Australian Academy of Science .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Mahler.html .

  42. Konrad Knopp (1882-1957)
    • His wife Helene (1857-1923) was the daughter of Otto Ostertun who was a head forester.
    • Konrad's school education was in Berlin, then he spent one semester at the University of Lausanne in 1901.
    • Following this he went to the University of Berlin where he was taught by Schwarz, Frobenius, Schottky, Landau and Schur, receiving a qualification to teach in 1906 and a doctorate in 1907.
    • Knopp left Germany in the spring of 1908 and travelled to Japan where he taught in Nagasaki in western Kyushu, at the Handelshochschule during 1908-09.
    • In 1910 he returned to Germany and married the painter Gertrud Kressner (1879-1974), the daughter of Colonel Karl Kressner and Hedwig Rebling; they had one son and one daughter.
    • Konrad and Gertrud Knopp then moved to Tsingtao, eastern Shantung province, China where he taught at the German-Chinese academy during 1910-11.
    • Knopp became an officer in the army during World War I, being wounded in action near the beginning of the war.
    • After being wounded he was discharged from the army and by the autumn of 1914 he was teaching at Berlin University.
    • Examples of some papers he published during this period are: Bemerkungen zur Struktur einer linearen perfekten nirgends dichten Punktmenge Ⓣ (1916); Ein einfaches Verfahren zur Bildung stetiger nirgends differenzierbarer Funktionen Ⓣ (1918); Mittelwertbildung und Reihentransformation Ⓣ (1920); and Uber das Eulersche Summierungsverfahren Ⓣ (1923).
    • He was appointed to a chair of mathematics at Tubingen University in 1926 and he remained there until he retired in 1950.
    • Examples of publications during this time are Zur Theorie der Limitlerungsverfahren (1930); and Uber die maximalen Abstande und Verhaltnisse verschiedener Mittelwerte Ⓣ (1935).
    • The chapters of the book are: .
    • Details of three further textbooks by Knopp are given in the article: Texts by Knopp.
    • He produced the sixth edition of Hans von Mangoldt's famous Hohere Mathematik: eine Einfuhrung fur Studierende und zum Selbststudium Ⓣ.
    • The book continued to appear as a jointly authored text by von Mangoldt and Knopp, and the three volumes which were reprinted in 1990 were the seventeenth, sixteenth and fifteenth editions of these volumes respectively.
    • Volume 1 covers numbers, functions, limits, analytic geometry, algebra, set theory; volume 2 covers differential calculus, infinite series, elements of differential geometry and of function theory; and volume 3 covers integral calculus and its applications, function theory, differential equations.
    • The review of the 1990 reprint states:- .
    • This famous and comprehensive introduction to analysis by von Mangoldt and Knopp has been popular for generations of German-speaking students, in mathematics, physics and other natural sciences, and engineering.
    • He was the co-founder of Mathematische Zeitschrift in 1918, being the editor from 1934 to 1952.
    • After he retired Knopp continued to publish interesting papers such as Zwei Abelsche Satze (1952) in which he proved abelian theorems for Laplace and Abel transforms which are closely related to the well-known Tauberian theorems of Karamata.
    • He was invited to lecture in March 1952 at a meeting held in conjunction with the first meeting of the International Mathematical Union.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (6 books/articles) .
    • A Poster of Konrad Knopp .
    • History Topics: Overview of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Knopp.html .

  43. Ernst Witt biography
    • Heinrich Witt, Ernst's father, was the seventh of thirteen children and his very religious upbringing led him to study theology at Halle University.
    • It was during this two years leave that Ernst was born on the island of Alsen.
    • Alsen together with the rest of North Schleswig became part of Germany in 1864.
    • The island is separated from the Sundeved peninsula of southern Jutland by a narrow piece of water called Als Sound.
    • Heinrich Witt became head of the Liebenzell Mission in Changsha.
    • Ernst spent the next nine years of his life in China with his elder sister and four younger siblings.
    • He learnt Chinese from the Chinese nannies employed to look after the children, and he also learn arithmetic from his father.
    • Then in the spring of 1920 Ernst and his younger brother Otto were also sent to live with their uncle in Mullheim.
    • His uncle was a preacher with eight children of his own and he ran a home for children of missionaries - usually there were at least 30 children in total.
    • However he was now able to attend the Realschule in Mullheim where his enthusiasm for mathematics soon became evident as did his liking for chemistry.
    • At this school Witt was fortunate to have a talented mathematics teacher, Karl Ottinger, who quickly realised the extraordinary talent of his pupil and did everything he could to allow him to progress rapidly to advanced topics.
    • Witt took his Abitur examination in the same year, then entered the University of Freiburg to study mathematics and physics.
    • It was usually for students at German universities at this time to move between institutions and after two terms at Freiburg, Witt moved to Gottingen for the start of the 1930 summer term.
    • Having seen a remarkably simple proof by Witt of Wedderburn's theorem that every finite skew field is commutative, Herglotz encouraged him to submit it for publication and it became Witt's first paper appearing in 1931.
    • At Artin's invitation he spent some time in Hamburg studying the class field theory of number fields.
    • Witt joined the Nazi Party on 1 May 1933 and also the SA, the military wing of the Party.
    • At one of the lectures Witt turned up wearing his SA (Sturm Abteilung: Storm Section) uniform.
    • The oral exemination was held at the end of July 1933; the committee comprising Herglotz, Weyl and the physicist Robert Pohl.
    • He had written up the thesis in the first week of July and submitted it on the 7th.
    • Herglotz wrote in 1946 about Witt's activities in the SA (see for example [',' I Kersten, Biography of Ernst Witt (1911-1991), in Quadratic forms and their applications, Dublin, 1999 (Amer.
    • I asked him about his impression of his comrades, whose ideas, as I suspected, would often come into conflict with his devotion to science.
    • The way he looked at the time caused quite a bit of worry.
    • This view that Witt put mathematics before Nazi ideology is born out by many, both staunch Nazi supporters and those opposed to the Nazi ideas.
    • Witt once explained that all sciences can reshape themselves in accordance with the spirit of the times; only in mathematics must everything remain as before.
    • Oswald Teichmuller and Ludwig Schmid were also members of the seminar, and Schmid collaborated with Witt on ideas which would lead to the Witt vector calculus.
    • In February of that year he took an oral examination and gave his habilitation lecture in June 1936.
    • Kersten writes [',' I Kersten, Biography of Ernst Witt (1911-1991), in Quadratic forms and their applications, Dublin, 1999 (Amer.
    • His habilitation on the theory of quadratic forms in arbitrary fields ranks as one of his most famous works.
    • In it he introduced what was later named the 'Witt ring' of quadratic forms.
    • Shortly after that, Witt introduced the ring of 'Witt vectors', which had a great influence on the development of modern algebraic geometry (see [',' G Harder, Wittvektoren, Jahresber.
    • In the following year Herglotz argued strongly in support of Witt becoming Toeplitz's successor but the chair was not filled until 1939 when Wolfgang Krull was appointed.
    • Witt published Treue Darstellung Liescher Ringe Ⓣ in 1937, which was inspired by earlier work of Wilhelm Magnus on free Lie algebras.
    • This, together with results of Poincare from 1899 and Birkhoff in 1937 (independently of Witt), led to the famous Poincare-Birkhoff-Witt theorem (see [',' P-P, Grivel, Une histoire du theoreme de Poincare-Birkhoff-Witt, Expo.
    • 22 (2) (2004), 145-184.','3] for the history of the theorem).
    • The Poincare-Birkhoff-Witt theorem gives an explicit description of the universal associative enveloping algebra of any Lie algebra over any field, and thereby establishes a remarkable relation between associative and nonassociative algebras.
    • It is one of the most important theorems in mathematics, connecting such diverse areas as representation theory, differential geometry, and universal algebra.
    • In August 1937 he attended the compulsory National Socialist course for lecturers and was given the follow assessment (see [',' I Kersten, Biography of Ernst Witt (1911-1991), in Quadratic forms and their applications, Dublin, 1999 (Amer.
    • Physical capabilities: weakly-built, cannot be established because of a sporting injury.
    • Description of his character: Witt has shown himself to be quiet, modest and restrained, with a tendency to keep to himself; characteristic features are a certain naivety and eccentricity.
    • He dedicates himself to his work with dogged tenacity, continuously brooding and thinking and thus represents the typical, politically indifferent researcher and scientist, who will probably be successful in his subject, but who, at least for the time being, is lacking any of the qualities of a leader or educator.
    • Emil Artin was not a Jew but his wife was a Jew so when the "New Official's Law" was passed by the Nazis in 1937 affecting those who were related to Jews by marriage he was forced from his post at the University of Hamburg.
    • At the end of the war he was taken prisoner, but still in 1945 was freed to return to Hamburg.
    • However, he was dismissed from his professorship at Hamburg because of his association with the Nazis.
    • Many of his colleagues wrote supporting his reinstatement, all testifying that Witt, despite his membership of the Party, had quickly realised that the aims of the Nazi Party were incompatible with scientific progress which was always the most important thing in his life.
    • He lectured in Spain and, as a consequence of these lectures, published two expository papers in Spanish: On Zorn's theorem (1950), and Intuitionistic mathematics (1951).
    • Witt was always completely honest and often rather naive, and he showed this side of his character during his time in Princeton.
    • Kurosh had lectured about a theorem of Witt's and, when told that, Witt smiled and said "I proved that theorem when I was in the USSR".
    • He is best known for his introduction of Witt vectors which appeared in his paper in 1936 in J.
    • The original construction of Witt vectors is given in the articles [',' G Harder, Wittvektoren, Jahresber.
    • 99 (1) (1997), 18-48.','4] is a German translation of [',' G Harder, Witt vectors, in Ernst Witt : gesammelte Abhandlungen (Berlin, 1996).','5]).
    • 95 (4) (1993), 166-180.','6], [',' I Kersten, Biography of Ernst Witt (1911-1991), in Quadratic forms and their applications, Dublin, 1999 (Amer.
    • Soc., Providence, RI, 2000), 155-171.','7] and [',' I Kersten, Ernst Witt - ein grosser Mathematiker, DMV-Mitteilungen 2 (1995), 28-30.','8] are written by Ina Kersten who was one of Witt's pupils at Hamburg and his assistant during the two years before he retired.
    • Kersten describes Witt's final years [',' I Kersten, Biography of Ernst Witt (1911-1991), in Quadratic forms and their applications, Dublin, 1999 (Amer.
    • Because of these troubles, he had to decline several invitations for talks at home and abroad, and in 1975, he experienced a slight stroke.
    • Moreover, because of his allergies, he could not join the move of the mathematics department to a modern high-rise building, in which, because of some kind of air conditioning, the windows could not be opened.
    • On his account the colloquia of the mathematics department were held in another building, and he could attend them until shortly before his death.
    • Witt received many honours such as membership of the German Mathematical Society (Deutsche Mathematiker-Vereinigung) in 1937, the Hamburg Mathematical Society, the oldest mathematical society in the world which still exists today, in 1954, and the Gottingen Academy of Sciences in 1978.
    • Ernst Witt seems to have actually suited a usual caricature of a mathematician - both heedless and ignorant of the world, somewhat naive, self-absorbed in his mathematical universe, truly unpolitical.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (8 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Witt.html .

  44. Xu Yue biography
    • We know a little of Xu Yue but the main text which bears his name, the Shushu jiyi (Notes on Traditions of Arithmetic Methods), is probably the work of a later author trying to claim a certain respectability for his writings.
    • Xu Yue was a pupil of Liu Hong and he studied mathematics under the famous calendar expert.
    • Liu Hong worked at the Imperial Observatory and it was there that Xu Yue held discussions with him and also with the head of the Astronomical Bureau.
    • Mathematics was used by Liu Hong and others at the Observatory in their studies of astronomy and the related work on the calendar which, of course, was based on the apparent motion of the sun and the moon.
    • This preceded the major commentary written by Liu Hui in the second half of the third century, and it would appear that Liu Hui commented on a version of the text which did not include Xu Yue's comments.
    • Whether Xu Yue wrote the Shushu jiyi (Notes on Traditions of Arithmetic Methods) is uncertain.
    • Fourteen old methods of calculation are mentioned in the text.
    • One of these uses a device resembling the abacus called ball-arithmetic.
    • Three others also uses balls, one involving balls in columns, one involving two balls of different colours which move at right angles to each other suggesting almost the idea of Cartesian coordinates.
    • shows an interesting appreciation of coordinate relationships.
    • There is no doubt that one of the main aims of the text is to introduce a notation which will allow the representation of large numbers.
    • Most historians believe that the aim of the author was to suggest that it was possible to represent any number, no matter how large.
    • Three systems of powers of 10 are given.
    • The lower system is based on the sequence of powers of 10 .
    • the middle system on powers of 104 .
    • the upper system being based on powers of 10, each being the square of its predecessor .
    • Finally, Xu Yue talks about calculations of the nine balls.
    • Despite being an obscure text, after editing by Li Chunfeng, the Shushu jiyi (Notes on Traditions of Arithmetic Methods) was selected as a text for the Imperial examinations in 656 and became one of The Ten Classics in 1084.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (6 books/articles) .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: The Ten Mathematical Classics .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Xu_Yue.html .

  45. Takakazu Seki (1642-1708)
    • Seki was an infant prodigy in mathematics.
    • He was self-educated in mathematics having been introduced to the topic by a servant in the household who, when Seki was nine years old, realised the talent of the young boy.
    • Seki soon built up a library of Japanese and Chinese books on mathematics and became acknowledged as an expert.
    • His position in life is described in [',' D E Smith and Y Mikami, A History of Japanese Mathematics (Chicago, 1914), 90-127.','18] as follows:- .
    • In due time he, as a descendant of the samurai class, served in public capacity, his office being that of examiner of accounts to the Lord of Koshu, just as Newton became master of the mint under Queen Anne.
    • When his lord became heir to the Shogun, Seki became Shogunate samurai and in 1704 was given a position of honor as master of ceremonies in the Shogun's household.
    • The work is remarkable for the careful analysis of the problems which Seki made and this certainly was one of the reasons for his great success as a teacher.
    • Seki anticipated many of the discoveries of Western mathematics.
    • He studied equations treating both positive and negative roots but had no concept of complex numbers.
    • He wrote on magic squares, again in his work of 1683, having studied a Chinese work by Yank Hui on the topic in 1661.
    • This was the first treatment of the topic in Japan.
    • He discovered the Newton or Newton-Raphson method for solving equations and also had a version of the Newton interpolation formula.
    • For example, in 1683, he considered integer solutions of ax - by = 1 where a, b are integers.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (20 books/articles) .
    • A Poster of Takakazu Seki .
    • History Topics: Matrices and determinants .
    • History Topics: A chronology of pi .
    • Japanese Association of Mathematical Sciences .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Seki.html .

  46. George Berkeley biography
    • William Berkeley was a gentleman farmer whose family originally came from Staffordshire in England while Elisabeth Southerne was the daughter of a Dublin brewer.
    • He entered the Duke of Ormonde's School in Kilkenny in July 1696 and studied there until January 1700 and then, although still not fifteen years of age, he entered Trinity College, Dublin.
    • He matriculated in March 1700, just after he reached the age of fifteen as a pensioner, meaning that he did not have a scholarship and paid for his own keep in College.
    • in the spring of 1704.
    • After graduating he prepared an elementary textbook in which he explored the basis of arithmetical notation and the principal arithmetical processes as functions of that notation, explaining these without resort to algebraic or geometrical techniques.
    • He published this in 1707 as "Arithmetica", jointly with a further set of studies entitled "Miscellanea mathematica" ..
    • and indicated that mathematics had been his primary interest for three years.
    • Berkeley was working on these mathematics texts, whose full title is Arithmetica absque Algebra aut Euclide demonstrata (Arithmetic demonstrated without algebra or Euclid), waiting for the chance to compete for a fellowship.
    • In 1706 a College Fellowship became available and, after taking some extremely demanding competitive examinations, he became a Junior Fellow of Trinity College, Dublin on 9 June 1707.
    • Later in that year, on 19 November, he read his article Of infinites to the Dublin Philosophical Society, but this mathematical and philosophical work was only published after his death.
    • During this time he published two important works, An Essay towards a New Theory of Vision in 1709 and A Treatise concerning the Principles of Human Knowledge in 1710.
    • The purpose of the first, as stated by Berkeley, is:- .
    • to show the manner wherein we perceive by sight the distance, magnitude, and situation of objects.
    • Also to consider the difference there is between the ideas of sight and touch, and whether there is any idea common to both senses.
    • It is agreed on all hands, that the qualities or modes of things do never really exist each of them apart by itself, and separated from all others, but are mixed, as it were, and blended together, several in the same object.
    • Not that it is possible for colour or motion to exist without extension: but only that the mind can frame to itself by abstraction the idea of colour exclusive of extension, and of motion exclusive of both colour and extension.
    • Berkeley sent A Treatise concerning the Principles of Human Knowledge to Samuel Clarke and William Whiston.
    • Although Berkeley continued to hold his fellowship at Trinity College, Dublin, until 1724, he spent most of the period from 1713 to 1724 away from Dublin.
    • He went to London in January 1713 where he arranged publication of some of his works.
    • In November of that year he set off for Italy as chaplain to Lord Peterborough.
    • He returned to England in August 1714 and towards the end of the year he had a fever which Arbuthnot described with a joke aimed at Berkeley's philosophy:- .
    • poor philosopher Berkeley has now 'the idea of health', which was very hard to produce in him; for he had 'an idea' of a strange fever on him so strong, that it was very hard to destroy it by introducing a contrary one.
    • Berkeley returned to Italy in 1716 with George Ashe, son of the Trinity College provost, and he spent four years there.
    • He gives a vivid description of the eruption of Vesuvius in 1717:- .
    • with much difficulty I reached the top of Mount Vesuvius, in which I saw a vast aperture full of smoke, which hindered the seeing its depth and figure.
    • I heard within that horrid gulf certain odd sounds, which seemed to proceed from the belly of the mountain; a sort of murmuring, sighing, throbbing, churning ..
    • June 5, after a horrid noise, the mountain was seen at Naples to spew a little out of the crater.
    • The seventh, nothing was observed till within two hours of night, when it began a hideous bellowing, which continued all that night and the next day till noon, causing the windows, and, as some affirm, the very houses in Naples to shake.
    • From that time it spewed vast quantities of molten stuff to the South, which streamed down the side of the mountain like a great pot boiling over.
    • On the journey back to England Berkeley spent some time in Lyon where he wrote the essay De motu (On motion) which he submitted for the Grand Prix of the Academy of Sciences.
    • He reached London early in 1721 and by the start of the 1721-22 academic year he was in Dublin where he was now a Senior Fellow (a position to which he had been appointed in 1717 while in Italy).
    • In May 1724 Berkeley resigned his position at Trinity College to become Anglican Dean of Londonderry, but he never resided in the city spending most of the next four years in London.
    • Over these years he planned to establish a College in Bermuda to train the sons of colonists and Native Americans whom he wished to convert to:- .
    • Funds of 10000 pounds for Berkeley's project were promised after the House of Commons in London voted:- .
    • That an humble address be presented to his majesty, that out of the lands in St Christopher's, yielded by France to Great Britain by the treaty of Utrecht, his majesty would be graciously pleased to make such grant for the use of the president and fellows of the college of St Paul, in Bermuda, as his majesty shall think proper.
    • However, by the middle of 1731 it became obvious that he would not receive the grant, and he returned to London in October.
    • He wrote a number of articles during his time in America which he published in the two or three years after his return.
    • In January 1734 Berkeley was appointed Bishop of Cloyne and was consecrated in St Paul's Church, Dublin, on 19 May 1734.
    • In this office he devoted himself to the social and economic plight of Ireland, doing his best as an Anglican bishop to help the conditions of all in the predominantly Roman Catholic country.
    • In 1745, at the time of the Jacobite rebellion, Berkeley addressed Roman Catholics in his diocese, and in 1749 he addressed to the Roman Catholic Clergy A Word to the Wise.
    • He received a reply in the Dublin Journal of 18 November 1749 in which the Catholic Clergy expressed:- .
    • their sincere and hearty thanks to the worthy author, assuring him that they are determined to comply with every particular recommended in his address to the utmost of their power.
    • in every page [Berkeley's address] contains a proof of the author's extensive charity; his views are only towards the public good; the means he prescribes are easily complied with; and his manner of treating persons in their circumstances so very singular, that they plainly show the good man, the polite gentleman, and the true patriot.
    • He was devoted to Cloyne and he always purchased his clothes locally even though this meant that they were not of top quality, but he wanted to support employment.
    • He set up a school to teach spinning to children, and he wanted to make possible the manufacture of linen.
    • I am building a workhouse for sturdy vagrants, and design to raise about two acres of hemp for employing them.
    • Berkeley is best known in the world of mathematics for his attack on the logical foundation of the calculus as developed by Newton.
    • In his tract The analyst: or a discourse addressed to an infidel mathematician, published in 1734, he tried to argue that although the calculus led to true results its foundations were no more secure than those of religion.
    • He declared that the calculus involved a logical fallacy of a shift in the hypothesis.
    • And what are these fluxions? The velocities of evanescent increments.
    • May we not call them ghosts of departed quantities? .
    • Berkeley's criticisms were well founded and important in that they focused the attention of mathematicians on a logical clarification of the calculus.
    • He developed an ingenious theory to explain the correct results obtained, claiming that it was the result of two compensating errors.
    • Ren writes in [',' B Russell, History of Western Philosophy (London, 1961), 623-633.','30]:- .
    • By reviewing Berkeley's lifetime and the content of the "Analysts", we conclude that his critique was correct and that it impelled the improvement of the foundations of calculus objectively.
    • It is helpful for the normal development of mathematics to accept various forms of critique positively.
    • Many of the other references which we give also discuss Berkeley's attack on the calculus; see [',' J Dancy, Berkeley, an Introduction (1987).','5], [',' A A Luce, The Life of George Berkeley, Bishop of Cloyne (1968).','11], [',' S Bonk, George Berkeleys Theorie der Zeit: ’a total disaster’?, Studia Leibnitiana 29 (2) (1997), 198-210.','19], [',' F Cajori, A history of the conceptions of limits and fluxions in Great Britain from Newton to Woodhouse (Chicago, 1919), 57-95.','21], [',' M Hughes, Newton, Hermes and Berkeley, British J.
    • 43 (1) (1992), 1-19.','26], [',' B Russell, History of Western Philosophy (London, 1961), 623-633.','30], and [',' M A Stewart, Berkeley and the Rankenian Club, Hermathena 139 (1985), 25-45.','33].
    • De Moivre, Taylor, Maclaurin, Lagrange, Jacob Bernoulli and Johann Bernoulli all made attempts to bring the rigorous arguments of the Greeks into the calculus.
    • In fact his intention was to remain at Oxford for the rest of his life, which he anticipated would not be very long.
    • He died of a heart attack on the evening of Sunday 14 January 1753, sitting with his family listening to his wife reading.
    • List of References (35 books/articles) .
    • A Poster of George Berkeley .
    • Multiple entries in The Mathematical Gazetteer of the British Isles .
    • History Topics: The rise of calculus .
    • History Topics: Infinity .
    • History Topics: Mathematics and the physical world .
    • History Topics: Newton's bucket .
    • Dictionary of Scientific Biography .
    • Dictionary of National Biography .
    • Internet Encyclopedia of Philosophy .
    • Stanford Encyclopedia of Philosophy .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Berkeley.html .

  47. Emma Lehmer biography
    • Although Emma Trotskaia only became Emma Lehmer in 1928 after she married, for the sake of simplicity we will use her married name throughout this article.
    • Emma Lehmer later wrote about the town of her birth:- .
    • I was born in a town with the lovely sounding name of Samara on the great Volga river..
    • In fact Samara, which was an industrial town and the administrative centre of Samara province, was renamed Kuybyshev in 1935.
    • Emma wrote many years later that she lost all interest in returning to the town of her birth after it was given such an unpleasant sounding name as Kuybyshev, but it reverted to its original name of Samara in 1991, a move which doubtless was strongly approved by Lehmer.
    • In fact Lehmer spent less than four years in Samara before her father was appointed as the Far East representative of the sugar company and posted to Harbin, Manchuria.
    • Why, one might ask, was a Russian sugar company keen to develop its business in Harbin? Fourteen years earlier it had been a small fishing village but it became the construction centre for the Chinese Eastern Railway by the Russians.
    • It was the Russian military base for operations in Manchuria during the Russian-Japanese War of 1904-05, but at the end of the war had come under joint Chinese-Japanese control.
    • It was at this school that Lehmer first developed her love of mathematics, encouraged by a superb mathematics teacher who had been an engineer in Moscow.
    • The Russian revolution which began in 1917 led the a large influx of Russian refugees to Harbin.
    • On the one hand this led to the creation of the new school, but on the other hand it made Lehmer realise that returning to Russia to study at university was going to be impossible.
    • She decided instead to go to the United States for her university studies and after completing her schooling in Harbin, she worked for a year doing a number of different jobs such as translating and baby-sitting as well as coaching children in both mathematics and music (she was an accomplished pianist).
    • She made the natural first step of going to Japan; since Harbin was jointly under Japanese control this was straightforward.
    • From there she travelled to Canada but not having a student visa on which to enter the United States, she had to spend several weeks in Canada waiting for the process of obtaining one to be completed.
    • Having allowed what should have been plenty of time to reach Berkeley by the start of term, by the time she arrived there classes had already been running for a week.
    • She enrolled for an engineering degree, taking classes in mathematics, physics, chemistry, English and engineering.
    • This was a heavy load for a student to take but this was not what proved a problem, rather it was that she discovered that dexterity with her hands was not one of her strong points making engineering less attractive.
    • After a couple of years she dropped engineering and moved to mathematics as her main subject.
    • Her favourite mathematics lecturer at Berkeley was Derrick Norman Lehmer and as well as taking several of his courses she did a research project with him on finding pseudosquares.
    • degree with honours in Mathematics in 1928 and in the same year she married Dick Lehmer (as Derrick Lehmer was called).
    • from Brown University, Emma continued working on her Master's Degree while she helped Dick by typing his thesis and working with him reading mathematics papers.
    • She also helped the family finances by coaching mathematics.
    • She received her Master's Degree in 1930 for a thesis entitled A Numerical Function Applied to Cyclotomy and in the same year the results were published in the Bulletin of the American Mathematical Society.
    • The two Lehmers' life over the next few years involved moving from place to place as Dick Lehmer sought a permanent university post in the particularly difficult times of the Depression.
    • They spent 1930-31 at the California Institute of Technology and 1931-32 at Stanford.
    • The Lehmers remained at Lehigh until 1940 except for the year 1938-39 which they spent in England visiting both the University of Cambridge and the University of Manchester.
    • Back in the United States after the outbreak of World War II, they spent another year at Lehigh before returning to Berkeley in 1940.
    • Although the computer worked most of the time computing trajectories for ballistics problems, some weekends Lehmer and her husband used it to solve certain number theory problems using the sieve methods that they were working on [',' T Perl, Emma Trotskaia Lehmer, in Notable Women in Mathematics (Westport, 1998), 123-128.','2]:- .
    • When they could arrange child care, they often stayed at the lab all night long while the ENIAC processed one of their problems.
    • They would return home at the break of dawn.
    • She wrote around 60 papers on different aspects of number theory, about 20 of these being joint publications with her husband.
    • As an example of other contributions Lehmer made to Fermat's Last Theorem she published On a resultant connected with Fermat's Last Theorem in 1935, then, jointly with her husband, On the first case of Fermat's Last Theorem in 1941.
    • She corresponded with Helmut Hasse on the topic and he encouraged her to publish her article On the Quadratic Character of Some Quadratic Surds in Crelle's Journal in 1971.
    • In an article On the advantages of not having a Ph.D.
    • First of all there are lower expectations.
    • In fact as well as the advantages in not holding faculty position, such as having more time to devote to research, she also had many of the advantages which would have gone with such a post [',' J Brillhart, Derrick Henry Lehmer, Acta Arith.
    • As a faculty wife she was a member of the university community, had library privileges, and never felt excluded from the mathematics community.
    • She was able to travel with her husband to attend mathematics conferences around the world.
    • In the sixty years during which they collaborated, the Lehmers were a research team who personally influenced a large number of people with their knowledge, their courtesy and sociability, and their fine mathematical work.
    • There is little doubt that one of their most enduring contributions to the world of mathematicians is their founding of the West Coast Number Theory Meeting in 1969.
    • This meeting, which has been held in the western United States in December of every year since that time, has provided an informal and often merry environment in which old friends can meet and younger mathematicians can present their work.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (2 books/articles) .
    • An entry in The Mathematical Gazetteer of the British Isles .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Lehmer_Emma.html .

  48. Abraham Seidenberg biography
    • Abraham Seidenberg studied at the University of Maryland and was awarded his B.A.
    • thesis Valuation Ideals in Rings of Polynomials in Two Variables he was awarded his doctorate in 1943.
    • In 1945 Seidenberg was appointed as an instructor in mathematics at the University of California at Berkeley.
    • He was promoted rapidly and in 1958 reached the rank of full professor.
    • She was born in Ancona, Italy, on 23 February 1915 into a family of Jewish origins.
    • She left Italy with the other members of her family in 1938 after racial persecution and they emigrated to the United States.
    • Ebe was the author of novels on the exile of the Jews during Fascism.
    • He held a Visiting Professorship at the University of Milan and he gave several series of lectures there.
    • In fact he was in Milan in the middle of giving a lecture series at the time of his death.
    • Ebe Seidenberg died in a clinic in Rome at the age of 87.
    • M A Rosenlicht, G P Hochschild, and P Lieber in an obituary, describe other features of their colleague Seidenberg's career at Berkeley:- .
    • His career included a Guggenheim Fellowship [awarded 1953], visiting Professorships at Harvard and at the University of Milan, and numerous invited addresses, including several series of lectures at the University of Milan, the National University of Mexico, and at the Accademia dei Lincei in Rome.
    • At the time of his death, he was in the midst of another series of lectures at the University of Milan.
    • Seidenberg contributed important research to commutative algebra, algebraic geometry, differential algebra, and the history of mathematics.
    • In the following year he published Prime ideals and integral dependence written jointly with I S Cohen which greatly simplified the existing proofs of the going-up and going-down theorems of ideal theory.
    • An example of one of his papers on algebraic geometry is The hyperplane sections of normal varieties (1950) which has proved fundamental in later advances.
    • He also wrote a book Elements of the theory of algebraic curves (1968).
    • a well-written text on the theory of algebraic curves.
    • [T]he leisurely style, with plenty of motivational discussion, makes it especially useful for an introduction to the subject.
    • Novel features are a chapter on ground fields of positive characteristic and one on "infinitely near points".
    • Kolchin writes the following in a review of this paper:- .
    • In the first part he shows that the usual definition of "(differentially) algebraic" is equivalent to one using induction on the number of derivation operators.
    • In the subsequent parts he proves that, in a separable differential field extension, every differential transcendence basis is separating, a result previously proved by him in the case of ordinary differential fields; and he also discusses the connection between the condition that every finitely generated extension of a differential field F be simply generated and the condition that 0 be the only differential polynomial over F vanishing identically on F.
    • Throughout his career Seidenberg published important papers on the history of mathematics.
    • For example Peg and cord in ancient Greek geometry (1959) in which he argues that the whole of Greek geometry had a ritual origin.
    • In The diffusion of counting practices (1960) Seidenberg argues that counting was diffused from one centre and was not discovered again and again as is commonly believed.
    • History of mathematics papers published after he retired include The zero in the Mayan numerical notation (1986) and On the volume of a sphere (1988).
    • In this latter paper Seidenberg compares the methods for calculating the volume of a sphere: in Greek mathematics, namely that by Archimedes; in Chinese mathematics, namely in the Nine Chapters on the Mathematical Art ; in Babylonian mathematics; and in Egyptian mathematics.
    • He argues, as he does in other papers, that there were two traditions in ancient mathematics, see [',' J Mathews, A Neolithic oral tradition for the van der Waerden/Seidenberg origin of mathematics, Arch.
    • These, he claims, originated from a common source prior to Greek, Babylonian, Chinese, and Vedic mathematics.
    • He also argues that the use methods of the Cavalieri type to determine volume go back to this common source.
    • In Geometry and Algebra in Ancient Civilizations Van der Waerden puts forward similar views for which he gives credit to Seidenberg, saying that Seidenberg made him look at the history of mathematics a new way.
    • We must not suppose that Seidenberg neglected his algebraic research in the latter part of his career.
    • When does the Lasker-Noether decomposition theorem, which says that an ideal in a commutative Noetherian ring is the intersection of a finite number of primary ideals, hold in a constructive sense? .
    • , xn] then there is an algorithm to compute generators of the primary ideals and of their associated prime ideals.
    • He had a number of very dear friends.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (3 books/articles) .
    • University of California .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Seidenberg.html .

  49. Edward Collingwood (1900-1970)
    • Colonel Collingwood had a career in the army, commanding the Lancashire Fusiliers in the battle of Omdurman in September 1898.
    • Colonel Collingwood retired from the army in 1899, the year before his son Edward was born on the family estate of Lilburn Tower.
    • The estate is in Northumberland in the north of England about 7 km from Wooler on the road to Alnwick (which is about 20 km to the south east).
    • Edward was the oldest of his parents four children, all boys, and he was brought up on the family estate enjoying [',' M L Cartwright, Sir Edward Foyle Collingwood, Dictionary of National Biography 1961-1970 (London, 1976), 232-234.','2]:- .
    • shooting and fishing and the social life of the country.
    • We should note that Edward's mother Dorothy came from a well off family, and the name "Foyle" is the name of their family estate.
    • Dorothy [',' M L Cartwright, Sir Edward Foyle Collingwood, Dictionary of National Biography 1961-1970 (London, 1976), 232-234.','2]:- .
    • This ship was named after Vice-Admiral Cuthbert Collingwood who was Nelson's second in command at the battle of Trafalgar.
    • Vice-Admiral Collingwood was the brother of Edward Collingwood's great-grandfather, and it was no coincidence that Collingwood served on HMS Collingwood for special arrangements had been made for this to happen.
    • However Collingwood's naval career came to an end when he fell down a hatchway on board ship, broke his wrist and damaged his knee, just before the Battle of Jutland.
    • He was transferred to the hospital ship, then invalided out of the Navy.
    • Attempting to go to Woolwich he failed the medical examination so, in 1918, he entered Trinity College, Cambridge to study mathematics.
    • Collingwood was influenced by his advisor of studies, Hardy at Cambridge and decided early on that he would undertake research in mathematics.
    • A friend, Gilbert Ashton, writing of these days, wrote that Collingwood was:- .
    • always known by his friends and contemporaries of Trinity as 'The Admiral' ..
    • Cartwright describes how Collingwood behaved as an undergraduate in [',' M L Cartwright, Sir Edward Foyle Collingwood, Dictionary of National Biography 1961-1970 (London, 1976), 232-234.','2]:- .
    • he kept somewhat aloof from his mathematical contemporaries, and had a full, but entirely separate, social life among a group, most of whom had served in the forces ..
    • In 1922 Collingwood went to Aberystwyth at the invitation of W H Young.
    • He received his doctorate for a thesis entitled Contributions to the theory of integral functions of finite order in 1929.
    • From 1930 he was appointed Steward of Trinity and gave advanced courses on integral and meromorphic functions but gave no undergraduate courses.
    • In 1937 Collingwood left Cambridge and became High Sheriff of Northumberland.
    • As stated in [',' M L Cartwright and W K Hayman, Edward Foyle Collingwood, Biographical Memoirs of Fellows of the Royal Society of London 17 (1971), 139-158.','3]:- .
    • He was in charge of the Sweeping Division in 1943, then Chief Scientist in the Admiralty Mine Design Department in 1945.
    • and received the Legion of Merit from the USA in 1946.
    • He then undertook research work with Mary Cartwright on the theory of cluster sets.
    • I found myself quite unable to grasp the deep results in the theory of sets of points on which much of Collingwood's later work in this field depended.
    • Collingwood became involved with hospital boards in Newcastle, being a founder member of the Newcastle Regional Hospital Board and its chairmen from 1953 to 1968, then later he was involved with medical affairs on a national and international level.
    • He was vice-president of the International Hospital Federation from 1959 to 1967, a member of the medical research council from 1960 to 1968, and he served on the royal commission on medical education from 1965 to 1968.
    • He was chairman of the Council of Durham University for most of the 1950's and 1960's.
    • He was elected to a fellowship of the Royal Society in 1965.
    • He also served the London Mathematical Society in many ways, as a member of the Council and as Treasurer.
    • He wrote an article in 1951 to mark the centenary of the Society.
    • In particular he had a fine collection of eighteenth century paintings, and a collection of Chinese porcelain.
    • As with all his activities Collingwood made a deep study of his hobbies and became a recognised expert on Chinese porcelain.
    • Collingwood was loved and admired both for his achievements and for the delight of his company.
    • Born in Glendale in Northumberland he remained a countrymen at heart with practical knowledge of forestry, farming and gardening.
    • He remained a bachelor to the grief of the many dancing partners who had been entranced by his waltzing! .
    • His appearance is described in [',' M L Cartwright, Sir Edward Foyle Collingwood, Dictionary of National Biography 1961-1970 (London, 1976), 232-234.','2]:- .
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (5 books/articles) .
    • A Poster of Edward Collingwood .
    • 2.nFellow of the Royal Society of Edinburghn1954 .
    • 4.nFellow of the Royal Societyn1965 .
    • Dictionary of National Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Collingwood.html .

  50. Gan De biography
    • We know nothing of Gan De's life and very little about his work.
    • He was one of the earliest of Chinese astronomers and made observations which he recorded giving dates and coordinates.
    • We know of two books which he wrote, namely the Suixing Jing (Treatise on Jupiter) and the Tianwen Xingzhan (Astrological Predictions), but sadly both texts have been lost.
    • This records an observation which he made in the summer of 365 BC:- .
    • Of course the intriguing question is what was the small reddish star which Gan De saw? Could it have been one of Jupiter's satellites? This has intrigued those interested in the history of astronomy.
    • If he did see one of the Galilean satellites, then it would have been Ganymede which is the brightest of the four.
    • Ganymede can have a magnitude of 4.6 which means that it is within the range which anyone might expect to see.
    • The only thing in the description which would have one doubt this is the fact that Gan De records the small star on the side of Jupiter to be reddish.
    • However, we must remember that he observed under conditions of absolutely no light pollution, something impossible today.
    • Evidence on the positive side suggesting that he did indeed observe Ganymede comes from the accuracy of his observations in general.
    • He gave the following description of Jupiter's journey through the constellations:- .
    • Every 12 years Jupiter returns to the same position in the sky; every 370 days it disappears in the fire of the Sun in the evening to the west, 30 days later it reappears in the morning to the east ..
    • and also gave accurately observed details of the planet's movements throughout its 12 year cycle.
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Gan_De.html .

  51. Nikoloz Muskhelishvili biography
    • He was also known by the Russian version of his name, namely Nikolay Ivanovich Muskhelisvili.
    • His father, Ivan Levanovich Muskhelisvili, was a General in the Engineering Corps of the Imperial Russian Army but his ancestors were well-known historians of Georgia [',' I N Vekua, Nikolai Ivanovich Muskhelishvili (on his seventieth birthday), Russian Math.
    • Ivan Levanovich had a great interest in the exact sciences and clearly foresaw the significant role they were to play in the development of civilisation in the twentieth century.
    • He also had interesting ideas on the education of children and spent a good deal of time teaching his own children, particularly Niko whose mathematical ability he recognised at an early age.
    • Muskhelishvili speaks about his father with great warmth and affection and in dedicating his 'Course of Theoretical Mechanics" (1926) to him he describes him as his first great teacher.
    • Muskhelishvili's mother, too, was a person of culture as well as charm.
    • Daria was the only child of Alexandre Saginashvili and his wife Elizabed Chavchavadze.
    • Elizabed was the sister of Ilia Chavchavadze, a famous Georgian writer who was canonized by the Georgian Orthodox Church in 1987 as "St Ilia the Righteous." .
    • Niko's childhood was spent in the village of Matsevani, near Tbilisi.
    • Later in the same year he entered the Faculty of Physics and Mathematics of the University of St Petersburg.
    • Among his teachers at university was the Professor of Mechanics Aleksei Nikolaevich Krylov who was so impressed with the young Muskhelishvili that he said to a colleague, "Mark the name Muskhelishvili; it is one you will hear often in the future!" Muskhelishvili graduated with a first degree in 1914 and continued his studies at St Petersburg in the Department of Mechanics, preparing for a professorship.
    • There he worked with Gury Vasilievich Kolosov who worked on the theory of elasticity and had discovered formulas expressing the components of the stress tensor and the displacement vector in terms of two analytic functions of one complex variable which he had published in his 1909 paper An application of the theory of functions of a complex variable to a planar problem in the mathematical theory of elasticity (Russian).
    • His first paper was On the equilibrium of circular elastic discs (Russian) (1915) written in collaboration with Kolosov [',' M L Williams, In memoriam Nikolai Ivanovich Muskhelishvili, Internat.
    • This gives the first explicit solution for the fundamental boundary problem of plane elasticity for circular regions.
    • Muskhelishvili then pursued systematic investigations on this subject and published a series of articles devoted to various boundary problems of the plane theory of elasticity and to other problems of mathematical physics.
    • His next papers were On heat stresses in the plane problem of the theory of elasticity (Russian) (1916), On the definition of harmonic functions by means of data on a contour (Russian) (1917) and Sur l'integration de l'equation biharmonique Ⓣ (1917).
    • It was renamed again in 1924 when it became Leningrad, but today it has returned to the original name of St Petersburg.
    • After the Russian Revolution of February 1917 Georgia was ruled from St Petersburg but in May 1918 they set up an independent state.
    • The independence of Georgia, however, was short lived for in early 1921 the Red Army entered Georgia and installed a Soviet regime.
    • We will give some details of Nikoladze and his family since Muskhelishvili married his sister Tamara.
    • The family were living in St Petersburg (named Petrograd at this time) when the Russian Revolution broke out in February 1917 [',' P Kapodanno, On a neglected work of N I Muskhelishvili (Russian), in Studies in the history of physics and mechanics, 1986 (Russian) (’’Nauka’’, Moscow, 1986), 239-242; 271.','7]:- .
    • Considering the political orientation of the family, it is hardly surprising that they greeted the news of the February Revolution with enthusiasm.
    • Even more important, the sisters took part in the work of the Society for the Study of the Russian Revolution ..
    • He published Sur l'equilibre des corps elastiques soumis a l'action de la chaleur Ⓣ (1923), The solution of an integral equation encountered in the theory of black body radiation (Russian) (1924) and in 1925 he published, jointly with George Nikoladze and Archil Kirillovich Kharadze (1895-1976), a dictionary of Russian-Georgian, Georgian-Russian mathematical terms.
    • Despite the mathematical background of both sides of his family, Guram chose not to become a mathematician.
    • Let us look at three different aspects of Muskhelishvili's contributions, namely the areas of his research, his monographs and textbooks, and his contributions to the development of mathematics and science in Georgia.
    • We have already seen some of the areas that Muskhelishvili worked on in the early part of his career, particularly on plane problems of the theory of elasticity.
    • In fact this was the area in which he produced the largest number of papers.
    • The second area on which his research concentrated was on the problems of torsion and bending of homogeneous and heterogeneous bars where he obtained fundamental results.
    • A third area was on boundary problems for harmonic and biharmonic functions while a fourth was the study of singular integral equations and boundary problems for analytic functions [',' M L Williams, In memoriam Nikolai Ivanovich Muskhelishvili, Internat.
    • In a series of articles Mushkelishvili worked out a method of solving a variety of boundary problems for analytic functions of a single complex variable.
    • Up to the time of Mushhelishvili's work the problems were studied for the most part for domains bounded by smooth closed curves.
    • One must mention above all the fundamental investigations of Hilbert himself.
    • However, the solution of Hilbert's problem for a single closed contour was first obtained in an explicit form by the Soviet mathematician F D Gakhov.
    • In the works of Muskhelishvili formulae were obtained expressing the solution of Hilbert's boundary problem for a plane when there are a finite number of discontinuities along a broken curve.
    • The results obtained were applied to the theory of singular integral equations on broken contours .These results have important applications in technical and physical problems.
    • Another of Mushhelishvili's major contributions was the monographs and textbooks that he published.
    • The editors of Soviet Applied Mathematics write in [',' E Obolashvili, Academician N I Muskhelishvili (Russian), Trudy Tbiliss.
    • 218 (1981), 7-13.','18] (we have modified the quote by adding the titles of the books):- .
    • One of his great accomplishments was the writing of textbooks on theoretical mechanics ['Theoretical Mechanics Part 1, Statics' (Russian) (1933) and 'Theoretical Mechanics Part 2, Kinematics' (Russian) (1932)] and analytic geometry ['Lectures on Analytical Geometry Presented at Tiflis Polytechnic Institute' (Russian) (1922), 'Course in Analytic Geometry Part 1, Course in Analytic Geometry in Vector Formulation' (Russian) (1933), 'Course in Analytic Geometry Part 2' (Russian) (1934), 'Course in Analytic Geometry' (Russian) (1939), 'Course in Analytic Geometry' (Georgian) (1951), 'Course in Analytic Geometry' (Chinese) (1955)], and also the production of a Russian-Georgian dictionary of mathematical terminology [(1925) mentioned above].
    • In 1933 he published a fundamental monograph, 'Some Fundamental Problems in the Mathematical Theory of Elasticity' (Russian) [First edition 1933, Second edition 1935, Third edition 1949, Fourth edition 1954, Firth edition 1966)].
    • This book marks an epoch in the development of the mathematical theory of elasticity; up to the present time it has gone through four editions and has been translated into English [(1963)], Romanian [',' Academician N I Muskhelishvili (Russian), Trudy Tbiliss.
    • 218 (1981), 7-13.','1956], and Chinese [',' Trudy Tbiliss.
    • For the second edition of this monograph N I Muskhelishvili, in 1941, was awarded the State Prize of the USSR of the first order.
    • His second well-known monograph 'Singular Integral Equations, Boundary-Value Problems in the Theory of Functions, and Some Applications to Mathematical Physics' (1946, 1962, 1968), went through three editions and was translated into English [(1953)].
    • The first edition of this book was awarded the State Prize of the USSR of the second order.
    • Perhaps his greatest contribution to the development of mathematics and science in general in Georgia was his work for the Georgian Academy of Sciences.
    • He was the person who declared the opening of the Academy at its first meeting on 27 February 1941.
    • He was its first President and for 31 years he remained the President of the Academy contributing not only to the mathematical work of the Academy but in the role of President he also had a major impact on the development of Georgian science in general.
    • Of course, the first years of the Academy were during World War II and Muskhelishvili organised the Academy to undertake work on national defence.
    • For more information on Muskhelishvili's contribution as President of the Georgian Academy of Sciences, see our English translation of [',' G K Mikhailov, G F Mandzhavidze, A Yu Ishlinskii and S A Khristianovich, Academician N I Muskhelishvili (on the centennial of his birth) (Russian) 5 (Vestnik Akad.
    • It circled the earth every 96 minutes in an elliptical orbit which took it a maximum distance of 940 km from the earth and a minimum distance of 230 km.
    • Naturally the Soviet Union was very proud of this achievement and the newspaper "Pravda" reported at length.
    • Muskhelishvili had several major university roles: Dean of the Polytechnical Faculty of Tbilisi University (1926-28); Pro-Rector of the Georgian Polytechnical Institute (1928-30); Director of Mathematical, Physics and Mechanical Institute of Tbilisi State University (1933-35); and Dean of Faculty of Physics and Mathematics of Tbilisi State University (1933-36).
    • He was President of the Georgian Academy of Sciences (1941-72) and then Honorary President of Georgian Academy of Sciences (1972-76).
    • He was a Member of the Bureau of the International Union of Theoretical and Applied Mechanics (1960-76).
    • He was also an academician and honorary professor of the Armenian Academy of Sciences (1961) and the Azerbaijani Academy of Sciences (1961), an honorary member of the Bulgarian Academy of Sciences (1952), an honorary member of the Polish Academy of Sciences (1960) and an honorary member of the Berlin Academy of Science (1967).
    • In 1969 he was awarded the International prize of Turin Academy of Sciences "Modesto Panetti".
    • In 1970 he was awarded the Gold Medal of the Czechoslovak Academy of Sciences and, in the same year, he was awarded the Order of "Cyrill and Mephodius" of the first degree by the Bulgarian Academy of Sciences.
    • Muskhelishvili received five Orders of Lenin in the years 1941, 1952, 1961, 1966 and 1975.
    • In 1944 he received the Order of the Red Banner for services in training specialists in improving the national economy and the national culture.
    • In 1944 he was awarded the military Medal "For Defence of Caucasus" and in 1946 the Medal "For Valiant Labour in Great Patriotic War 1941-1945".
    • We mentioned above his contribution to the launching of Sputnik 1 and in 1961 he received a Medal for his contribution to the launching of the first artificial satellite.
    • In 1971 he received the Order of October Revolution.
    • Muskhelishvili retired when he reached the age of 80.
    • He died five years later and was buried at the Mtatsminda Pantheon of Writers and Public Figures in Tbilisi.
    • The article [',' G K Mikhailov, G F Mandzhavidze, A Yu Ishlinskii and S A Khristianovich, Academician N I Muskhelishvili (on the centennial of his birth) (Russian) 5 (Vestnik Akad.
    • Nauk SSSR, 1991), 68-81.','12] gives further information about Muskhelishvili, particularly his contributions to the Georgian Academy of Sciences and his personality, and we give an English translation of it at THIS LINK.
    • List of References (21 books/articles) .
    • N I Muskhelishvili: President of the Georgian Academy of Sciences .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Muskhelishvili.html .

  52. Herbert Turnbull biography
    • Herbert Turnbull's father, William Peveril Turnbull (born in Hackness, Yorkshire about 1842), was an H M Inspector of Schools.
    • Turnbull's father was interested in mathematics and transmitted his enthusiasm for the subject to his young son Herbert.
    • In the Preface to The Mathematical Discoveries of Newton (1945) Turnbull thanks his father for a quotation which:- .
    • is taken from a lecture on Newton which he gave to a group of Nottinghamshire miners seventy years ago.
    • Turnbull also dates his own interest in the history of mathematics to his childhood, in particular writing in the same Preface:- .
    • My own interest in Newton dates from childhood: his mathematical prowess was as well known at the family breakfast table as the batsmanship of W G Grace.
    • He went up to Trinity College, Cambridge where he had a career of great distinction being placed Second Wrangler in the Mathematical Tripos (meaning that he was ranked second among those being awarded a First Class degree) and, in 1909, he was the winner of the Smith's Prize.
    • After graduating, Turnbull taught at St Catharine's College, Cambridge (1909), and then at the University of Liverpool (1910).
    • I naturally look back on my start at lecturing in thinking of your present position.
    • My first post outside Cambridge (where I gave one course a term) was at Liverpool and involved at least three sets of new lectures to be prepared, and in one term four sets.
    • She was the daughter of Canon H D Williamson.
    • After his year as a lecturer at the University of Liverpool, Turnbull taught at the Hong Kong University, becoming master at St Stephen's College in Hong Kong in 1911, and warden of the University Hostel two years later.
    • The University Hostel was run by the Church Missionary Society as part of Hong Kong University.
    • The Society, founded by Evangelical clergy of the Church of England in 1799, stressed biblical faith, personal conversion, and piety.
    • Turnbull had duties as a mathematics lecturer at the University in addition to being warden of the hostel.
    • On his return to England, Turnbull worked as a school teacher for three years in the leading independent school at Repton in the county of Derbyshire in the north of England.
    • He taught at the famous boys' school of Repton which had a long history, being founded in 1556 in buildings which included a restored Augustinian priory established in 1172.
    • Following this Turnbull became an a Schools Inspector, entering at this stage the same profession as that of his father.
    • As an undergraduate at Cambridge Turnbull had become fascinated by the topic of invariant theory.
    • Turnbull was appointed Regius Professor of Mathematics in the United College of St Salvator and St Leonard at the University of St Andrews in 1921.
    • He held the Regius Chair until he retired in 1950 when he was succeeded as Regius Professor of Mathematics by Copson.
    • In [',' W Ledermann, Memoir : Two Mathematical Cultures, British Society for the History Mathematics Newsletter 41 (Spring 2000), 5-10.','3] Ledermann describes an interesting incident which occurred soon after he arrived in St Andrews in 1934.
    • Ledermann writes [',' W Ledermann, Memoir : Two Mathematical Cultures, British Society for the History Mathematics Newsletter 41 (Spring 2000), 5-10.','3]:- .
    • [Turnbull] was very kind and most patient when our communications were hindered by my poor command of English.
    • Before he was appointed to the Regius Chair of Mathematics at St Andrews Professor Turnbull had been a missionary in China.
    • He had picked up some of the local language there.
    • Would it help you if I spoke to you in Chinese?" I thanked him for his offer, but asked him, with apologies, to persevere with English.
    • Turnbull was interested in algebra, particularly invariant theory building on work of Gordan and Clebsch.
    • As to Turnbull's approach to mathematics it was [',' W Ledermann, Herbert Westren Turnbull, J.
    • concrete and formal in the sense that he sought to solve problems by an effective formalism rather than by a conceptual analysis of the underlying structures.
    • His topics were algebraical, but he was fond of presenting them against a geometrical background.
    • Turnbull was also interested in the history of mathematics.
    • He explains in the Preface to his little book The Great Mathematicians his attitude towards historical study in mathematics.
    • Firstly it tells us something of Turnbull's character and attitude towards mathematics.
    • Secondly we relate it because this web archive is run from a server which we have named after Turnbull and in this archive of the history of mathematics we have tried to follow Turnbull's thinking on the subject:- .
    • The usefulness of mathematics in furthering the sciences is commonly acknowledged: but outside the ranks of the experts there is little inquiry into its nature and purpose as a deliberate human activity.
    • Fully conscious of the difficulties in the undertaking, I have written this little book in the hope that it will help to reveal something of the spirit of mathematics, without unduly burdening the reader with intricate symbolism.
    • I have tried to show how a mathematician thinks, how his imagination, as well as his reason, leads him to new aspects of the truth.
    • Turnbull published his own historical research into mathematics in the James Gregory Tercentenary Volume (1939).
    • In our library in St Andrews the copy of this volume is inscribed in Turnbull's own hand:- .
    • in a bundle of remarkable original documents in the Library of the University of St Andrews ..
    • I first examined the documents at St Andrews in 1932, when it was discovered that Gregory, the original recipient of the letters, had used their blank spaces for recording his own mathematical thoughts.
    • As a result of careful scrutiny it has been established that Gregory made several remarkable and unsuspected discoveries, particularly in the calculus and the theory of numbers, which he never published.
    • He was, for example, employing Taylor and Maclaurin expansions more than forty years in advance of anyone else.
    • Turnbull's major beautifully written works include The Theory of Determinants, Matrices, and Invariants (1928), The Great Mathematicians (1929), Theory of Equations (1939), The Mathematical Discoveries of Newton (1945), and An Introduction to the Theory of Canonical Matrices (1945), which was jointly written with Aitken.
    • The Mathematical Discoveries of Newton arose from two lectures which Turnbull gave on Newton.
    • The first was given at a meeting of the Edinburgh Mathematical Society in December 1942 to commemorate the 300th anniversary of Newton's birth.
    • The second was given at a meeting of Edinburgh University's Mathematical and Physical Society.
    • Without going into too much detail I have tried to explain - as far as the work of geniuses can be explained - what led Newton to these discoveries.
    • It is so typical of Turnbull that he chose to emphasise the extraordinarily positive aspects of Newton's life and work.
    • After he retired in 1950 Turnbull, at the request of the Royal Society, began to work on the Correspondence of Isaac Newton.
    • Two volumes of this important work were published before his death.
    • Turnbull received many honours for his work, the most major being his election as a Fellow of the Royal Society in 1932.
    • He was also elected to the Royal Society of Edinburgh, receiving their Keith Medal and Gunning Victoria Jubilee Prize.
    • Outside mathematics Turnbull had several major interests.
    • One of these was music, where he was an excellent pianist, playing in a chamber orchestra.
    • Another of his loves was mountaineering and as a member of the Alpine Club he made many ascents without the help of a guide.
    • Nearer his home opportunities for practice were provided on the cliffs of St Andrews bay.
    • The mastery of the "Rock and Spindle" was not exactly part of the mathematical syllabus, but many a student experienced on this striking formation his first thrill of rock climbing under the guidance of his professor of mathematics.
    • They are close to the cliffs on the south side of St Andrews bay.
    • Ledermann [',' W Ledermann, Private communication (2000).','4] writes of Turnbull's:- .
    • The inevitable shyness of the younger guests was overcome by drawing room games, but the highlight of the evening, for those who could appreciate it, was the performance on two pianos by Professor and Mrs Turnbull.
    • Their playing, highly musical and exquisitely blended, was a beautiful expression of a harmonious partnership.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (5 books/articles) .
    • A Poster of Herbert Turnbull .
    • Papers in the Proceedings and Notes of the EMS .
    • The Tercentenary of the birth of James Gregory .
    • The Upper Hall of the University Library .
    • An entry in The Mathematical Gazetteer of the British Isles .
    • 2.nFellow of the Royal Society of Edinburghn1922 .
    • 4.nFellow of the Royal Societyn1932 .
    • 7.nHonorary Fellow of the Edinburgh Maths Societyn1954 .
    • History Topics: Matrices and determinants .
    • History Topics: Overview of Chinese mathematics .
    • Royal Society of Edinburgh .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Turnbull.html .

  53. Liu Hong biography
    • Liu Hong was of noble birth, descended from the Imperial family of the Eastern Han Dynasty.
    • This dynasty was established in 25 AD after the brief 15 year reign of Wang Mang's Hsin dynasty.
    • In Encyclopaedia Britannica the aims and achievements of the Han rulers are described:- .
    • the Han came to require cultural accomplishment from their public servants, making mastery of classical texts a condition of employment.
    • The title list of the enormous imperial library is China's first bibliography.
    • Its text included works on practical matters such as mathematics and medicine, as well as treatises on philosophy and religion and the arts.
    • Two works which he wrote, namely the Qi Yao Shu (The Art of Seven Planets) and a new version entitled the Ba Yuan Shu (The Art of Eight Elements) have been lost so we know little of their contents.
    • Perhaps the greatest of Liu's achievements was his work which led to a new calendar.
    • This calendar was published in 187 and described the motion of the moon far more accurately than any previous Chinese calendar.
    • His measurements of the length of the shadow of a pole at the summer and at the winter solstices give results which are accurate to within 1% of their true value.
    • List of References (3 books/articles) .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Liu_Hong.html .

  54. Paul Dirac biography
    • Charles Dirac was a Swiss citizen born in Monthey, in the Valais Canton of Switzerland, while his mother came from Cornwall in England.
    • Charles had been educated at the University of Geneva, then came to England in around 1888 and taught French in Bristol.
    • Charles and Florence married in 1899 and they moved into a house in Bishopston, Bristol, which they named Monthey after the town of Charles's birth.
    • Paul was one of three children, his older brother being Reginald Charles Felix Dirac and his younger sister being Beatrice Isabelle Marguerite Walla Dirac.
    • The first school which Paul attended was Bishop Primary school and already in this school his exceptional ability in mathematics became clear to his teachers.
    • When he was twelve years old he entered secondary school, attending the secondary school where his father taught which was part of the Merchant Venturers Technical College.
    • There was no Latin or Greek, something of which I was rather glad, because I did not appreciate the value of old cultures.
    • I was rushed through the lower forms, and was introduced at an especially early age to the basis of mathematics, physics and chemistry in the higher forms.
    • In mathematics I was studying from books which mostly were ahead of the rest of the class.
    • He completed his school education in 1918 and then studied electrical engineering at the University of Bristol.
    • Although mathematics was his favourite subject he chose to study an engineering course at university since he thought that the only possible career for a mathematician was school teaching and he certainly wanted to avoid that profession.
    • By this time he was developing a real passion for mathematics but his attempts to study at Cambridge failed for rather strange reasons.
    • Taking the Cambridge scholarship examinations in June 1921 he was awarded a scholarship to study mathematics at St John's College, Cambridge, but it did not provide enough to support him.
    • Dirac was offered the chance to study mathematics at Bristol without paying fees and he did so being awarded first class honours in 1923.
    • Dirac had been hoping to have his research supervised by Ebenezer Cunningham, for by this time Dirac had become fascinated in the general theory of relativity and wanted to undertake research on this topic.
    • The authors of [',' R H Dalitz and R Peierls, Paul Adrien Maurice Dirac, 8 August 1902 - 20 October 1984, Biographical Memoirs of Fellows of the Royal Society of London 32 (1986), 139-180.','13] write:- .
    • Fowler was then the leading theoretician in Cambridge, well versed in the quantum theory of atoms; his own research was mostly on statistical mechanics.
    • He recognised in Dirac a student of unusual ability.
    • Within six months of arriving in Cambridge he wrote two papers on these problems.
    • Already a person who had few friends, this personal tragedy had the effect of making him even more withdrawn.
    • This was as a result of Dirac being given proofs of a paper by Heisenberg to read in the summer of 1925.
    • The significance of the algebraic properties of Heisenberg's commutators struck Dirac when he was out for a walk in the country.
    • He realised that Heisenberg's uncertainty principle was a statement of the noncommutativity of the quantum mechanical observables.
    • This similarity provided the clue which led him to formulate for the first time a mathematically consistent general theory of quantum mechanics in correspondence with Hamiltonian mechanics.
    • Following the award of the degree he went to Copenhagen to work with Niels Bohr, moving on to Gottingen in February 1927 where he interacted with Robert Oppenheimer, Max Born, James Franck and the Russian Igor Tamm.
    • He was elected a Fellow of St John's College, Cambridge in 1927.
    • It was the first of many visits for he went again in 1929, 1930, 1932, 1933, 1935, 1936, 1937, 1957, 1965, and 1973.
    • In 1929 he made his first visit to the United States, lecturing at the Universities of Wisconsin and Michigan.
    • In 1930 Dirac published The principles of Quantum Mechanics and for this work he was awarded the Nobel Prize for Physics in 1933.
    • De Facio, reviewing [',' R H Dalitz (ed.), The collected works of P A M Dirac : 1924-1948 (Cambridge, 1995).','4], says of this book:- .
    • Dirac was not influenced by the feeding frenzy in experimental phenomenology of the time.
    • The authors of [',' R H Dalitz and R Peierls, Paul Adrien Maurice Dirac, 8 August 1902 - 20 October 1984, Biographical Memoirs of Fellows of the Royal Society of London 32 (1986), 139-180.','13] comment that the book:- .
    • His lectures at Cambridge were closely modelled on [The principles of Quantum Mechanics], and they conveyed to generations of students a powerful impression of the coherence and elegance of quantum theory.
    • Also in 1930 Dirac was elected a Fellow of the Royal Society.
    • This honour came on the first occasion that his name was put forward, in itself quite an unusual event which says much about the extremely high opinion that Dirac's fellow scientists had of him.
    • Dirac was appointed Lucasian professor of mathematics at the University of Cambridge in 1932, a post he held for 37 years.
    • In 1933 he published a pioneering paper on Lagrangian quantum mechanics which became the foundation on which Feynman later built his ideas of the path integral.
    • Both children adopted the name Dirac and Gabriel Andrew Dirac went on the became a famous pure mathematician, particularly contributing to graph theory, becoming professor of pure mathematics at the University of Aarhus in Denmark.
    • This association led to Dirac being prevented by the British government from visiting the Soviet Union after the end of the war; he was not able to visit again until 1957.
    • We noted above that Dirac was elected a fellow of the Royal Society in 1930.
    • in recognition of his remarkable contributions to relativistic dynamics of a particle in quantum mechanics.
    • In 1969 Dirac retired from the Lucasian chair of mathematics at Cambridge and went with his family to Florida in the United States.
    • He held visiting appointments at the University of Miami and at Florida State University.
    • Then, in 1971, Dirac was appointed professor of physics at Florida State University where he continued his research.
    • In these lectures he spoke about the problems of cosmology or, to be more precise, to the problems of non-dimensional combinations of world constants.
    • Although Dirac made vastly important contributions to physics, it is important to realise that he was always motivated by principles of mathematical beauty.
    • Dirac unified the theories of quantum mechanics and relativity theory, but he also is remembered for his outstanding work on the magnetic monopole, fundamental length, antimatter, the d-function, bra-kets, etc.
    • There is a standard folklore of Dirac stories, mostly revolving around Dirac saying exactly what he meant and no more.
    • It has been said in jest that his spoken vocabulary consisted of "Yes", "No", and "I don't know".
    • I was taught at school never to start a sentence without knowing the end of it.
    • Dirac received many honours for his work, some of which we have mentioned above.
    • He refused to accept honorary degrees but he did accept honorary membership of academies and learned societies.
    • The list of these is long but among them are USSR Academy of Sciences (1931), Indian Academy of Sciences (1939), Chinese Physical Society (1943), Royal Irish Academy (1944), Royal Society of Edinburgh (1946), Institut de France (1946), National Institute of Sciences of India (1947), American Physical Society (1948), National Academy of Sciences (1949), National Academy of Arts and Sciences (1950), Accademia delle Scienze di Torino (1951), Academia das Ciencias de Lisboa (1953), Pontifical Academy of Sciences, Vatican City (1958), Accademia Nazionale dei Lincei, Rome (1960), Royal Danish Academy of Sciences (1962), and Academie des Sciences Paris (1963).
    • He was appointed to the Order of Merit in 1973.
    • A memorial meeting was held at the University of Cambridge on 19 April 1985 and the papers presented at this meeting were published in Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987).
    • The papers [',' R H Dalitz, A biographical sketch of the life of Professor P A M Dirac, OM, FRS, in J G Taylor (ed.), Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987), 3-28.','12], [',' C J Eliezer, Some reminiscences of Professor P A M Dirac, in J G Taylor (ed.), Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987), 58-60.','15], [',' J E Lannutti, Eulogy for Paul A M Dirac, 19 November 1984: ’Who was this guy?’, in J G Taylor (ed.), Tributes to Paul Dirac (Bristol, 1987), 43-47.','25], [',' J Mehra, Dirac’s contribution to the early development of quantum mechanics, in Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987), 63-75.
    • ','30], [',' R Peierls, Address to Dirac memorial meeting, Cambridge, in Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987), 35-37.','37], [',' J C Polkinghorne, A brief reminiscence of Dirac, in Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987), 33-34.','39] and [',' S Shanmugadhasan, Dirac as research supervisor and other remembrances, in Tributes to Paul Dirac, Cambridge, 1985 (Bristol, 1987), 48-57.','40] come from this volume.
    • we vividly see everywhere the brilliant imprints of Dirac, unifier of quantum mechanics and relativity theory.
    • Each of the pieces not only is in praise of an exceptionally gifted intellect but also places on record how deeply and abidingly the human mind can delve into the realms of mathematical insight and modelling, keeping intact the spirit of beauty and clarity of a creative genius.
    • Only a few Nobel laureates ever can compare as well with this giant of mathematical sciences in whose demise the world of original thinking certainly has lost one of the most precious souls retaining fortunately still the glory for others to sing and emulate for a long time to come.
    • In November 1995 of a plaque was unveiled in Westminster Abbey commemorating Paul Dirac.
    • The volume [',' A Pais, M Jacob, D I Olive, and M F Atiyah, Paul Dirac : The man and his work (Cambridge, 1998).','9] consists of lectures presented to the Royal Society on this occasion.
    • The memorial address was presented by Stephen Hawking who was Dirac's successor in the Lucasian chair of mathematics at Cambridge which was also Newton's chair.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (46 books/articles) .
    • A Poster of Paul Dirac .
    • Multiple entries in The Mathematical Gazetteer of the British Isles .
    • 2.nFellow of the Royal Societyn1930 .
    • 6.nFellow of the Royal Society of Edinburghn1946 .
    • History Topics: A history of time: 20th century time .
    • Dictionary of National Biography .
    • Nobel prizes site (A biography of Dirac and his Nobel prize presentation speech) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Dirac.html .

  55. Lai-Sang Young biography
    • Entering the University of Wisconsin, Madison, Young received her B.A.
    • She then moved to the University of California at Berkeley where she was awarded an M.A.
    • After the award of her doctorate Young taught at Northwestern University for a year beginning in 1978.
    • We mention below the time she spent at the University of Warwick, England, and following this, in 1980, she also taught at Michigan State University.
    • Following this she held a position at the University of Arizona, and concurrently she worked at at the University of California at Los Angeles where she became Professor of Mathematics.
    • Following this she was awarded an Alfred P Sloan Fellowship for 1985-86 which enabled her to spend this year at the University of Bielefeld in Germany.
    • Young's first paper was published in 1977 and gives results which she proved at the beginning of her graduate studies.
    • The paper Entropy of continuous flows on compact 2-manifolds was published in Topology and shows that a continuous one-dimensional flow on a 2-dimensional manifold has zero topological entropy.
    • In each of 1979, 1980 and 1981 one of Young's papers appeared, these being: A closing lemma on the interval; Non absolutely continuous foliations for an Anosov diffeomorphism; and On the prevalence of horseshoes.
    • In 1979-80 a symposium was held at the University of Warwick in England on Dynamical systems and turbulence.
    • Young attended this and was one of the two editors, the other being David Rand, of the Proceedings of the Symposium which was published in 1981.
    • It would be best for us give Young's own description of the area of dynamical systems as she describes it at the beginning of her clearly written article Developments in chaotic dynamics which appeared in the Notices of the American Mathematical Society in 1998:- .
    • Today it stands at the crossroads of several areas of mathematics, including analysis, geometry, topology, probability, and mathematical physics.
    • It is generally regarded as a study of the iteration of maps, of time evolution of differential equations, and of group actions on manifolds.
    • In 1993 Young was honoured with the award of the Ruth Lyttle Satter Prize.
    • Young has played a leading role in the investigation of the statistical (or ergodic) properties of dynamical systems and has developed important and difficult techniques which have done much to clarify the subject.
    • In one major paper she established the exponential decay of correlations for a certain class of quadratic maps, which are one of the simplest kinds of nonuniformly hyperbolic systems.
    • this implies that the limit theorems of probability hold in this case.
    • A second outstanding piece of work is her joint paper with Benedicks, in which they study the statistical properties of the Henon attractor.
    • They show that orbits from a subset of the basin of attraction of positive measure have a common distribution in the limit ..
    • The joint paper referred to here is one Young wrote with Michael Benedicks which they published in the Ergodic Theory of Dynamical Systems in 1992.
    • Young's response to the presentation of the Ruth Lyttle Satter Prize was to thank Joan Birman for creating the prize and the committee who selected her which was chaired by the previous winner Dusa McDuff.
    • For the last ten years one of my projects has been to study the dynamics of strange sttractors.
    • Numerically it has been observed time and time again that if we randomly pick a point near an attractor and plot the first n points of its orbit, then the same picture emerges independently of the initial condition.
    • This suggests the existence of a natural invariant measure, one that governs the asymptotic distribution of almost all points in the basin of attraction.
    • After giving details of how this measure was discovered, and her own contributions, but she explained that there had been an embarrassing lack of examples.
    • These are of course only the first examples.
    • We have a long way to go before we understand the ergodic theory of strange attractors.
    • Young then spoke about women in mathematics:- .
    • I feel that more institutional support is still needed for women who try to juggle career and family, and a conscious effort on our part is necessary if we are to rid ourselves of the cultural prejudices that have existed for so long.
    • Young has published several excellent surveys of her area and her own contributions to it.
    • Her contribution Ergodic theory of differentiable dynamical systems appeared in the resulting 1995 publication presents:- .
    • A comprehensive self-contained survey of the results on ergodic theory of differentiable dynamical systems achieved in the last two decades..
    • More recent surveys include her invited lecture to the International Congress of Mathematicians in 1994.
    • An article based on her lecture Ergodic theory of attractors was published in the Proceedings.
    • recent developments in the study of the ergodic properties of attractors of certain dissipative surface diffeomorphisms.
    • The motivating prototype is the Henon family of plane diffeomorphisms ..
    • Young had been an invited plenary speaker at further meetings over the last few years including: the International Congress of Mathematical Physics (1997); the American Mathematical Society Annual Meeting (1998); and the Society for Industrial and Applied Mathematics Annual Meeting (1999).
    • At the first of these she lectured on the Ergodic theory of chaotic dynamical systems.
    • Young gives the following list of her current research interests: .
    • measurements of dynamical complexity, including entropy, Lyapunov exponents and fractal dimension; .
    • analysis of strange attractors; .
    • cumulative effects of small random perturbations (or "noise") on the long term behaviour of dynamical systems; .
    • Young, who is currently Henry & Lucy Moses Professor of Science at New York University's Courant Institute of Mathematical Sciences, was elected as a Fellow of the American Academy of Arts and Sciences in 2004.
    • In the following year she was the Association for Women in Mathematics' Noether Lecturer at the Joint Mathematics Meetings in Atlanta, Georgia.
    • In 2007 the Association for Women in Mathematics and the Society for Industrial and Applied Mathematics selected Young to give the Sonia Kovalevsky Lecture at the 2007 SIAM Conference on Applications of Dynamical Systems held at the end of May in Snowbird, Utah.
    • The Kovalevsky Lecture recognizes her fundamental contributions in the field of ergodic theory and dynamical systems.
    • Her pioneering research has had a significant impact in the investigation of dynamical complexity, strange attractors and probabilistic laws of chaotic systems.
    • She is an inspiration to the entire mathematics community, especially to the women's mathematics community.
    • Barbara Keyfitz, Past President of the Association for Women in Mathematics, said that:- .
    • with the choice of Lai-Sang Young to give the Kovalevsky lecture within two years of her Noether lecture, two independent selection committees have recognized the importance of Young's work.
    • Cathy Kessel, President of the Association for Women in Mathematics, commented that:- .
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Young_Lai-Sang.html .

  56. Terence Tao biography
    • His father, Billy Tao, is a Chinese-born paediatrician who has undertaken research on educating gifted children and on autism.
    • Terry's mother, Grace, was born in Hong Kong and has a university degree in physics and mathematics.
    • Billy and Grace met while they were studying at the University of Hong Kong and they emigrated to Australia in 1972.
    • Grace Tao taught physics, chemistry, science and mathematics in various secondary schools in Hong Kong before she emigrated to Australia and, once in Australia, also taught in secondary schools there.
    • Terry, the subject of this biography, is their eldest child, having two younger brothers Trevor and Nigel.
    • The article [',' M A (Ken) Clements, Terence Tao, Educational Studies in Mathematics 15 (3) (1984), 213-238.','4] is an evaluation of Terry's mathematical abilities just before his eighth birthday by which time he was attending Blackwood High School, Adelaide.
    • sitting in the far corner of a room reading a hardback book with the title 'Calculus'.
    • Clements discovered that Terry knew the definition of a group and could solve graph sketching problems using the differential calculus.
    • He wondered how much his mother was teaching him but found that her role [',' M A (Ken) Clements, Terence Tao, Educational Studies in Mathematics 15 (3) (1984), 213-238.','4] .
    • is more one of guiding and stimulating Terence's development than one of teaching him.
    • She said that Terence likes to read mathematics by himself, and he often spent three or four hours after school reading mathematics textbooks.
    • By the time Terry reached the age of eleven, he was dividing his time between his studies at Blackwood High School and taking classes at Flinders University in Adelaide where he was taught by Garth Gaudry.
    • Even earlier, at the age of ten, he began participating in International Mathematical Olympiads.
    • At the age of fourteen he began full-time university studies at Flinders University and was awarded a B.Sc.
    • These are: Weak-type endpoint bounds for Riesz means; (with Andrew C Millard) On the structure of projective group representations in quaternionic Hilbert space; On the almost everywhere convergence of wavelet summation methods; and Convolution operators on Lipschitz graphs with harmonic kernels.
    • Following the award of his doctorate, Tao was appointed Hedrick Assistant Professor at the University of California at Los Angeles, a position he held from 1996 to 1998.
    • He continued as an assistant professor at the University of California at Los Angeles where, at the age of twenty-four, he was promoted to full professor in 2000.
    • It is very difficult to write a biography of someone who is at the height of their creative powers as Tao is.
    • Anything that one writes about his research contributions will be quickly outdated as he is contributing major results in such a wide range of different areas.
    • Yet he has produced such a fantastic collection of results, leading to the award of all the top prizes in mathematics, that one must try to at least give a vague picture of the work of this remarkable mathematician.
    • These include: the Salem Prize (2000); the Bocher Memorial Prize from the American Mathematical Society (2002); the Clay Research Award from the Clay Mathematical Institute (2003); the Levi L Conant Award from the American Mathematical Society (2005); the Australian Mathematical Society Medal (2005); the ISAAC Award from the International Society of Analysis, its Application and Computation (2005); the SASTRA Ramanujan Prize (2006); the Fields Medal (2006); the Ostrowski Prize from the Ostrowski Foundation (2007); the Alan T Waterman Award from the National Science Foundation (2008); the Onsager Medal(2008); the Information Theory Society Paper Award (2008); the Convocation Award from Flinders University Alumni Association (2008); the King Faisal International Prize (Mathematics) (2010); the Nemmers Prize in Mathematics from Northwestern University (2010); and the George Polya Prize from the Society for Industrial and Applied Mathematics (2010).
    • He has been elected to the Australian Academy of Sciences (2006), to a fellowship of the Royal Society (2007), to the National Academy of Sciences (2008), and to the American Academy of Arts and Sciences (2009).
    • He was a finalist in Australian of the Year in 2007.
    • Monthly 53 (9) (2006), 1037-1044.','1], describing the award of the Fields Medal, gives this overview:- .
    • He combines sheer technical power, an other-worldly ingenuity for hitting upon new ideas, and a startlingly natural point of view that leaves other mathematicians wondering, " Why didn't anyone see that before?" At 31 years of age, Tao has written over eighty research papers, with over thirty collaborators, and his interests range over a wide swath of mathematics, including harmonic analysis, nonlinear partial differential equations, and combinatorics.
    • " I work in a number of areas, but I don't view them as being disconnected," he said in an interview published in the Clay Mathematics Institute Annual Report.
    • " I tend to view mathematics as a unified subject and am particularly happy when I get the opportunity to work on a project that involves several fields at once." .
    • Let us note here that he has now greatly exceeded the eighty research papers mentioned in the 2006 article with MathSciNet recording a list of 224 publications between 1996 and 2010.
    • The Press Release which announced the award of the Fields Medal to Tao listed his accomplishments in a number of areas which had led to the award of this most prestigious mathematical award.
    • First it describes his work with Ben Green on the distribution of prime numbers.
    • They proved the remarkable result that the primes contain arithmetic progressions of any length.
    • An area to which Tao has made many contributions is that of the Kakeya problem.
    • This problem, originally posed in 2 dimensions, asked for the minimum area of a shape in which one can rotate a needle through 180° .
    • Tao has worked on the n-dimensional Kakeya problem where again the minimum volume can be made as small as one chooses, but the fractal dimension of the shape is unknown.
    • Another area in which Tao has worked is solving special cases of the equations of general relativity describing gravity.
    • Imposing cylindrical symmetry on the equations leads to the "wave maps" problem where, although it has yet to be solved, Tao's contributions have led to a great resurgence of interest since his ideas seem to have made a solution possible.
    • Another area where Tao has introduced novel ideas, giving the subject a whole new look, is the theory of the nonlinear Schrodinger equations.
    • These equations have considerable practical applications and again Tao's insights have shed considerable light on the behaviour of a particular Schrodinger equation.
    • One might imagine that with his remarkable output of research papers, Tao would not find time to write books.
    • The material starts at the very beginning - the construction of the number systems and set theory, and then goes on to the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several-variable calculus and Fourier analysis, and finally the Lebesgue integral.
    • These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces.
    • Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the various tactics involved in solving mathematical problems at the Mathematical Olympiad level.
    • Assuming only basic high-school mathematics, the text is ideal for general readers and students of 14 years and above with an interest in pure mathematics.
    • But, amazingly, this still does not complete the list of Tao's 2006 books for in that year, in collaboration with Van Vu, he published Additive combinatorics.
    • Serge Konyagin and Ilya Shkredov begin their detailed review of the book by describing the area:- .
    • The subject of the book under review is additive combinatorics - a young and extensively developing area in mathematics with many applications, especially to number theory.
    • Modern additive combinatorics studies various groups, from the classical group of integers to abstract groups of arbitrary nature.
    • However, everybody who intends to read this book should be ready to study tools and ideas from different areas of mathematics, which are concentrated in the book and presented in an accessible, coherent, and intuitively clear manner and provided with immediate applications to problems in additive combinatorics.
    • It will come as no surprise to learn that Tao, who is such an innovator in everything he does, has created a new style of book.
    • The textbook and research monographs described above are innovative in their approach but are traditional type of books.
    • Pages from year one of a mathematical blog and, in 2009, two similar books Poincare's legacies, pages from year two of a mathematical blog Part I and Part II.
    • Textbooks and popular science are still the two obvious niches for mathematics in the book market, but the advent of the Internet has brought about a sudden change in the possibilities for mathematical exposition, because now anybody can put anything they like on the Web.
    • As a result, there has been a rapid rise in a form of mathematical exposition that is too technical for the layperson, but much easier to read and enjoy for mathematicians than a textbook.
    • A medium that is particularly well suited to this is the blog, and the undisputed king of all mathematics blogs, with thousands of regular readers, is that of Terence Tao.
    • Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics as diverse as partial differential equations, analytic number theory, the geometry of 3-manifolds, nonstandard analysis, group theory, model theory, quantum mechanics, probability, ergodic theory, combinatorics, harmonic analysis, image processing, functional analysis, and many others.
    • Some of these are areas to which he has made fundamental contributions.
    • Others are areas that he appears to understand at the deep intuitive level of an expert despite officially not working in those areas.
    • It has been said that Hilbert was the last person to know all of mathematics, but it is not easy to find gaps in Tao's knowledge, and if you do then you may well find that the gaps have been filled a year later.
    • Now, in an interesting experiment, several of Tao's blog posts have been tidied up (partly in response to comments from others on the posts) and published as books.
    • In 2010 the next in Tao's series was published An epsilon of room, I: real analysis.
    • Pages from year three of a mathematical blog.
    • One can anticipate a long and fascinating series of books that will appear over the next years.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (7 books/articles) .
    • 8.nFellow of the Royal Societyn2007 .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Tao.html .

  57. Jean-Louis Koszul biography
    • Jean-Louis was the youngest of his parents four children, having three older sisters, Marie Andree, Antoinette, and Jeanne.
    • He was educated at the Lycee Fustel-de-Coulanges in Strasbourg before studying at the Faculty of Science in Strasbourg and the Faculty of Science in Paris.
    • The first was Sur le troisieme nombre de Betti des espaces de groupes de Lie compacts Ⓣ in which he completed the proof that the third Betti number of a simple compact Lie group is one by studying certain of the exceptional groups.
    • The second of the three papers was Sur les operateurs de derivation dans un anneau Ⓣ in which he studied rings having a derivation operator with the formal properties of the coboundary operator of algebraic topology.
    • The third of the three papers was Sur l'homologie des espaces homogenes Ⓣ in which he applied the ideas in his second paper to cohomology rings.
    • He became a second generation member of Bourbaki along with J Dixmier, R Godement, S Eilenberg, P Samuel, J P Serre and L Schwartz.
    • Koszul was appointed as Maitre de Conferences at the University of Strasbourg in 1949.
    • He was promoted to professor there in 1956, and remained in Strasbourg until he appointed professor in the Faculty of Science at Grenoble in 1963.
    • The main topics on which Koszul undertook research included: homology and cohomology of Lie algebra; relative cohomology; reductive subalgebras and the transgression theorem; the formalism of spectral sequences; "Koszul complexes"; proper and differentiable actions of Lie groups; slices; hermitian forms on complex homogeneous domains; bounded domains; locally flat manifolds; convex homogeneous domains; simplicial spaces; themes related to Gelfand-Fuks theory and supergeometry.
    • In 1950 Koszul published a major 62 page paper Homologie et cohomologie des algebres de Lie Ⓣ in which he studied the connections between the homology and cohomology (with real coefficients) of a compact connected Lie group G and purely algebraic problems on the Lie algebra associated with G.
    • The superb lecture notes were published in 1957 and covered: Čech cohomology with coefficients in a sheaf; resolutions; a theorem concerning the cohomology with coefficients in a sheaf for a paracompact space; isomorphism of ordinary Čech cohomology with de Rham-cohomology, Alexander-Spanier- cohomology, and singular cohomology.
    • In the autumn of 1958 he again held a seminar series in Sao Paulo, this time on symmetric spaces.
    • Apart from the more or less standard theorems on symmetric spaces, the author discusses the geometry of geodesics, the Bergmann metric, and finally investigates the bounded domains in considerable detail.
    • In the mid 1960s Kosul lectured at the Tata Institute of Fundamental Research in Bombay On groups of transformations and On fibre bundles and differential geometry.
    • The second course was on the theory of connections and the lecture notes were first published in 1965 and reprinted in 1986.
    • After a number of further important publications which appeared in the proceedings of various conferences that Koszul attended such as Convegno sui Gruppi Topologici e Gruppi di Lie in Rome (1974), Symplectic geometry in Toulouse (1981), an international meeting on geometry and physics in Florence (1982), he published Introduction to symplectic geometry in Chinese in 1986.
    • This work has coincided with developments in the field of analytic mechanics.
    • Many new ideas have also been derived with the help of a great variety of notions from modern algebra, differential geometry, Lie groups, functional analysis, differentiable manifolds and representation theory.
    • [Koszul's book] emphasizes the differential-geometric and topological properties of symplectic manifolds.
    • It gives a modern treatment of the subject that is useful for beginners as well as for experts.
    • Koszul was honoured with election to the Academy of Sciences in Paris on 28 January 1980.
    • In 1994 a volume containing 24 articles by Koszul were published under the title Selected papers of J-L Koszul.
    • Among Koszul's hobbies was mountaineering and he was a member of the Club Alpin Francais.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • History Topics: Bourbaki: the post-war years .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Koszul.html .

  58. Heron of Alexandria (about 10-about 75)
    • Heron of Alexandria .
    • Sometimes called Hero, Heron of Alexandria was an important geometer and worker in mechanics.
    • Perhaps the first comment worth making is how common the name Heron was around this time and it is a difficult problem in the history of mathematics to identify which references to Heron are to the mathematician described in this article and which are to others of the same name.
    • There are additional problems of identification which we discuss below.
    • There were two main schools of thought on this, one believing that he lived around 150 BC and the second believing that he lived around 250 AD.
    • The first of these was based mainly on the fact that Heron does not quote from any work later than Archimedes.
    • Both of these arguments have been shown to be wrong.
    • There was a third date proposed which was based on the belief that Heron was a contemporary of Columella.
    • Columella, in a text written in about 62 AD [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1931).','5]:- .
    • gave measurements of plane figures which agree with the formulas used by Heron, notably those for the equilateral triangle, the regular hexagon (in this case not only the formula but the actual figures agree with Heron's) and the segment of a circle which is less than a semicircle ..
    • We now know that those who believed that Heron lived around the time of Columella were in fact correct, for Neugebauer in 1938 discovered that Heron referred to a recent eclipse in one of his works which, from the information given by Heron, he was able to identify with one which took place in Alexandria at 23.00 hours on 13 March 62.
    • His works look like lecture notes from courses he must have given there on mathematics, physics, pneumatics, and mechanics.
    • Some are clearly textbooks while others are perhaps drafts of lecture notes not yet worked into final form for a student textbook.
    • Pappus describes the contribution of Heron in Book VIII of his Mathematical Collection.
    • Pappus writes (see for example [',' I Thomas, Selections illustrating the history of Greek mathematics II (London, 1941).','8]):- .
    • The mechanicians of Heron's school say that mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals, architecture, carpentering and painting and anything involving skill with the hands.
    • the ancients also describe as mechanicians the wonder-workers, of whom some work by means of pneumatics, as Heron in his Pneumatica, some by using strings and ropes, thinking to imitate the movements of living things, as Heron in his Automata and Balancings, ..
    • or by using water to tell the time, as Heron in his Hydria, which appears to have affinities with the science of sundials.
    • A large number of works by Heron have survived, although the authorship of some is disputed.
    • We will discuss some of the disagreements in our list of Heron's works below.
    • It contains a chapter on astronomy giving a method to find the distance between Alexandria and Rome using the difference between local times at which an eclipse of the moon is observed at each cities.
    • The fact that Ptolemy does not appear to have known of this method led historians to mistakenly believe Heron lived after Ptolemy; .
    • Belopoeica describing how to construct engines of war.
    • The cheirobalistra about catapults is thought to be part of a dictionary of catapults but was almost certainly not written by Heron; .
    • Metrica which gives methods of measurement.
    • Definitiones contains 133 definitions of geometrical terms beginning with points, lines etc.
    • In [',' W R Knorr, ’Arithmetike stoicheiosis’ : on Diophantus and Hero of Alexandria, Historia Math.
    • Geometria seems to be a different version of the first chapter of the Metrica based entirely on examples.
    • Stereometrica measures three-dimensional objects and is at least in part based on the second chapter of the Metrica again based on examples.
    • Mensurae measures a whole variety of different objects and is connected with parts of Stereometrica and Metrica although it must be mainly the work of a later author; .
    • Catoptrica deals with mirrors and is attributed by some historians to Ptolemy although most now seem to believe that this is a genuine work of Heron.
    • Let us examine some of Heron's work in a little more depth.
    • Book I of his treatise Metrica deals with areas of triangles, quadrilaterals, regular polygons of between 3 and 12 sides, surfaces of cones, cylinders, prisms, pyramids, spheres etc.
    • A method, known to the Babylonians 2000 years before, is also given for approximating the square root of a number.
    • Heron gives this in the following form (see for example [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1931).','5]):- .
    • The side of 720 will therefore be very nearly 265/6.
    • If we desire to make the difference smaller still than 1/36, we shall take 7201/36 instead of 729 (or rather we should take 265/6 instead of 27), and by proceeding in the same way we shall find the resulting difference much less than 1/36.
    • Heron also proves his famous formula in Book I of the Metrica : .
    • if A is the area of a triangle with sides a, b and c and s = (a + b + c)/2 then .
    • In Book II of Metrica, Heron considers the measurement of volumes of various three dimensional figures such as spheres, cylinders, cones, prisms, pyramids etc.
    • His preface is interesting, partly because knowledge of the work of Archimedes does not seem to be as widely known as one might expect (see for example [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1931).','5]):- .
    • After the measurement of surfaces, rectilinear or not, it is proper to proceed to solid bodies, the surfaces of which we have already measured in the preceding book, surfaces plane and spherical, conical and cylindrical, and irregular surfaces as well.
    • The methods of dealing with these solids are, in view of their surprising character, referred to Archimedes by certain writers who give the traditional account of their origin.
    • But whether they belong to Archimedes or another, it is necessary to give a sketch of these results as well.
    • Book III of Metrica deals with dividing areas and volumes according to a given ratio.
    • This was a problem which Euclid investigated in his work On divisions of figures and Heron's Book III has a lot in common with the work of Euclid.
    • Also in Book III, Heron gives a method to find the cube root of a number.
    • In particular Heron finds the cube root of 100 and the authors of [',' G Deslauriers and S Dubuc, Le calcul de la racine cubique selon Heron, Elem.
    • 51 (1) (1996), 28-34.','9] give a general formula for the cube root of N which Heron seems to have used in his calculation: .
    • Heron begins with a theoretical consideration of pressure in fluids.
    • Some of this theory is right but, not surprisingly, some is quite wrong.
    • Then there follows a description of a whole collection of what might best be described as mechanical toys for children [',' A G Drachmann, M S Mahoney, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • It seems to be an attempt to make scientific theories relevant to everyday items that students of the time would be familiar with.
    • There is, rather remarkably, descriptions of over 100 machines such as a fire engine, a wind organ, a coin-operated machine, and a steam-powered engine called an aeolipile.
    • The aeolipile was a hollow sphere mounted so that it could turn on a pair of hollow tubes that provided steam to the sphere from a cauldron.
    • Heron wrote a number of important treatises on mechanics.
    • They give methods of lifting heavy weights and describe simple mechanical machines.
    • It also examines the theory of motion, certain statics problems, and the theory of the balance.
    • There is a discussion on centres of gravity of plane figures.
    • Book III examines methods of transporting objects by such means as sledges, the use of cranes, and looks at wine presses.
    • Other works have been attributed to Heron, and for some of these we have fragments, for others there are only references.
    • The works for which fragments survive include one on Water clocks in four books, and Commentary on Euclid's Elements which must have covered at least the first eight books of the Elements.
    • Also in the Fihrist, a tenth century survey of Islamic culture, a work by Heron on how to use an astrolabe is mentioned.
    • Finally it is interesting to look at the opinions that various writers have expressed as to the quality and importance of Heron.
    • Neugebauer writes [',' O Neugebauer, A history of ancient mathematical astronomy (New York, 1975).','7]:- .
    • The decipherment of the mathematical cuneiform texts made it clear that much of the "Heronic" type of Greek mathematics is simply the last phase of the Babylonian mathematical tradition which extends over 1800 years.
    • Some have considered Heron to be an ignorant artisan who copied the contents of his books without understanding what he wrote.
    • This in particular has been levelled against the Pneumatica but Drachmann, writing in [',' A G Drachmann, M S Mahoney, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Some scholars have approved of Heron's practical skills as a surveyor but claimed that his knowledge of science was negligible.
    • However, Mahony writes in [',' A G Drachmann, M S Mahoney, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • In the light of recent scholarship, he now appears as a well-educated and often ingenious applied mathematician, as well as a vital link in a continuous tradition of practical mathematics from the Babylonians, through the Arabs, to Renaissance Europe.
    • Finally Heath writes in [',' T L Heath, A history of Greek mathematics I, II (Oxford, 1931).','5]:- .
    • The practical utility of Heron's manuals being so great, it was natural that they should have great vogue, and equally natural that the most popular of them at any rate should be re-edited, altered and added to by later writers; this was inevitable with books which, like the "Elements" of Euclid, were in regular use in Greek, Byzantine, Roman, and Arabian education for centuries.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (19 books/articles) .
    • A Poster of Heron of Alexandria .
    • Honours awarded to Heron of Alexandria .
    • History Topics: Greek Astronomy .
    • History Topics: Doubling the cube .
    • History Topics: Pythagoras's theorem in Babylonian mathematics .
    • History Topics: The Golden ratio .
    • History Topics: Light through the ages: Ancient Greece to Maxwell .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: Euclid's definitions .
    • Dictionary of Scientific Biography .
    • University of Iowa .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Heron.html .

  59. Fibonacci (1170-1250)
    • He was the son of Guilielmo and a member of the Bonacci family.
    • As stated in [',' K Vogel, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Did his countrymen wish to express by this epithet their disdain for a man who concerned himself with questions of no practical value, or does the word in the Tuscan dialect mean a much-travelled man, which he was? .
    • His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia.
    • The town lies at the mouth of the Wadi Soummam near Mount Gouraya and Cape Carbon.
    • Fibonacci was taught mathematics in Bugia and travelled widely with his father and recognised the enormous advantages of the mathematical systems used in the countries they visited.
    • When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting.
    • There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily and Provence, in all its various forms.
    • There he wrote a number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own.
    • Fibonacci lived in the days before printing, so his books were hand written and the only way to have a copy of one of his books was to have another hand-written copy made.
    • Of his books we still have copies of Liber abaci (1202), Practica geometriae (1220), Flos (1225), and Liber quadratorum.
    • His book on commercial arithmetic Di minor guisa is lost as is his commentary on Book X of Euclid's Elements which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view.
    • Fibonacci was a contemporary of Jordanus but he was a far more sophisticated mathematician and his achievements were clearly recognised, although it was the practical applications rather than the abstract theorems that made him famous to his contemporaries.
    • He had been crowned king of Germany in 1212 and then crowned Holy Roman emperor by the Pope in St Peter's Church in Rome in November 1220.
    • State control was introduced on trade and manufacture, and civil servants to oversee this monopoly were trained at the University of Naples which Frederick founded for this purpose in 1224.
    • Frederick became aware of Fibonacci's work through the scholars at his court who had corresponded with Fibonacci since his return to Pisa around 1200.
    • Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci.
    • Three of these problems were solved by Fibonacci and he gives solutions in Flos which he sent to Frederick II.
    • We give some details of one of these problems below.
    • This is a decree made by the Republic of Pisa in 1240 in which a salary is awarded to:- .
    • This salary was given to Fibonacci in recognition for the services that he had given to the city, advising on matters of accounting and teaching the citizens.
    • The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe.
    • Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work.
    • Certainly many of the problems that Fibonacci considers in Liber abaci were similar to those appearing in Arab sources.
    • The second section of Liber abaci contains a large collection of problems aimed at merchants.
    • They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China.
    • A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today:- .
    • A certain man put a pair of rabbits in a place surrounded on all sides by a wall.
    • How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? .
    • This sequence, in which each number is the sum of the two preceding numbers, has proved extremely fruitful and appears in many different areas of mathematics and science.
    • The Fibonacci Quarterly is a modern journal devoted to studying mathematics related to this sequence.
    • Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.
    • There are also problems involving perfect numbers, problems involving the Chinese remainder theorem and problems involving summing arithmetic and geometric series.
    • A second edition of Liber abaci was produced by Fibonacci in 1228 with a preface, typical of so many second editions of books, stating that:- .
    • Another of Fibonacci's books is Practica geometriae written in 1220 which is dedicated to Dominicus Hispanus whom we mentioned above.
    • It contains a large collection of geometry problems arranged into eight chapters with theorems based on Euclid's Elements and Euclid's On Divisions.
    • In addition to geometrical theorems with precise proofs, the book includes practical information for surveyors, including a chapter on how to calculate the height of tall objects using similar triangles.
    • The final chapter presents what Fibonacci called geometrical subtleties [',' K Vogel, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Among those included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles; the inverse calculation is also given, as well as that of the sides from the surfaces.
    • In Flos Fibonacci gives an accurate approximation to a root of 10x + 2x2 + x3 = 20, one of the problems that he was challenged to solve by Johannes of Palermo.
    • This problem was not made up by Johannes of Palermo, rather he took it from Omar Khayyam's algebra book where it is solved by means of the intersection of a circle and a hyperbola.
    • Fibonacci proves that the root of the equation is neither an integer nor a fraction, nor the square root of a fraction.
    • And because it was not possible to solve this equation in any other of the above ways, I worked to reduce the solution to an approximation.
    • Liber quadratorum, written in 1225, is Fibonacci's most impressive piece of work, although not the work for which he is most famous.
    • The book's name means the book of squares and it is a number theory book which, among other things, examines methods to find Pythogorean triples.
    • Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n2 + (2n+1) = (n+1)2.
    • I thought about the origin of all square numbers and discovered that they arose from the regular ascent of odd numbers.
    • For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers.
    • Thus when I wish to find two square numbers whose addition produces a square number, I take any odd square number as one of the two square numbers and I find the other square number by the addition of all the odd numbers from unity up to but excluding the odd square number.
    • For example, I take 9 as one of the two squares mentioned; the remaining square will be obtained by the addition of all the odd numbers below 9, namely 1, 3, 5, 7, whose sum is 16, a square number, which when added to 9 gives 25, a square number.
    • He defined the concept of a congruum, a number of the form ab(a + b)(a - b), if a + b is even, and 4 times this if a + b is odd.
    • Fibonacci's influence was more limited than one might have hoped and apart from his role in spreading the use of the Hindu-Arabic numerals and his rabbit problem, Fibonacci's contribution to mathematics has been largely overlooked.
    • As explained in [',' K Vogel, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Direct influence was exerted only by those portions of the "Liber abaci" and of the "Practica" that served to introduce Indian-Arabic numerals and methods and contributed to the mastering of the problems of daily life.
    • Here Fibonacci became the teacher of the masters of computation and of the surveyors, as one learns from the "Summa" of Luca Pacioli ..
    • Fibonacci was also the teacher of the "Cossists", who took their name from the word 'causa' which was first used in the West by Fibonacci in place of 'res' or 'radix'.
    • Three hundred years later we find the same results appearing in the work of Maurolico.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (22 books/articles) .
    • A Poster of Fibonacci .
    • T C Scott and P Marketos The origin of the Fibonacci sequence .
    • History Topics: Mathematical games and recreations .
    • History Topics: A chronology of pi .
    • History Topics: An overview of the history of mathematics .
    • History Topics: The trigonometric functions .
    • History Topics: Prime numbers .
    • History Topics: A history of Zero .
    • History Topics: Arabic numerals .
    • History Topics: The Golden ratio .
    • History Topics: The real numbers: Pythagoras to Stevin .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Fibonacci.html .

  60. Blaise Pascal (1623-1662)
    • Blaise Pascal was the third of Etienne Pascal's children and his only son.
    • Etienne Pascal decided that Blaise was not to study mathematics before the age of 15 and all mathematics texts were removed from their house.
    • Blaise however, his curiosity raised by this, started to work on geometry himself at the age of 12.
    • He discovered that the sum of the angles of a triangle are two right angles and, when his father found out, he relented and allowed Blaise a copy of Euclid.
    • At the age of 14 Blaise Pascal started to accompany his father to Mersenne's meetings.
    • Mersenne belonged to the religious order of the Minims, and his cell in Paris was a frequent meeting place for Gassendi, Roberval, Carcavi, Auzout, Mydorge, Mylon, Desargues and others.
    • Soon, certainly by the time he was 15, Blaise came to admire the work of Desargues.
    • At the age of sixteen, Pascal presented a single piece of paper to one of Mersenne's meetings in June 1639.
    • It contained a number of projective geometry theorems, including Pascal's mystic hexagon.
    • You can see pictures of the Mystic Hexagram at THIS LINK.
    • The device, called the Pascaline, resembled a mechanical calculator of the 1940s.
    • You can see pictures of the Pascaline at THIS LINK and at THIS LINK.
    • There were problems faced by Pascal in the design of the calculator which were due to the design of the French currency at that time.
    • Pascal had to solve much harder technical problems to work with this division of the livre into 240 than he would have had if the division had been 100.
    • However production of the machines started in 1642 but, as Adamson writes in [',' D Adamson, Blaise Pascal : mathematician, physicist and thinker about God (Basingstoke, 1995).','3], .
    • By 1652 fifty prototypes had been produced, but few machines were sold, and manufacture of Pascal's arithmetical calculator ceased in that year.
    • Events of 1646 were very significant for the young Pascal.
    • From about this time Pascal began a series of experiments on atmospheric pressure.
    • In August of 1648 Pascal observed that the pressure of the atmosphere decreases with height and deduced that a vacuum existed above the atmosphere.
    • It was I who two years ago advised him to do it, for although I have not performed it myself, I did not doubt of its success ..
    • In October 1647 Pascal wrote New Experiments Concerning Vacuums which led to disputes with a number of scientists who, like Descartes, did not believe in a vacuum.
    • Etienne Pascal died in September 1651 and following this Blaise wrote to one of his sisters giving a deeply Christian meaning to death in general and his father's death in particular.
    • From May 1653 Pascal worked on mathematics and physics writing Treatise on the Equilibrium of Liquids (1653) in which he explains Pascal's law of pressure.
    • This treatise is a complete outline of a system of hydrostatics, the first in the history of science, it embodies his most distinctive and important contribution to physical theory.
    • In The Generation of Conic Sections (mostly completed by March 1648 but worked on again in 1653 and 1654) Pascal considered conics generated by central projection of a circle.
    • This was meant to be the first part of a treatise on conics which Pascal never completed.
    • The work is now lost but Leibniz and Tschirnhaus made notes from it and it is through these notes that a fairly complete picture of the work is now possible.
    • Although Pascal was not the first to study the Pascal triangle, his work on the topic in Treatise on the Arithmetical Triangle was the most important on this topic and, through the work of Wallis, Pascal's work on the binomial coefficients was to lead Newton to his discovery of the general binomial theorem for fractional and negative powers.
    • In correspondence with Fermat he laid the foundation for the theory of probability.
    • This correspondence consisted of five letters and occurred in the summer of 1654.
    • They considered the dice problem, already studied by Cardan, and the problem of points also considered by Cardan and, around the same time, Pacioli and Tartaglia.
    • The dice problem asks how many times one must throw a pair of dice before one expects a double six while the problem of points asks how to divide the stakes if a game of dice is incomplete.
    • They solved the problem of points for a two player game but did not develop powerful enough mathematical methods to solve it for three or more players.
    • Through the period of this correspondence Pascal was unwell.
    • In one of the letters to Fermat written in July 1654 he writes .
    • After this time Pascal made visits to the Jansenist monastery Port-Royal des Champs about 30 km south west of Paris.
    • These were written in defence of his friend Antoine Arnauld, an opponent of the Jesuits and a defender of Jansenism, who was on trial before the faculty of theology in Paris for his controversial religious works.
    • Pascal's most famous work in philosophy is Pensees Ⓣ, a collection of personal thoughts on human suffering and faith in God which he began in late 1656 and continued to work on during 1657 and 1658.
    • His last work was on the cycloid, the curve traced by a point on the circumference of a rolling circle.
    • He applied Cavalieri's calculus of indivisibles to the problem of the area of any segment of the cycloid and the centre of gravity of any segment.
    • He also solved the problems of the volume and surface area of the solid of revolution formed by rotating the cycloid about the x-axis.
    • Wren had been working on Pascal's challenge and he in turn challenged Pascal, Fermat and Roberval to find the arc length, the length of the arch, of the cycloid.
    • Pascal died at the age of 39 in intense pain after a malignant growth in his stomach spread to the brain.
    • a man of slight build with a loud voice and somewhat overbearing manner.
    • he lived most of his adult life in great pain.
    • precocious, stubbornly persevering, a perfectionist, pugnacious to the point of bullying ruthlessness yet seeking to be meek and humble ..
    • In [',' R Taton, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Pascal was embarrassed by the very abundance of his talents.
    • It has been suggested that it was his too concrete turn of mind that prevented his discovering the infinitesimal calculus, and in some of the Provinciales the mysterious relations of human beings with God are treated as if they were a geometrical problem.
    • But these considerations are far outweighed by the profit that he drew from the multiplicity of his gifts, his religious writings are rigorous because of his scientific training..
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (53 books/articles) .
    • A Poster of Blaise Pascal .
    • Gilberte Pascal: The life of Pascal .
    • Another picture of it .
    • History Topics: An overview of the history of mathematics .
    • History Topics: Jaina mathematics .
    • History Topics: Infinity .
    • History Topics: The brachistochrone problem .
    • History Topics: Cubic surfaces .
    • History Topics: The mathematician and the forger .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: The Weil family .
    • Famous Curves: Pearls of Sluze .
    • Famous Curves: Limacon of Pascal .
    • Dictionary of Scientific Biography .
    • Stanford Encyclopedia of Philosophy (Pascal's wager) .
    • History of Computing Project .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Pascal.html .

  61. Sheila Power Tinney biography
    • Let us note at this point that she became Sheila Tinney after her marriage but we shall refer to her by the name of Power until we reach the time when she married.
    • Michael Power had been appointed to the Chair of Mathematics at University College Galway in 1912 and he held this position for over forty years until he retired in 1955.
    • Sheila was the fourth of her parents' six children.
    • One of Sheila's fondest memories was of her mother playing the piano at home, and she became an accomplished pianist herself ..
    • Sheila was educated at the Dominican College of Taylor's Hill in Galway.
    • This girls' Roman Catholic school had been founded by the Dominican Sisters in 1858 and, in keeping with the Dominican philosophy of education, aimed to give an education for the whole person.
    • From the Dominican College of Taylor's Hill in Galway, Sheila went to the Dominican women's house in Cabra on the outskirts of Dublin.
    • She sat the Leaving Certificate in 1935 and was awarded Honours in Mathematics.
    • This may not look like a particularly great achievement but, to put it in perspective, we note that she was one of only 8 girls to receive this qualification in mathematics in Ireland in 1935 while 126 boys were successful.
    • Even more surprising for a girl at that time, she entered the Arts Faculty of University College, Galway in 1935 to study mathematics.
    • with First Class Honours in Mathematics.
    • She was awarded a National University of Ireland travelling studentship prize which enabled her to undertake research for her doctorate with Max Born at the University of Edinburgh in Scotland.
    • Born had been forced to flee from Germany when the Nazis came to power in 1933 and, after spells in north Italy and in Cambridge, he had been appointed to the Tait Chair of Applied Mathematics at Edinburgh in 1936.
    • He had a number of Ph.D.
    • When Born arrived in the morning he first used to make the round of his research students, asking them whether they had any progress to report, and giving them advice, sometimes presenting them with sheets of elaborate calculations concerning their problems which he had himself done the day before.
    • Born and his students were involved in research on the stability of crystal lattices.
    • Born wrote the first in a series of papers on this topic with the title On the stability of crystal lattices I which was published in the Mathematical Proceedings of the Cambridge Philosophical Society in 1940.
    • The stability of lattices is discussed from the standpoint of the method of small vibrations.
    • Reinhold Henry Furth (1893-1979) was a Czechoslovakian physics professor at the German University of Prague before World War II but had come to the University of Edinburgh as a research fellow and lecturer.
    • All six of these papers appeared in quick succession in the Mathematical Proceedings of the Cambridge Philosophical Society in 1940 and 1941.
    • The next in the series, published in the same journal in 1942, was by Power and it was entitled On the stability of crystal lattices.
    • I take this opportunity of expressing my sincere thanks to Professor Born for much valuable advice.
    • A review of the paper by Lothar Wolfgang Nordheim (1899-1985) states:- .
    • The next paper in the series, also in the same journal in 1942, was On the stability of crystal lattices.
    • Stability of rhombohedral Bravais lattices by H W Peng and S C Power.
    • This is part of a thesis presented by Miss S C Power for the Ph.D.
    • degree in the Faculty of Arts, The University, Edinburgh.
    • Huan Wu Peng was a Chinese student who had come to the University of Edinburgh to study under Max Born.
    • The stability of the Bravais lattices with rhombohedral unit cell of arbitrary angle is investigated under the assumption that the potential contains two terms, each proportional to a reciprocal power of distance.
    • It is shown that among the cubic Bravais lattices contained in this group the face and body centred ones correspond to a minimum of potential energy, but the simple cubic lattice to a maximum.
    • By numerical calculation of the energy of intermediate lattices it is shown that no other extrema for the potential energy exist and that the face centred lattice corresponds to the absolute minimum.
    • Finally, equilibrium conditions for compound lattices with a number of parameters are formulated and it is shown, under assumption of the above form of potential energy, that these conditions can be divided into one set for change of volume and an independent set for change of shape.
    • by the University of Edinburgh in 1941 and in the same year she was appointed as an assistant lecturer at University College, Dublin.
    • Paul Dirac gave lectures on quantum electrodynamics and Arthur Eddington lectured on the unification of relativity and quantum theory.
    • The two in the photograph who are not at present included in this archive are Monsignor Padraig de Brun (1889-1960), the Professor of Mathematics at St Patrick's College, Maynooth and Albert Joseph McConnell (1903-1993), the Professor of Natural Philosophy at Trinity College Dublin.
    • This was not unusual at this time since most departments had large numbers of students taught by a small number of lecturers.
    • She taught honours students studying mathematical sciences as well as teaching mathematics to large first year classes of engineers.
    • In 1944 Power published Note on the Influence of Damping on the Compton Scattering in the Proceedings of the Royal Irish Academy.
    • She was awarded a fellowship to go to the Institute for Advanced Study at Princeton and, after obtaining permission for leave of absence from University College Dublin, she went to the United States.
    • The records at the Princeton Institute for Advanced Study indicate that her visit started on 1 September 1948 and lasted until 30 June 1949 and that her field of study there was Nuclear Physics.
    • Over the years, a small number of honorary women members were elected but no woman was allowed to become a full member until 1931 when, after taking legal advice, the Academy ruled that "in the existing state of the law women are eligible".
    • Sheila Power has the distinction of being one of these four.
    • Despite her outstanding achievements, Power was not promoted at University College, Dublin, as one would have expected and many of her colleagues sympathised with her on not being made a professor.
    • In 1951 Sheila Power attended the Edinburgh Mathematical Society Colloquium held at the University of St Andrews.
    • Her father Michael Power also attended this Colloquium as he had the 1938 Edinburgh Mathematical Society Colloquium held at the University of St Andrews, one year before his daughter went to undertake research in Edinburgh.
    • Also at the 1951 Colloquium in St Andrews is Miss N Power who, we assume, must be a sister of Sheila Power.
    • Of those in the 1942 Dublin Institute photograph above, Arthur Conway, Padraig de Brun and A J McConnell were also in St Andrews at the 1938 Edinburgh Mathematical Society Colloquium.
    • In 1952 Power married Sean Tinney who had been one of her former engineering students.
    • Sean, who later became President of the Royal Dublin Society, shared his wife's love of music and had a fine singing voice.
    • One of these graduate students, Philip McShane, writes [',' P McShane, Cantower XXVI: Refined Woman and Feynman.','1]:- .
    • I think now of my best graduate teacher, Sheila Tinney, who, as it happens, lectured in Feynman's home-zone, quantum electrodynamics.
    • And, I fondly remember her diagraming, not on the board, but in the air between her and her two students! Luckily, a lot of the stuff was mirror-invariant.
    • The other class-member was Lochlainn O'Raifertaigh (1932-2000) who later worked in the Institute of Theoretical Physics, Dublin, where Schrodinger worked before him.
    • She enjoyed the sociability of hill-walking, and more strenuous sports such as skiing and horse riding, and pursued them all wherever she happened to be.
    • She had a deep love of music and of literature, both of which she shared with her husband Sean ..
    • Tinney continued her association with the Dublin Institute for Advanced Studies and she was appointed as a Research Associate in the School of Theoretical Physics of the Dublin Institute for Advanced Studies for the three year period from October 1954 to September 1957.
    • She was eventually promoted to associate professor of mathematical physics (quantum theory) at University College, Dublin, in 1966.
    • He struggled to choose between mathematics, sport or music.
    • His initial choice was mathematics which he studied at University for two years before deciding that he would change to music which he studied in London.
    • Sheila Tinney retired in 1978 when she reached the age of sixty.
    • This forced her gradual withdrawal into the privacy of family life, and eventually to the Molyneux Home, where she spent the last nine years of her life.
    • (very peacefully) in the wonderful care of the Matron and all the staff of the Molyneux Home, Leeson Park, Dublin 6, after a long illness borne with fortitude and grace.
    • Her funeral was held on 30 March 2010 after Mass at the Church of the Three Patrons, Rathgar, Dublin.
    • List of References (3 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Tinney.html .

  62. Walter Feit biography
    • Esther had lived in Vienna from the age of one year, but Paul had only moved there when he married Esther.
    • Feit's family were Jewish and this turn of events proved disasterous for them.
    • Krystalnacht was the night of 9 November 1938 when anti-Jewish violence, organised by the Nazis, broke out across the whole of Germany and Austria.
    • The only option was the KinderTransport which allowed children between the ages of 5 and 17 to reach Britain as refugees.
    • Walter's parents had put him on that last train which left Vienna on one of the last days in August 1939.
    • He was only eight years old and, on arriving in England, went to live with Frieda, a sister of his mother, who had come to London to work as a maid.
    • Soon after he arrived the evacuation of children from London began.
    • He then was moved to what was considered a safer area and there followed a number of moves before he finally was sent to a hostel in Oxford.
    • His teachers were very encouraging and he recorded that it was at this time that he became "passionately interested in mathematics." .
    • I have many vivid memories of Oxford since I spent the formative years of my life there.
    • It was a 35th Anniversary and a victory banquet given in honor of the ex-servicemen in the family.
    • On Monday I was outfitted for Miami; I now possess five new pairs of trousers, two new jackets plus new shoes and lots of new underwear.
    • After graduating with a high school diploma in Miami he entered the University of Chicago in September 1947.
    • He had developed a strong interest in group theory and was advised to go to the University of Michigan at Ann Arbor to study for his doctorate under Richard Brauer.
    • Feit attended Brauer's seminar which was on the modular representations of finite groups and also took an informal reading course from Brauer on class field theory.
    • Feit remained at the University of Michigan and Brauer continued to supervise his doctoral thesis.
    • Feit graduated with a doctorate in 1955, awarded for the thesis Topics in the Theory of Group Characters.
    • They had a son Paul, who became a professor of mathematics, and a daughter Alexandra who became an artist.
    • Rapid progress began to be made in the study of finite groups.
    • John Thompson proved significant results in his thesis presented to the University of Chicago in 1959, and character theoretic results proved by Feit were seen to be relevant.
    • Adrian Albert, chairmen of the Chicago Mathematics Department, decided to facilitate the ongoing work by organising a 'Finite Group Theory Year' in 1960-61.
    • This brought together many leading group theorists, and in particular it provided the opportunity for Feit and Thompson to enbark on the ambitious project of attempting to prove the conjecture that all groups of odd order are soluble.
    • They achieved their aim and the result appeared in their joint 250 page paper Solvability of groups of odd order published in the Pacific Journal of Mathematics in 1963.
    • I think there are only a few who understood the precision and subtlety with which Walter handled a variety of character-theoretic situations.
    • Suzuki and, of course, Brauer appreciated Walter's strength.
    • But only Walter and I knew just how intertwined our thinking was over a period of more than a year.
    • There was a false dawn of a few days when we thought the thing was done.
    • Walter then discovered that there was one case that our techniques did not cover, and he told me of this.
    • If that had happened, it is doubtful that we could have generated a new head of steam to bust the difficulty, which in fact took us several additional months of thought and nail biting.
    • Solomon, in [',' R Solomon, A brief history of the classification of the finite simple groups, Bull.
    • a moment in the evolution of finite group theory analogous to the emergence of fish onto dry land.
    • It defined the monumental scale of the classification project for finite simple groups and threw down a gauntlet to other researchers in the field.
    • It resolved a seemingly intractable case of the problem and offered entirely new and powerful ways of thinking about finite simple groups - ways of thinking that proved powerful enough to complete the entire project.
    • He served the Yale mathematics department in several administrative roles, acting as Director of Undergraduate Studies, Director of Graduate Studies, and Chairman.
    • His standing in the mathematics community was marked by award of the American Mathematical Society Cole Prize in Algebra, election to the National Academy of Sciences and the American Academy of Arts and Sciences, editorship of various journals, and Vice-Presidency of the International Mathematical Union.
    • We have said nothing of Feit's achievements so far, other than the odd order paper.
    • Towards the end of his career he added an interest in Galois theory to this list of interests.
    • I [EFR] heard him address the colloquium on p-adic and modular representations of finite groups.
    • In 1990 he again addressed the British Mathematical Colloquium, this time in East Anglia on The construction of Galois groups.
    • He addressed the International Congresses of Mathematicians in Nice in 1970 on The Current Situation in the Theory of Finite Simple Groups.
    • The delegation was supposed to evaluate the condition of China's mathematicians.
    • At that time, the infamous "Gang of 4" was still in power.
    • This allowed the delegation to write an honest report without endangering the welfare or lives of many Chinese mathematicians who had been dispersed to factories or sent to Outer Mongolia.
    • His retirement from Yale in October 2003 was marked with the holding of a 'Conference on Groups, Representations and Galois Theory' in his honour.
    • 52 (7) 728-735.','2] of Feit's character:- .
    • who ever discussed world affairs with Walter knows what a history buff he was.
    • He knew, in detail, the history of every country, ancient or modern, as far as I could tell.
    • So it is significant that he refrained from telling his own history to his children until begged by [his daughter] Alexandra ten years ago to relate his family history.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (4 books/articles) .
    • A Poster of Walter Feit .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Feit.html .

  63. Isaac Newton (1643-1727)
    • Isaac Newton was born in the manor house of Woolsthorpe, near Grantham in Lincolnshire.
    • Although by the calendar in use at the time of his birth he was born on Christmas Day 1642, we give the date of 4 January 1643 in this biography which is the "corrected" Gregorian calendar date bringing it into line with our present calendar.
    • (The Gregorian calendar was not adopted in England until 1752.) Isaac Newton came from a family of farmers but never knew his father, also named Isaac Newton, who died in October 1642, three months before his son was born.
    • You can see a picture of Woolsthorpe Manor as it is now at THIS LINK.
    • Isaac's mother Hannah Ayscough remarried Barnabas Smith the minister of the church at North Witham, a nearby village, when Isaac was two years old.
    • The young child was then left in the care of his grandmother Margery Ayscough at Woolsthorpe.
    • Upon the death of his stepfather in 1653, Newton lived in an extended family consisting of his mother, his grandmother, one half-brother, and two half-sisters.
    • His mother, by now a lady of reasonable wealth and property, thought that her eldest son was the right person to manage her affairs and her estate.
    • This time he lodged with Stokes, who was the headmaster of the school, and it would appear that, despite suggestions that he had previously shown no academic promise, Isaac must have convinced some of those around him that he had academic promise.
    • Another piece of evidence comes from Isaac's list of sins referred to above.
    • He lists one of his sins as:- .
    • There is no evidence that he learnt any mathematics, but we cannot rule out Stokes introducing him to Euclid's Elements which he was well capable of teaching (although there is evidence mentioned below that Newton did not read Euclid before 1663).
    • Anecdotes abound about a mechanical ability which Isaac displayed at the school and stories are told of his skill in making models of machines, in particular of clocks and windmills.
    • However, when biographers seek information about famous people there is always a tendency for people to report what they think is expected of them, and these anecdotes may simply be made up later by those who felt that the most famous scientist in the world ought to have had these skills at school.
    • He was older than most of his fellow students but, despite the fact that his mother was financially well off, he entered as a sizar.
    • Westfall (see [',' R S Westfall, Never at Rest: A Biography of Isaac Newton (1990).','23] or [',' R S Westfall, The Life of Isaac Newton (Cambridge, 1993).','24]) has suggested that Newton may have had Humphrey Babington, a distant relative who was a Fellow of Trinity, as his patron.
    • This reasonable explanation would fit well with what is known and mean that his mother did not subject him unnecessarily to hardship as some of his biographers claim.
    • Instruction at Cambridge was dominated by the philosophy of Aristotle but some freedom of study was allowed in the third year of the course.
    • Newton studied the philosophy of Descartes, Gassendi, Hobbes, and in particular Boyle.
    • The mechanics of the Copernican astronomy of Galileo attracted him and he also studied Kepler's Optics.
    • It is a fascinating account of how Newton's ideas were already forming around 1664.
    • How Newton was introduced to the most advanced mathematical texts of his day is slightly less clear.
    • According to de Moivre, Newton's interest in mathematics began in the autumn of 1663 when he bought an astrology book at a fair in Cambridge and found that he could not understand the mathematics in it.
    • Attempting to read a trigonometry book, he found that he lacked knowledge of geometry and so decided to read Barrow's edition of Euclid's Elements.
    • The new algebra and analytical geometry of Viete was read by Newton from Frans van Schooten's edition of Viete's collected works published in 1646.
    • Other major works of mathematics which he studied around this time was the newly published major work by van Schooten Geometria a Renato Des Cartes Ⓣ which appeared in two volumes in 1659-1661.
    • The book contained important appendices by three of van Schooten's disciples, Jan de Witt, Johan Hudde, and Hendrick van Heuraet.
    • Newton also studied Wallis's Algebra and it appears that his first original mathematical work came from his study of this text.
    • He read Wallis's method for finding a square of equal area to a parabola and a hyperbola which used indivisibles.
    • Newton made notes on Wallis's treatment of series but also devised his own proofs of the theorems writing:- .
    • It would be easy to think that Newton's talent began to emerge on the arrival of Barrow to the Lucasian chair at Cambridge in 1663 when he became a Fellow at Trinity College.
    • Certainly the date matches the beginnings of Newton's deep mathematical studies.
    • It would appear that his scientific genius had still not emerged, but it did so suddenly when the plague closed the University in the summer of 1665 and he had to return to Lincolnshire.
    • There, in a period of less than two years, while Newton was still under 25 years old, he began revolutionary advances in mathematics, optics, physics, and astronomy.
    • The 'method of fluxions', as he termed it, was based on his crucial insight that the integration of a function is merely the inverse procedure to differentiating it.
    • Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions.
    • When the University of Cambridge reopened after the plague in 1667, Newton put himself forward as a candidate for a fellowship.
    • [Newton] brought me the other day some papers, wherein he set down methods of calculating the dimensions of magnitudes like that of Mr Mercator concerning the hyperbola, but very general; as also of resolving equations; which I suppose will please you; and I shall send you them by the next.
    • Collins corresponded with all the leading mathematicians of the day so Barrow's action should have led to quick recognition.
    • Collins showed Brouncker, the President of the Royal Society, Newton's results (with the author's permission) but after this Newton requested that his manuscript be returned.
    • Collins could not give a detailed account but de Sluze and Gregory learnt something of Newton's work through Collins.
    • Newton's first work as Lucasian Professor was on optics and this was the topic of his first lecture course begun in January 1670.
    • When he passed a thin beam of sunlight through a glass prism Newton noted the spectrum of colours that was formed.
    • He argued that white light is really a mixture of many different types of rays which are refracted at slightly different angles, and that each different type of ray produces a different spectral colour.
    • In 1672 Newton was elected a fellow of the Royal Society after donating a reflecting telescope.
    • Also in 1672 Newton published his first scientific paper on light and colour in the Philosophical Transactions of the Royal Society.
    • The paper was generally well received but Hooke and Huygens objected to Newton's attempt to prove, by experiment alone, that light consists of the motion of small particles rather than waves.
    • He was always pulled in two directions, there was something in his nature which wanted fame and recognition yet another side of him feared criticism and the easiest way to avoid being criticised was to publish nothing.
    • Certainly one could say that his reaction to criticism was irrational, and certainly his aim to humiliate Hooke in public because of his opinions was abnormal.
    • However, perhaps because of Newton's already high reputation, his corpuscular theory reigned until the wave theory was revived in the 19th century.
    • Newton's relations with Hooke deteriorated further when, in 1675, Hooke claimed that Newton had stolen some of his optical results.
    • Although the two men made their peace with an exchange of polite letters, Newton turned in on himself and away from the Royal Society which he associated with Hooke as one of its leaders.
    • He delayed the publication of a full account of his optical researches until after the death of Hooke in 1703.
    • It dealt with the theory of light and colour and with .
    • investigations of the colours of thin sheets .
    • diffraction of light.
    • To explain some of his observations he had to use a wave theory of light in conjunction with his corpuscular theory.
    • Another argument, this time with the English Jesuits in Liege over his theory of colour, led to a violent exchange of letters, then in 1678 Newton appears to have suffered a nervous breakdown.
    • His mother died in the following year and he withdrew further into his shell, mixing as little as possible with people for a number of years.
    • Newton's greatest achievement was his work in physics and celestial mechanics, which culminated in the theory of universal gravitation.
    • By 1666 Newton had early versions of his three laws of motion.
    • However he did not have a correct understanding of the mechanics of circular motion.
    • Newton's novel idea of 1666 was to imagine that the Earth's gravity influenced the Moon, counter- balancing its centrifugal force.
    • From his law of centrifugal force and Kepler's third law of planetary motion, Newton deduced the inverse-square law.
    • M Nauenberg writes an account of the next events:- .
    • After his 1679 correspondence with Hooke, Newton, by his own account, found a proof that Kepler's areal law was a consequence of centripetal forces, and he also showed that if the orbital curve is an ellipse under the action of central forces then the radial dependence of the force is inverse square with the distance from the centre.
    • This discovery showed the physical significance of Kepler's second law.
    • In 1684 Halley, tired of Hooke's boasting [M Nauenberg]:- .
    • ' he only gave a proof of the converse theorem that if the orbit is an ellipse the force is inverse square.
    • 13 in Book 1 of the second and third editions of the 'Principia', but not in the first edition.
    • Halley persuaded Newton to write a full treatment of his new physics and its application to astronomy.
    • Newton analysed the motion of bodies in resisting and non-resisting media under the action of centripetal forces.
    • He further demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalised that all heavenly bodies mutually attract one another.
    • Further generalisation led Newton to the law of universal gravitation:- .
    • all matter attracts all other matter with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
    • Newton explained a wide range of previously unrelated phenomena: the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis, and motion of the Moon as perturbed by the gravity of the Sun.
    • The Continental scientists certainly did not accept the idea of action at a distance and continued to believe in Descartes' vortex theory where forces work through contact.
    • James II became king of Great Britain on 6 February 1685.
    • He then went further, appointing only Catholics as judges and officers of state.
    • Newton was a staunch Protestant and strongly opposed to what he saw as an attack on the University of Cambridge.
    • However William of Orange had been invited by many leaders to bring an army to England to defeat James.
    • The University of Cambridge elected Newton, now famous for his strong defence of the university, as one of their two members to the Convention Parliament on 15 January 1689.
    • Newton was at the height of his standing - seen as a leader of the university and one of the most eminent mathematicians in the world.
    • The reasons for this breakdown have been discussed by his biographers and many theories have been proposed: chemical poisoning as a result of his alchemy experiments; frustration with his researches; the ending of a personal friendship with Fatio de Duillier, a Swiss-born mathematician resident in London; and problems resulting from his religious beliefs.
    • Newton himself blamed lack of sleep but this was almost certainly a symptom of the illness rather than the cause of it.
    • There seems little reason to suppose that the illness was anything other than depression, a mental illness he must have suffered from throughout most of his life, perhaps made worse by some of the events we have just listed.
    • Newton decided to leave Cambridge to take up a government position in London becoming Warden of the Royal Mint in 1696 and Master in 1699.
    • As Master of the Mint, adding the income from his estates, we see that Newton became a very rich man.
    • For many people a position such as Master of the Mint would have been treated as simply a reward for their scientific achievements.
    • Newton did not treat it as such and he made a strong contribution to the work of the Mint.
    • He led it through the difficult period of recoinage and he was particularly active in measures to prevent counterfeiting of the coinage.
    • In 1703 he was elected president of the Royal Society and was re-elected each year until his death.
    • However the last portion of his life was not an easy one, dominated in many ways with the controversy with Leibniz over which of them had invented the calculus.
    • We have given details of this controversy in Leibniz's biography and refer the reader to that article for details.
    • Perhaps all that is worth relating here is how Newton used his position as President of the Royal Society.
    • In this capacity he appointed an "impartial" committee to decide whether he or Leibniz was the inventor of the calculus.
    • He wrote the official report of the committee (although of course it did not appear under his name) which was published by the Royal Society, and he then wrote a review (again anonymously) which appeared in the Philosophical Transactions of the Royal Society.
    • Newton was of the most fearful, cautious and suspicious temper that I ever knew.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (271 books/articles) .
    • A Poster of Isaac Newton .
    • The title page of Philosophiae naturalis principia mathematica (The Principia 1687) .
    • The title page of Analysis per quantitatum series fluxiones (1711) .
    • Multiple entries in The Mathematical Gazetteer of the British Isles .
    • Astronomy: The Reaches of the Milky Way .
    • Astronomy: The Dynamics of the Solar System .
    • 2.nFellow of the Royal Societyn1672 .
    • 3.nPresident of the Royal Societyn1703-1727 .
    • Famous Curves: Trident of Newton .
    • History Topics: Mathematics and Architecture .
    • History Topics: The Bakhshali manuscript .
    • History Topics: The brachistochrone problem .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: London Coffee houses and mathematics .
    • History Topics: A brief history of cosmology .
    • History Topics: Elliptic functions.
    • History Topics: The mathematician and the forger .
    • History Topics: The Berlin Academy and forgery .
    • History Topics: General relativity .
    • History Topics: Theories of gravitation .
    • History Topics: Christianity and the Mathematical Sciences - the Heliocentric Hypothesis .
    • History Topics: An overview of the history of mathematics .
    • History Topics: An overview of Indian mathematics .
    • History Topics: Infinity .
    • History Topics: Light through the ages: Ancient Greece to Maxwell .
    • History Topics: Longitude and the Academie Royale .
    • History Topics: English attack on the Longitude Problem .
    • History Topics: Newton's bucket .
    • History Topics: Newton poetry .
    • History Topics: Orbits and gravitation .
    • History Topics: A chronology of pi .
    • History Topics: Special relativity .
    • History Topics: The rise of the calculus .
    • History Topics: A history of time: Classical time .
    • History Topics: A history of time: 20th century time .
    • History Topics: Mathematics and the physical world .
    • Dictionary of Scientific Biography .
    • Dictionary of National Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Newton.html .

  64. Paul Turán (1910-1976)
    • Turan entered Pazmany Peter University of Budapest already showing his potential for research.
    • Erdős writes [',' P Erdős, Some personal reminiscences of the mathematical work of Paul Turan.
    • We first met at the University of Budapest in September 1930 and immediately discovered our common interest in number theory.
    • In 1933 Turan was awarded his diploma which qualified him to teach mathematics and science, and he continued working for his doctorate.
    • One was A problem in the elementary theory of numbers which appeared in the American Mathematical Monthly.
    • The second was On a theorem of Hardy and Ramanujan which was published in the Journal of the London Mathematical Society.
    • Rather it was the method of proof which, although it does not use probabilistic terminology, in fact became one of the foundations of probabilistic number theory.
    • His thesis On the number of prime divisors of integers, written in Hungarian, had been published in 1934 and contained his new proof of the theorem of Hardy and Ramanujan referred to above.
    • Even at this early stage he had built up an impressive international reputation and had seven papers in print by the end of 1935, three of which had appeared in the Journal of the London Mathematical Society.
    • However, this was far from the case since the severe discrimination against him because of his Jewish origins meant that he could not even obtain a post as a school teacher.
    • In order to support himself financially, and give himself the chance to continue his mathematical researches, he had to make a living as a private mathematics tutor.
    • By the end of 1938, five years after his first paper appeared, he had sixteen papers in print in internationally important journals world-wide.
    • At last he managed to get a position as a school teacher when, in 1938, he was appointed as an assistant teacher of mathematics at the Hungarian Rabbinical Training School in Budapest.
    • Erdős writes in [',' P Erdős, Some personal reminiscences of the mathematical work of Paul Turan.
    • Probably the most important, most enduring and most original of Turan's results are in his power sum method and its applications.
    • Their importance first of all is that they lead to interesting deep problems of a completely new type; they have quite unexpectedly surprising consequences in many branches of mathematics - differential equations, numerical algebra, and various branches of function theory.
    • In fact Turan invented the power sum method while investigating the zeta function and he first used the method to prove results about the zeros of the zeta function.
    • Later Turan and S Knapowski [',' L Alpar, In memory of Paul Turan, J.
    • investigated the distribution of primes in the reduced residue classes mod k.
    • Turan had grown up during the years of World War I which had proved a time of great hardship.
    • After the war ended, the Treaty of Trianon of 1921 saw Hungary's territory reduced to about one third of its previous size.
    • After the German invasion of Poland which began World War II, Hungary was not involved at first but was still greatly influenced by Nazi policies.
    • In 1940 Turan was sent to a labour camp, and he was in and out of various forced labour camps throughout the war.
    • This proved an horrific experience but, as we remark below, perhaps in the end his life was saved because of it.
    • Alpar writes [',' L Alpar, In memory of Paul Turan, J.
    • In every situation, making use of the smallest opportunity, he carried on his research without books and journals, missing the company of colleagues, jotting down his ideas and results on scraps of paper.
    • Several of his new ideas, problems and now famous theorems, originate from that period.
    • Erdős wrote to his father, Lajos Erdős, in Budapest, who then wrote to Turan, copying out the relevant parts of his son's letters.
    • Remarkably, even some of these letters have survived and they are reproduced in English translation in [',' V T Sos, Turbulent years : Erdős in his correspondence with Turan from 1934 to 1940, in Paul Erdős and his mathematics I, Budapest, 1999 (Budapest, 2002), 85-146.','20], but there is no record of any correspondence between June 1941 and Spring 1945.
    • We note that Vera T Sos, the author of [',' V T Sos, Turbulent years : Erdős in his correspondence with Turan from 1934 to 1940, in Paul Erdős and his mathematics I, Budapest, 1999 (Budapest, 2002), 85-146.','20], was Turan's wife and he wrote a number of joint papers with her.
    • Another remarkable fact is that extremal graph theory, an area which Turan founded, was one of the "best ideas" that he had while in the labour camps.
    • In March 1944 Hungary fully cooperated with Nazi aims and Jews were forced to wear a yellow star, robbed of their property, and forced into ghettos as in other Nazi-occupied areas.
    • Except for the Jews in the forced-labour camps, like Turan, others were sent to the gas chambers of German concentration camps.
    • It is estimated that 550,000 of Hungary's 750,000 Jews were killed during the war.
    • After World War II ended Turan was appointed as a Privatdozent at the Eotvos Lorand University of Budapest (it had formerly been called the Pazmany Peter University of Budapest).
    • On his return to Hungary he was elected to the Hungarian Academy of Sciences in 1948, and received the Kossuth Prize from the Hungarian government in the same year.
    • In 1949 he was appointed to the Chair of Algebra and Number Theory at Eotvos Lorand University of Budapest, a position he held until his death.
    • From 1955 he was Head of the Complex Function Theory Department in the Mathematical Institute of the Hungarian Academy of Sciences.
    • Erdős in [',' P Erdős, Some personal reminiscences of the mathematical work of Paul Turan.
    • She told me that I should visit him as soon as possible and that I should be careful in talking to him because he did not know the true nature of his illness.
    • I am now fairly sure that her decision was right, since he clearly never tried to find out the true nature of his illness.
    • in fact a few days before his death [his wife] and their son George (also a mathematician) tried to persuade him to dictate some parts of his book to Halasz or Pintz.
    • The book mentioned here is On a new method of analysis and its applications which was published in 1984.
    • In 1953 the author published a book, A new method of analysis and its applications ..
    • giving a systematic account of his methods for estimating "power sums", which he had developed (1941-53) into a versatile and powerful technique with numerous applications to Diophantine approximations, zero-free regions for the Riemann zeta function and the error term in the prime number theorem, and to problems in other parts of classical analysis.
    • As regards the latter, Turan found new approaches to such topics as quasi-analytic classes, Fabry's gap theorem and the theory of lacunary series, amongst others.
    • The book was revised (with improved estimates) in a second edition, but this had a limited mathematical audience since it was only available in Chinese.
    • In 1959 Turan embarked on the preparation of a new, greatly expanded version of the book.
    • Constant rewriting became necessary in the light of the new improvements and applications, and, at the time of his death in 1976, the project had still not been completed to Turan's total satisfaction.
    • The book under review represents the culmination of all this work ..
    • Odoni ends his review with this tribute to Turan's mathematics:- .
    • In the opinion of the reviewer this book renders a great service to mathematicians working in a wide area of classical analysis, particularly analytic number theorists; Turan's methods are still of great relevance in current research, and it is particularly gratifying to have all this material within the confines of a single volume.
    • The book is a fitting tribute to Turan's remarkable achievements in analysis, and the editors of the manuscript deserve high praise for their efforts in bringing it to publication.
    • We have mentioned some of Turan's mathematics above.
    • However, it is impossible to do justice to the huge amount of work which he did, publishing around 150 papers.
    • We mention, however, his work on statistical group theory, much of which was undertaken jointly with Erdős.
    • Of course conjugacy classes of the symmetric group Sn on n letters are characterized by partitions of n, so the connection with number theory is clear.
    • Most questions discussed by Turan and Erdős on this topic concern the distribution of the order of random elements of the symmetric group Sn .
    • In some of the problems they considered, all permutations are taken to be equally probable, some others are about the set of conjugacy classes, all equally probable.
    • Turan and Erdős also proved that in a group of order n, at least n log log n of the n2 pairs of elements commute.
    • outstanding in analytic number theory but not a good manager of a department.
    • However he did outstanding work for both the Hungarian Academy of Sciences, serving on numerous committees.
    • Another major contribution made by Turan was his editing of the papers of Renyi and Fejer which is the main point made by Askey in the article [',' R Askey, Dedication: remembering Paul Turan, J.
    • He commented on many of the papers, setting them in context and telling what happened to the ideas Fejer introduced.
    • But this is only a part of the editorial work Turan undertook, being on the editorial boards of Acta Arithmetica, Archiv fur Mathematik, Analysis Mathematica, Compositio Mathematica, Journal of Number Theory, and essentially all Hungarian mathematical journals.
    • He was also elected a member of the American Mathematical Society, the Austrian Mathematical Society, and the Polish Mathematical Society.
    • A special issue of Acta Mathematica devoted to Paul Turan was published in 1980.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (22 books/articles) .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Turan.html .

  65. Jack Kiefer biography
    • Jack Kiefer was the son of Carl J Kiefer and Marguerite K Rosenau.
    • He entered the Massachusetts Institute of Technology in 1942 but after one year of study of engineering and economics he left to undertake war work.
    • Air Force and spent part of the time teaching radar.
    • After a break of three years he returned to MIT in the spring of 1946 but it was not clear at this stage in which area Kiefer wanted to make a career.
    • The theatre was one of Kiefer's passions and he had the opportunity at MIT to become involved in writing, directing and producing plays.
    • If that's true, it was certainly one of the best things I ever did for statistics.
    • His Master's thesis Sequential determination of the maximum of a function had been supervised by Harold Freeman.
    • After graduating Kiefer attended a summer school at Berkeley where courses of lectures were given by Michel Loeve and Abraham Wald.
    • In the autumn he began research for his doctorate in the Department of Mathematical Statistics at Columbia University working under Abraham Wald and Jacob Wolfowitz.
    • In December 1950 Wald died and Wolfowitz was appointed professor of mathematics at Cornell University in 1951.
    • Kiefer moved to Cornell as an instructor in the Department of Mathematics and continued working for his doctorate with Wolfowitz.
    • He received his doctorate from Columbia University in 1952 for his thesis Contributions to the Theory of Games and Statistical Decision Functions.
    • After holding this position from 1952 to 1955 he was promoted to associate professor, becoming Professor of Mathematics in 1959.
    • Before this, on 15 September 1957, he had married Dooley Sciple who had been one of his undergraduate students at Cornell [',' R Bechhofer, Jack Carl Kiefer, 1924-1981, Amer.
    • Dooley and Jack shared a love of music.
    • Although he practiced conscientiously, he eventually abandoned that goal with reluctance because of other demanding and conflicting interests.
    • Dooley and Jack were devoted parents of two children, Sarah Elizabeth and Daniel Jonathan Baird, with whom they shared their love of music, stamp collecting, and mushroom hunting.
    • In July 1973 Kiefer was elected the first Horace White Professor at Cornell University, a position he held until 1979 when he retired and joined the faculty at the University of California at Berkeley.
    • Kiefer's main research area was the optimal design of experiments, and about half his 100 publications dealt with that topic.
    • However he also wrote papers on a whole variety of topics in mathematical statistics including decision theory, inventory theory, stochastic approximation, queuing theory, nonparametric inference, estimation, sequential analysis, and conditional inference.
    • A paper Sequential minimax search for a maximum which Kiefer published in the Proceedings of the American Mathematical Society in 1953 was based on his master's thesis.
    • The method of search proposed in the paper, namely the Fibonacci Search, became a widely used tool.
    • Let us note a few other examples of Kiefer's work.
    • In 1958 he published On the nonrandomized optimality and randomized nonoptimality of symmetrical designs.
    • various criteria of optimality of experimental designs.
    • Extending rather special results due to A Wald and S Ehrenfeld he shows that many commonly employed symmetrical designs (such as balanced incomplete block designs, Latin squares, Youden squares, etc.) possess optimum properties among the class of non-randomized designs.
    • In this paper Kiefer generalised the notions of a balanced incomplete block design and a Youden square, to the balanced block design and generalized Youden square.
    • In the same year, while on a research visit to Oxford University, he wrote Optimum experimental designs which was published in the Journal of the Royal Statistical Society.
    • the approach being in the spirit of Wald's decision theory.
    • In 1975 he published Optimal design: variation in structure and performance under change of criterion which discussed the robustness of designs against a chosen optimality criterion.
    • In 1980 Kiefer went to China as part of Berkeley's China Exchange Program.
    • He gave eight lectures at Beijing University covering topics such as multivariate analysis, sequential methods, nonparametric estimation, robustness and efficiency of nonparametric methods, fundamentals of experimental design, complete class and regression design, factorial experiment, and nonlinear models, sequential design, and robust design.
    • He was a Fellow of the Institute of Mathematical Statistics and the American Statistical Association, President of the Institute of Mathematical Statistics (1969-70), the Wald lecturer in 1962, and a Guggenheim fellow at Stanford (1962-63).
    • He was elected to the American Academy of Arts and Sciences (1972) and to the National Academy of Sciences (United States) in 1975.
    • This interest led to him being a member of the Mycological Society of America and the North American Mycological Association.
    • His passion for human rights saw him become a member of the American Civil Liberties Union.
    • E L Lehmann, D R Brillinger, and I M Singer, give details of his character in an obituary:- .
    • Jack Kiefer was a gentle man with a great capacity for love and friendship; this was combined, however, with a fierce and sometimes combative determination to uphold the exacting standards he demanded of others, but especially of himself.
    • His collaborator Zvi Galil writes (see [',' R Bechhofer, Jack Carl Kiefer (1924-1981) (Chinese), Math.
    • Jack was unbelievably humble, especially for a man of his stature.
    • He was very kind and considerate and spent a lot of time with his students.
    • But mostly he was critical, probably too critical and demanding, of himself.
    • This was a great success and enabled him to resume most of his normal activities.
    • However he was only 57 when he died of a heart attack not long after he had been appointed as Miller Research Professor at Berkeley.
    • The two volume proceeding [',' L M Le Cam and R A Olshen (eds.), Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer (2 vols.) (Hayward, CA, 1985).','4] was the result of a conference held in his honour.
    • Memorial sessions were held in [his] honour at various meetings of learned societies, as is fitting for scientists of [his] rank.
    • However, for reasons of heart as well as science, we felt that a special tribute should be paid to [him] at Berkeley, where [he] had so many friends.
    • It presents the central ideas of mathematical statistics as developed by Neyman and Wald in a decision-theoretic framework, referring to the earlier work of Fisher and others as based on "intuitive" considerations and misleading in certain situations.
    • List of References (13 books/articles) .
    • University of California .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Kiefer.html .

  66. Simon Stevin (1548-1620)
    • Simon Stevin's father was Anthuenis (Anton) Stevin who, it is believed, was a cadet son of a mayor of Veurne.
    • His mother was Cathelijne (or Catelyne) van der Poort who was the daughter of a burgher family of Ypres.
    • Nothing is known of Simon's early years or of his education although one assumes he was brought up in the Calvinist tradition.
    • After this he moved to Leiden in 1581 where he first attended the Latin school, then he entered the University of Leiden in 1583 (at the age of 35).
    • To understand these we need to look briefly at the history of the period.
    • The Union of Utrecht on 23 January 1579 was designed to form a block (known as the States-General) within the larger union of the Low Countries which would resist Spanish rule.
    • It produced a union in the north Netherlands, still officially under the rule of the King of Spain, but distinct from the south.
    • The strong reaction against the Spanish followed the start of a reign of terror by the Spanish occupation in the south beginning around 1567.
    • The north was predominantly Calvinist and effectively ruled by William, Prince of Orange.
    • Stevin's move to the north Netherlands certainly coincided with their move to independence from the King of Spain.
    • While Stevin was at the University of Leiden he met Maurits (Maurice), the Count Of Nassau, who was William of Orange's second son.
    • The two became close friends and Stevin became mathematics tutor to the Prince as well as a close advisor.
    • William of Orange was assassinated on 10 July 1584 at Delft by a Roman Catholic who believed that by assassinating William he would prevent the rebellion against Catholic Spain.
    • William's eldest son Philip William was loyal to Spain so it was Maurits who was appointed stadholder of Holland and Zeeland, or the United Provinces of the Netherlands, in 1584.
    • With Prince Maurits now head of the army of the republic, and with Stevin as an advisor in his service, a series of military triumphs over the Spanish forces followed.
    • Maurits understood the importance of military strategy, tactics, and engineering in military success.
    • In 1600 he asked Stevin to set up an engineering school within the University of Leiden.
    • Certainly Prince Maurits saw his friend Stevin as having major importance in his success and the recent discovery of a journal in the Public Record Office of The Hague recording Stevin's salary as 600 Dutch guilders in 1604 confirms his high position.
    • It is believed that from 1604 Stevin was quartermaster-general of the army of the States-General.
    • He invented a way of flooding the lowlands in the path of an invading army by opening selected sluices in dikes.
    • He advised Prince Maurits on building fortifications for the war against Spain and wrote detailed descriptions of the military innovations adopted by the army.
    • The army of the States-General reclaimed from Spanish rule essentially the territory which is today The Netherlands, and the States-General became officially recognized by England and France as an independent state.
    • Stevin bought a house at the Raamstraat in The Hague in 1612 for 3800 Dutch guilders (another sign of his high status and wealth).
    • Hendrik, their second child, went on to attend the University of Leiden and, becoming a famous scientist in his own right, was the editor of his father's collected works.
    • The author of 11 books, Simon Stevin made significant contributions to trigonometry, mechanics, architecture, musical theory, geography, fortification, and navigation.
    • Before presenting the numerical tables, Stevin gave rules for simple and compound interest and also gave many examples of their use.
    • Stevin gave an interesting account in this work of constructions related to polygons and polyhedra, using the concept of similarity, and a study of regular and semi-regular polyhedra.
    • It was written in Latin, and is the only one of his books to be first published in that language.
    • He became a strong advocate of writing his scientific works in Dutch and he gives clear reasons for this choice in a text written in 1586.
    • In 1585 he published La Theinde Ⓣ, a twenty-nine page booklet in which he presented an elementary and thorough account of decimal fractions.
    • He wrote this small book for the benefit of:- .
    • stargazers, surveyors, carpet-makers, wine-gaugers, mint-masters and all kind of merchants.
    • Although he did not invent decimals (they had been used by the Arabs and the Chinese long before Stevin's time) he did introduce their use in mathematics in Europe.
    • Stevin states that the universal introduction of decimal coinage, measures and weights would only be a matter of time (but he probably would be amazed to know that in the 21st century some countries still resist adopting decimal systems).
    • Robert Norton published an English translation of La Theinde Ⓣ in London in 1608.
    • It was titled Disme, The Arts of Tenths or Decimal Arithmetike and it was this translation which inspired Thomas Jefferson to propose a decimal currency for the United States (note that one tenth of a dollar is still called a dime).
    • In the latter Stevin presented a unified treatment for solving quadratic equations and a method for finding approximate solutions to algebraic equations of all degrees.
    • Stevin's notion of a real number was accepted by essentially all later scientists.
    • Particularly important was Stevin's acceptance of negative numbers but he did not accept the 'new' imaginary numbers and this was to hold back their development.
    • It is famous for containing the theorem of the triangle of forces which gave impetus to statics.
    • In the same year his treatise De Beghinselen des Waterwichts Ⓣ on hydrostatics contained notable improvements to the work of Archimedes on this topic.
    • Many consider that he founded the science of hydrostatics with this work by showing that the pressure exerted by a liquid upon a given surface depends on the height of the liquid and the area of the surface.
    • His experiments were conducted using two lead balls, one being ten times the weight of the other, which he dropped thirty feet from the church tower in Delft.
    • In De Hemelloop Ⓣ, published in 1608, he wrote on astronomy and strongly defended the sun centred system of Copernicus.
    • Although he undertook his mathematical work earlier in his life, Stevin collected together some of his mathematical writings which he edited and published during the years 1605 to 1608 in Wiskonstighe Ghedachtenissen Ⓣ (Mathematical Memoirs).
    • The work on perspective looks at a number of innovations such as the case of calculating the perspective for making a drawing on a canvas which is not perpendicular to the ground, and the case of inverse perspective.
    • This calculates where the eye of the observer should be placed if an object and a perspective drawing of that object are given.
    • In Het Burgherlick leven Ⓣ Stevin discusses how a citizen of a state should comply with the rules of the authorities (even when they appear unjust) and, in particular, he advises citizens how to behave in times of civil unrest.
    • In De Sterktenbouwing Ⓣ Stevin takes an Italian method of fortification and modifies it for Dutch use.
    • The work De Havenvinding Ⓣ literally means 'finding the harbour' and presents a method of finding the position of a ship by determining its longitude using the magnetic variation of the compass needle.
    • In the first of the final double work that we mentioned above, Stevin describes the establishment, layout and setting up of a military camp.
    • Particularly fascinating is his description of Prince Maurits camp which he set up prior to the Battle of Juliers in 1610.
    • The second of the two works deals with sluices Stevin had designed to put into fortifications to keep a moat at the correct depth.
    • This is usually seen as the first correct theory of the division of the octave into twelve equal intervals, see for example [',' M Minnaert, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1].
    • Cohen in [',' H F Cohen, Simon Stevin’s equal division of the octave, Ann.
    • of Sci.
    • 44 (5) (1987), 471-488.','13] explains the importance of the problem to scientists of the period:- .
    • Many pioneers of the Scientific Revolution, such as Galileo, Kepler, Stevin, Descartes, Mersenne, and others, wrote extensively about music theory.
    • This was not a chance interest of a few individual scientists.
    • Rather, it reflects a continuing concern of scientists from Pythagorean times onwards to solve certain quantifiable problems in music theory.
    • One of the issues involved was technically known as 'the division of the octave', the problem, that is, with which notes to make music.
    • Cohen argues that this was not, as is commonly believed (see [',' M Minnaert, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]), the purpose of Stevin's treatise:- .
    • A careful analysis of the problem situation in the science of music around 1600, reveals that Stevin's treatise highlights a particular stage in the history of what has always been the core issue of the science of music, namely, the problem of consonance.
    • This is the search for an explanation, on scientific principles, of Pythagoras's law: Why is it that those few musical intervals which affect our ear in a sweet and pleasing manner, correspond to the ratios of the first few integers?.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (19 books/articles) .
    • A Poster of Simon Stevin .
    • A page from the preface of La Disme (1634) a French translation of De Thiende .
    • History Topics: An overview of the history of mathematics .
    • History Topics: The real numbers: Pythagoras to Stevin .
    • History Topics: The real numbers: Stevin to Hilbert .
    • Dictionary of Scientific Biography .
    • Amsterdam (An English translation of La Theinde) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Stevin.html .

  67. al-Kashi biography
    • Details of Jamshid al-Kashi's life and works are better known than many others from this period although details of his life are sketchy.
    • One of the reasons we is that he dated many of his works with the exact date on which they were completed, another reason is that a number of letters which he wrote to his father have survived and give fascinating information.
    • Al-Kashi was born in Kashan which lies in a desert at the eastern foot of the Central Iranian Range.
    • He had proclaimed himself sovereign and restorer of the Mongol empire at Samarkand in 1370 and, in 1383, Timur began his conquests in Persia with the capture of Herat.
    • Timur died in 1405 and his empire was divided between his two sons, one of whom was Shah Rokh.
    • al-Kashi lived in poverty, like so many others at this time, and devoted himself to astronomy and mathematics while moving from town to town.
    • The first event in al-Kashi's life which we can date accurately is his observation of an eclipse of the moon which he made in Kashan on 2 June 1406.
    • He was certainly in his home town on 1 March 1407 when he completed Sullam Al-sama the text of which has survived.
    • The full title of the work means The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes (of the heavenly bodies).
    • Al-Kashi played this card to his advantage and brought himself into favour in the new era where patronage of the arts and sciences became popular.
    • His Compendium of the Science of Astronomy written during 1410-11 was dedicated to one of the descendants of the ruling Timurid dynasty.
    • Samarkand, in Uzbekistan, is one of the oldest cities of Central Asia.
    • The city became the capital of Timur's empire and Shah Rokh made his own son, Ulugh Beg, ruler of the city.
    • It was to Ulugh Beg that Al-Kashi dedicated his important book of astronomical tables Khaqani Zij which was based on the tables of Nasir al-Tusi.
    • In the introduction al-Kashi says that without the support of Ulugh Beg he could not have been able to complete it.
    • In this work there are trigonometric tables giving values of the sine function to four sexagesimal digits for each degree of argument with differences to be added for each minute.
    • See [',' J Hamadanizadeh, The trigonometric tables of al-Kashi in his ’Zij-i Khaqani’, Historia Math.
    • 7 (1) (1980), 38-45.','14] for a detailed discussion of this work.
    • The Khaqani Zij also contains [',' B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • detailed tables of the longitudinal motion of the sun, the moon, and the planets.
    • Al-Kashi also gives the tables of the longitudinal and latitudinal parallaxes for certain geographical latitudes, tables of eclipses, and tables of the visibility of the moon.
    • Al-Kashi had certainly found the right patron in Ulugh Beg since he founded a university for the study of theology and science at Samarkand in about 1420 and he sought out the best scientists to help with his project.
    • Ulugh Beg invited Al-Kashi to join him at this school of learning in Samarkand, as well as around sixty other scientists including Qadi Zada.
    • These were written from Samarkand and give a wonderful description of the scientific life there.
    • In 1424 Ulugh Beg began the construction of an observatory in Samarkand and, although the letters by al-Kashi are undated they were written at a time when construction of the observatory had begun.
    • The contents of one of these letters has only recently been published, see [',' M Bagheri, A newly found letter of al-Kashi on scientific life in Samarkand, Historia Math.
    • In the letters al-Kashi praises the mathematical abilities of Ulugh Beg but of the other scientists in Samarkand, only Qadi Zada earned his respect.
    • Usually these problems were too difficult for all except al-Kashi and Qadi Zada and on a couple of occasions only al-Kashi succeeded.
    • It is clear that al-Kashi was the best scientist and closest collaborator of Ulugh Beg at Samarkand and, despite al-Kashi's ignorance of the correct court behaviour and lack of polished manners, he was highly respected by Ulugh Beg.
    • After Al-Kashi's death, Ulugh Beg described him as (see for example [',' B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]):- .
    • a remarkable scientist, one of the most famous in the world, who had a perfect command of the science of the ancients, who contributed to its development, and who could solve the most difficult problems.
    • This was an achievement far beyond anything which had been obtained before, either by the ancient Greeks or by the Chinese (who achieved 6 decimal places in the 5th century).
    • The work is a major text intended to be used in teaching students in Samarkand, in particular al-Kashi tries to give the necessary mathematics for those studying astronomy, surveying, architecture, accounting and trading.
    • The authors of [',' B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1] describe the work as follows:- .
    • In the richness of its contents and in the application of arithmetical and algebraic methods to the solution of various problems, including several geometric ones, and in the clarity and elegance of exposition, this voluminous textbook is one of the best in the whole of medieval literature; it attests to both the author's erudition and his pedagogical ability.
    • Dold-Samplonius has discussed several aspects of al-Kashi's Key to Arithmetic in [',' Y Dold-Samplonius, The 15th century Timurid mathematician Ghiyath al-Din Jamshid al-Kashi and his computation of the Qubba, in S S Demidov et al.
    • (eds), Amphora : Festschrift for Hans Wussing on the occasion of his 65th birthday (Basel- Boston- Berlin, 1992), 171-181.','11], [',' Y Dold-Samplonius, Practical Arabic mathematics : measuring the muqarnas by al-Kashi, Centaurus 35 (3-4) (1992), 193-242.','12], and [',' Y Dold-Samplonius, al-Kashi’s measurement of Muqarnas, in Deuxieme Colloque Maghrebin sur l’Histoire des Mathematiques Arabes (Tunis, 1990), 74-84.','13].
    • For example the measurement of the muqarnas refers to a type of decoration used to hide the edges and joints in buildings such as mosques and palaces.
    • The decoration resembles a stalactite and consists of three-dimensional polygons, some with plane surfaces, and some with curved surfaces.
    • Al-Kashi uses decimal fractions in calculating the total surface area of types of muqarnas.
    • The qubba is the dome of a funerary monument for a famous person.
    • Al-Kashi finds good methods to approximate the surface area and the volume of the shell forming the dome of the qubba.
    • We mentioned above al-Kashi's use of decimal fractions and it is through his use of these that he has attained considerable fame.
    • Masud al-Kasi (Wiesbaden, 1951).','4]) showed that in the Key to Arithmetic al-Kashi gives as clear a description of decimal fractions as Stevin does.
    • However, to claim that al-Kashi is the inventor of decimal fractions, as was done by many mathematicians following the work of Luckey, would be far from the truth since the idea had been present in the work of several mathematicians of al-Karaji's school, in particular al-Samawal.
    • Rashed (see [',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','5] or [',' R Rashed, Entre arithmetique et algebre: Recherches sur l’histoire des mathematiques arabes (Paris, 1984).','6]) puts al-Kashi's important contribution into perspective.
    • (1) The analogy between both systems of fractions; the sexagesimal and the decimal systems.
    • (2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π.
    • Rashed also writes (see [',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','5] or [',' R Rashed, Entre arithmetique et algebre: Recherches sur l’histoire des mathematiques arabes (Paris, 1984).','6]):- .
    • Al-Kashi can no longer be considered as the inventor of decimal fractions; it remains nonetheless, that in his exposition the mathematician, far from being a simple compiler, went one step beyond al-Samawal and represents an important dimension in the history of decimal fractions.
    • There are other major results in the work of al-Kashi which were pointed out by Luckey.
    • He found that al-Kashi had an algorithm for calculating nth roots which was a special case of the methods given many centuries later by Ruffini and Horner.
    • In later work Rashed shows (see for example [',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','5] or [',' R Rashed, Entre arithmetique et algebre: Recherches sur l’histoire des mathematiques arabes (Paris, 1984).','6]) that Al-Kashi was again describing methods which were present in the work of mathematicians of al-Karaji's school, in particular al-Samawal.
    • The last work by al-Kashi was The Treatise on the Chord and Sine which may have been unfinished at the time of his death and then completed by Qadi Zada.
    • He also considered the equation associated with the problem of trisecting an angle, namely a cubic equation.
    • However, the iterative method proposed by al-Kashi was [',' B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • one of the best achievements in medieval algebra.
    • But all these discoveries of al-Kashi's were long unknown in Europe and were studied only in the nineteenth and twentieth centuries by ..
    • historians of science..
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (17 books/articles) .
    • A Poster of al-Kashi .
    • Muslim extraction of roots .
    • History Topics: Pi through the ages .
    • History Topics: A chronology of pi .
    • History Topics: Arabic mathematics : forgotten brilliance? .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Kashi.html .

  68. Chuan-Chih Hsiung biography
    • Chuan-Chih Hsiung describes his ancestors in [',' C-C Hsiung, Personal and professional history, in C-C Hsiung, Selected papers of Chuan-Chih Hsiung (World Scientific Publishing Co., Inc., River Edge, NJ, 2001), xi-xii.','2]:- .
    • For many generations my ancestors lived in a small village called Shefong, which is about ten miles from Nanchang, the capital of the province of Jiangsi.
    • My ancestors were all farmers until my grandfather's generation when his only brother and he studied the works of Confucius and other literary scholars.
    • Unfortunately they died within two years of each other, both in their mid thirties.
    • At that time, my father [Mu-Han Hsiung], the second of three sons, was only eight years old.
    • Mu-Han Hsiung received a good education at the Advanced School in Nanchang and there he studied mathematics from English textbooks.
    • After four years of study he graduated and was appointed as vice-principal and mathematics teacher at the new Jiangsi Provincial First High School.
    • Chuan-Chih was the third of the four sons.
    • The second of Mu-Han and Tu Shih's sons, C Y Hsiung, also went on the become a professor of mathematics writing books such as Elementary theory of numbers (1992).
    • Chuan-Chih was taught mathematics by his father who gave him a great love of the subject.
    • He graduated from the Jiangsi Provincial First High School in Nanchang and, after sitting a highly competitive university entrance examination, became a mathematics student at the National Chekiang University (founded in 1897) in Hangchow.
    • He began undertaking research in mathematics on topics suggested by Buchin Su.
    • For example he published On the curvature form and the projective curvatures of curves in space of four dimensions (1940).
    • In this Hsiung expresses certain invariants of curves in a space of four dimensions found by Buchin Su, and published by him in 1937, in terms of double ratios of covariant points.
    • Hsiung's paper On the plane sections of the tangent surface of a space curve (1940) was reviewed by Guy Grove of Michigan State University who wrote:- .
    • This paper concerns itself with the study of the sections of the tangent developable surface of a twisted curve through the tangent line of the curve.
    • The problem was previously studied by Buchin Su [(1933) and (1937)], making use of certain osculants introduced by Bompiani [(1926)].
    • It is found among other things that among the plane sections of the tangent surface there exists one and only one whose seven-point cuspidal cubic also has eight-point contact; there are only two sections for which the eight-point cubic through the cusp of the seven-point cuspidal cubic has nine-point contact with the plane section; as the sectioning plane revolves about the tangent line the locus of the cusp of the seven-point cuspidal cubic is a twisted cubic; the locus of the line of cusps of the six-point cuspidal cubics is a quadric cone; the locus of the cuspidal tangent is a cubic ruled surface having the tangent to the space curve as a triple line; the locus of the seven-point cuspidal cubic is a surface of order six.
    • Guy Grove not only reviewed this paper but also Hsiung's papers Sopra il contatto di due curve piane (1940), The canonical lines (1941), A graphical construction of the sphere osculating a space curve (1941), On the curvature form and the projective curvatures of a space curve (1942), Asymptotic ruled surfaces (1943), Projective differential geometry of a pair of plane curves (1943), Theory of intersection of two plane curves (1943), An invariant of intersection of two surfaces (1943), and Projective invariants of a pair of surfaces (1943).
    • Grove was very impressed with Hsiung's mathematics and tried to arrange for him to come to Michigan State University to study for a doctorate.
    • By 1938 the Japanese occupied much of Chekiang province and Hsiung could not travel to the United States.
    • Although we listed several of Hsiung's papers between 1940 and 1943 there was one which we did not mention since it was rather different from the rest.
    • This was joint work with his colleague Fu Traing Wang at the National University of Chekiang and involved a study of Tangrams, an ancient Chinese puzzle consisting of seven tiles which can be assembled into a variety of geometric and decorative forms.
    • By this time China was receiving considerable support from the United States in its war effort but Japan still held most of eastern China.
    • In fact he had the distinction of becoming the first student to be awarded a Ph.D.
    • After the award of his doctorate, Hsiung was appointed as an instructor at Michigan State University and held this for two years until 1950 after which he spent the winter semester as a visiting lecturer at Northwestern University in Evanston.
    • Hassler Whitney then invited Hsiung to become his research assistant at Harvard and Hsiung took up this position in the spring of 1951.
    • He wrote [',' C-C Hsiung, Personal and professional history, in C-C Hsiung, Selected papers of Chuan-Chih Hsiung (World Scientific Publishing Co., Inc., River Edge, NJ, 2001), xi-xii.','2]:- .
    • I benefited very much from my visit to Harvard; I was able to learn the latest developments in mathematics.
    • Whitney went to the Institute for Advanced Study in the autumn of 1952 and Hsiung was offered an assistant professorship at Lehigh University in Bethlehem, Pennsylvania.
    • The university had been founded in 1865 by Asa Packer, an industrialist and philanthropist, and Hsiung was to spend the rest of his career there.
    • We have already indicated the range of Hsiung's early work on projective geometry.
    • Topics he then investigated include two-dimensional Riemannian manifolds with boundary (uniqueness and isoperimetic inequalities), groups of conformal transformations of a compact Riemannian manifold, curvature and characteristic classes, complex structure, and isospectral almost-L-manifolds.
    • is designed as a course of classical differential geometry for beginning graduate students or advanced undergraduate students.
    • In fact, the title of the book might well be A first course in differential and integral geometry.
    • The first of these is a monograph which presents many of Hsiung's own results, in particular including his important work on the nonexistence of a complex structure on S6.
    • The second of these books has been reviewed by Man Chun Leung who writes:- .
    • The book is designed to introduce differential geometry via the study of curves and surfaces in E3.
    • With the fundamental material in place, the author discusses the theory of smooth curves, including the Frenet formulas, the isoperimetric inequality and the four-vertex theorem.
    • Local theory of surfaces is presented, with the emphasis on fundamental forms, curvatures, ruled and minimal surfaces.
    • The global theory of surfaces in E3 is studied in the last chapter.
    • Most sections end with a carefully selected set of exercises, some of which supplement the content of the section.
    • The book provides a solid introduction to differential geometry of curves and surfaces.
    • It also includes many short and valuable remarks on the classical literature of the subject.
    • This is his founding of the Journal of Differential Geometry in 1967.
    • He acted as editor-in-chief of the journal from the time of its foundation.
    • Hsiung writes [',' C-C Hsiung, Personal and professional history, in C-C Hsiung, Selected papers of Chuan-Chih Hsiung (World Scientific Publishing Co., Inc., River Edge, NJ, 2001), xi-xii.','2]:- .
    • Under the influence of this journal, differential geometry has become a very active branch of mathematics, with scope far exceeding its former classical one.
    • It is not necessary to heap accolades upon this contribution - the journal has long been recognized as a forum of publication of the finest papers in differential geometry.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (3 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Hsiung.html .

  69. Claude Hardy (1598-1678)
    • Claude Hardy's father was Sebastien Hardy, seigneur of Estour and of Tabaize.
    • He married Marie Belot Despontis in Paris around 1596 and their son Claude, one of their seven children, was born in Le Mans.
    • The date of his birth is uncertain, some experts giving 1598 and others giving 1604.
    • In 1604 the family were living in rue Quinquempoix, in the parish of St Jacques de la Boucherie but by 1610 they had moved to a home in rue St Honore, in the parish of St Germain de l'Auxerrois.
    • We know nothing of Claude's upbringing other than the locations of the homes in which he lived.
    • However, in 1613 he published a translation of a work by Erasmus under the title De la civilite morale des enfants Ⓣ.
    • This, of course, suggests that Claude Hardy was indeed born in 1604 (being nine years old in 1613), but this seems somewhat at odds with when his father was known to be in Le Mans.
    • It is possible, of course, that Hardy did translate the work when nine years old but it was only published some six years later.
    • Although it might seem impossible for so young a child to translate from Latin to French, we do know that Hardy had the reputation for being a remarkable linguist with knowledge of thirty-six languages.
    • His friends claimed that some of these languages took him no more than one day of study to master them.
    • In 1614 he published another translation, this time the work of the poet Michel Verin who died in Florence in 1487 at the age of 19.
    • Perhaps given the young age of the translator, it is not too surprising that the translation has been described as "lively, sometimes obscure, and full of dubious and antiquated expressions." In the Preface, Hardy pays a fine tribute to his father.
    • Nothing of Hardy's education in Paris is known and the first information about him after 1614 is not until 1622 when records show that Claude Hardy married Perrette Presche in Paris.
    • These records give his date of birth as 1604, making him eighteen years old when he married.
    • A year later he was certainly attached to the court of justice in Paris as a counsellor.
    • He edited the Greek edition of Euclid and provided a Latin translation of the work and the commentary by Marin Mersenne.
    • He became involved with a group of mathematicians working in Paris at this time, in particular becoming a friend of Claude Mydorge.
    • He was in Mydorge's home when he was introduced to Rene Descartes who was a friend of Mydorge.
    • This was the beginning of a close friendship between Hardy and Descartes during which they exchanged views on all the scientific and philosophical issues of the day.
    • And I assure you, that if you give Mr Hardy a good dictionary in Chinese, or any other language whatsoever, and a book written in the same language, he will undertake to make sense of it.
    • A translation into French of Viete's book on algebra, originally written in Latin, appeared around 1630 with Antoine Vasset as the translator.
    • These works dealt with the problem of the duplication of the cube and in them Hardy pointed out a fallacy which had arisen regarding this problem [',' H L L Busard, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Several writers of the seventeenth century suggested methods for the duplication of the cube, including Viete, Descartes, Fermat, and Newton.
    • Among the less well-known persons who also occupied themselves with this problem was Paul Yvon, lord of Laleu, who claimed that he had found the construction of the two mean proportionals, required in solving the problem.
    • In addition to Mydorge and J de Beaugrand, Hardy exposed the fallacy of Yvon's construction ..
    • In 1637 Mersenne gave Descartes a copy of Fermat's De Maximis et Minimis et de Tangentibus Ⓣ to review.
    • Descartes wrote a strong criticism of Fermat's work which he gave to Mersenne with instructions to forward his review to Fermat.
    • Mersenne also let Roberval and Etienne Pascal, both friends of Fermat, see Descartes' critical attack.
    • Descartes wrote to Mersenne making sure that others, particularly his friend Hardy, be given details of the controversy.
    • Of course, Fermat's methods as given in De Maximis et Minimis et de Tangentibus Ⓣ are perfectly correct but he had not stated his innovative results rigorously enough to satisfy his opponents.
    • One must also realise that Descartes believed that only with his methods could really innovative mathematical ideas be discovered so he was set against ideas from anyone who had not followed his way of thinking.
    • After the death of Descartes in 1650, information about Hardy vanishes and we know nothing about the last 25 years of his life.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (12 books/articles) .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Hardy_Claude.html .

  70. Nasir al-Din al-Tusi biography
    • In fact al-Tusi was known by a number of different names during his lifetime such as Muhaqqiq-i Tusi, Khwaja-yi Tusi and Khwaja Nasir.
    • Al-Tusi was born in Tus, which lies close to Meshed in northeastern Iran high up in the valley of the Kashaf River.
    • He was born at the beginning of a century which would see conquests across the whole of the Islamic world from close to China in the east to Europe in the west.
    • It was the era when the vast military power of the Mongols would sweep across the vast areas of the Islamic world displaying a bitter animosity towards Islam and cruelly massacring people.
    • The Twelfth Imam was the main sect of Shi'ite Muslims and the school where al-Tusi was educated was mainly a religious establishment.
    • These topics included logic, physics and metaphysics while he also studied with other teachers learning mathematics, in particular algebra and geometry.
    • In 1214, when al-Tusi was 13 years old, Genghis Khan, who was the leader of the Mongols, turned away from his conquests in China and began his rapid advance towards the west.
    • It would not be too long before al-Tusi would see the effects of these conquests on his own regions, but before that happened he was able to study more advanced topics.
    • From Tus, al-Tusi went to Nishapur which is 75 km west of Tus.
    • Nishapur was a good choice for al-Tusi to complete his education since it was an important centre of learning.
    • There al-Tusi studied philosophy, medicine and mathematics.
    • In particular he was taught mathematics by Kamal al-Din ibn Yunus, who himself had been a pupil of Sharaf al-Din al-Tusi.
    • The Mongol invasion reached the area of Tus around 1220 and there was much destruction.
    • The Assassins, who practised an intellectual form of extremist Shi'ism, controlled the castle of Alamut in the Elburz Mountains, and other similar impregnable forts in the mountains.
    • When invited by the Isma'ili ruler Nasir ad-Din 'Abd ar-Rahim to join the service of the Assassins, al-Tusi accepted and became a highly regarded member of the Isma'ili Court.
    • However, al-Tusi did some of his best work while moving round the different strongholds, and during this period he wrote important works on logic, philosophy, mathematics and astronomy.
    • The first of these works, Akhlaq-i nasiri, was written in 1232.
    • In 1256 al-Tusi was in the castle of Alamut when it was attacked by the forces of the Mongol leader Hulegu, a grandson of Genghis Khan, who was at that time set on extending Mongol power in Islamic areas.
    • Some claim that al-Tusi betrayed the defences of Alamut to the invading Mongols.
    • He was also put in charge of religious affairs and was with the Mongol forces under Hulegu when they attacked Baghdad in 1258.
    • After having laid siege to the city, the Mongols entered it in February 1258 and al-Musta'sim together with 300 of his officials were murdered.
    • Hulegu had little sympathy with a city after his armies had won a battle, so he burned and plundered the city and killed many of its inhabitants.
    • Certainly al-Tusi had made the right move as far as his own safety was concerned, and he would also profit scientifically by his change of allegiance.
    • Hulegu was very pleased with his conquest of Baghdad and also pleased that such an eminent scholar as al-Tusi had joined him.
    • So, when al-Tusi presented Hulegu with plans for the construction of a fine Observatory, Hulegu was happy to agree.
    • Maragheh was in the Azerbaijan region of northwestern Iran, and it was at Maragheh that the Observatory was to be built.
    • Construction of the Observatory began in 1259 west of Maragheh, and traces of it can still be seen there today.
    • Interestingly the Persians were assisted by Chinese astronomers in the construction and operation of the observatory.
    • It had various instruments such as a 4 metre wall quadrant made from copper and an azimuth quadrant which was the invention of Al-Tusi himself.
    • It possessed a fine library with books on a wide range of scientific topics, while work on science, mathematics and philosophy were vigorously pursued there.
    • Al-Tusi put his Observatory to good use, making very accurate tables of planetary movements.
    • This work contains tables for computing the positions of the planets, and it also contains a star catalogue.
    • It is fair to say that al-Tusi made the most significant development of Ptolemy's model of the planetary system up to the development of the heliocentric model in the time of Copernicus.
    • devised a new model of lunar motion, essentially different from Ptolemy's.
    • Abolishing the eccentric and the centre of prosneusis, he founded it exclusively on the principle of eight uniformly rotating spheres and thereby succeeded in representing the irregularities of lunar motion with the same exactness as the "Almagest" Ⓣ.
    • In his model Nasir, for the first time in the history of astronomy, employed a theorem invented by himself which, 250 years later, occurred again in Copernicus, "De Revolutionibus", III 4.
    • The theorem referred to in this quotation concerns the famous "Tusi-couple" which resolves linear motion into the sum of two circular motions.
    • The aim of al-Tusi with this result was to remove all parts of Ptolemy's system that were not based on the principle of uniform circular motion.
    • 4 (2) (1973), 128-130.','38] where it is claimed that Copernicus took the result from Proclus's Commentary on the first book of Euclid and not from al-Tusi.
    • Among numerous other contributions to astronomy, al-Tusi calculated the value of 51' for the precession of the equinoxes.
    • In logic al-Tusi followed the teachings of ibn Sina.
    • He wrote five works on the subject, the most important of which is one on inference.
    • These included revised Arabic versions of works by Autolycus, Aristarchus, Euclid, Apollonius, Archimedes, Hypsicles, Theodosius, Menelaus and Ptolemy.
    • Nauk (6) (1990), 40-43, 80.','41] for details), and Archimedes' On the sphere and cylinder (see [',' A Kubesov, The commentaries of Nasi ad-Din at-Tusi on the treatise of Archimedes ’On the sphere and cylinder’ (Russian), Voprosy Istor.
    • In the latter work al-Tusi discussed objections raised by earlier mathematicians to comparing lengths of straight lines and of curved lines.
    • Ptolemy's Almagest Ⓣ was one of the works which Arabic scientists studied intently.
    • In 1247 al-Tusi wrote Tahrir al-Majisti (Commentary on the Almagest) in which he introduced various trigonometrical techniques to calculate tables of sines; see [',' N A Abdulkasumova, The ’Tahrir al-Majisti [Commentary on the Almagest]’ of Nasir ad-Din at-Tusi (first book) (Russian), Izv.
    • As in the Zij-i Ilkhahi al-Tusi gave tables of sines with entries calculated to three sexagesimal places for each half degree of the argument.
    • One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications.
    • In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry.
    • As stated in [',' S H Nasr, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth.
    • Another mathematical contribution was al-Tusi's manuscript, dated 1265, concerning the calculation of n-th roots of an integer; see [',' S A Ahmedov, Extraction of a root of any order and the binomial formula in the work of Nasir ad-Din at-Tusi (Russian), Mat.
    • v Skole (5) (1970), 80-82.','6] for details of a copy of this manuscript made in 1413.
    • This work by al-Tusi is almost certainly not original but rather it is his version of methods developed by al-Karaji's school.
    • In the manuscript al-Tusi determined the coefficients of the expansion of a binomial to any power giving the binomial formula and the Pascal triangle relations between binomial coefficients.
    • He wrote a famous work on minerals which contains an interesting theory of colour based on mixtures of black and white, and included chapters on jewels and perfumes.
    • In particular in philosophy he asked important questions on the nature of space.
    • Al-Tusi had a number of pupils, one of the better known being Nizam al-a'Raj who also wrote a commentary on the Almagest Ⓣ.
    • Another of his pupils Qutb ad-Din ash-Shirazi gave the first satisfactory mathematical explanation of the rainbow.
    • al-Tusi's influence, which continued through these pupils, is summed up in [',' S H Nasr, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1] as follows:- .
    • Probably, if we take all fields into account, he was more responsible for the revival of the Islamic sciences than any other individual.
    • His bringing together so many competent scholars and scientists at Maragheh resulted not only in the revival of mathematics and astronomy but also in the renewal of Islamic philosophy and even theology.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (41 books/articles) .
    • A Poster of Nasir al-Din al-Tusi .
    • History Topics: Arabic mathematics : forgotten brilliance? .
    • Dictionary of Scientific Biography .
    • Internet Encyclopedia of Philosophy .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Nasir.html .

  71. Claude Berge biography
    • He was the son of the Rene Berge, a mining engineer, and Antoinette Faure.
    • Felix Francois Faure (1841-1899) was Antoinette Faure's father; he was President of France from 1895 to 1899.
    • Andre Berge married Genevieve in 1924 and Claude, the subject of this biography, was the second of their six children.
    • Claude attended the Ecole des Roches near Verneuil-sur-Avre about 110 km west of Paris.
    • I wasn't quite sure that I wanted to do mathematics.
    • His love of literature and other non-mathematical subjects never left him and we shall discuss below how they played a large role in his life.
    • However, he decided to study mathematics at the University of Paris.
    • After the award of his first degree, he continued to undertake research for his doctorate, advised by Andre Lichnerowicz.
    • He began publishing mathematics papers in 1950.
    • In that year two of his papers appeared, the short paper Sur l'isovalence et la regularite des transformateurs Ⓣ and the major 30-page paper Sur un nouveau calcul symbolique et ses applications Ⓣ.
    • The symbolic calculus which he discussed in this major paper is a combination of generating functions and Laplace transforms.
    • In this thesis he examined games where perfect information is available in which, at each move, there are possibly an infinite number of choices.
    • Berge examined properties of such games with a thorough analysis.
    • In 1952, before the award of his doctorate, Berge was appointed as a research assistant at the Centre National de la Recherche Scientifique.
    • He took part in the Economics Research Project there which was under contract with the Office of Naval Research.
    • While in Princeton he undertook work which was presented in the paper Two theorems in graph theory published in the Proceedings of the National Academy of Sciences of the United States of America.
    • This was one of his first papers on graph theory, his earlier work being on the theory of games and combinatorics.
    • He was writing his famous book Theorie des graphes et ses applications Ⓣ at this time and had just published his book on the theory of games Theorie generale des jeux a n personnes Ⓣ (1957).
    • Returning to France from the United States, Berge took up the position on Director of research at the Centre national de la recherche scientifique.
    • Also in 1957 he was appointed as a professor in the Institute of Statistics of the University of Paris.
    • In the highly entertaining article [',' V Chvatal, In praise of Claude Berge, in Graphs and combinatorics, Marseille, 1995, Discrete Math.
    • I used to think of the book on game theory and the book on topology as a couple of false starts from the days before Claude found his true calling in graph theory and combinatorics.
    • A computer search through Mathematical Reviews changed my mind: with each of these two books, Claude left a lasting mark on the subject.
    • I was pleased to learn that the notions of 'Berge equilibrium' and 'Berge strategies' were being studied by game theorists thirty years after the publication of Claude's book; I was pleased to learn that the 'maximum theorem of Berge' and 'Berge upper semicontinuity' were being studied by economists thirty years after the publication of Claude's book on topology.
    • I was pleased to read Mark Walker's words: "The maximum theorem and its generalizations have become one of the most useful tools in economic theory.
    • It is amusing to speculate that, just as Claude Berge is a combinatorist to many of us combinatorists, he may be a game theorist to some game theorists and he may be a topologist to some economists.
    • For reviews of these books by Berge see THIS LINK .
    • Of course, today it is difficult to realise that 50 years ago graph theory was an unknown topic to most mathematicians.
    • Graph Theory has its origin in a great number of old problems (in the work of Euler, Kirchhoff et al.) and in recent years its range has become vastly greater.
    • We draw a graph each time we want to represent by points a number of individuals, cities, chemical substances, strategic positions, and to join certain pairs of them by arrows, symbolizing a definite relationship.
    • Thanks to the process of abstraction, which is so characteristic of twentieth-century mathematics, the properties of all these diagrams have been systematically analysed and a uniform theory has arisen which is applicable to all these fields.
    • It was held 20-22 October 1959 in Dobogoko, in the mountains to the north of Budapest, and organised by the Bolyai Mathematical Society.
    • In 1960 Berge attended a conference at the Martin Luther University of Halle-Wittenberg in Halle am Saale, Germany.
    • In a lecture he gave at the conference he made his famous 'strong perfect graph' conjecture but he only published the conjecture three years later in his paper Some classes of perfect graphs (1963).
    • Stefan Hougardy, writing in 2006, looks at the impact of this conjecture.
    • It [',' S Hougardy, Classes of perfect graphs, Discrete Math.
    • has had a major impact on the development of graph theory over the last 40 years.
    • It has led to the definition and study of many new classes of graphs for which the strong perfect graph conjecture has been verified.
    • In this paper we survey 120 of these classes, list their fundamental algorithmic properties and present all known relations between them.
    • Also in 1960 Berge became a founder member of Oulipo, an organisation which was particularly well suited to combining two of Berge's interests, namely mathematics and literature.
    • The authors of [',' D Bouyssou, D de Werra and O Hudry, Claude Berge and the ’Oulipo’, EURO Newsletter 6 (2006).','6] describe Oulipo:- .
    • Oulipo was a group of writers and mathematicians aiming at exploring in a systematic way formal constraints on the production of literary texts.
    • Although poets had explored for years the formal constraints of versification, the originality of the group was to develop new or rarely studied constraints and to explore them systematically.
    • In this short story Who killed the Duke of Densmore (1995), the Duke of Densmore has been murdered by one of his six mistresses, and Holmes and Watson are summoned to solve the case.
    • He then applies a theorem of Gyorgy Hajos to the graph which produces the name of the murderer.
    • Other clever contributions of Berge to Oulipo are described in [',' D Bouyssou, D de Werra and O Hudry, Claude Berge and the ’Oulipo’, EURO Newsletter 6 (2006).','6].
    • Another of Berge's interests was in art and sculpture.
    • Bjarne Toft writes [',' B Toft, Claude Berge - sculptor of graph theory, in Graph theory in Paris (Birkhauser, Basel, 2007), 1-9.','21]:- .
    • Berge catches again something general and essential, as he did in his mathematics.
    • Sculpture was certainly not his only interest in art for he wrote the small book L'Art Asmat Ⓣ describing the art of the Asmat, an ethnic group of people living in Papua New Guinea.
    • After this excursion into Berge's interests outside mathematics, we should return to his mathematical contributions.
    • He was Director of the International Computing Centre in Rome (1965-1967).
    • The announcement of his appointment gave details of his previous experience in the United States:- .
    • He has participated in colloquia at Harvard University and the Massachusetts Institute of Technology and has lectured at a number of institutions in the United States, including Stanford University and the University of California, Berkeley.
    • The first of these, Principes de combinatoire Ⓣ, was essentially lecture notes of a course Berge gave at the Faculty of Science in Paris in 1967-68.
    • It was translated into English and published as Principles of combinatorics in 1971.
    • Introduction a la theorie des hypergraphes Ⓣ was also notes from a lecture course, this time a course given by Berge at the University of Montreal in the summer of 1971.
    • The book is a classic in its area and has been instrumental in introducing mathematicians to graphs and taking them to the frontiers of current research.
    • Combinatoire des ensembles finis Ⓣ (1987), are revised versions of the first half and the second half (respectively) of the original text.
    • The style and content of the book reflect throughout the influence of the author's own work and the distinctive flavour of his personal approach to the subject.
    • Berge has received many honours for his mathematical contributions including the EURO Gold Medal from the Association of European Operational Research Societies in 1989 and the Euler Medal from the Institute of Combinatorics and Its Applications in 1995 (awarded jointly with Ron Graham).
    • Berge was still alive when the news of the resolution of his more difficult conjecture came in, although he was reportedly very ill.
    • He must have felt happy to see the end of this four-decade long adventure during his lifetime, although I do not think that the proof itself would have given him any great pleasure.
    • Berge was very much a product of the old school and 'clean' and 'pretty' proofs mattered to him almost as much as the proof itself.
    • Bhogle was one of Berge's research students and recalls some charming personal details of his advisor [',' S Bhogle, Claude Berge, Current Science 83 (7) (2002), 906-907.','5]:- .
    • His residence at 10 rue Galvani in Paris' 17th arrondissement was like an unkempt museum with his own sculpture competing for space with the many Chinese works of art that he so adored.
    • Being one of Berge's foreign students in France, it was my privilege to be invited to his house for Christmas dinners.
    • I have the most wonderful memories of these dinners.
    • Mathematician, combinatorist, sculptor, amateur anthropologist, world traveller, collector of Indonesian art, Oulipist, chess player and player of exotic game pieces, author of manuals, professor renowned for his 30-minute lectures, 35-year member of the Centre de Mathematique Sociale, CRNS researcher emeritus at the EHESS until June 30, 2002: such was Claude Berge.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (23 books/articles) .
    • Reviews of Claude Berge's books .
    • Mathematical Society of Philippines .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Berge.html .

  72. Hideo Tanaka biography
    • Hideo Tanaka studied at Kobe University in the city of Kobe, a neighbouring city of Osaka, the city of his birth.
    • In April 1964 he enrolled in the Graduate School of Engineering at Osaka City University and was awarded an M.S.
    • His research topic was 'Sensitivity analysis in control theory' but it is not for his work in that area that we have included Tanaka in this mathematics archive, rather it is for his work on Fuzzy Operations Research.
    • He was a professor at the University of California, Berkeley, from 1959 and published the paper Fuzzy Sets in 1965.
    • A fuzzy set is a class of objects with a continuum of grades of membership.
    • Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one.
    • The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established.
    • After graduating, Tanaka was appointed to the Department of Industrial Engineering at Osaka Prefecture University as a research associate in July 1969 and he worked there until he retired in March 2000.
    • In March 1967 Kiyoji Asai went to the University of California, Berkeley, to spend a year working in the Computer Science Division with Lotfi Zadeh.
    • While having a permanent position at Osaka Prefecture University, Tanaka held a number of other visiting positions during these years.
    • From 1972 to 1973, he was a Visiting Research Associate in the Computer Science Division of the University of California, Berkeley, where he was able to work with the Head of the Division, L A Zadeh, whose 1965 paper had inspired him to work on fuzzy theory.
    • Zimmermann, who worked in the Department for Operations Research, was pursuing research in fuzzy set theory and its applications and his interests were close to those of Tanaka.
    • There he spent the year as a research associate working with Liang-tseng Fan (1929-) who was a professor in the Department of Chemical Engineering.
    • As an indication of the topics on which Tanaka undertook research, we quote the titles of some of his papers.
    • For example there is: Synthesis of low-sensitivity closed-loop systems by iterative method (1969); Synthesis of linear feedback control systems based on sensitivity considerations (1969); Equivalence condition of optimal control problems (1971); On fuzzy-mathematical programming (1973); Decision-making and information in fuzzy events (1974); Decision-making and its goal in a fuzzy environment (1974); Applications of fuzzy sets to decision making and control (1975); A formulation of fuzzy decision problems and its application to an investment problem (1976); Fuzzy information and decision in statistical model (1979); and Fuzzy linear programming based on fuzzy functions (1980).
    • Tanaka published a number of books in Japanese including: Fuzzy modelling and its applications (1990); Fuzzy OR (1993); Data analysis software (1995); and Reasoning and knowledge acquisition from the data (2004).
    • The publisher writes of the cover of the book:- .
    • This unique monograph provides new theories and techniques of possibility theory for data analysis in operations research.
    • In this book the basic concept of an exponential possibility distribution and its properties are introduced.
    • As applications of exponential possibility distributions, possibility regression, possibility discriminant and possibility portfolio selection problems and other related theories are presented.
    • Chapter 1 outlines the content of this book.
    • Chapter 2 covers the fundamentals of possibility theory, such as possibility distributions and the associated operations, possibility and necessity measures and possibilistic linear systems.
    • Chapter 3 focuses on the theory of possibilistic systems based on exponential possibility distributions.
    • A month after Tanaka retired in March 2000, he was appointed as a Professor in the Graduate School of Management and Information Science at Toyohashi-Sozo College.
    • He was also appointed as a Professor in the Department of Kansei Design in the Faculty of Psychological Science at Hiroshima International University, which had been founded two years earlier in 1998.
    • Kansei Design aims to improve products and services by studying the psychological feelings of the users.
    • He was made an Honorary Professor of Chongqing University of Posts and Telecommunications on 9 October 2003.
    • This Chinese university specialised in information science and technology, with coordinated development in the fields of engineering, science, management, and liberal arts.
    • He received the Contribution Award from The Japanese Institute of Industrial Engineering (1991), the Literary Award from the Japanese Society for Fuzzy Theory and Systems (1993), the Achievement Award from Japanese Society for Fuzzy Theory and Systems (1999), and the IEEE Fuzzy Systems Pioneer Award (2010).
    • The first is that curiosity is necessary, that is what led me to the fuzzy title of the paper by Professor Zadeh.
    • I have enjoyed the process of studying fuzzy systems for a long time.
    • Probably I think that my works will end up being a nice little tree of collaborations of my friends, my colleagues and many other researchers.
    • So, I would like to share this award with all of you.
    • Making 11 (2012), 353-361.','2], describe what is was like to be one of his students:- .
    • We have fond memories of Professor Hideo Tanaka, Mrs Fuku Tanaka and their daughters.
    • Prof Tanaka's students have happy memories of these experiences.
    • Mrs Fuku Tanaka was always very hospitable and provided a variety of succulent dishes.
    • He offered explanations that provided easier comprehension of difficult subjects.
    • Professor Tanaka was an exceptionally good and benevolent person who developed a personal friendship with many of his colleagues, in Japan and abroad.
    • He behaved like a good father to his students and juniors, helping them in every way he could, both in physical and spiritual distress, and helping them towards the development of better personalities, in their academic and private lives.
    • The authors of [',' M Inuiguchi and H Ichihashi, Obituary: On behalf of the pupils of Hideo Tanaka, Fuzzy Sets and Systems 213 (2013), 1-5.','1], also students of Tanaka, write:- .
    • Tanaka suffered from interstitial pneumonia and he struggled for several years with the disease before it finally led to his death at the age of 74.
    • List of References (2 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Tanaka.html .

  73. Mikhail Moiseevich Lesokhin biography
    • His father was a shoemaker in Leningrad and Mikhail Moiseevich had a very hard childhood as a result of World War II.
    • In 1941 the Germans invaded Russia and by September of that year German troops were on the outskirts of Leningrad cutting the city off from the rest of Russia.
    • There was appalling suffering from shortages of supplies.
    • The exceptionally bitter winter of 1941-42, when temperatures fell to -40° C, was one of extreme hardship.
    • The difficulties continued after the end of the war and Mikhail Moiseevich had to suffer not only the general hardships of the time but also the loss of his father who died from starvation in 1944.
    • It was through a piece of good fortune that Mikhail Moiseevich survived these difficult times.
    • The siege was broken in January 1944 when a Soviet offensive drove off the besieging Germans from the southern outskirts of the city.
    • In 1951 Lesokhin entered the Leningrad Herzen Pedagogical Institute where he studies mathematics and physics.
    • He was awarded a Master's Degree with distinction in 1956 and became a doctoral student of Evgeniy Sergeyevich Lyapin.
    • Lyapin was one of the founders of the algebraic theory of semigroups and one of the leading researchers on the topic.
    • This deals with an abstract generalization of some features of the theory of characters of semigroups.
    • Lesokhin published a number of papers on this topic beginning in 1958 and 1961, followed by Regularity of systems with exterior multiplication with regular first component (Russian) (1963) and On the completeness of systems with exterior multiplication (Russian) (1963).
    • All of these papers and his dissertation consider triples of semigroups (A, B, C) with an external multiplication which is a bilinear map from A × B to C.
    • In fact most of the time Lesokhin was undertaking research for his doctorate he was working at the Pedagogical Institute in Khabarovsk in the east of the USSR about 30 km from the Chinese border.
    • He worked there for three years, returning to take up a post as an assistant in the Department of Higher Algebra in the Leningrad Herzen Pedagogical Institute in 1962.
    • He worked in this department for the next 36 years being a full professor in the last part of this period.
    • In addition to the theory of characters of semigroups which he began to work on from the time he was a research student, he also made major contributions to the theory of approximations of semigroups.
    • A semigroup A is "approximately" in a class of semigroups P if A is imbeddable in the product of the members of P.
    • In this paper Lesokhin looks only at commutative semigroups where he finds necessary and sufficient conditions that a semigroup be approximately in the class of regular semigroups, the class of groups, the class of idempotent semigroups, and the class of finite groups.
    • Eugene Schenkman, himself the author of a famous group theory textbook, writes:- .
    • This book is meant to give the reader practice with the techniques and methods of algebra.
    • Elements of language and speech system analysis as well as application peculiarities of fundamentals of modern mathematics to the setting of linguistic models are considered, major attention being focused on the arrangement of thesaurus semantic nets.
    • The book is intended for linguists, specialists in the fields of cybernetics and computing machinery and postgraduates and students training in structural, applied and mathematical linguistics.
    • S I Kublanovsky, the auther if [',' M M Lesokhin (1933-1998), Semigroup Forum 59 (2) (1999), 159-166.','1], was a student of Lesokhin.
    • Mikhail Moiseevich followed the mathematical growth of his students with great care and was sincerely pleased by their success.
    • We became surer and surer of our strength, of our ability to do something in mathematics.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Lesokhin.html .

  74. Gaston Tarry biography
    • He attended the Lycee Saint-Louis in Paris where he became interested in mathematics.
    • He spent the whole of his working life in Algeria.
    • Tarry was part of the French administration in Algeria, retiring in 1902.
    • One has simply to feel amazement at some of the problems he solved using purely combinatorial and calculating skills.
    • We give some examples below to illustrate both the type of problem which interested him and also to illustrate his undoubted genius.
    • Even more surprising is the fact that his mathematical achievements came after the age of fifty.
    • He published a solution to the problem of finding the way out of a maze in 1895, a problem which had been of interest from classical times.
    • 15 (2) (1886), 49-53.','4], Tarry gave a general method for finding the number of Euler circuits.
    • Tarry also solved Euler's 36 Officer Problem, proving that two orthogonal Latin squares of order 6 do not exist.
    • How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, a major, a captain, a lieutenant, and a sub-lieutenant be arranged in a regular 6 × 6 array such that no row or column duplicates a rank or a regiment? .
    • First we note that a magic square of order n contains the numbers 1 to n2 in an n × n array such that each row, each column, and the two main diagonals all have the same sum.
    • For about three years, a friend of Alger [Brutus Portier], a keen magician, never stopped titillating me in order that I take an interest in magic squares.
    • I disregarded him, because I didn't like what I considered as a Chinese game giving me a headache.
    • One day, my magic teacher (he forced himself to be my teacher in spite of me), claimed to me that it was not possible to construct diabolic squares (panmagic) of order 3n, with n being not divisible be 3.
    • Panmagic squares of order 3n, where n is not divisible by 3, do exist and indeed Tarry was the first to prove this when he constructed the first panmagic square of order 15.
    • Before explaining another major achievement that he accomplished on magic squares we need to explain another couple of terms.
    • First a bimagic square is a magic square with the property that the rows, columns and main diagonals add to the same number when each of the entries is squared.
    • A bimagic square is trimagic if also the cubes of the entries in the rows, columns and main diagonals add to the same constant.
    • Tarry was the first to find an example of a trimagic square.
    • He called his method of construction the "cabalistic condensator" and wrote:- .
    • Poincare was so impressed by Tarry's results that he presented them to the Academy of Sciences.
    • In the same paper Tarry introduced the term tetramagic square for a square which is trimagic and also the fourth powers of the entries in the rows, columns and main diagonals add to the same constant.
    • are equal for each value of k between 0 and 10.
    • both simplify to the same expression for each of the values of k from 0 to 5.
    • He invented what is today called the Tarry point of a triangle which is related to the Brocard triangle and lies on the circumcircle opposite the Steiner point.
    • Lucas was fascinated by many of Tarry's results.
    • He devoted a chapter entitled La Geometrie des Reseaux et le Probleme des Dominos Ⓣ in volume 4 of Recreations Arithmetiques Ⓣ to Tarry's work.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (7 books/articles) .
    • A Poster of Gaston Tarry .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Tarry.html .

  75. Herbert Wilf biography
    • Even as a child, Herb (as he was known to his friends and colleagues) was fascinated by mathematics and he would read mathematics books in his bed after he had been put down to sleep by his parents.
    • Alexander Wilf, known as Alex, had right-wing Zionist views and was a passionate supporter of the idea of creating a Jewish state in the middle east.
    • He founded the local branch of the Committee for a Jewish Army in 1941.
    • Three years later he decided to devote his life to that cause, sold his home in Wynnefield together with the family business, and moved to New York to become the executive director of the American League for a Free Palestine.
    • Herb continued to study at the Central High School in Philadelphia and lived with some of his relatives.
    • Alexander Wilf was involved in smuggling refugees and arms into Palestine, using ships in which he had invested most of his wealth.
    • Alex Wilf also put money into the Altalena which sailed for Tel Aviv in June 1948 with 940 fighters and a large quantity of arms and ammunition.
    • However, there was great differences of opinion among the leaders in Israel about accepting this cargo when a ceasefire was in operation.
    • Herb Wilf completed his schooling in 1948 and, following his father's advice, applied to study at the Massachusetts Institute of Technology.
    • Before he had begun his studies, however, with the state of Israel now set up, Alexander Wilf decided to take his family to Israel and start up a newspaper.
    • He pressed Herb to come to Israel with the rest of the family, but he resisted [',' M F Bernstein, Proof and beauty, The Pennsylvania Gazette (May, 1998), 14-18.','1]:- .
    • in mathematics from the Massachusetts Institute of Technology in 1952.
    • I went to the headquarters of the American Contract Bridge League, in the west 50's in Manhattan, to find a list of tournaments that I might play in.
    • supervised by Herbert Ellis Robbins, who was the Professor of Mathematical Statistics.
    • However, although Robbins was his official advisor, in fact the inspiration behind his doctoral thesis came from Gerald Goertzel, Professor of Physics at New York University [',' H S Wilf, Gerald Goertzel (1920-2002), As I Knew Him (25 October 2005).','8]:- .
    • In 1952, when I met Jerry, it was, for me anyway, the dawn of the computer era, and I was doing as much programming as I could because I enjoyed it.
    • [The] technical work for my thesis was done with the inspiration and guidance of Jerry Goertzel.
    • The title of the thesis was "The transmission of neutrons in multilayered slab geometry." It solved the transport equation in multilayered geometry by regarding each homogeneous layer as a little black box with prescribed inputs and outputs (which point of view was Jerry's hallmark), and it wired them together by representing each by a matrix.
    • While he was undertaking graduate studies at Columbia University, Wilf married Ruth Tumen; they had a daughter Susan (who became a Chinese scholar), and two sons David (who became a lawyer) and Peter (who became a palaeontologist).
    • With a wife and young family to support, Wilf needed to earn money and took on a number of jobs such as designing jet engines with the Fairchild Engine Division in 1954 and as head of the Computing Section of Nuclear Development Associates 1955-59.
    • in 1958 and, in the following year, he was appointed as an Assistant Professor of Mathematics at the University of Illinois.
    • After three years in this position, Wilf was appointed as Associate Professor of Mathematics at the University of Pennsylvania.
    • He worked for the University of Pennsylvania for the rest of his career being promoted to Professor of Mathematics in 1965.
    • One of the earliest endowed chairs at the University of Pennsylvania was the Thomas A Scott Professorship established by the Trustees in June 1881.
    • Even before the award of his doctorate, Wilf had written a remarkable range of papers: (with M Kalos) Monte Carlo solves reactor problems (1957); An open formula for the numerical integration of first order differential equations (1957); An open formula for the numerical integration of first order differential equations.
    • II (1958); Curve-fitting matrices (1958); A stability criterion for numerical integration (1959); and Matrix inversion by the annihilation of rank (1959).
    • The first, Mathematical Methods for Digital Computers (1960), written with A Ralston, came out of the work that he had done for Nuclear Development Associates while he was still a research student.
    • His next book, Mathematics for the Physical Sciences (1962) was intended for:- .
    • physicists, engineers and other natural scientists in their first or second year of graduate study.
    • My name was on the book, but Jerry Goertzel's ideas and inspiration were the core of the presentation of orthogonal polynomials and Gauss Quadrature via tridiagonal matrices and their spectra.
    • However, Wilf is best known for his remarkable contributions to combinatorics [',' Herbert S Wilf, Penn Mathematics: In Memoriam.','5]:- .
    • In the 1960's, Wilf became interested in the newly developing field of combinatorial analysis.
    • He wrote fundamental research papers, forming the foundation of today's work in discrete mathematics with its applications to computer algorithms and its close interconnections with the mathematical fields of algebra and probability theory.
    • For this work Wilf and Zeilberger were jointly awarded the Leroy P Steele Prize by the American Mathematical Society at the January 1998 meeting of the Society in Baltimore.
    • The remarkably simple idea of the work of Wilf and Zeilberger has already changed a part of mathematics for the experts, for the high-level users outside the area, and the area itself.
    • Each semester, after my final grades have been turned in and all is quiet, it is my habit to leave the light off in my office, leave the door closed, and sit by the window catching up on reading the stack of preprints and reprints that have arrived during the semester.
    • That year, one of the preprints was by Zeilberger, and it was a 21st century proof of one of the major hypergeometric identities, found by computer, or more precisely, found by Zeilberger using his computer.
    • I remember feeling that I was about to connect to a parallel universe that had always existed but which had until then remained well hidden, and I was about to find out what sorts of creatures lived there.
    • I also learned that such results emerge only after the efforts of many people have been exerted, in this case, of Sister Mary Celine Fasenmyer, Bill Gosper, Doron Zeilberger and others.
    • We must also mention Wilf's contributions to mathematics teaching.
    • The quality of this is seen from the wards that he received: the Christian and Mary Lindback Award for excellence of undergraduate teaching from the University of Pennsylvania (1973); the Award for Distinguished College or University Teaching of Mathematics, Eastern Pennsylvania and Delaware Section of the Mathematical Association of America (1995); and the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics from the Mathematical Association of America in January, 1996.
    • A lot of attention is paid in this country to the programmatic aspects of education; that is to say, to curricula.
    • The cost to society of regimenting kids so they all listen to their teacher and go home and do homework for four hours is tremendous.
    • I want my kids to be free and untrammelled to screw up their mathematics and flunk all those standardized tests.
    • The biggest single problem we have to block understanding in mathematics is the fact that instructors don't push students hard enough to force them to verbalize it.
    • Yes, mathematics is a universal language, but you've got to speak it.
    • what is at the highest premium is the ability of students to wrap complete English sentences around their mathematical thoughts.
    • Let us look briefly at the books that Wilf has published, in addition to Mathematics for the Physical Sciences (1962) which we mentioned above.
    • In 1970 he published Finite sections of some classical inequalities which studied spectral theory of finite sections of infinite-dimensional operators.
    • The author is to be commended for having given us a connected and clear account of the interesting work developed mainly by M Kac, G Szegő, H Widom, and N de Bruijn during the last two decades.
    • This book presented a collection of programs, written in FORTRAN, for various combinatorial algorithms.
    • This is an introductory textbook on the design and analysis of algorithms.
    • The main strengths of the book are its very clear writing style and strong emphasis on motivation.
    • This book contains an introduction to the topic of generating functions that is suitable for an advanced undergraduate.
    • This is not surprising - they seem to be functions of x yet we pay more attention to their coefficients than their values.
    • We just do not take the time in a typical undergraduate combinatorics course to give a slow, careful treatment of generating functions.
    • During the past several years an important part of mathematics has been transformed from an Art to a Science: No longer do we need to get a brilliant insight in order to evaluate sums of binomial coefficients, and many similar formulas that arise frequently in practice; we can now follow a mechanical procedure and discover the answers quite systematically.
    • Finally we mention that Wilf founded two journals, The Journal of Algorithms in 1980 with co-founder Donald Knuth, and The Electronic Journal of Combinatorics in 1994 with co-founder Neil Calkin.
    • He also served as the editor-in-chief of the American Mathematical Monthly from 1987 to 1992 and on the editorial board of Discrete Mathematics and Theoretical Computer Science from 1999-2011.
    • Outside of mathematics Wilf had a passion for flying:- .
    • It gave my wife and me a great deal of pleasure, as well as a few hair raising moments, and we saw a great deal of North America as a result.
    • Wilf died of a progressive neuromuscular disease.
    • The traditional days of mourning, shivah, were observed at his home in Wynnewood and a memorial service was held on 15 January 2012 at Temple Beth Hillel-Beth El in Wynnewood.
    • The Herbert S Wilf Award, recognising outstanding student achievement, is being set up at the Department of Mathematics, The University of Pennsylvania.
    • List of References (9 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Wilf.html .

  76. Michael Atiyah biography
    • He was a strong supporter of the Palestinian cause.
    • Michael Atiyah, when interviewed in [',' J Meek, I’m a bit of a jack of all trades, The Guardian (Wednesday 21 April 2004).','32], spoke about his father:- .
    • When he came back to Sudan, he found he wasn't part of the English class structure, he was regarded as one of the lower classes, although he was Oxford-educated and regarded himself as culturally English.
    • Michael's mother Jean, although of Scottish descent, was the daughter of a minister of a church in Yorkshire.
    • They had four children, three sons Michael (the eldest and subject of this biography), Patrick Selim (born 5 March 1931, who went on to become an English lawyer and academic) and Joseph (known as Joe, the youngest of the four children who after a mathematics degree from Cambridge University, became a computer scientist working in computer software and telecommunications), and a daughter Selma (who studied English at an American University and lives in America).
    • Michael's primary school education was at the Diocesan school in Khartoum which he entered in 1934 at the age of five.
    • Lebanon had been controlled by the French and, after the fall of France in 1940, it came under the control of the Vichy government.
    • However, just after this began, the British and Free French began fighting to gain control of the Lebanon.
    • I survived by helping bigger boys with their homework and so was protected by them from the inevitable bullying of a boarding school.
    • Intelligencer 6 (1) (1984), 9-19.','35] about how he came to chose mathematics:- .
    • I was always interested in mathematics from a very young age.
    • But there was a stage [at Victoria College in Cairo] when I got very interested in chemistry, and I thought that would be a great thing; after about a year of advanced chemistry I decided that it wasn't what I wanted to do and I went back to mathematics.
    • He gave a somewhat fuller description of his decision between chemistry and mathematics in the interview.
    • It was how to make sulphuric acid and all that sort of stuff.
    • Lists of facts, just facts, you had to memorize a vast amount of material.
    • Organic chemistry was more interesting, there was a bit of structure to it.
    • But inorganic chemistry was just a mountain of facts in books like this.
    • It's true that in mathematics you don't really need an enormous memory.
    • If you want to do other things, you've got to work hard to learn a lot of facts.
    • Michael Atiyah attended Manchester Grammar School, one of the best schools for mathematics in the country.
    • Although he was only sixteen years old, he had already taken his A-level examinations having been two years ahead of his age groups in Victoria College, Cairo.
    • We had an old-fashioned but inspiring teacher who had graduated from Oxford in 1912 and from him I acquired a love of projective geometry, with its elegant synthetic proofs, which has never left me.
    • He served as a clerical officer and took the opportunity to read mathematics books and articles.
    • He was granted special permission to cut short the final year of his military service and spend it at Cambridge.
    • There he played a lot of tennis and avidly studied mathematics on his own in the library.
    • He matriculated at Trinity College in the autumn of 1949.
    • Many of his fellow students had decided to postpone their National Service, so Atiyah was one of the older of the students in his year.
    • While still an undergraduate, he wrote his first paper A note on the tangents of a twisted cubic (1952).
    • After graduating with his BA in 1952, Atiyah continued to undertake research at Trinity College, Cambridge obtaining his doctorate in 1955 with his thesis Some Applications of Topological Methods in Algebraic Geometry.
    • Speaking of the work for his thesis, Atiyah said [',' R Minio, An interview with Michael Atiyah, Math.
    • I'd come up to Cambridge at a time when the emphasis in geometry was on classical projective algebraic geometry of the old-fashioned type, which I thoroughly enjoyed.
    • I would have gone on working in that area except that Hodge represented a more modern point of view - differential geometry in relation to topology; I recognized that.
    • I got interested in it, he got interested in it, and we worked together and wrote a joint paper which was part of my thesis.
    • Atiyah published two joint papers with his thesis advisor William Hodge, Formes de seconde espece sur une variete algebrique Ⓣ (1954) and Integrals of the second kind on an algebraic variety (1955).
    • He was made a fellow of Trinity College, Cambridge in 1954.
    • Lily, born in Edinburgh in 1928, was the daughter of a dock worker at the Rosyth naval yard.
    • She had studied mathematics first at the University of Edinburgh and then took the Cambridge Tripos.
    • Returning to Cambridge, he was a college lecturer from 1957 and a Fellow of Pembroke College from 1958.
    • He remained at Cambridge until 1961 when he moved to a readership at the University of Oxford where he became a Fellow of St Catherine's College.
    • Atiyah was soon to fill the highly prestigious Savilian Chair of Geometry at Oxford from 1963, holding this chair until 1969 when he was appointed professor of mathematics at the Institute for Advanced Study in Princeton.
    • He was also elected a Fellow of St Catherine's College, Oxford.
    • Oxford was to remain Atiyah's base until 1990 when he became Master of Trinity College, Cambridge and Director of the newly opened Isaac Newton Institute for Mathematical Sciences in Cambridge.
    • Atiyah showed how the study of vector bundles on spaces could be regarded as the study of cohomology theory, called K-theory.
    • Grothendieck also contributed substantially to the development of K-theory.
    • Michael Atiyah has contributed to a wide range of topics in mathematics centring around the interaction between geometry and analysis.
    • His first major contribution (in collaboration with F Hirzebruch) was the development of a new and powerful technique in topology (K-theory) which led to the solution of many outstanding difficult problems.
    • Subsequently (in collaboration with I M Singer) he established an important theorem dealing with the number of solutions of elliptic differential equations.
    • Combined with considerations of symmetry it led (jointly with Raoul Bott) to a new and refined 'fixed point theorem' with wide applicability.
    • The K-theory and the index theorem are studied in Atiyah's book K-theory (1967, reprinted 1989) and his joint work with G B Segal, The Index of Elliptic Operators I-V, in the Annals of Mathematics, volumes 88 and 93 (1968, 1971).
    • Atiyah also described his work on the index theorem in The index of elliptic operators given as an American Mathematical Society Colloquium Lecture in 1973.
    • The ideas which led to Atiyah being awarded a Fields Medal were later seen to be relevant to gauge theories of elementary particles.
    • The index theorem could be interpreted in terms of quantum theory and has proved a useful tool for theoretical physicists.
    • Atiyah initiated much of the early work in this field and his student Simon Donaldson went on to make spectacular use of these ideas in 4-dimensional geometry.
    • More recently Atiyah has been influential in stressing the role of topology in quantum field theory and in bringing the work of theoretical physicists, notably E Witten, to the attention of the mathematical community.
    • The theories of superspace and supergravity and the string theory of fundamental particles, which involves the theory of Riemann surfaces in novel and unexpected ways, were all areas of theoretical physics which developed using the ideas which Atiyah was introducing.
    • Atiyah has published a number of highly influential books: K-theory (1967); (with I G Macdonald) Introduction to commutative algebra (1969); Vector fields on manifolds (1970); Elliptic operators and compact groups (1974); Geometry on Yang-Mills fields (1979); (with N J Hitchin) The geometry and dynamics of magnetic monopoles (1988); The geometry and physics of knots (1990); (Video) The mysteries of space (1992); Siamo tutti Matematici Ⓣ (2007); and Edinburgh Lectures on Geometry, Analysis and Physics (2010).
    • We give extracts from some reviews of these books, some extracts from Prefaces and some Publisher's descriptions at THIS LINK.
    • Atiyah and John Tate described the Clay Mathematics Institute Millennium Prize Problems in a lecture in Paris on 24 May 2000.
    • He also discussed the implications for various fields of mathematics and physics if solutions to these problems were found.
    • A 60-minute video of the lecture is available entitled The millennium prize problems.
    • Six volumes of Atiyah's Collected Works have been published.
    • These contain a commentary by Atiyah and in the Preface he comments on the practice of publishing 'collected works' during the lifetime of their author:- .
    • There are several clear advantages to all parties: posterity is saved the trouble of undertaking the collection, while the author can add some personal touches by way of a commentary.
    • Another important aspect of Atiyah's contribution is the remarkable collection of doctoral students he supervised.
    • We have listed his students with the title and date of their thesis and, for those who we know have gone on to an academic career, a university at which they have taught, at THIS LINK.
    • He was elected a Fellow of the Royal Society of London in 1962 at the age of 32.
    • He received the Royal Medal of the Society in 1968 and its Copley Medal in 1988.
    • He gave the Royal Society's Bakerian Lecture on Global geometry in 1975 and was President of the Royal Society from 1990 to 1995.
    • Among the prizes that he has received are the Feltrinelli Prize from the Accademia Nazionale dei Lincei in 1981, the King Faisal International Prize for Science in 1987, the Gunning Victoria Jubilee Prize from the Royal Society of Edinburgh in 1990, the Benjamin Franklin Medal in 1993, the Jawaharlal Nehru Memorial Medal in 1993, the Order of Andres Bello (1st Class) from the Republic of Venezuela in 1997, the Royal Medal from the Royal Society of Edinburgh in 2003, the Order of Merit (Gold) from the Lebanon in 2005, and the President's Medal from the Institute of Physics in 2008.
    • In 2004 Atiyah and Isadore Singer were awarded the Neils Abel prize of £480 000 by the Norwegian Academy of Science and Letters:- .
    • for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.
    • They were presented with the prize by King Harald V of Norway at a ceremony in Oslo.
    • He was President of the London Mathematical Society in 1974-76 receiving its De Morgan Medal in 1980.
    • Atiyah was knighted in 1983 and made a member of the Order of Merit in 1992.
    • He has been elected a foreign member of many national academies including: the American Academy of Arts and Sciences (1969), Royal Swedish Academy of Sciences (1972), German Academy of Scientist Leopoldina (1977), Academie des Sciences, Paris (1978), United States National Academy of Sciences (1978), Royal Irish Academy (1979), Third World Academy of Science (1983), Australian Academy of Sciences (1992), Ukrainian Academy of Sciences (1992), Indian National Science Academy (1993), Russian Academy of Sciences (1994), Georgian Academy of Sciences (1996), Academy of Physical, Mathematical and Natural Sciences of Venezuela (1997), American Philosophical Society (1998), Accademia Nazionale dei Lincei, Rome (1999), Royal Norwegian Society of Sciences and Letters (2001), Czechoslovakia Union of Mathematics (2001), Moscow Mathematical Society (2001), Spanish Royal Academy of Sciences (2002), Lebanese Academy of Sciences (2008), Norwegian Academy of Science and Letters (2009).
    • He has been made an Honorary Fellow or Member of: Trinity College, University of Cambridge (1976), Pembroke College, University of Cambridge (1983), Royal Institution (1991), St Catherine's College, University of Oxford, (1991), Darwin College, University of Cambridge (1992), Royal Academy of Engineering (1993), New College, University of Oxford (1999), Faculty of Actuaries (1999), Academy of Medical Sciences (2000).
    • Many universities have awarded him an honorary degree including: Bonn (1968), Warwick (1969), Durham (1979), St Andrews (1981), Trinity College Dublin (1983), Chicago (1983), Edinburgh (1984), Cambridge (1984), Essex (1985), London (1985), Sussex (1986), Ghent (1987), Reading (1990), Helsinki (1990), Leicester (1991), Rutgers (1992), Salamanca (1992), Montreal (1993), Waterloo (1993), Wales (1993), Queen's-Kingston (1994), Keele (1994), Birmingham (1994), Open University (1995), Manchester (1996), Chinese University of Hong Kong (1996), Brown University (1997), Oxford (1998), University of Wales Swansea (1998), Charles University Prague (1998), Heriot-Watt University (1999), University of Mexico (2001), American University of Beirut (2004), York (2005), Harvard University (2006), Scuola Normale Pisa (2007), Universitat Politecnica de Catalunya (2008).
    • Let us end this biography by recording the sad facts that Atiyah's eldest son John died on 24 June 2002 while on a walking holiday in the Pyrenees with his wife, while Jeremy, the youngest son of Atiyah's brother Patrick, died on 12 April 2006 while walking in Italy.
    • Lily Atiyah died on 13 March 2018 at the age of 90.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (41 books/articles) .
    • A Poster of Michael Atiyah .
    • Reviews of Michael Atiyah's books .
    • Michael Atiyah on beauty and mathematics .
    • Multiple entries in The Mathematical Gazetteer of the British Isles .
    • 5.nFellow of the Royal Societyn1962 .
    • 13.nHonorary Fellow of the Edinburgh Maths Societyn1979 .
    • 15.nFellow of the Royal Society of Edinburghn1985 .
    • 20.nPresident of the Royal Societyn1990-1995 .
    • Paintings of Atiyah .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Atiyah.html .

  77. Mary Warner biography
    • Her parents were Sydney and Esther Davies and Mary was the eldest of their two children, having one younger sister.
    • It was in this in agricultural market town that Sydney had been appointed as headmaster of the grammar school, and this was the town in which Mary spent the next ten years of her life.
    • An old-fashioned Welsh grammar school headmaster, one of the last of the kind, his standards were very high.
    • Mary took her school certificate examinations while at her father's school, achieving the best results of anyone in her year in Wales.
    • By this time her interests were very firmly in the area of mathematics but she had to move to a new school in order to study physics which she wanted to take to complement her mathematical studies.
    • After sitting the School Certificate examinations she entered Howell's Boarding School in Denbigh which is in North Wales, and in 1948 her father accepted the position of headmaster at the grammar school at Holywell, which is about 12 miles north east of Denbigh close to the coast of North Wales.
    • This was certainly a step up for Sydney Davies, for this move brought him to a larger more influential school and he went on to reach the position of the leading headmaster in Welsh secondary schools.
    • In 1951 Mary Davies entered the University of Oxford to read mathematics.
    • One of her fellow students gave this description of her as an undergraduate (see [',' I M James and A R Pears, Mary Wynne Warner, Bull.
    • Although she achieved distinction in her years as an undergraduate, being awarded both college and university prizes, she only received a Second Class degree in mathematics when she graduated in 1953.
    • Her supervisor was Henry Whitehead who had been appointed to the Waynflete Chair of Pure Mathematics at Oxford some six years earlier.
    • She published her first research paper A note on Borsuk's antipodal point theorem in the Oxford Quarterly Journal of Mathematics in 1956.
    • She had become friendly with a history student at Oxford, Gerald Warner, who graduated in 1954 and joined the Diplomatic Service in the Intelligence Branch.
    • Mary Warner now fitted into the role of diplomat's wife, a role which she certainly enjoyed, but she remained a committed mathematician taking every opportunity to continue to pursue her studies.
    • In Beijing she met up with the Chinese topologist Chang Su-chen who had also been a student of Henry Whitehead.
    • The period that the Warners spent in China was one of change.
    • The country had been following a line of peaceful coexistence but began to take a more military line beginning in 1957.
    • A policy of socialist construction was adopted based largely on ideological principles.
    • Conditions became more tense for the Warners and eventually Chang Su-chen told Mary that in meeting to discuss mathematics they were creating suspicions which would cause them difficulties [',' Obituary in The Times','1]:- .
    • [Chang Su-chen] came to [the Warners] flat one day, crouched beside the sofa to avoid any eavesdropping microphones, and told her that, although he was a liberal, the beginning of Chairman Mao's Great Leap Forward meant that he could see her no longer.
    • Warner now lived in London where she was appointed to an part-time lectureship in mathematics at Bedford College.
    • The third of Warner's children, Rachel, was born in Rangoon in 1961 where, despite being a diplomat's wife and the mother of three young children, she continued to pursue her mathematical career with an appointment as Senior Lecturer in Mathematics at Rangoon University.
    • One of her main tasks was to set up an M.Sc.
    • Borsuk led a seminar in which he developed a unique atmosphere of successful international cooperation and Warner was welcomed as a Visiting Research Fellow.
    • The effect of this active group on Warner was to make her begin work on a doctoral thesis under the supervision of one of Borsuk's colleagues.
    • After two years in Warsaw, Warner spent two further years in Geneva during which time she completed her doctoral dissertation The homology of Cartesian product spaces which she submitted to the Polish Academy of Sciences.
    • A successful defence of the thesis, which was examined by Borsuk and Kuratowski, saw Warner complete, fifteen years after she became a research student at Oxford, the task she had set out on.
    • It is remarkable that she had the tenacity to keep up her mathematical interests throughout her life on the move often under the most difficult of circumstances.
    • In 1968 Warner was back in London where she was appointed as a Lecturer in Mathematics at the City University.
    • The following year her second paper appeared The homology of tensor products and her next two papers, published in 1975 and 1976, were the results of work she undertook while in Malaysia, the last overseas posting her husband was to have, during 1974-76.
    • The City University had agreed to give her leave of absence for two years so on her return to London Warner again took up her post.
    • In Warner's case, however, it only became possible for her to concentrate fully on mathematics from this time on and the result was a rapid climb to become [',' I M James and A R Pears, Mary Wynne Warner, Bull.
    • one of the foremost researchers in fuzzy mathematics, highly respected by all her colleagues in the field.
    • She then generalised both concepts with the introduction of the notion of a lattice-valued relation in 1984.
    • Perhaps the best way to describe the ideas of tolerance space and lattice-valued relation is to quote the introduction from Warner's paper Some thoughts on lattice valued functions and relations published in 1985:- .
    • In fact, every real-valued function is lattice-valued by virtue of the usual max, min lattice on the ordered set of reals.
    • We concentrate on some areas where the actual lattice structure of L plays a major part in a topological theory.
    • Continuous real-valued functions from a topological space are thus excluded per se, but feature within the context of fuzzy topological spaces.
    • Some of the formal transition from ordinary to fuzzy spaces is largely mechanical.
    • More interesting are the difficulties encountered, and on some of these we shall concentrate.
    • The term 'fuzzy' has been used by Poston in his thesis and C T J Dodson [in 1974] to describe a set with a reflexive, symmetric relation, elsewhere [in E C Zeeman's paper" Topology of 3-manifolds and related topics" of 1962] called a tolerance space.
    • Her great mathematical achievements brought her great satisfaction, but the latter part of her life was also filled with tragedy.
    • The level of her activity at this stage can be seen from the fact that she published three papers in 1996 and three more in 1997.
    • This charming story, recounted in [',' Obituary in The Times','1], tells us quite of lot about her character:- .
    • Once when they were giving a diplomatic dinner party in a Geneva restaurant noted for its tartes a la creme, a guest was mocking Welsh poetry, of which Mary Warner was very fond.
    • Becoming more and more indignant, but prevented from fighting back, she finally turned illogically but effectively on her husband, throwing at him one of the specialities of the house and bringing the conversation to a full stop.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (2 books/articles) .
    • Dictionary of National Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Warner.html .

  78. Bonaventura Cavalieri biography
    • Born: 1598 in Milan, Duchy of Milan, Habsburg Empire (now Italy) .
    • His father's name was Bonaventura Cavalieri but when Francesco joined the religious order of the Jesuati in Milan in 1615 he took the name Bonaventura.
    • The religious order of the Jesuati, which he joined, was founded by Giovanni Colombini of Siena and his friend Francesco Miani in 1360.
    • The order was named Jesuati because their sermons always began and ended with the name of 'Jesus' being shouted out.
    • In Pisa, Cavalieri was taught mathematics by Benedetto Antonio Castelli, a lecturer in mathematics at the University of Pisa.
    • He taught Cavalieri geometry and introduced him to the ideas of Galileo.
    • Cavalieri's interest in mathematics had been stimulated by Euclid's Elements and, after meeting Galileo, he considered himself a disciple of the astronomer.
    • The Cardinal himself saw clearly the genius in Cavalieri while he was at the monastery in Milan [',' K Anderson, Cavalieri’s method of indivisibles, Archive for the History of Exact Sciences 31 (4) (1985), 291-367.','12]:- .
    • Galileo did not answer all of them, but sent an occasional letter to Cavalieri; of these all but a very few have disappeared.
    • Urbano Diviso, Cavalieri's pupil and first biographer writing about 30 years after Cavalieri's death, claimed that Castelli told Cavalieri to study of mathematics since that would cure him of depression.
    • However, there is no other evidence for this claim and certainly some checkable claims in Diviso's account of Cavalieri's life are incorrect.
    • In 1619 Cavalieri applied for the chair of mathematics in Bologna, which had become vacant following the death of Giovanni Antonio Magini, but was not successful since he was considered too young for a position of this seniority.
    • He also failed to get the chair of mathematics at several other universities including Rome and Pisa when Castelli left for Rome in 1626.
    • Cavalieri himself blamed the fact that he was in the Jesuati order as the reason for his lack of success in these applications.
    • In 1621 Cavalieri became a deacon and assistant to Cardinal Federico Borromeo at the monastery of San Girolamo in Milan.
    • It was during his time in Milan that he began to develop his method of indivisibles for which he is famed today.
    • He taught theology in Milan until 1623 when he became prior of St Peter's at Lodi.
    • After three years at Lodi he went to the Jesuati monastery in Parma where he was the prior [',' E Carruccio, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • In the autumn of 1626, during a trip from Parma to Milan, he fell ill with the gout, from which he had suffered since childhood and which was to plague him to the end of his life.
    • This illness kept him at Milan for a number of months.
    • This contains the method of indivisibles which became a factor in the development of the integral calculus.
    • In 1629 Cavalieri was appointed to the chair of mathematics at Bologna.
    • His application had been supported by Galileo who [',' E Carruccio, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • in 1629, wrote to Cesare Marsili, a gentleman of Bologna and member of the Accademia dei Lincei, who had been commissioned to find a new lecturer in mathematics.
    • In his letter, Galileo said of Cavalieri, "few, if any, since Archimedes, have delved as far and as deep into the science of geometry." In support of his application to the Bologna position, Cavalieri sent Marsili his geometry manuscript and a small treatise on conic sections and their applications in optics.
    • Galileo's testimonial, as Marsili wrote him, induced the "Gentlemen of the Regiment" to entrust the first chair in mathematics to Cavalieri, who held it continuously from 1629 to his death.
    • The chair of mathematics at Bologna was not the only position he received for he was also appointed prior of the Jesuati convent in Bologna attached to the Church of Santa Maria della Mascarella.
    • This was an ideal situation for Cavalieri who now had the peace to undertake mathematics research at the Jesuati convent while teaching mathematics at the university where he could have contacts with other mathematicians.
    • However, his health deteriorated around the time of his appointment to Bologna, and he suffered from problems with his legs which persisted throughout the rest of his life.
    • The theory of indivisibles, presented in his Geometria indivisibilibus continuorum nova quadam ratione promota of 1635, was a development of Archimedes' method of exhaustion incorporating Kepler's theory of infinitesimally small geometric quantities.
    • This theory allowed Cavalieri to find simply and rapidly the area and volume of various geometric figures.
    • Howard Eves writes [',' H Eves, Two Surprising Theorems on Cavalieri Congruence, The College Mathematics Journal 22 (2) (1991), 118-124.','26]:- .
    • Cavalieri's treatise on the method of indivisibles is voluble and not clearly written, and it is not easy to learn from it precisely what Cavalieri meant by an "indivisible." It seems that an indivisible of a given planar piece is a chord of the piece, and a planar piece can be considered as made up of an infinite parallel set of such indivisibles.
    • Similarly, it seems that an indivisible of a given solid is a planar section of that solid, and a solid can be considered as made up of an infinite parallel set of this kind of indivisible.
    • Now, Cavalieri argued, if we slide each member of a parallel set of indivisibles of some planar piece along its own axis, so that the endpoints of the indivisibles still trace a continuous boundary, then the area of the new planar piece so formed is the same as that of the original planar piece, inasmuch as the two pieces are made up of the same indivisibles.
    • A similar sliding of the members of a parallel set of indivisibles of a given solid will yield another solid having the same volume as the original one.
    • (This last result can be strikingly illustrated by taking a vertical stack of cards and then pushing the sides of the stack into curved surfaces; the volume of the disarranged stack is the same as that of the original stack.) These results give the so-called Cavalieri principles: .
    • If two planar pieces are included between a pair of parallel lines, and if the lengths of the two segments cut by them on any line parallel to the including lines are always equal, then the areas of the two planar pieces are also equal.
    • If two solids are included between a pair of parallel planes, and if the areas of the two sections cut by them on any plane parallel to the including planes are always equal, then the volumes of the two solids are also equal.
    • The method of indivisibles was not put on a rigorous basis and his book was widely attacked.
    • In particular, Paul Guldin attacked Cavalieri [',' P Mancosu, Philosophy of mathematics and mathematical practice in the seventeenth century (Oxford University Press, Oxford, 1996).','8]:- .
    • The debate between Cavalieri and Guldin is usually mentioned in connection with the objections made by Guldin to Cavalieri's use of indivisibles.
    • Although that is probably the main issue between Cavalieri and Guldin, a more careful reading of the debate will allow us to indicate the existence of other interesting issues ..
    • The argument really centres around the fact that Guldin was a classical geometer following the methods of the ancient Greek mathematicians.
    • His first point, however, was to accuse Cavalieri of plagiarising Kepler's Stereometria Doliorum (1615) and Sover's Curvi ac Recti Proportio (1630).
    • There is something in his argument relating to Kepler since in that work Kepler does regard a circle as an infinite polygon composed of infinitesimals.
    • Guldin attacked Cavalieri's indivisibles by arguing that when a surface is generated by rotating a line about the axis, the surface is not just a set of lines.
    • He writes (see [',' P Mancosu, Philosophy of mathematics and mathematical practice in the seventeenth century (Oxford University Press, Oxford, 1996).','8]):- .
    • In my opinion no geometer will grant Cavalieri that the surface is, and could, in geometrical language be called "all the lines of such a figure"; never in fact can several lines, or all the lines, be called surfaces; for, the multitude of lines, however great that might be, cannot compose even the smallest surface.
    • As Mancosu writes [',' P Mancosu, Philosophy of mathematics and mathematical practice in the seventeenth century (Oxford University Press, Oxford, 1996).','8]:- .
    • Guldin was a "classicist" geometer, steeped in the idea of explicit construction, sceptical of considerations of infinity in the domain of geometry, and wary of the risk of ending up with an atomistic theory of the continuum.
    • This book of logarithms was published by Cavalieri as part of his successful application to have the position extended.
    • The tables of logarithms which he published included logarithms of trigonometric functions for use by astronomers [',' T Hockey, Bonaventura Cavalieri, Biographical Encyclopedia of Astronomers (Springer, New York, 2007), 210-211.','31]:- .
    • In addition to noteworthy innovations in terminology, the work includes important demonstrations of John Napier's rules of the spherical triangle and the theorem of the squaring of each spherical triangle that, attributed to Albert Girard, was later claimed by Joseph Lagrange.
    • Galileo praised Cavalieri for his work on logarithms, in particular the book he wrote entitled A hundred varied problems to illustrate the use of logarithms (1639).
    • He developed a general rule for the focal length of lenses and described a reflecting telescope.
    • He also worked on a number of problems of motion.
    • Cavalieri's work of interest is his 'Specchio ustorio', printed in 1632 and reprinted in 1650.
    • In this work Cavalieri concerned himself with reflecting mirrors for the express purpose of resolving the age-long dispute of how Archimedes allegedly burned the Roman fleet that was besieging Syracuse in 212 B.C.
    • The book, however, goes well beyond the stated purpose and systematically treats the properties of conic sections, reflection of light, sound, heat (and cold!), kinematic and dynamic problems, and the idea of the reflecting telescope.
    • He even published a number of books on astrology, one in 1639 entitled Nuova pratica astromlogica and another, his last work, Trattato della ruota planetaria perpetua in 1646.
    • However, although these use the terminology of astrology, they are serious astronomical works.
    • Torricelli was full of praise for Cavalieri's methods writing (see [',' E Carruccio, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]):- .
    • I should not dare affirm that this geometry of indivisibles is actually a new discovery.
    • I should rather believe that the ancient geometricians availed themselves of this method in order to discover the more difficult theorems, although in their demonstration they may have preferred another way, either to conceal the secret of their art or to afford no occasion for criticism by invidious detractors.
    • Whatever it was, it is certain that this geometry represents a marvellous economy of labour in the demonstrations and establishes innumerable, almost inscrutable, theorems by means of brief, direct, and affirmative demonstrations, which the doctrine of the ancients was incapable of.
    • The geometry of indivisibles was indeed, in the mathematical briar bush, the so-called royal road, and one that Cavalieri first opened and laid out for the public as a device of marvellous invention.
    • Angeli wrote many of the letters which Cavalieri sent to his fellow mathematicians during his time of study.
    • This was at a time when Galileo was living under house arrest in Arcetri, and Cavalieri spent the summer discussing mathematics with him.
    • His health had not improved and he was being pressed by the university authorities to work on astronomy rather than on mathematics, the topic that Cavalieri loved.
    • He had the chance to leave Bologna when he was offered the chair of mathematics at Pisa, but he turned it down.
    • By the time of his death in the following year he was totally crippled and unable to walk.
    • He was buried in the church of Santa Maria della Mascarella in Bologna.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (46 books/articles) .
    • A Poster of Bonaventura Cavalieri .
    • Title page of Exercitationes geometricae sex (1647) .
    • History Topics: The rise of calculus .
    • History Topics: An overview of the history of mathematics .
    • History Topics: The trigonometric functions .
    • History Topics: Infinity .
    • History Topics: Light through the ages: Ancient Greece to Maxwell .
    • History Topics: Overview of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Cavalieri.html .

  79. al-Nasawi biography
    • Al-Nasawi had prepared an original version of it in Persian for the library of the Iranian prince Majd al-Dawlah, of the Buyid dynasty.
    • Before the work was completed, however, Majd al-Dawlah was deposed as ruler so, on completion of the work, al-Nasawi presented it to Sharaf al-Muluk who was the vizier of Jalal ad-Dawlah (Jalal ad-Dawlah was the ruler of Baghdad from 1025 to 1044).
    • From this description, and from the fact that al-Nasawi dedicated another work to a Shi'ite leader in Baghdad, we at least can deduce that al-Nasawi worked for part of his life in Baghdad.
    • A few more details of his life have become known recently.
    • The authors of [',' J Ragep and E S Kennedy, A description of Zahiriyya (Damascus) MS 4871 : a philosophical and scientific collection, J.
    • ','3] give an analysis of this mid-12th century manuscript which once contained 80 tracts, but of these only 43 survive.
    • Tract 26 is a summary of Euclid's Elements by al-Nasawi.
    • He does not meet the first of these aims very successfully for the tract is nothing more than a copy of the first six books of the Elements together with Book XI.
    • All al-Nasawi appears to have done is to omit some constructions and change a few of the proofs.
    • This work is interesting historically for our understanding of the way that the Elements was transmitted in Arabic countries but has little significance for its contributions to mathematics.
    • There were three different types of arithmetic used in Arab countries around this period: (i) a system derived from counting on the fingers with the numerals written entirely in words; this finger-reckoning arithmetic was the system used by the business community, (ii) the sexagesimal system with numerals denoted by letters of the Arabic alphabet, and (iii) the arithmetic of the Indian numerals and fractions with the decimal place-value system.
    • The arithmetic book by al-Nasawi is of this third "Indian numeral" type.
    • The book is composed of four separate treatises, each dealing with a particular class of numbers.
    • In each of the four cases al-Nasawi explains the four elementary arithmetical operations.
    • Each method for each of the four types is illustrated with worked examples and a checking procedure is explained which usually involves usually casting out nines The method al-Nasawi gives for taking cube roots is the same as the method described in the Chinese Mathematics in Nine Books, but quite how he learnt of this method is unknown.
    • Al-Nasawi is critical of works on arithmetic written by earlier authors.
    • There seems nothing original in any of his works and, more significantly, there are several places where al-Nasawi presents pieces of mathematics which he fails to properly understand.
    • For example he fails to understand the principle of "borrowing" when doing subtraction.
    • One discusses the theorem of Menelaus while the other is [',' A S Saidan, Biography in Dictionary of Scientific Biography (New York 1970-1990).','1]:- .
    • a corrected version of Archimedes' Lemmata as translated into Arabic by Thabit ibn Qurra, which was last revised by Nasir al-Din al-Tusi.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (3 books/articles) .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Nasawi.html .

  80. Germund Dahlquist biography
    • Germund Dahlquist's father was a minister in the Church of Sweden, the established Lutheran church in Sweden.
    • Germund's mother was a poet, but also wrote a number of well-known hymns.
    • Dahlquist entered Stockholm University in 1942 to begin his study of mathematics.
    • Bohr took the time to discuss mathematics with his young student and inspired Dahlquist's early interests, which centred on analytic number theory, complex analysis, and analytical mechanics.
    • Dahlquist would later refer to the profound influence on his view of mathematics of that early time with Bohr.
    • Dahlquist received a first degree from Stockholm University in 1949 having written a thesis On the Analytic Continuation of Eulerian Products which he published in 1952.
    • Deciding against continuing to undertake research for his doctorate, he joined the Swedish Board of Computer Machinery as an applied mathematician and programmer.
    • This led him to a deep study of numerical analysis.
    • Their methods led to a speeding up of the process of reducing the data and in September 1954 they produced the first 24 hour forecast produced on the day in which the input data was collected.
    • During this time Dahlquist wrote a number of papers such as The Monte Carlo-method (1954), Convergence and stability for a hyperbolic difference equation with analytic initial-values (1954), and Convergence and stability in the numerical integration of ordinary differential equations (1956).
    • He submitted his doctoral thesis Stability and error bounds in the numerical integration of ordinary differential equations to Stockholm University in 1958, defending it in a viva in December.
    • introduced the logarithmic norm (also introduced independently by Lozinskii in 1958), which he used to derive differential inequalities that discriminated between forward and reverse time of integration.
    • From 1956 to 1959 Dahlquist had been head of Mathematical Analysis and Programming Development at the Swedish Board of Computer Machinery.
    • Following the award of his doctorate in 1959 he was appointed to the Royal Institute of Technology in Stockholm.
    • He spent the rest of his career at this institution where the Department of Numerical Analysis, an offshoot the Department of Applied Mathematics, was founded in 1962.
    • In the following year Dahlquist became Sweden's first professorship in Numerical Analysis when he became the professorial head of the Department which at this stage had six members of the academic staff.
    • In the same year of 1963 he published Stability questions for some numerical methods for ordinary differential equations, an expository paper on his fundamental results concerning stability of difference approximations for ordinary differential equations.
    • In the same year he published A special stability problem for linear multistep methods which introduced A-stability and became one of the most cited papers in numerical analysis.
    • He served as an editor of BIT for over 30 years.
    • In 1969 a collaboration with Ake Bjorck led to the publication of Numeriska metoder (Numerical methods) published in Swedish:- .
    • This is a substantial, detailed and rigorous textbook of numerical analysis, in which an excellent balance is struck between the theory, on the one hand, and the needs of practitioners (i.e., the selection of the best methods - for both large-scale and small-scale computing) on the other.
    • The prerequisites are slight (calculus and linear algebra and preferably some acquaintance with computer programming) so that some of the finer theoretical points (those at which numerical analysis becomes applied functional analysis, for example) are outside the scope of the book.
    • However, the class of readers for whom the book is intended are admirably served.
    • The importance of the book is easily seen from the fact that a German translation appeared in 1972 under the title Numerische Methoden, an English translation was published two years later under the title Numerical methods, and a Polish translation was published as Metody numeryczne in 1983.
    • Several later editions and reprints of the various translations of the classic text continued to appear as well as a Chinese translation in 1990.
    • In 1984 the authors were invited by Prentice-Hall to prepare a new edition of [Numerical Methods].
    • After some attempts it soon became apparent that, because of the rapid development of the field, one volume would no longer suffice to cover the topics treated in the 1974 book.
    • Thus a large part of the new book would have to be written more or less from scratch.
    • The present volume is the result of several revisions worked out during the past 10 years.
    • Fortunately the gaps left in his parts of the manuscript were relatively few.
    • Encouraged by his family, I decided to carry on and I have tried to the best of my ability to fill in the missing parts.
    • I hope that I have managed to convey some of his originality and enthusiasm for his subject.
    • It is sad that he could never enjoy the fruits of his labour on this book.
    • Nick Higham, in a review of the book, writes:- .
    • This work is a monumental undertaking and represents the most comprehensive textbook survey of numerical analysis to date.
    • In 1990 Dahlquist retired from the Royal Institute of Technology in Stockholm, but remained very active in research.
    • As an example let us note the publication of On summation formulas due to Plana, Lindelof and Abel, and related Gauss-Christoffel rules in BIT in three parts (1997, 1997, 1999).
    • Gauss quadrature rules are designed for each of them.
    • They are so efficient that they should be considered for the development of software for special functions.
    • Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation of power series outside the circle of convergence, and to ill-conditioned power series.
    • [The second part] is mainly concerned with the derivation, analysis and applications of a summation formula, due to Lindelof, for alternating series and complex power series, including ill-conditioned power series.
    • The authors of [',' A Bjorck, C W Gear and G Soderlind, Obituary: Germund Dahlquist.','1] tell of two other aspects of Dahlquist's life outside of mathematical research, namely his work for Amnesty International and his love of music.
    • As an active member of Amnesty International during the 1970s, Dahlquist worked to help scientists who were politically persecuted, in some cases travelling to offer his encouragement and recognition in person.
    • He used to tell the story of his intervention on behalf of a Russian mathematician who, in despair, had made a thoughtless public statement to the effect that the Soviet Union was "a land of alcoholics." Guriy I Marchuk, who had visited Stockholm University in the 1960s, was then president of the USSR Academy of Sciences and vice-chair of the USSR Council of Ministers.
    • After a long time with no response, two staff members of the Soviet Embassy called at Germund's office one day, bringing greetings from Marchuk and a package, that turned out to contain ..
    • two bottles of vodka! .
    • Next the story regarding his love of music:- .
    • He would often happily sit down at the piano and entertain his colleagues with a few old standards, starting with "On the Sunny Side of the Street" and ending with "As Time Goes By." But his knowledge went much deeper.
    • On one visit to the USA, with a few colleagues in a fine restaurant, Germund heard a female bar pianist whose music was obviously the highlight of the evening for him.
    • When it was time to leave, Germund told the pianist how much he had enjoyed her stylish playing, adding that it had reminded him of one of his favourites, the great jazz pianist Art Tatum.
    • He was elected into the Royal Swedish Academy of Engineering Sciences in 1965.
    • He was a plenary speaker at the International Congress of Mathematicians at Berkeley in 1986.
    • The Society for Industrial and Applied Mathematics (SIAM) named him their John von Neumann lecturer in 1988.
    • In 1995, on the occasion of his 70th birthday, SIAM established the 'Germund Dahlquist Prize' to be awarded biennially:- .
    • Awarded to a young scientist (normally under 45) for original contributions to fields associated with Germund Dahlquist, especially the numerical solution of differential equations and numerical methods for scientific computing.
    • The following announcement of the award appeared:- .
    • Yesterday, the Eidgenossische Technische Hochschule Zurich and Society for Industrial and Applied Mathematics presented 'The 1999 Peter Henrici Prize' to Germund Dahlquist for his outstanding research and leadership in numerical analysis.
    • He has created the fundamental concepts of stability, A-stability and the nonlinear G-stability for the numerical solution of ordinary differential equations.
    • His interests, like Henrici's, are very broad, and he contributed significantly to many parts of numerical analysis.
    • As a human being and scientist, he gives freely of his talent and knowledge to others and remains a model for many generations of scientists to come.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (3 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Dahlquist.html .

  81. Heisuke Hironaka biography
    • However, his second wife then died and he married for the third time, to a younger sister of his second wife who, by this time, was widowed with one child.
    • Heisuke's parents then had ten children of their own (six boys and four girls), so Heisuke grew up in a family of fifteen children.
    • Of his parents' ten children, Heisuke was the second eldest having one older sister.
    • Two of Heisuke's half-brothers were killed, one fighting against the Americans and one fighting against the Chinese.
    • Heisuke attended elementary school and then middle school where he began to develop a liking for mathematics.
    • The town of Yamaguchi is about 80 km from Hiroshima and on Monday, 6 August 1945, at 8.15 in the morning, Heisuke's father witnessed the dropping of the atomic bomb on Hiroshima.
    • He started to learn to play the piano and became very keen but was advised by his teachers not to think of becoming a professional musician.
    • After a mathematics professor from Hiroshima University gave a lecture at the junior high school, Heisuke became enthusiastic and applied to study mathematics at Hiroshima University.
    • He was able to live with one of his sisters who had married and was living in Kyoto.
    • Kyoto University was founded in 1897 to train small numbers of selected students as academics.
    • In his second year, however, he began to realise that he was best suited to mathematics.
    • By his third year as an undergraduate he had moved completely to courses in mathematics.
    • Yasuo Akizuki, a pioneer of modern algebra in Japan, was a major influence on Hironaka during his time at Kyoto.
    • He received his Bachelor of Science in 1954 and his Master of Science in 1956, both from Kyoto University.
    • Hironaka went to the United States in the summer of 1957 where he continued his studies at Harvard.
    • from Harvard in 1960 for his thesis On the Theory of Birational Blowing-up.
    • He had already published three papers before submitting his thesis, On the arithmetic genera and the effective genera of algebraic curves (1957), A note on algebraic geometry over ground rings.
    • The invariance of Hilbert characteristic functions under the specialization process (1958), and A generalized theorem of Krull-Seidenberg on parameterized algebras of finite type (1960).
    • She entered Japanese politics being first elected to the House of Councillors in 1986.
    • She has held high positions in the Democratic Party of Japan and in the Hosokawa Cabinet.
    • In 1970 he had the distinction of being awarded a Fields Medal at the International Congress at Nice.
    • Classical algebraic geometry studies properties of varieties which are invariant under birational transformations.
    • Difficulties that arise as a result of the presence of singularities are avoided by using birational correspondences instead of biregular ones.
    • Hironaka gave a general solution of this problem in any dimension in 1964 in Resolution of singularities of an algebraic variety over a field of characteristic zero.
    • His work generalised that of Zariski who had proved the theorem concerning the resolution of singularities on an algebraic variety for dimension not exceeding 3.
    • These tools have proved useful for attacking many other problems quite far removed from the resolution of singularities.
    • Hironaka talked about his solution in his lecture On resolution of singularities (characteristic zero) to the International Congress of Mathematicians in Stockholm in 1962.
    • In 1975 Hironaka returned to Japan where he was appointed a professor in the Research Institute for Mathematical Sciences of Kyoto University.
    • He gave a course on the theory of several complex variables in 1977 and his lecture notes were written up and published in the book Introduction to analytic spaces (Japanese).
    • Some fundamental theorems in the theory of several complex variables and of the geometry of complex manifolds are proved in a simple but rigorous form.
    • Throughout this book one recognizes again the importance of the preparation theorem, and one finds good introductions to the study of the advanced theory of Stein spaces, the works of A Douady, and the theory of the resolution of singularities, to which Hironaka has contributed deeply.
    • Hironaka was Director of the Research Institute in Kyoto from 1983 to 1985, retiring in 1991 when he was made Professor Emeritus.
    • However, in1996 he became president of Yamaguchi University, being inaugurated on 16 May.
    • He then became Academic Director of the University of Creation, a private university in Takasaki, Gunma, Japan.
    • Hironaka has contributed much time and effort to encouraging young people interested in mathematics.
    • Among the many honours he has received, in addition to the Fields Medal, we mention the Japan Academy Award in 1970 and the Order of Culture from Japan in 1975.
    • He has been elected to the Japan Academy, the American Academy of Arts and Sciences and academies in France, Russia, Korea and Spain.
    • On 4-5 May 2009 the Clay Mathematics Institute held its 2009 Clay Research Conference in Harvard Science Center.
    • Hironaka was invited to give one of the featured lectures on recent research advances and he spoke on Resolution of Singularities in Algebraic Geometry.
    • I will present my way of proving resolution of singularities of an algebraic variety of any dimension over a field of any characteristic.
    • There are some points of general interest, I hope, technically and conceptually more than just the end result.
    • The resolution problem for all arithmetic varieties (meaning algebraic schemes of finite type over the ring of integers) is reduced to the question of how to extend the result from modulo pm to modulo pm+1 after a resolution of singularities over Q.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (8 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Hironaka.html .

  82. János Bolyai biography
    • Janos was born in Zsuzsanna's parents home in Kolozsvar (now renamed Cluj in Romania) but soon went to Marosvasarhely where his father Farkas had a job at the Calvinist College teaching mathematics, physics and chemistry.
    • It was clear from early on, however, that Janos was an extremely bright and observant child [',' B Szenassy, History of Mathematics in Hungary until the 20th Century (Berlin-Heidelberg-New York, 1992).','7]:- .
    • At the age of seven he took up playing the violin and made such good progress that he was soon playing difficult concert pieces.
    • It is important to understand that although Farkas had a lecturing post he was not well paid and even although he earned extra money from a variety of different sources, Janos was still brought up in poor financial circumstances.
    • Until Janos was nine years old the best students from the Marosvasarhely College taught him all the usual school subjects except mathematics, which he was taught by his father.
    • Only from the age of nine did he attend school.
    • By the time Bolyai was 13, he had mastered the calculus and other forms of analytical mechanics, his father continuing to give him instruction.
    • Certainly it would have been a wonderful education for Janos and it is interesting to speculate what benefits might have come to the world of mathematics if he had accepted the plan.
    • Neither of the universities at Pest or Vienna offered a good quality mathematical education at this time, and Farkas could not afford to send his son to a more prestigious university abroad.
    • The decision that Janos would study military engineering at the Academy of Engineering at Vienna was not taken without a lot of heartache and soul-searching but in the end this route was chosen as the least bad of the options.
    • This is not to say that the Academy did not excel in teaching mathematics, for indeed the subject was emphasised throughout the course.
    • He was an outstanding student and from his second year of study on he came top in most of the subjects he studied.
    • Of course he had received military training during his time in Vienna, for the summer months were devoted to this, but Bolyai's nature did not allow him to accept easily the strict military discipline.
    • He spent a total of 11 years in military service and was reputed to be the best swordsman and dancer in the Austro-Hungarian Imperial Army.
    • He neither smoked nor drank, not even coffee, and at the age of 23 he was reported to still retain the modesty of innocence.
    • He was also an accomplished linguist speaking nine foreign languages including Chinese and Tibetan.
    • In fact he gave up this approach within a year for still in 1820, as his notebooks now show, he began to develop the basic ideas of hyperbolic geometry.
    • created a new, another world out of nothing..
    • By 1824, however, there is evidence to suggest that he had developed most of what would appear in his treatise as a complete system of non-Euclidean geometry.
    • Bolyai was posted to Arad in 1826 and there he found that Captain Wolther von Eckwehr, one of his old teachers of mathematics from the Academy in Vienna, was also stationed.
    • Bolyai gave him a draft of the materials which he was writing on the theory of geometry, probably because he hoped for some constructive comments from him.
    • By now Farkas had come to understand the full significance of what his son had accomplished and strongly encouraged him to write up the work for publication as an Appendix to the Tentamen which was close to publication.
    • Had my father not happened to urge or even force me at Marosvasarhely, on my way to duty in Lemberg, to immediately put things to paper, possibly the contents of the Appendix would never have seen the light of day.
    • What was contained in this mathematical masterpiece? After setting up his own definitions of 'parallel' and showing that if the Fifth Postulate held in one region of space it held throughout, and vice versa, he then stated clearly the different systems he would consider:- .
    • denote by Σ the system of geometry based on the hypothesis that Euclid's Fifth Postulate is true, and by S the system based on the opposite hypothesis.
    • All theorems we state without explicitly specifying the system Σ or S in which the theorem is valid are meant to be absolute, that is, valid independently of whether Σ or S is true.
    • Most of the Appendix deals with absolute geometry.
    • I regard this young geometer Bolyai as a genius of the first order .
    • For the entire content of the work ..
    • The clearest reference in Gauss's letters to his work on non-euclidean geometry, which shows the depth of his understanding, occurs in a letter he wrote to Taurinus on 8 November 1824 when he wrote:- .
    • The assumption that the sum of the three angles of a triangle is less than 180° leads to a curious geometry, quite different from ours [i.e.
    • Euclidean geometry] but thoroughly consistent, which I have developed to my entire satisfaction, so that I can solve every problem in it excepting the determination of a constant, which cannot be fixed a priori.
    • the three angles of a triangle become as small as one wishes, if only the sides are taken large enough, yet the area of the triangle can never exceed, or even attain a certain limit, regardless of how great the sides are.
    • The discovery that Gauss had anticipated much of his work, however, greatly upset Bolyai who took it as a severe blow.
    • It seems that Farkas Bolyai did not approve of Rozalia, was unhappy about his son's financial position, was unhappy that the family estate at Domald was not being properly cared for, and was unhappy that his son was damaging his good name for Farkas was a highly respected member of the community.
    • Certainly Bolyai continued to develop mathematical theories while he lived at Domald, but being isolated from the rest of the world of mathematics much of what he attempted was of little value.
    • His one major undertaking, to attempt to develop all of mathematics based on axiom systems, was begun in 1834, for he wrote the preface in that year, but he never completed the work.
    • What he did write concerned geometry and there are several ideas in this unpublished work which were ahead of their time such as notions of topological invariance.
    • In addition to his work on geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers.
    • Janos's paper was called Responsio and it was written to answer the question of whether the imaginary quantities used in geometry could be constructed.
    • In 1848 Bolyai discovered that Lobachevsky had published a similar piece of work in 1829.
    • Kagan writes [',' V F Kagan, The construction of non-Euclidean geometry by Lobachevsky, Gauss and Bolyai (Russian), Proc.
    • History of Science II (1948), 323-399.','14]:- .
    • The 'Comments' to the 'Geometrical Examinations' are more than a critical analysis of the work.
    • They express the thoughts and anxieties of Janos provoked by the perusal of the book.
    • They include his complaint that he was wronged, his suspicion that Lobachevsky did not exist at all, and that everything was the spiteful machinations of Gauss: it is the tragic lament of an ingenious geometrician who was aware of the significance of his discovery but failed to get support from the only person who could have appreciated his merits.
    • In spite of his mental agitation amidst which Janos put observations to paper, he preserved enough objectivity to highly appreciate the work of his rival.
    • In his comment to Theorem 35 he remarks that the proofs of Lobachevsky concerning spherical trigonometry bear the impress of genius and his work should be esteemed as a masterly achievement.
    • In 1852 Bolyai left Rozalia, whom he had married on 18 May 1849 believing that the law had now changed due to the Hungatian declaration of independance, and splitting with Rozalia at least had the advantage that relations with his father improved.
    • He gave up working on mathematics in his last years and instead tried to construct a theory of all knowledge.
    • Although he never published more than the few pages of the Appendix he left more than 20000 pages of manuscript of mathematical work when he died of pneumonia at the age of 57.
    • The Romanian University of Cluj which had been the King Ferdinand I University, was renamed Babeș University (after the Romanian natural scientist Victor Babeș).
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (27 books/articles) .
    • A Poster of Janos Bolyai .
    • Preface of Elemer Kiss Mathematical Gems from the Bolyai Chests .
    • History Topics: Non-Euclidean geometry .
    • History Topics: The development of group theory .
    • History Topics: An overview of the history of mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Bolyai.html .

  83. Sijue Wu biography
    • Even before the award of the Master's Degree, she had a paper published, namely Hilbert transforms for convex curves in Rn.She then went to the United States to undertake research.
    • This thesis is composed of three interrelated parts: w-Calderon-Zygmund operators, a wavelet characterization for weighted Hardy spaces, and the analytic dependence of minimal surfaces on their boundaries.
    • After the award of her doctorate, Wu was appointed as Courant Instructor at the Courant Institute, New York University.
    • She was a member at the Institute for Advanced Study at Princeton in the autumn of 1992 and was then she was appointed Assistant Professor at Northwestern University, holding this position for four years until 1996.
    • Her publications during this period included: A wavelet characterization for weighted Hardy spaces (1992); (with Italo Vecchi) On L1-vorticity for 2-D incompressible flow (1993); Analytic dependence of Riemann mappings for bounded domains and minimal surfaces (1993) and w-Calderon-Zygmund operators (1995).
    • After spending the year 1996-97 as a member of the Institute for Advanced Study at Princeton, she was appointed as Assistant Professor at the University of Iowa.
    • In 1997 she published the important paper Well-posedness in Sobolev spaces of the full water wave problem in 2-D.
    • Everyone is familiar with the motion of water waves in everyday experience, and there has been an extremely rich variety of phenomena observed in the motion of such waves.
    • However, the full equations governing the motion of the waves are notoriously difficult to work with because of the free boundary and the inherent nonlinearity, which are non-standard and non-local.
    • Although many approximate treatments, such as linear theory and shallow-water theory as well as numerical computations, have been used to explain many important phenomena, it is certainly of importance to study the solutions of the equations which include the effects neglected by approximate models.
    • The well-posedness of the fully nonlinear problem is one of the main mathematical problems in fluid dynamics.
    • Here, the motion of two-dimensional irrotational, incompressible, inviscid water waves under the influence of gravity is considered.
    • Promoted to Associate Professor at Iowa in 1998, Wu was appointed as an Associate Professor at the University of Maryland, College Park, in 1998.
    • Sijue Wu comes to us from the University of Iowa.
    • Her recent work concerns the full nonlinear water wave problem and the motion of general two-fluid flows.
    • At the 107th Annual Meeting of the American Mathematical Society in January 2001 in New Orleans, Wu was awarded the 2001 Satter Prize.
    • The Ruth Lyttle Satter Prize in Mathematics is awarded to Sijue Wu for her work on a long-standing problem in the water wave equation, in particular for the results in her papers (1) "Well-posedness in Sovolev spaces of the full water wave problem in 2-D" (1997); and (2) "Well-posedness in Sobolev spaces of the full water wave problem in 3-D" (1999).
    • Of the paper (2) Emmanuel Grenier writes:- .
    • In this very important paper the author investigates the motion of the interface of a 3D inviscid, incompressible, irrotational water wave, with an air region above a water region and surface tension zero.
    • In her response Wu thanked her teachers, friends, and colleagues, making special mention of her thesis advisor Ronald Coifman for the constant support he had given her and Lihe Wang for his friendship and his help.
    • Also in 2001 Wu received a Silver Morningside Medal at the International Congress of Chinese Mathematicians held in Taiwan in December:- .
    • for her establishment of local well-posedness of the water wave problems in a Sobolev class in arbitrary space dimensions.
    • In August 2002 Wu was an invited speaker at the International Congress of Mathematicians held in Beijing where she delivered the lecture Recent progress in mathematical analysis of vortex sheets.
    • She gave the following summary of her lecture:- .
    • We consider the motion of the interface separating two domains of the same fluid that move with different velocities along the tangential direction of the interface.
    • We assume that the fluids occupying the two domains are of constant densities that are equal, are inviscid, incompressible and irrotational, and that the surface tension is zero.
    • We discuss results on the existence and uniqueness of solutions for given data, the regularity of solutions, singularity formation and the nature of the solutions after the singularity formation time.
    • Her project Mathematical Analysis of Vortex Dynamics was described in an announcement of the award [',' 2002-2003 Radcliffe Institute Fellows, Sijue Wu, Mathematics, Mathematical Analysis of Vortex Dynamics.','2]:- .
    • Using harmonic analysis technique, she has established the local well-posedness of the full two- and three-dimensional waterwave problem.
    • As a Radcliffe fellow, Wu will continue her study of vortex sheet dynamics, a phenomenon that arises from the mixing of fluids, such as occurs during aircraft takeoffs.
    • A vortex sheet is the interface separating two domains of the same fluid across which the tangential component of the velocity field is discontinuous.
    • Achieving a better understanding of the motion of a vortex sheet requires proper mathematical modelling; Wu's long-term goal is to establish a successful model.
    • One outcome of this project, and of an NFS grant she was awarded for 2004-2009, was the paper Mathematical analysis of vortex sheets (2006).
    • Helena Nussenzveig Lopes begins a review of this paper by explaining what vortex sheets are:- .
    • Vortex sheets are an idealized model of flows undergoing intense shear.
    • Vortex sheets arise in a wide range of physical problems, and hence it is of fundamental importance to understand their evolution.
    • The Birkhoff-Rott equations provide a mathematical description of the evolution of a vortex sheet.
    • Wu was named Robert W and Lynne H Browne Professor of Mathematics at the University of Michigan and delivered her inaugural lecture Mathematical Analysis of Water Waves on 29 October 2008.
    • Finally, let us mention her recent important paper Almost global wellposedness of the 2-D full water wave problem (2009).
    • List of References (2 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Wu.html .

  84. Ivo Babuska biography
    • The German invasion of the west of the country in September 1938 was followed six months later by the whole country being taken over by Germany at the start of World War II.
    • Babuska was thirteen years old at this time and most of his secondary education, therefore, took place under German occupation.
    • by the Faculty of Civil Engineering of the Czech Technical University in 1951.
    • However, simultaneously with this research in engineering, Babuska was a mathematics student at the Central Mathematical Institute in Prague studying under Vladimir Knichal.
    • While Babuska was studying at the Institute, its name changed in 1953 to the Mathematical Institute of the Czechoslovak Academy of Sciences.
    • At the Mathematical Institute, in addition to Knichal, he was strongly influenced by Eduard Čech who was appointed Director of the Central Mathematical Institute in 1950, and Director of the Czechoslovak Academy of Sciences in 1952.
    • Given Babuska's training, coming first to engineering and, slightly later to mathematics, it is no surprise to see his publications being in the engineering area but become more slanted towards advanced mathematical techniques to solve engineering problems.
    • His first papers, all written in Czech, were Welding stresses and deformations (1952), Plane elasticity problem (1952), A contribution to the theoretical solution of welding stresses and some experimental results (1953), A contribution to one method of solution of the biharmonic problem (1954), Solution of the elastic problem of a half-plane loaded by a sequence of singular forces (1954), (with L Mejzlik) The stresses in a gravity dam on a soft bottom (1954), On plane biharmonic problems in regions with corners (1955), (with L Mejzlik) The method of finite differences for solving of problems of partial differential equations (1955), and Numerical solution of complete regular systems of linear algebraic equations and some applications in the theory of frameworks (1955).
    • His 1954 paper investigating stresses in a gravity dam was a direct consequence of a project that he led between 1953 and 1956 on using computational techniques to examine the technology involved in building the Orlik Dam on the Vltava River about 80 km from Prague.
    • This river, the longest in the Czech Republic, is a major source of hydroelectric power with several important dams creating artificial lakes.
    • The group, consisting of civil engineers, material scientists, mathematicians, and desk calculator operators, concentrated on the technology without artificial cooling, which is usually used to remove the heat created during the hardening of concrete.
    • In 1955, after the award of his doctorate, Babuska was appointed as head of the Department of Constructive Methods of Mathematical Analysis of the Mathematical Institute of the Czechoslovak Academy of Sciences.
    • In the same year, in collaboration with Karel Rektorys and Frantisek Vycichlo, he published his first book The mathematical theory of plane elasticity (Czech).
    • Frantisek Kroupa writes in a review of the original Czech edition:- .
    • The book is devoted to the application of the theory of functions of a complex variable to solving plane problems of the classical mathematical theory of elasticity (for static problems without the effect of body forces).
    • From the mathematical point of view it deals with the special method of solving a biharmonic equation for given boundary conditions.
    • The book gives and further develops some of the results of N I Muskelishvili and collaborators.
    • An original contribution is the axiomatic construction of the fundamentals of plane elasticity, the accuracy and generality of the mathematical procedures and some new numerical methods of solution.
    • In the following year, 1956, Babuska founded the journal Applications of Mathematics (Applikace Matematiky).
    • (the highest possible degree in Czechoslovakia, equivalent to a D.Sc.) by the Czechoslovak Academy of Sciences.
    • His next important book, published in collaboration with Milan Prager and Emil Vitasek in 1964, was Numerical Solution of Differential Equations (Czech).
    • This book shows both how much mathematics has to contribute to computing when competent mathematicians actually look at what computing is (rather than treating it as if it were a branch of mathematics), and how much they can miss the current temper of computing.
    • They make frequent experimental verifications of their theories, thus showing that they regard computing as a science whose results are to be accepted or rejected by the final authority of experience.
    • The Communists had seized control of the country in 1948 and it was under strong Soviet influence over the following years.
    • Mathematics was allowed to develop without interference, however, and the applied and computational methods developed by Babuska found favour.
    • Babuska had just been appointed as a professor at the Charles University of Prague but, given the political situation, he travelled with his family to the United States where he spent a year as a visiting professor at the Institute for Fluid Dynamics and Applied Mathematics at the University of Maryland at College Park.
    • He was given a permanent appointment as a professor at the University of Maryland in the following year and he held this position until 1995.
    • He was then appointed Professor of Aerospace Engineering and Engineering Mechanics, Professor of Mathematics, and appointed to the Robert Trull Chair in Engineering at the University of Texas at Austin.
    • Although now over 85 years of age, he continues to hold these positions.
    • The authors of [',' L Demkowicz, B Guo, J Osborn and M Vogelius, Ivo Babuska - mathematician and engineer, Comput.
    • During his 27 year career at the University of Maryland, Professor Babuska established himself as the unquestionable leader of the international finite element community.
    • In his landmark paper in 1971, Ivo introduced the discrete inf-sup condition, generalizing the results of J Cea and R Varga, and setting the theoretical framework for stability and convergence analysis of arbitrary linear problems.
    • Three years later, F Brezzi reinforced the formalism for problems with constraints, and the name of the famous discrete BB condition was coined.
    • Ivo has had a unique ability to foresee the development of the field of finite elements.
    • In a landmark paper in 1979 with B Kellogg and J Pitkaranta, the effect of h-adaptivity on the convergence rates for problems with singularities was explained.
    • In the late seventies, Barna Szabo convinced Ivo to reexamine the then established concept of higher order methods, and the p-version of the Finite Element Method was born.
    • The p-method turned out to be much less sensitive to incompressibility constraints (work with M Vogelius) and locking effects in the analysis of thin-walled structures.
    • The monograph on the p-method with B Szabo (1991) reaches far outside of the constraints of the mathematical community and has become a standard reference for engineers practicing higher order methods.
    • Our purpose in writing this book is to introduce the finite element method to engineers and engineering students in the context of the engineering decision-making process.
    • The principles that guide the construction of mathematical models are described and illustrated by examples.
    • The reviewer [',' L R S, Review: Finite Element Analysis by Barna Szabo and Ivo Babuska, Mathematics of Computation 60 (201) (1993), 432-433.','5] writes:- .
    • Numerous books on the finite element method with a variety of objectives have appeared recently, so many that it would be quite lengthy to compare even a representative number of them.
    • The present book has extensive detail with regard to examples, and its coverage of topics in linear elasticity is exhaustive.
    • Both of these are essential for the book to be successful with an engineering audience.
    • It is a very nice, and somewhat unique, blend of theory and engineering practice.
    • The reliability of a given numerical approximation is one essential task in applied science and engineering.
    • Here, two leading scientists devote six chapters on eight hundred pages to it and fix the state of the art of rigorous 'a posteriori' finite element error analysis.
    • Among the many other services to mathematics which Babuska has given, we mention the many journals which have benefited by his accepting a position on their editorial board: Communications in Applied Analysis; Communications in Numerical Methods in Engineering; Computer & Mathematics; Computer Methods in Applied Mechanics and Engineering; Computers and Structures; Communications in Applied Analysis; International Journal for Numerical Methods in Engineering; Modelling and Scientific Computing; Numerical Mathematics - A Journal of Chinese Universities; Numerical Methods for Partial Differential Equations; and Siberian Journal of Computer Mathematics.
    • Babuska has received many awards for his contributions: the Czechoslovak State Prize for Mathematics (1968); the Humbolt Senior US Scientist Award of Federal Republic of Germany (1976); the Medal for Merit in the Development of Mechanics of the Czech Society for Mechanics (1993); and the George David Birkhoff Prize in Applied Mathematics awarded jointly by the American Mathematical Society and the Society for Industrial and Applied Mathematics (1994):- .
    • for important contributions to the reliability of finite element methods, the development of a general framework for finite element error estimation, and the development of p and h-p finite element methods..
    • extraordinary contributions and the breadth and depth of his work, and their importance to the broad fields of computational mechanics .
    • Babuska ended his Acceptance Speech, which examined the legacy of von Neumann, with these words:- .
    • Mr President, I would like to thank you again for the great honour that has been bestowed upon me and to express my opinion that nearly 40 years after the death of von Neumann, a towering scientific figure of the 20th Century, we, who work in computational mechanics, can still learn tremendously from the legacy, ideas and philosophy of John von Neumann.
    • The Union of Czech Mathematicians and Physicists made him an honorary member in 1996 and, in the same year, presented him with their Commemorative Medal.
    • He received the Bolzano Medal from the Czech Academy of Sciences in 1997 and the Honorary Medal "De Scientia et Humanitate Optime Meritis" from the same Academy in 2006.
    • He has received honorary degrees from the University of Westminster, London (1996), Brunel University, London (1996), Charles University, Prague (1997), and the Helsinki University of Technology (2000).
    • He was elected a fellow of the International Association of Computational Mechanics in 2002 and of the World Innovation Foundation in 2004.
    • He was elected a member of the European Academy of Sciences in 2003 and of the National Academy of Engineering in 2005.
    • List of References (8 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Babuska.html .

  85. Louis Castel biography
    • Originally from Bearn, a mountainous region of south west France, Guillaume Castel practiced as a surgeon in Montpellier.
    • He studied mathematics and philosophy in the Order and decided that he wished to go on a mission to China where his Order was very active following an edict of toleration which had been proclaimed in 1692.
    • Jesuit communities had been established in many cities of south and central China, and a church had been built in Peking under Imperial patronage.
    • Castel, however, was not allowed to be part of a Chinese mission since his superiors decided that his health was not sufficiently good for such a strenuous undertaking and he remained in Toulouse.
    • Indeed late in 1720 Castel did go to Paris and taught physics and mathematics at the Jesuit school in the rue Saint-Jaques which was later to become the Lycee Louis-le-Grand.
    • Castel was never to leave Paris again except for a short visit to the South of France near the end of his life.
    • Immediately on arriving in Paris, Castel was appointed as an associate editor of the Journal de Trevoux, remaining on the editorial board of this monthly publication until 1745.
    • Castel was a strong opponent of Newton's views on science and he made these views clear in his two volume work Traite de physique sur la pesanteur universelle des corps Ⓣ (1724).
    • On nationalistic grounds he supported the views of Descartes and his opposition delayed the acceptance of Newton's theories in France.
    • He gave his own alternative system to replace Newton's system but it is of little importance [',' A R Desautels, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • It was an attempt to harmonise philosophy, scientific curiosity, and religious dogma by methods of rationalism.
    • Science, he believed, had to be accessible to everyone and so both higher mathematics and costly experimentations had to be excluded from scientific methods.
    • the experiments capable of perfecting physics, ought to be easy to make and to repeat at any time, and almost by everyone.
    • He published Vrai systeme de physique de M Isaac Newton in 1743 in which he claimed (with some degree of exaggeration!) that:- .
    • in order to make these experiments on the refraction of light correctly one must be a millionaire.
    • Castel's ideas on the accessibility of science were also held by others such as Diderot, a friend of Castel.
    • Castel's system to replace the theories of Newton did not bring him fame.
    • In the November 1725 issue of Mercure de France he set out his ideas for an instrument, the clavecin oculair, which made colours and musical tones correspond.
    • Two articles in the Journal de Trevoux in 1735, namely Nouvelles experiences d'optique et d'acoustique Ⓣ and L'optique des couleurs fondee sur les simples observations Ⓣ, took the idea further describing an instrument to accomplish the colour-tone correspondence, namely the ocular harpsichord [',' A R Desautels, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • By 1742 the fame of Castel and of his invention had reached as far as St Petersburg and had been brought to the attention of the empress.
    • the instrument was completed in July 1754, and on 21 December of the same year Castel gave a private demonstration of it before fifty guests.
    • Castel was elected to the Royal Society of London in 1730.
    • He was also elected to the Bordeaux Academy (1746), the Academy of Rouen (1748) and the Academy of Lyon (1748).
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (2 books/articles) .
    • 1.nFellow of the Royal Societyn1730 .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Castel.html .

  86. Pythagoras (about 569 BC-about 475 BC)
    • Pythagoras of Samos .
    • Pythagoras of Samos is often described as the first pure mathematician.
    • He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements.
    • Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings.
    • The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.
    • We do have details of Pythagoras's life from early biographies which use important original sources yet are written by authors who attribute divine powers to him, and whose aim was to present him as a god-like figure.
    • What we present below is an attempt to collect together the most reliable sources to reconstruct an account of Pythagoras's life.
    • There is fairly good agreement on the main events of his life but most of the dates are disputed with different scholars giving dates which differ by 20 years.
    • Some historians treat all this information as merely legends but, even if the reader treats it in this way, being such an early record it is of historical importance.
    • Pythagoras's father was Mnesarchus ([',' Porphyry, Vita Pythagorae (Leipzig, 1886),','12] and [',' Porphyry, Life of Pythagoras in M Hadas and M Smith, Heroes and Gods (London, 1965)..
    • ','13]), while his mother was Pythais [',' Iamblichus, Life of Pythagoras (translated into English by T Taylor) (London, 1818).','8] and she was a native of Samos.
    • Mnesarchus was a merchant who came from Tyre, and there is a story ([',' Porphyry, Vita Pythagorae (Leipzig, 1886),','12] and [',' Porphyry, Life of Pythagoras in M Hadas and M Smith, Heroes and Gods (London, 1965)..
    • ','13]) that he brought corn to Samos at a time of famine and was granted citizenship of Samos as a mark of gratitude.
    • There are accounts of Mnesarchus returning to Tyre with Pythagoras and that he was taught there by the Chaldaeans and the learned men of Syria.
    • Little is known of Pythagoras's childhood.
    • All accounts of his physical appearance are likely to be fictitious except the description of a striking birthmark which Pythagoras had on his thigh.
    • One of the most important was Pherekydes who many describe as the teacher of Pythagoras.
    • In [',' Iamblichus, Life of Pythagoras (translated into English by T Taylor) (London, 1818).','8] it is said that Pythagoras visited Thales in Miletus when he was between 18 and 20 years old.
    • However he did contribute to Pythagoras's interest in mathematics and astronomy, and advised him to travel to Egypt to learn more of these subjects.
    • Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras's own views.
    • This happened a few years after the tyrant Polycrates seized control of the city of Samos.
    • There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed [',' Diogenes Laertius, Lives of eminent philosophers (New York, 1925).','5] that Pythagoras went to Egypt with a letter of introduction written by Polycrates.
    • The accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests.
    • According to Porphyry ([',' Porphyry, Vita Pythagorae (Leipzig, 1886),','12] and [',' Porphyry, Life of Pythagoras in M Hadas and M Smith, Heroes and Gods (London, 1965)..
    • It is not difficult to relate many of Pythagoras's beliefs, ones he would later impose on the society that he set up in Italy, to the customs that he came across in Egypt.
    • For example the secrecy of the Egyptian priests, their refusal to eat beans, their refusal to wear even cloths made from animal skins, and their striving for purity were all customs that Pythagoras would later adopt.
    • Porphyry in [',' Porphyry, Vita Pythagorae (Leipzig, 1886),','12] and [',' Porphyry, Life of Pythagoras in M Hadas and M Smith, Heroes and Gods (London, 1965)..
    • In 525 BC Cambyses II, the king of Persia, invaded Egypt.
    • After Cambyses had won the Battle of Pelusium in the Nile Delta and had captured Heliopolis and Memphis, Egyptian resistance collapsed.
    • Iamblichus writes that Pythagoras (see [',' Iamblichus, Life of Pythagoras (translated into English by T Taylor) (London, 1818).','8]):- .
    • was transported by the followers of Cambyses as a prisoner of war.
    • and was instructed in their sacred rites and learnt about a very mystical worship of the gods.
    • He also reached the acme of perfection in arithmetic and music and the other mathematical sciences taught by the Babylonians..
    • Polycrates had been killed in about 522 BC and Cambyses died in the summer of 522 BC, either by committing suicide or as the result of an accident.
    • The deaths of these rulers may have been a factor in Pythagoras's return to Samos but it is nowhere explained how Pythagoras obtained his freedom.
    • Darius of Persia had taken control of Samos after Polycrates' death and he would have controlled the island on Pythagoras's return.
    • This conflicts with the accounts of Porphyry and Diogenes Laertius who state that Polycrates was still in control of Samos when Pythagoras returned there.
    • Pythagoras made a journey to Crete shortly after his return to Samos to study the system of laws there.
    • Iamblichus [',' Iamblichus, Life of Pythagoras (translated into English by T Taylor) (London, 1818).','8] writes in the third century AD that:- .
    • he formed a school in the city [of Samos], the 'semicircle' of Pythagoras, which is known by that name even today, in which the Samians hold political meetings.
    • Outside the city he made a cave the private site of his own philosophical teaching, spending most of the night and daytime there and doing research into the uses of mathematics..
    • Iamblichus [',' Iamblichus, Life of Pythagoras (translated into English by T Taylor) (London, 1818).','8] gives some reasons for him leaving.
    • he tried to use his symbolic method of teaching which was similar in all respects to the lessons he had learnt in Egypt.
    • Pythagoras was dragged into all sorts of diplomatic missions by his fellow citizens and forced to participate in public affairs.
    • Pythagoras founded a philosophical and religious school in Croton (now Crotone, on the east of the heel of southern Italy) that had many followers.
    • Pythagoras was the head of the society with an inner circle of followers known as mathematikoi.
    • (5) that all brothers of the order should observe strict loyalty and secrecy.
    • Both men and women were permitted to become members of the Society, in fact several later women Pythagoreans became famous philosophers.
    • The outer circle of the Society were known as the akousmatics and they lived in their own houses, only coming to the Society during the day.
    • Of Pythagoras's actual work nothing is known.
    • His school practised secrecy and communalism making it hard to distinguish between the work of Pythagoras and that of his followers.
    • Certainly his school made outstanding contributions to mathematics, and it is possible to be fairly certain about some of Pythagoras's mathematical contributions.
    • First we should be clear in what sense Pythagoras and the mathematikoi were studying mathematics.
    • They were not acting as a mathematics research group does in a modern university or other institution.
    • Rather Pythagoras was interested in the principles of mathematics, the concept of number, the concept of a triangle or other mathematical figure and the abstract idea of a proof.
    • As Brumbaugh writes in [',' R S Brumbaugh, The philosophers of Greece (Albany, N.Y., 1981).','3]:- .
    • It is hard for us today, familiar as we are with pure mathematical abstraction and with the mental act of generalisation, to appreciate the originality of this Pythagorean contribution.
    • There is another step to see that the abstract notion of 2 is itself a thing, in some sense every bit as real as a ship or a house.
    • having been brought up in the study of mathematics, thought that things are numbers ..
    • This generalisation stemmed from Pythagoras's observations in music, mathematics and astronomy.
    • Pythagoras noticed that vibrating strings produce harmonious tones when the ratios of the lengths of the strings are whole numbers, and that these ratios could be extended to other instruments.
    • In fact Pythagoras made remarkable contributions to the mathematical theory of music.
    • Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc.
    • However to Pythagoras numbers had personalities which we hardly recognise as mathematics today [',' R S Brumbaugh, The philosophers of Greece (Albany, N.Y., 1981).','3]:- .
    • This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry.
    • Ten was the very best number: it contained in itself the first four integers - one, two, three, and four [',' K von Fritz, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Of course today we particularly remember Pythagoras for his famous geometry theorem.
    • Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [',' T L Heath, A history of Greek mathematics 1 (Oxford, 1931).','7]):- .
    • After [Thales, etc.] Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.
    • Again Proclus, writing of geometry, said:- .
    • I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.
    • Heath [',' T L Heath, A history of Greek mathematics 1 (Oxford, 1931).','7] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.
    • (i) The sum of the angles of a triangle is equal to two right angles.
    • Also the Pythagoreans knew the generalisation which states that a polygon with n sides has sum of interior angles 2n - 4 right angles and sum of exterior angles equal to four right angles.
    • (ii) The theorem of Pythagoras - for a right angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.
    • We should note here that to Pythagoras the square on the hypotenuse would certainly not be thought of as a number multiplied by itself, but rather as a geometrical square constructed on the side.
    • To say that the sum of two squares is equal to a third square meant that the two squares could be cut up and reassembled to form a square identical to the third square.
    • (iii) Constructing figures of a given area and geometrical algebra.
    • (iv) The discovery of irrationals.
    • This went against Pythagoras's philosophy the all things are numbers, since by a number he meant the ratio of two whole numbers.
    • However, because of his belief that all things are numbers it would be a natural task to try to prove that the hypotenuse of an isosceles right angled triangle had a length corresponding to a number.
    • (vi) In astronomy Pythagoras taught that the Earth was a sphere at the centre of the Universe.
    • He also recognised that the orbit of the Moon was inclined to the equator of the Earth and he was one of the first to realise that Venus as an evening star was the same planet as Venus as a morning star.
    • the dependence of the dynamics of world structure on the interaction of contraries, or pairs of opposites; the viewing of the soul as a self-moving number experiencing a form of metempsychosis, or successive reincarnation in different species until its eventual purification (particularly through the intellectual life of the ethically rigorous Pythagoreans); and the understanding ..
    • .that all existing objects were fundamentally composed of form and not of material substance.
    • identified the brain as the locus of the soul; and prescribed certain secret cultic practices.
    • In [',' R S Brumbaugh, The philosophers of Greece (Albany, N.Y., 1981).','3] their practical ethics are also described:- .
    • Pythagoras's Society at Croton was not unaffected by political events despite his desire to stay out of politics.
    • He remained there for a few months until the death of his friend and teacher and then returned to Croton.
    • Pythagoras escaped to Metapontium and the most authors say he died there, some claiming that he committed suicide because of the attack on his Society.
    • Iamblichus in [',' Iamblichus, Life of Pythagoras (translated into English by T Taylor) (London, 1818).','8] quotes one version of events:- .
    • Cylon, a Crotoniate and leading citizen by birth, fame and riches, but otherwise a difficult, violent, disturbing and tyrannically disposed man, eagerly desired to participate in the Pythagorean way of life.
    • He approached Pythagoras, then an old man, but was rejected because of the character defects just described.
    • Because of this Pythagoras left for Metapontium and there is said to have ended his days.
    • Gorman [',' P Gorman, Pythagoras, a life (1979).','6] argues that this is a strong reason to believe that Pythagoras returned to Croton and quotes other evidence such as the widely reported age of Pythagoras as around 100 at the time of his death and the fact that many sources say that Pythagoras taught Empedokles to claim that he must have lived well after 480 BC.
    • The evidence is unclear as to when and where the death of Pythagoras occurred.
    • Certainly the Pythagorean Society expanded rapidly after 500 BC, became political in nature and also spilt into a number of factions.
    • Its meeting houses were everywhere sacked and burned; mention is made in particular of "the house of Milo" in Croton, where 50 or 60 Pythagoreans were surprised and slain.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (27 books/articles) .
    • A Poster of Pythagoras .
    • An entry in The Mathematical Gazetteer of the British Isles .
    • Astronomy: The Structure of the Solar System .
    • History Topics: Greek Astronomy .
    • History Topics: Perfect numbers .
    • History Topics: Prime numbers .
    • History Topics: The Indian Sulbasutras .
    • History Topics: The history of cartography .
    • History Topics: Pythagoras's theorem in Babylonian mathematics .
    • History Topics: The Golden ratio .
    • History Topics: Mathematics and Architecture .
    • History Topics: Infinity .
    • History Topics: Christianity and the Mathematical Sciences - the Heliocentric Hypothesis .
    • History Topics: A history of time: Classical time .
    • History Topics: Mathematics and the physical world .
    • History Topics: Overview of Chinese mathematics .
    • History Topics: The Ten Mathematical Classics .
    • History Topics: Nine Chapters on the Mathematical Art .
    • History Topics: The real numbers: Pythagoras to Stevin .
    • Dictionary of Scientific Biography .
    • Internet Encyclopedia of Philosophy .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Pythagoras.html .

  87. William Horner (1786-1837)
    • John Wesley, a founder of Methodism, encouraged William Horner senior to come to England and join the Methodist Society as a minister.
    • At this time Methodists were members of the Church of England, the break coming later in 1795.
    • William junior, the subject of this biography, was educated at Kingswood School Bristol.
    • At the almost unbelievable age of 14 he became an assistant master at Kingswood school in 1800 and headmaster four years later.
    • Horner is largely remembered only for the method, Horner's method, of solving algebraic equations ascribed to him by Augustus De Morgan and others.
    • He published on the subject in the Philosophical Transactions of the Royal Society of London in 1819, submitting his article on 1 July.
    • But Fuller [',' A T Fuller, Horner versus Holdred: an episode in the history of root computation, Historia Math.
    • However, he was apparently of an eccentric and obsessive nature ..
    • Horner made other mathematical contributions, however, publishing a series of papers on transforming and solving algebraic equations, and he also applied similar techniques to functional equations.
    • It is also worth noting that he gave a solution to what has come to be known as the "butterfly problem" which appeared in The Gentleman's Diary for 1815 [',' L Bankoff, The metamorphoses of the Butterfly Problem, Math.
    • Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn.
    • Prove that M is also the midpoint of XY.
    • The butterfly problem, whose name becomes clear on looking at the figure, has led to a wide range of interesting solutions.
    • We mention that Horner published Natural magic, a familiar exposition of a forgotten fact in optics (1832) and the modern invention of the zoetrope, under the name Daedaleum in 1834, has been attributed to him.
    • The zoetrope is a device which gives the impression of a moving picture by displaying a sequence of drawings or photographs.
    • Neither the date of Horner's marriage nor the name of the woman he married are known, but it is recorded that they had several children.
    • After Horner died in his home in Grosvenor Place, Bath, of a stroke in 1837, one of his sons, also called William, carried on running the school The Seminary in Bath.
    • List of References (7 books/articles) .
    • History Topics: Arabic mathematics : forgotten brilliance? .
    • History Topics: Overview of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • Dictionary of National Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Horner.html .

  88. Ibn al-Haytham biography
    • Ibn al-Haytham is sometimes called al-Basri, meaning from the city of Basra in Iraq, and sometimes called al-Misri, meaning that he came from Egypt.
    • He is often known as Alhazen which is the Latinised version of his first name "al-Hasan".
    • In particular this name occurs in the naming of the problem for which he is best remembered, namely Alhazen's problem: .
    • Given a light source and a spherical mirror, find the point on the mirror where the light will be reflected to the eye of an observer.
    • In contrast to our lack of knowledge of the lives of many of the Arabic mathematicians, we have quite a number of details of ibn al-Haytham's life.
    • It is worth commenting that an autobiography written by ibn al-Haytham in 1027 survives, but it says nothing of the events his life and concentrates on his intellectual development.
    • Since the main events that we know of in ibn al-Haytham's life involve his time in Egypt, we should set the scene regarding that country.
    • The Fatimid political and religious dynasty took its name from Fatimah, the daughter of the Prophet Muhammad.
    • The Fatimids headed a religious movement dedicated to taking over the whole of the political and religious world of Islam.
    • The Fatimid caliphs ruled North Africa and Sicily during the first half of the 10th century, but after a number of unsuccessful attempts to defeat Egypt, they began a major advance into that country in 969 conquering the Nile Valley.
    • They founded the city of Cairo as the capital of their new empire.
    • We know little of ibn al-Haytham's years in Basra.
    • In his autobiography he explains how, as a youth, he thought about the conflicting religious views of the various religious movements and came to the conclusion that none of them represented the truth.
    • It appears that he did not devote himself to the study of mathematics and other academic topics at a young age but trained for what might be best described as a civil service job.
    • However, ibn al-Haytham became increasingly unhappy with his deep studies of religion and made a decision to devote himself entirely to a study of science which he found most clearly described in the writings of Aristotle.
    • Having made this decision, ibn al-Haytham kept to it for the rest of his life devoting all his energies to mathematics, physics, and other sciences.
    • Al-Hakim was the second of the Fatimid caliphs to begin his reign in Egypt; al-Aziz was the first of the Fatimid caliphs to do so.
    • Al-Aziz became Caliph in 975 on the death of his father al-Mu'izz.
    • For most of his 20 year reign he worked towards this aim.
    • Al-Hakim, despite being a cruel leader who murdered his enemies, was a patron of the sciences employing top quality scientists such as the astronomer ibn Yunus.
    • His support for science may have been partly because of his interest in astrology.
    • Al-Hakim was highly eccentric, for example he ordered the sacking of the city of al-Fustat, he ordered the killing of all dogs since their barking annoyed him, and he banned certain vegetables and shellfish.
    • However al-Hakim kept astronomical instruments in his house overlooking Cairo and built up a library which was only second in importance to that of the House of Wisdom over 150 years earlier.
    • Our knowledge of ibn al-Haytham's interaction with al-Hakim comes from a number of sources, the most important of which is the writings of al-Qifti.
    • We are told that al-Hakim learnt of a proposal by ibn al-Haytham to regulate the flow of water down the Nile.
    • However, as the team travelled further and further up the Nile, ibn al-Haytham realised that his idea to regulate the flow of water with large constructions would not work.
    • According to al-Qifti, ibn al-Haytham lived for the rest of his life near the Azhar Mosque in Cairo writing mathematics texts, teaching and making money by copying texts.
    • Since the Fatimids founded the University of Al-Azhar based on this mosque in 970, ibn al-Haytham must have been associated with this centre of learning.
    • A different report says that after failing in his mission to regulate the Nile, ibn al-Haytham fled from Egypt to Syria where he spent the rest of his life.
    • One further complication is the title of a work ibn al-Haytham wrote in 1027 which is entitled Ibn al-Haytham's answer to a geometrical question addressed to him in Baghdad.
    • Several different explanations are possible, the simplest of which being that he visited Baghdad for a short time before returning to Egypt.
    • He may also have spent some time in Syria which would partly explain the other version of the story.
    • He seems to have written around 92 works of which, remarkably, over 55 have survived.
    • The main topics on which he wrote were optics, including a theory of light and a theory of vision, astronomy, and mathematics, including geometry and number theory.
    • We will give at least an indication of his contributions to these areas.
    • The previous major work on optics had been Ptolemy's Almagest Ⓣ and although ibn al-Haytham's work did not have an influence to equal that of Ptolemy's, nevertheless it must be regarded as the next major contribution to the field.
    • Also in Book I, ibn al-Haytham makes it clear that his investigation of light will be based on experimental evidence rather than on abstract theory.
    • He notes that light is the same irrespective of the source and gives the examples of sunlight, light from a fire, or light reflected from a mirror which are all of the same nature.
    • He gives the first correct explanation of vision, showing that light is reflected from an object into the eye.
    • Most of the rest of Book I is devoted to the structure of the eye but here his explanations are necessarily in error since he does not have the concept of a lens which is necessary to understand the way the eye functions.
    • His studies of optics did led him, however, to propose the use of a camera obscura, and he was the first person to mention it.
    • Book II of the Optics discusses visual perception while Book III examines conditions necessary for good vision and how errors in vision are caused.
    • From a mathematical point of view Book IV is one of the most important since it discusses the theory of reflection.
    • Ibn al-Haytham gave [',' A I Sabra, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • experimental proof of the specular reflection of accidental as well as essential light, a complete formulation of the laws of reflection, and a description of the construction and use of a copper instrument for measuring reflections from plane, spherical, cylindrical, and conical mirrors, whether convex or concave.
    • Alhazen's problem, quoted near the beginning of this article, appears in Book V.
    • The paper [',' B A Rosenfeld, ’Geometric trigonometry’ in treatises of al-Khwarizmi, al-Mahani and ibn al-Haytham, in Vestigia mathematica (Amsterdam, 1993), 305-308.','36] gives a detailed description of six geometrical lemmas used by ibn al-Haytham in solving this problem.
    • To find the point of reflection on the surface of a spherical mirror, convex or concave, given the two points related to one another as eye and visible object.
    • Book VI of the Optics examines errors in vision due to reflection while the final book, Book VII, examines refraction [',' A I Sabra, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • Ibn al-Haytham does not give the impression that he was seeking a law which he failed to discover; but his "explanation" of refraction certainly forms part of the history of the formulation of the refraction law.
    • Ibn al-Haytham's study of refraction led him to propose that the atmosphere had a finite depth of about 15 km.
    • He explained twilight by refraction of sunlight once the Sun was less than 19° below the horizon.
    • Abu al-Qasim ibn Madan was an astronomer who proposed questions to ibn al-Haytham, raising doubts about some of Ptolemy's explanations of physical phenomena.
    • Ibn al-Haytham wrote a treatise Solution of doubts in which he gives his answers to these questions.
    • What should we think of Ptolemy's account in "Almagest" Ⓣ I.3 concerning the visible enlargement of celestial magnitudes (the stars and their mutual distances) on the horizon? Is the explanation apparently implied by this account correct, and if so, under what physical conditions? How should we understand the analogy Ptolemy draws in the same place between this celestial phenomenon and the apparent magnification of objects seen in water? ..
    • In Al-Shukuk ala Batlamyus (Doubts concerning Ptolemy), ibn al-Haytham is critical of Ptolemy's ideas yet in a popular work the Configuration, intended for the layman, ibn al-Haytham completely accepts Ptolemy's views without question.
    • One of the mathematical problems which ibn al-Haytham attacked was the problem of squaring the circle.
    • He wrote a work on the area of lunes, crescents formed from two intersecting circles, (see for example [',' B Steffens, Ibn al-Haytham: First Scientist (Greensboro, North Carolina, USA, 2007).','10]) and then wrote the first of two treatises on squaring the circle using lunes (see [',' A Al-Ayib, Precise measurements in ibn al-Haytham’s treatise on geography (Arabic), in Deuxieme Colloque Maghrebin sur l’Histoire des Mathematiques Arabes (Tunis, 1990), A68-A86, 202.','14]).
    • In Opuscula ibn al-Haytham considers the solution of a system of congruences.
    • In his own words (using the translation in [',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','7]):- .
    • Ibn al-Haytham gives two methods of solution:- .
    • The problem is indeterminate, that is it admits of many solutions.
    • One of them is the canonical method: we multiply the numbers mentioned that divide the number sought by each other; we add one to the product; this is the number sought.
    • Here ibn al-Haytham gives a general method of solution which, in the special case, gives the solution (7 - 1)! + 1.
    • Using Wilson's theorem, this is divisible by 7 and it clearly leaves a remainder of 1 when divided by 2, 3, 4, 5, and 6.
    • Ibn al-Haytham's second method gives all the solutions to systems of congruences of the type stated (which of course is a special case of the Chinese Remainder Theorem).
    • The converse of this result, namely that every even perfect number is of the form 2k-1(2k - 1) where 2k - 1 is prime, was proved by Euler.
    • Rashed ([',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','7], [',' R Rashed, Entre arithmetique et algebre: Recherches sur l’histoire des mathematiques arabes (Paris, 1984).','8] or [',' R Rashed, L’analyse et la synthese selon ibn al-Haytham, in Mathematiques et philosophie de l’antiquite a l’age classique (Paris, 1991), 131-162.','27]) claims that ibn al-Haytham was the first to state this converse (although the statement does not appear explicitly in ibn al-Haytham's work).
    • Rashed examines ibn al-Haytham's attempt to prove it in Analysis and synthesis which, as Rashed points out, is not entirely successful [',' R Rashed, The development of Arabic mathematics : between arithmetic and algebra (London, 1994).','7]:- .
    • But this partial failure should not eclipse the essential: a deliberate attempt to characterise the set of perfect numbers.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (51 books/articles) .
    • A Poster of Ibn al-Haytham .
    • History Topics: Squaring the circle .
    • History Topics: Perfect numbers .
    • History Topics: Arabic mathematics : forgotten brilliance? .
    • History Topics: Light through the ages: Ancient Greece to Maxwell .
    • History Topics: Mathematics and art - perspective .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Haytham.html .

  89. Tadashi Nakayama biography
    • Tadashi Nakayama's father was a leading scholar of Chinese classics.
    • However, perhaps even more of an influence on Nakayama than Takagi was his student Kenjiro Shoda who had studied in Germany with Issai Schur in Berlin and with Emmy Noether in Gottingen.
    • Shoda had been appointed as a professor in the Faculty of Science at Osaka University in 1933 so Nakayama's appointment there allowed them to easily continue their work together.
    • [After Nakayama's early death, many felt that this doctor was to blame for covering up his tuberculosis at a young age.] It was in September 1937 that Nakayama arrived in Princeton and there he met a number of leading algebraists such as Hermann Weyl, Emil Artin who had just emigrated to the United States, and Claude Chevalley who arrived in Princeton in 1938.
    • Richard Brauer was by this time a professor at Toronto and he invited Nakayama to make two research visits to Toronto during his time in the United States, When he visited Brauer, he become inspired to work on group representations, publishing articles such as Some studies on regular representations, induced representations and modular representations (1938) and A remark on representations of groups (1938).
    • In 1939 Nakayama published the first part of his paper On Frobeniusean Algebras in the Annals of Mathematics.
    • In 1941 he submitted the two parts of the paper On Frobeniusean Algebras (the second part was published in the Annals of Mathematics in 1941) for the degree of Rigakuhakushi (doctorate) which was conferred on him in that year.
    • During the difficult times of World War II he continued his pioneering works in mathematics.
    • Together with Azumaya, Nakayama worked on the representation theory of algebras, particularly Frobenius algebras.
    • Theory of rings in Japanese in 1954.
    • is devoted to a systematic presentation of the theory of associative rings, a subject which has developed remarkably since the 1940's.
    • The first part, "general structural theory", written by the first author [Nakayama], consists of chapters 1-12 and is mainly concerned with the theory of rings without finiteness assumptions.
    • The second part "algebras and representation theory", written by the second author [Azumaya], consists of chapters 13-17 and is mainly devoted to the theory of algebras and their representations.
    • The first part of the book, written by Nakayama on the structure of rings, is similar to the famous text by Nathan Jacobson.
    • Although the second part was written by Azumaya, it included much of their joint work on the representation theory of algebras.
    • In 1948 and 1949 he was in the United States, at the University of Illinois, and, during 1953 and 1955, he spent time at Hamburg University in Germany, and spent a second period at the Institute for Advanced Study at Princeton.
    • For example, in 1947 he received (with his collaborator Goro Azumaya) a Chubu Nippon Bunka Sho in recognition of his research in the theory of infinite dimensional algebras.
    • It was for his research on the theory of rings and representations that the Japan Academy of Science awarded him a Gakushi-in shou (Japan Academy Prize) in 1953 and, ten years later, he was elected a member of the Japan Academy.
    • This level of productivity is all the more remarkable when one realises that he achieved this despite severe health problems.
    • As the tuberculosis became worse, he refused to give up his work and continued to come to the mathematics department at Nagoya University.
    • He was so short of breath that chairs were placed on each floor so the he could rest while moving around the building.
    • Even when he was eventually confined to bed he did not give up mathematics but read Grothendieck's work on algebraic geometry.
    • He continued to publish research up to the time of his death, his last paper Class group of cohomologically trivial modules and cyclotomic ideals appearing in the journal Acta Arithmetica in 1964, the year in which he died from tuberculosis.
    • In addition to his scientific and educational activities great importance must be attached to the indefatigable labors which he expended for the development of the Mathematical Institute of the Nagoya University.
    • Then he was one of the founders and the editor-in-chief of the 'Nagoya Mathematical Journal'.
    • He was also an always enthusiastic and active member of the Mathematical Society of Japan, under whose auspices he often lectured in many towns of this country.
    • His widespread activity also included his work as the cooperating editor of the 'Proceedings' of the Mathematical Society of Japan and 'Acta Arithmetica', and as a reviewer for the 'Mathematical Reviews' and for the 'Zentralblatt fur Mathematik'.
    • He was a man of rare nobility of mind and great kindness whose life was consumed in incessant labors.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Nakayama.html .

  90. Paolo Ruffini (1765-1822)
    • Died: 10 May 1822 in Modena, Duchy of Modena (now Italy) .
    • As a young child Paolo was [',' R G Ayoub, Paolo Ruffini’s Contributions to the Quintic, Archive for History of Exact Science 23 (1980), 253-277.','4]:- .
    • of a mystical temperament and appeared to be destined for the priesthood..
    • The family moved to Reggio, near Modena in the Emilia-Romagna region of northern Italy, when Paolo was a teenager.
    • He entered the University of Modena in 1783 where he studied mathematics, medicine, philosophy and literature.
    • Among his teachers of mathematics at Modena were Luigi Fantini, who taught Ruffini geometry, and Paolo Cassiani, who taught him calculus.
    • Cassiani's course at Modena on the foundations of analysis was taken over by Ruffini in 1787-88 although he was still a student at this time.
    • Soon after this he graduated with a mathematics degree.
    • Ruffini must have made a good job of the foundations of analysis course he took over from Cassiani for, on 15 October 1788, he was appointed professor of the foundations of analysis.
    • Ruffini was appointed to fill the position of Professor of the Elements of Mathematics in 1791.
    • He had trained in medicine and, also in 1791, he was granted a licence to practice medicine by the Collegiate Medical Court of Modena.
    • This was a time of wars following the French Revolution.
    • In March 1796 he was replaced by Napoleon Bonaparte who executed a brilliant campaign of manoeuvres.
    • The King of Sardinia asked for an armistice and Nice and Savoy were annexed to France.
    • Before Mantua fell to his armies he signed armistices with the duke of Parma and the duke of Modena.
    • Napoleon's troops occupied Modena and, much against his wishes, Ruffini found himself in the middle of the political upheaval.
    • Napoleon set up the Cisalpine Republic consisting of Lombardy, Emilia, Modena and Bologna.
    • Although not wishing to get involved, Ruffini found himself appointed as a representative to the Junior Council of the Cisalpine Republic.
    • However, he soon left this position and, in early 1798, he returned to his scientific work at the University of Modena.
    • He was required to swear an oath of allegiance to the republic and this Ruffini found he could not bring himself to do on religious grounds.
    • Ruffini did not seem greatly disturbed by the loss of his chair, in fact he was a very calm man who took all the dramatic events around him in his stride.
    • The fact that he could not teach mathematics meant that he had more time to practise medicine and therefore help his patients to whom he was extremely devoted.
    • On the other hand it gave him the chance to work on what was one of the most original of projects, namely to prove that the quintic equation cannot be solved by radicals.
    • To solve a polynomial equation by radicals meant finding a formula for its roots in terms of the coefficients so that the formula only involves the operations of addition, subtraction, multiplication, division and taking roots.
    • Quadratic equations (of degree 2) had been known to be soluble by radicals from the time of the Babylonians.
    • Ferrari had solved the quartic by radicals in 1540 and so 250 years had passed without anyone being able to solve the quintic by radicals despite the attempts of many mathematicians.
    • Certainly no mathematician has published such a claim and even Lagrange in his famous paper Reflections on the resolution of algebraic equations says he will return to the question of the solution of the quintic and, clearly, he still hoped to solve it by radicals.
    • In 1799 Ruffini published a book on the theory of equations with his claim that quintics could not be solved by radicals as the title shows: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.
    • The algebraic solution of general equations of degree greater than four is always impossible.
    • Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume.
    • The immortal Lagrange, with his sublime reflections, has provided the basis of my proof.
    • Lagrange had used permutations and one can argue that groups appear in Lagrange's work but since Lagrange never composed permutations it is rather with hindsight that we now see the beginnings of group theory in his paper.
    • Ruffini is the first to introduce the notion of the order of an element, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive.
    • He proved some remarkable theorems (not of course with the modern terminology quoted below):- .
    • The order of a permutation is the least common multiple of the lengths in the decomposition into disjoint cycles.
    • An element of S5 of order 5 is a 5-cycle.
    • If G < S5 has order divisible by 5 then G has an element of order 5.
    • S5 has no subgroups of index 3, 4 or 8.
    • The proof is given in modern notation in [',' R G Ayoub, Paolo Ruffini’s Contributions to the Quintic, Archive for History of Exact Science 23 (1980), 253-277.','4].
    • However there was a strange lack of response to Ruffini's work from mathematicians.
    • In 1801 Ruffini sent a copy of his book to Lagrange.
    • He received no response and so he sent a second copy with a covering letter [',' R G Ayoub, Paolo Ruffini’s Contributions to the Quintic, Archive for History of Exact Science 23 (1980), 253-277.','4]:- .
    • Because of the uncertainty that you may have received my book, I send you another copy.
    • to receive the book which I take the liberty of sending you.
    • In writing this book, I had principally in mind to give a proof of the impossibility of solving equations of degree higher than four.
    • Some mathematicians accepted Ruffini's proof although one would have to say that Pietro Paoli, the professor at Pisa, was influenced by patriotic motives when he wrote in 1799 [',' R G Ayoub, Paolo Ruffini’s Contributions to the Quintic, Archive for History of Exact Science 23 (1980), 253-277.','4]:- .
    • and recommend greatly the most important theorem which excludes the possibility of solving equations of degree greater than four.
    • To understand this quotation one has to realise that Lagrange was born in Turin which was part of Italy at the time.
    • This patriotic reaction apart, the world of mathematics seemed to almost ignore Ruffini's great result.
    • In the present memoir, I shall try to prove the same proposition [insolubility of the quintic] with, I hope, less abstruse reasoning and with complete rigour.
    • Of this last proof Ayoub writes in [',' R G Ayoub, Paolo Ruffini’s Contributions to the Quintic, Archive for History of Exact Science 23 (1980), 253-277.','4]:- .
    • Can anything be more elegant? This proof is essentially what was later called the Wantzel modification of Abel's proof and was published in 1845.
    • ."using works of Abel and Ruffini..
    • When Delambre wrote in a report on the state of mathematics since 1789:- .
    • Ruffini asked the Institute in Paris to pronounce on the correctness of his proof and Lagrange, Legendre and Lacroix were appointed to examine it.
    • if a thing is not of importance, no notice is taken of it and Lagrange himself, "with his coolness" found little in it worthy of attention.
    • The Royal Society were also asked to pronounce on the correctness and Ruffini received a somewhat kinder reply which said that although they did not give approval of particular pieces of work they were quite sure that it proved what was claimed.
    • This is all the more surprising since Cauchy was one of the worst of all mathematicians at giving credit to others.
    • He wrote to Ruffini in 1821, less than a year before Ruffini's death [',' E Carruccio, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the impossibility of solving algebraically equations of higher than the fourth degree.
    • In fact Cauchy had written a major work on permutation groups between 1813 and 1815 and in it he generalised some of Ruffini's results.
    • This influence through Cauchy is perhaps the only way in which Ruffini's work was to make an impact on the development of mathematics.
    • We left the story of Ruffini's career around 1799 when he began his publications on the quintic.
    • He left the University of Modena to spend 7 years teaching applied mathematics in the military school in Modena.
    • After the fall of Napoleon, Ruffini became rector of the University of Modena in 1814.
    • As well as the rectorship, Ruffini held a chair of applied mathematics, a chair of practical medicine and a chair of clinical medicine in the University of Modena.
    • Although he made a partial recovery, he never fully regained his health and in 1819 he gave up his chair of clinical medicine.
    • There are further aspects of Ruffini's work which should be mentioned.
    • He wrote several works on philosophy, one of which argues against some of Laplace's philosophical ideas.
    • He also wrote on probability and the application of probability to evidence in court cases.
    • Given the information in this article about the insolubility of the quintic, it is reasonable to ask why Abel has been credited with proving the theorem while Ruffini has not.
    • Ayoub suggests that [',' R G Ayoub, Paolo Ruffini’s Contributions to the Quintic, Archive for History of Exact Science 23 (1980), 253-277.','4]:- .
    • Then, too, the method of permutations was too exotic and, it must be conceeded, Ruffini's early account is not easy to follow.
    • between 1800 and 1820 say, the mood of the mathematical community ..
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (18 books/articles) .
    • A Poster of Paolo Ruffini .
    • History Topics: The development of group theory .
    • History Topics: Arabic mathematics : forgotten brilliance? .
    • History Topics: The abstract group concept .
    • History Topics: Overview of Chinese mathematics .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Ruffini.html .

  91. Juan Caramuel biography
    • Juan was a highly intelligent boy and by the age of twelve he was constructing his own astronomical tables.
    • After this he entered the University of Alcala (near Madrid) where he studied philosophy and the humanities.
    • from the University of Alcala having written a dissertation on Infinite Logic, after which he entered the Cistercian Order at the Monasterio de la Espina near Medina de Rioseco, Valladolid.
    • He studied philosophy at the Monasterio de Montederramo, Orense, before going to the Santa Maria del Destierro of Salamanca to study theology.
    • He was a brilliant scholar with an amazing flair for languages; he learnt to speak twenty languages including Latin, Greek, Arabic, Syriac, Hebrew and Chinese.
    • Fleming writes about this period spent in Salamanca [',' J A Fleming, Defending probabilism: the moral theology of Juan Caramuel (Georgetown University Press, 2006).','2]:- .
    • Years later, he would credit a "great man", the Cistercian scholar Angel Manrique, with teaching him to understand the essence of intrinsic probability.
    • He travelled widely, going to Portugal and to Belgium in 1632 where he spent time at the Monastery of Dunes in Flanders.
    • Settling in Louvain [',' J A Fleming, Defending probabilism: the moral theology of Juan Caramuel (Georgetown University Press, 2006).','2]:- .
    • He was appointed Abbot of Melrose, in Scotland, but this was only a nominal appointment and he did not visit Scotland.
    • He taught at Louvain until 1644 and, while there, he planned the defence of the city and published works on military engineering.
    • He also wrote at this time on other topics such as a work in which he argued that the King of Spain to had the right to rule Portugal.
    • Much of Caramuel's scientific work was done during the period in Louvain [',' J A Fleming, Defending probabilism: the moral theology of Juan Caramuel (Georgetown University Press, 2006).','2]:- .
    • He published works in mathematics and astronomy, corresponded with important scholars, and even experimented with a pendulum by hanging weights from a library roof.
    • At this time the Roman Catholic Church had many different Orders promoting their own version of the truth.
    • Caramuel was a Cistercian and in his writings he attacked Jansenism, a movement within the Catholic Church which considered itself following the teachings of St Augustus, but was attacked by its opponents, particularly the Jesuits, as having views close to Protestants.
    • He was appointed Abbot of Disibodenberg near Mainz and left Louvain on 9 February 1644 to journey to his new monastery.
    • He had been sent to this difficult area in an attempt to invigorate the Catholics in that part of Germany in their opposition to the spreading Protestant faith.
    • This proved an almost impossible task - when he arrived he found the monastery half in ruins and the fighting between Catholics and Protestants in the area meant that he had to flee in fear of his life on several occasions.
    • In 1647 he became Abbot of the Benedictine Monastery in Vienna and also Abbot of the Emaus Monastery in Prague, the residence of the Spanish Benedictines of Montserrat.
    • He lived in Prague and, on 26 July 1648, he helped defend the city from an attack by the Swedes, one of the last pieces of military action of the Thirty Years War.
    • He was honoured for this act with the award of a gold medal by the Emperor.
    • However, his support for the Peace of Westphalia, which declared Catholics and Protestants equal, angered Fabio Chigi and other leading Catholics who refused to recognise the Treaty.
    • However, in Rome Caramuel served as consulter to the Congregation of Rites and to the Holy Office.
    • He did amazing work during an outbreak of the plague, disregarding his own safety by looking after the many ill people and arranging for those who had died to be buried.
    • There were still problems concerning his views, and the Congregation of the Index banned some of his propositions and required him to "correct" Theologia moralis fundamentalis Ⓣ and issue a new edition.
    • In July 1657 Pope Alexander VII made Caramuel bishop of Satriano in southern Italy and he moved there in 1659.
    • When he did publish again, it was Apologema pro Antiquisima et Universalissima Doctrina de Probabilitate Ⓣ (1663) which was quickly added to the Index of prohibited books.
    • He reacted to the lack of support from the pope by setting up a printing press in Satriano to distribute his work.
    • He wrote the fascinating Syntagma de arte typographica Ⓣ (1664) describing his methods of printing and publishing.
    • In the encyclopaedia of mathematics Mathesis biceps, vetus et nova Ⓣ, published in 1670, he expounded the general principle of numbers to base n pointing out the benefits of some other bases than 10.
    • Donald Knuth writes in The Art of Computer Programming Volume 2:- .
    • The first published discussion of the binary system was given in a comparatively little-known work by a Spanish bishop, Juan Caramuel Lobkowitz, 'Mathesis biceps' (Campaniae, 1670) pp.
    • 45-48: Caramuel discusses the representation of numbers in radices 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, and 60 at some length, but gave no examples of arithmetic operations in nondecimal systems (except for the trivial operation of adding unity).
    • Caramuel still had to resist attacks against views he had put forward forty years earlier [',' J A Fleming, Defending probabilism: the moral theology of Juan Caramuel (Georgetown University Press, 2006).','2]:- .
    • In 1677, three members of the University of Louvain's theological faculty, journeying to Rome for the condemnation, made a detour to Vigevano to urge Caramuel to retract his propositions.
    • The unexpected confrontation reduced the elderly bishop to a short fit of weeping.
    • Caramuel's eyesight became poor in the last years of his life but he was able to continue with his duties as bishop.
    • We have described many of Caramuel's contributions above but let us end with a few more details.
    • Mathematical puzzles and games of chance form part of Mathesis biceps (1670).
    • He proposed a new method of trisecting an angle and developed a system of logarithms to base 109 where log 1010 = 0 and log 1 = 10.
    • Among Caramuel's other scientific work we mention a system he developed to determine longitude using the position of the moon.
    • He wrote widely on grammar, linguistics and rhetoric but perhaps his most interesting proposal in this area was to argue for the creation of a universal language.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (17 books/articles) .
    • A Poster of Juan Caramuel .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Caramuel.html .

  92. Marjorie Senechal biography
    • Her father was Abraham Wikler, born in New York on 12 October 1910, the son of the Jewish butcher Isaac Wikler who had emigrated from the Ukraine, and his wife Clara.
    • Ada Fay Fischer was the daughter of May Fischer and Flora M Samuels.
    • Abraham and Ada Fay Wikler had four children, the eldest being Marjorie, the subject of this biography.
    • She wrote [',' M Senechal, Adventures of an Amateur Crystallographer, American Crystallographic Association (2013).','21]:- .
    • We lived on a farm, twelve hundred acres in the heart of the Kentucky bluegrass country.
    • What you need to know is: they all were federal prisoners, serving time for possession of drugs.
    • Farm work, and housework - we had "houseboys" too - was part of the cure.
    • I talked with whomever I chose: Kentucky drifters, Chicago jazz musicians, New York street criminals, Chinese laundrymen.
    • And so I also learned that some problems, like drug addiction, are essentially intractable, and that any partial traction is achieved through the synthesis of multiple perspectives .
    • Senechal attended the Training School of the University of Kentucky in Lexington, Kentucky which was a small school.
    • She was not happy there, and she explained why in the interview [',' M Senechal, Narco Brat, in D Paty (ed.), Of human bondage, Smith College Studies in History 52 (2003), 173-200.','19]:- .
    • The guard dropped us off at a small ivy-covered building on the campus of the University of Kentucky.
    • When she was fifteen years old her parents moved from the Narcotic Farm to a house in Lexington [',' M Senechal, Narco Brat, in D Paty (ed.), Of human bondage, Smith College Studies in History 52 (2003), 173-200.','19]:- .
    • She said that the children at this school [',' M Senechal, Narco Brat, in D Paty (ed.), Of human bondage, Smith College Studies in History 52 (2003), 173-200.','19]:- .
    • For me, Lexington was a looking-glass prison, with walls of hypocrisy and conformity as strong as brick and stone.
    • After two years I escaped to the University of Chicago, which didn't require a high school diploma.
    • I erased all traces of my Kentucky accent, or thought I did, and didn't look back for a long, long time.
    • Entering the University of Chicago, Senechal had no idea what she wanted to study.
    • She did not enjoy the chemistry laboratories in the afternoon, but mathematics was more enjoyable so she majored in that subject [',' M Senechal, Adventures of an Amateur Crystallographer, American Crystallographic Association (2013).','21]:- .
    • But I didn't know, or think to ask myself, what I enjoyed about it: which areas of math appealed to me and why; what stirred my imagination.
    • Nor did I have any clear idea of a career.
    • from the University of Chicago in 1960 and then continued her study of mathematics, undertaking postgraduate work at the Illinois Institute of Technology.
    • in 1962 she continued undertaking research for a doctorate at the Illinois Institute of Technology advised by Abe Sklar.
    • Sklar's thesis advisor had been Tom Apostol at the California Institute of Technology where he had been awarded his Ph.D.
    • in 1956 for his thesis Summation formulas associated with a class of Dirichlet Series.
    • When Sklar became Senechal's thesis advisor she had no idea which area of mathematics attracted her most and she ended up undertaking research on analytic number theory, somewhat by accident [',' M Senechal, Adventures of an Amateur Crystallographer, American Crystallographic Association (2013).','21]:- .
    • Analytic number theory became my research topic because my advisor, an expert on summation formulas for divergent infinite series, had a grant from the Office of Naval Research that included support for a graduate student.
    • At the Illinois Institute of Technology Senechal met Lester John Senechal who, like her, was undertaking research for a Ph.D.
    • in mathematics.
    • Marjorie Wikler, as she then was, married Lester Senechal while both were undertaking research at the Illinois Institute of Technology.
    • After the award of his doctorate, Lester Senechal was appointed to the University of Arizona.
    • While in Tucson she completed the work and submitted her thesis Approximate Functional Equations and Probabilistic Inner Product Spaces to the Illinois Institute of Technology and was awarded a Ph.D.
    • She published the paper based on her thesis A summation formula and an identity for a class of Dirichlet series in 1966.
    • Unable to obtain a position at the University of Arizona since a husband and wife were not permitted posts at the same institution, Lester and Marjorie Senechal obtained a Fulbright scholarship and a Fulbright travel grant respectively, to teach at the University of Ceara in Fortaleza, Brazil, from June to December 1965.
    • In fact the Five College Consortium consisting of Amherst College, Hampshire College, Mount Holyoke College, Smith College and the University of Massachusetts at Amherst had been formally established in 1965.
    • Senechal's second daughter, Jenna Juliet, was born in the spring of 1967 but this had a more major impact on Senechal's career than one would expect all because of a chance occurrence.
    • She wrote [',' M Senechal, Adventures of an Amateur Crystallographer, American Crystallographic Association (2013).','21]:- .
    • In 1973 she was made Associate Professor and a full Professor of Mathematics in 1978.
    • Her discovery of the mathematics of crystallography and the group structure of tilings was exciting but she did not find anyone else at Smith College who was interested.
    • Via one of his articles, she was able to make contact with Arthur Loeb who worked at the Kennecott Copper Company's laboratory in Lexington, Massachusetts.
    • She studied books in the Smith College Art Library looking at the patterns illustrated in books like The Grammar of Ornament and tried to learn how to quickly identify which of the 17 space groups was the symmetry group of the pattern.
    • A professor of art, seeing her interest, asked her if she knew Dorothy Wrinch.
    • Senechal published her first paper in her new area of interest in 1975, Point groups and color symmetry in the journal Zeitschrift fur Kristallographie.
    • This was her first publication in ten years, but it was the first of a great wealth of papers and books on these topics.
    • She spent her first sabbatical year 1974-75 at the University of Groningen in The Netherlands.
    • She wrote [',' M Senechal, Adventures of an Amateur Crystallographer, American Crystallographic Association (2013).','21]:- .
    • For the first time, I was working in a real crystallography department, gaining a broader sense of the field and its denizens past and present.
    • She was able to spend the academic year 1979-80 at the Institute of Crystallography of the Academy of Sciences of the USSR in Moscow as an exchange scientist.
    • Charlene Morrow writes [',' C Morrow, Marjorie Wikler Senechal, in C Morrow and T Perl, Notable Women in Mathematics (Greenwood Press, Westport, Connecticut, 1998), 225-229.','13]:- .
    • This community of colleagues, particularly the mathematicians and crystallographers associated with Boris Delone, a leading researcher, soon became very important to her.
    • Nor Senechal was firmly grounded in an international community of researchers that would take her overseas many times.
    • For Senechal's description of the influence Delone had on her, see THIS LINK.
    • She wrote [',' M Senechal, Adventures of an Amateur Crystallographer, American Crystallographic Association (2013).','21]:- .
    • The girls braved the -45° cold - that merciless crossing point of the Fahrenheit and Centigrade scales [In fact -40°F = -40°C] - to trudge to their Russian schools.
    • They learned Russian and enough other things to skip a year of school when they came home; and they made Russian friends.
    • As well as international visits, Senechal also played a major role in organising various international conferences at Smith College, For example in 1973 an interdisciplinary symmetry meeting was held at Smith College and Senechal was one of the two editors of the proceedings along with George Fleck.
    • In April 1984, Smith College was the site of a Shaping Space Conference.
    • For extracts of reviews of the two books, see THIS LINK.
    • For a longer extract from this review and other reviews of this and other works by Senechal see THIS LINK.
    • There is a surprising aspect of Senechal's publication list, namely the fact that she has published articles on silk.
    • Fascinated with the story of silk in Albania, she began researching silk in America in the 1830s [',' M Senechal, Adventures of an Amateur Crystallographer, American Crystallographic Association (2013).','21]:- .
    • I remembered that Northampton, Massachusetts had been the home of the Corticelli Company, which made silk thread.
    • With a Smith colleague, I launched a town-gown history recovery project.
    • This interest in the history of silk was matched by an interest in the history of mathematics.
    • Senechal retired from teaching in July 2007 but this allowed her to spend more time on two major projects, namely editing The Mathematical Intelligencer and writing a biography of Dorothy Wrinch.
    • She offers a gripping portrait of an era and of a scientist whose complications acquire a tragic glamour.
    • This includes being a member of the Editorial Board of the Mathematical Association of America's Carus Mathematical Monographs (1987-1996), being editor from 1992 to 1996; on the Editorial Board of the journal "Discrete and Computational Geometry" 1985- 2006; being editor of the column "Mathematical Communites" for The Mathematical Intelligencer (1997-); being Co-Editor-in Chief of The Mathematical Intelligencer from 1 July 2005 and being the sole Editor-in-Chief from 1 January 2013.
    • Among the awards and honours that have been given to Senechal we mention: the Carl B Allendoerfer Award from the Mathematical Association of America in 1982, for her paper Which tetrahedra fill space?; the Honored Faculty Award from Smith College in May 2007; the Millia Davenport Publication Award from the American Costume Society in 2008; and elected a Fellow of the American Mathematical Society in 2012.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (22 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Senechal.html .

  93. Sun-Yung Alice Chang biography
    • Sun-Yung Alice Chang was born in China at the time when the Communists were rapidly taking control of the country.
    • By late 1949 the Communists were in control of almost all of China, the main exception being Taiwan which continued to be controlled by the Nationalist government.
    • Soon after this Chang's family moved from mainland China to the Republic of China in Taiwan.
    • She grew up in Taiwan, attending school there and then studying at the National University of Taiwan.
    • from the University of California, Berkeley, for a thesis written with Donald Sarason as her advisor.
    • In her thesis, Chang worked on problems in classical analysis, in particular the study of boundary behaviour of bounded analytic functions on the unit disc.
    • After receiving her doctorate, Chang was appointed as an assistant professor at the State University of New York at Buffalo for the academic year 1974-1975.
    • Following this she was appointed Hedrick Assistant Professor at the University of California at Los Angeles until 1977 when she moved to the University of Maryland as an assistant professor.
    • In 1980 Chang returned to the University of California at Los Angeles as an associate professor, being later promoted to full professor.
    • She was an invited speaker at the International Congress of Mathematicians at Berkeley in 1986.
    • During 1988-1989 she was also a full professor at the University of California, Berkeley.
    • Chang's research interests include the study of certain geometric types of nonlinear partial differential equations.
    • Perhaps Chang's greatest honour was the award of the 1995 Ruth Lyttle Satter Prize in Mathematics.
    • The prize is awarded every two years to a woman who has made an outstanding contribution to mathematics research in the previous five years.
    • The Ruth Lyttle Satter Prize is awarded to Sun-Yung Alice Chang for her deep contributions to the study of partial differential equations on Riemannian manifolds and in particular for her work on extremal problems in spectral geometry and the compactness of isospectral metrics within a fixed conformal class on a compact 3-manifold.
    • On receiving the prize Chang spoke about her work (see [',' 1995 Ruth Lyttle Satter Prize in Mathematics, Notices Amer.
    • The problems which I have been working on in the past few years are mainly connected with the study of extremal functions of Sobolev inequalities.
    • Such functions play an important role in the study of the blow-up phenomenon in a number of problems in geometry.
    • Following the early work of J Moser and influenced by the work of T Aubin and R Schoen on the Yamabe problem, P.
    • Yang and I have solved the partial differential equation of Gaussian/scalar curvatures on the sphere by studying the extremal functions for certain variation functionals.
    • This latter piece of work is a natural extension of the earlier work by Osgood-Phillips-Sarnak on the log-determinant functional on compact surfaces.
    • Chang also spoke about the position of women in mathematics research and how things are changing rapidly:- .
    • Since the Satter Prize is an award for women mathematicians, one cannot help but to reflect on the status of women in our profession now.
    • I can personally testify to the importance of having role models and the companionship of other women colleagues.
    • In 1998, in addition to her Professorship at the University of California, Los Angeles, Chang was appointed Professor at Princeton University.
    • The appointment at the University of California ended in 2000.
    • Chang is married to Paul Yang, also a Professor of Mathematics at Princeton University; they have one daughter and one son.
    • Chang has received many honours in addition to the Ruth Lyttle Satter Prize in Mathematics which we described above.
    • She was an Invited Speaker at the International Congress of Mathematicians held in Berkeley in 1986, and a Plenary Speaker at the International Congress of Mathematicians held in Beijing in 2002.
    • She held a John Simon Guggenheim Memorial Foundation Fellowship in 1999-2000 and served on the Steel Prize Selection Committee of the American Mathematical Society during 2001-2004.
    • On 28 April 2009 the National Academy of Sciences announced that Chang had been elected a member.
    • In June 2009 Chang was one of the organisers of '2009 the Program for Women and Mathematics' held at the Institute for Advanced Study at Princeton.
    • The course was entitled "Geometric PDE" and described using analytic tools like that of partial differential equations to solve problems in geometry.
    • model differential equations like that of the Gaussian curvature equations on compact surfaces, the prescribing curvature equations and the evolution equations related to the curvature flows.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • Index of Chinese mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Chang.html .

  94. Paul Erds (1913-1996)
    • Paul Erdős came from a Jewish family (the original family name being Englander) although neither of his parents observed the Jewish religion.
    • Paul's father Lajos and his mother Anna had two daughters, aged three and five, who died of scarlet fever just days before Paul was born.
    • This naturally had the effect making Lajos and Anna extremely protective of Paul.
    • He would be introduced to mathematics by his parents, themselves both teachers of mathematics.
    • Anna, excessively protective after the loss of her two daughters, kept Paul away from school for much of his early years and a tutor was provided to teach him at home.
    • The situation in Hungary was chaotic at the end of World War I.
    • Anna was at this time made head teacher of her school but when the Communists called for strike action against Kun's regime she continued working, not for political reasons but simply because she did not wish to see children's education suffer.
    • After four months in control of Hungary, Kun fled to Vienna when Romanian troops advanced on Budapest in July 1919.
    • Miklos Horthy, a right-wing nationalist, took over control of the country.
    • She was dismissed from her post and she was left in fear of her life as Horthy's men roamed the streets killing Jews and Communists.
    • He now set about teaching Paul to speak English, but the strange English accent which this gave Paul remained one of his characteristics throughout his life.
    • Despite the restrictions on Jews entering universities in Hungary, Erdős, as the winner of a national examination, was allowed to enter in 1930.
    • During his tenure of the fellowship, Erdős travelled widely in the UK.
    • The situation in Hungary by the late 1930s clearly made it impossible for someone of Jewish origins to return.
    • However he did visit Budapest three times a year during his tenure of the Manchester fellowship.
    • In March 1938 Hitler took control of Austria and Erdős had to cancel his intended spring visit to Budapest.
    • We shall return later to give further details of the strange life which Erdős lived from this time on, devoted exclusively to seeking out and solving good mathematical problems.
    • First we make some comments about his mathematics.
    • The contributions which Erdős made to mathematics were numerous and broad.
    • However, basically Erdős was a solver of problems, not a builder of theories.
    • To Erdős the proof had to provide insight into why the result was true, not just provide a complicated sequence of steps which would constitute a formal proof yet somehow fail to provide any understanding.
    • Chebyshev proved Bertrand's conjecture in 1850 but when Erdős was only an eighteen year old student in Budapest he found an elegant elementary proof of this result.
    • the number of primes ≤ n tends to ∞ as n/logen.
    • Erdős was not much concerned with the competitive aspect of mathematics and was philosophical about the episode.
    • This result was typical of the type of mathematics Erdős worked on.
    • Erdős did receive the Cole Prize of the American Mathematical Society in 1951 for his many papers on the theory of numbers, and in particular for the paper On a new method in elementary number theory which leads to an elementary proof of the prime number theorem published in the Proceedings of the National Academy of Sciences in 1949.
    • It was quite an innocent event with the three mathematicians being too absorbed in discussion of mathematics to notice a NO TRESPASSING sign.
    • Erdős was simply too honest in saying that he would wish to return to Budapest at the end of the war.
    • Although it was a difficult time with great uncertainty about the fate of his family in Hungary, yet mathematically Erdős flourished.
    • It is unlikely that the full extent of the horror was understood by Erdős in the United States at the time.
    • However, in August 1945, Erdős received a telegram giving details of his family.
    • His father had died of a heart attack in 1942.
    • The family had suffered terribly through the Nazi campaign against the Jews, however, and four of Erdős's uncles and aunts had been murdered.
    • Near the end of 1948 Erdős was able to return to Hungary for a visit and there he was reunited with his surviving family and friends.
    • For the next three years he travelled frequently between England and the United States before accepting a temporary post at the University of Notre Dame in 1952.
    • Erdős could not bring himself to accept the same generous offer on a permanent basis, which both the University of Notre Dame and Erdős's friends tried hard to encourage him to accept.
    • When asked by US immigration, as he returned after a conference in Amsterdam in 1954, what he thought of Marx, Erdős made the ill judged reply:- .
    • This was followed by a line of questioning about whether he would ever return to Hungary.
    • So, was it only the fear of not being let out of Hungary that stopped him going there.
    • Of course, my mother is there and I have many friends there.
    • The files indicate that the official reasons were not the answers Erdős gave to the above questions, but the fact that he had corresponded with a Chinese mathematician who had subsequently returned from the United States to China and also Erdős's 1941 FBI record.
    • He spent much of the next ten years in Israel.
    • By this time, however, Erdős had become a traveller moving from one university to another, and from the home of one mathematician to another.
    • However, he did have a home of sorts with his friend Ronald Graham.
    • He was one of the century's greatest mathematicians, who posed and solved thorny problems in number theory and other areas and founded the field of discrete mathematics, which is the foundation of computer science.
    • He was also one of the most prolific mathematicians in history, with more than 1,500 papers to his name.
    • And, his friends say, he was also one of the most unusual.
    • Erdős won many prizes including the Wolf Prize of 50 000 dollars in 1983.
    • most of the money he earned from lecturing at mathematics conferences, donating it to help students or as prizes for solving problems he had posed.
    • In 1976 Ulam gave this description of Erdős:- .
    • He had been a true child prodigy, publishing his first results at the age of eighteen in number theory and in combinatorial analysis.
    • In 1941 he was twenty-seven years old, homesick, unhappy, and constantly worried about the fate of his mother who remained in Hungary.
    • His eyes indicated he was always thinking about mathematics, a process interrupted only by his rather pessimistic statements on world affairs, politics, or human affairs in general, which he viewed darkly.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (18 books/articles) .
    • A Poster of Paul Erdős .
    • An entry in The Mathematical Gazetteer of the British Isles .
    • History Topics: Attempts to understand the real numbers .
    • Australian Academy of Science .
    • New York Academy of Sciences .
    • Dictionary of Scientific Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Erdos.html .

  95. John T Graves (1806-1870)
    • John T Graves was the son of John Crosbie Graves (2 July 1776 - 13 January 1835), a lawyer by training who became Chief Police Magistrate for Dublin, and Helena Perceval (1786-1850), the daughter of the Rev Charles Perceval (1751-1795) of Templehouse, County Sligo.
    • John and Helena Graves were married in 1806 and they had six children: John Thomas Graves, the subject of this biography; Helena Clarissa Graves (1808-1871); Robert Perceval Graves (1810-1893); James Perceval Graves (1811-1852); Charles Graves (1812-1899), who also has a biography in this archive; and Caroline Graves (1819-1855).
    • We note that Robert Perceval Graves was the author of the 3-volume work Life of Sir William Rowan Hamilton (Hodges, Figgis and Co., Dublin, 1882-1889).
    • The Graves family acquired number 12 and members of the family lived there throughout the 19th century.
    • One of his fellow students was William Rowan Hamilton and the two soon became friends.
    • He had also come to the attention of John Brinkley, the Andrews Professor of Astronomy at Trinity who was an excellent mathematician having studied the latest continental mathematics.
    • This friendship with Hamilton rapidly brought Graves into contact with the latest mathematics.
    • Bartholomew Lloyd had become professor of mathematics at Trinity in 1813 and, taking over a department in which the teaching had been very old-fashioned with no calculus taught, he had quickly introduced the continental approach to calculus teaching from Lacroix's textbook Traite elementaire de calcul differentiel et du calcul integral, from Poisson's Traite de mecanique, and from Laplace's Mecanique Celeste.
    • However, in 1826, while still an undergraduate, Graves had began to study complex logarithms and produced an extension of Euler's formulas for logarithms given in Introductio in Analysin Infinitorum (1748).
    • He did not publish this work until 1829 when his paper An Attempt to Rectify the Inaccuracy of some Logarithmic Formulae was published in the Philosophical Transactions of the Royal Society of London.
    • From the recent researches of MM Poisson and Poinsot on angular section, and their discovery of error in trigonometrical formulae usually considered complete, my attention has been drawn to analogous incorrectness in logarithmic series.
    • Accordingly, the end proposed in the present investigation is the exhibition in an amended form of two fundamental developments, as principles employed in their establishment admit of application in expanding by different methods various similar functions, and tend to elucidate other parts of the exponential theory.
    • This paper produced a considerable reaction from a number of mathematicians supporting his results, and a number who objected to them.
    • Both George Peacock and John Herschel expressed doubts concerning the validity of Graves' results and Peacock printed his objections in his Report to the British Association for the Advancement of Science in 1833.
    • Hamilton, however, strongly supported Graves' results and when he spoke at the 1834 meeting of the British Association for the Advancement of Science in Edinburgh, he offered another proof of the results.
    • Hamilton presented the paper On Conjugate Functions or Algebraic Couples, as tending to illustrate generally the Doctrine of Imaginary Quantities, and as confirming the Results of Mr Graves respecting the existence of Two independent Integers in the complete expression of an Imaginary Logarithm.
    • When translated into the language of imaginaries, they agree with the results respecting imaginary exponential functions, direct and inverse, which were published by Mr Graves in the 'Philosophical Transactions' for 1829, and it was in meditating on those results of Mr Graves that Mr Hamilton was led, several years ago, to this theory of conjugate functions, as tending to illustrate and confirm them.
    • For example, Mr Graves had found, for the logarithm of unity to the Napierian base, [an] expression ..
    • This result of Mr Graves appeared erroneous to the author of the excellent Report on Algebra, which was lately printed for the Association; but it is confirmed by Mr Hamilton's theory ..
    • Augustus De Morgan expressed doubts about the correctness of the results in 1836 and, in the same year Graves replied to De Morgan by publishing in the Philosophical Magazine an alternative and shorter proof in his paper On the lately proposed logarithms of unity, in reply to Professor De Morgan.
    • However, Peacock was still not convinced and had published a "proof" that Graves was wrong in his Treatise on Algebra of 1830.
    • Duncan Farquharson Gregory pointed out the error in Peacock's "proof" in 1837 so, in the second edition of his Treatise on Algebra in 1845, Peacock admitted his error and gave a proof of Graves' result.
    • We have moved well ahead to complete the story of Graves' 1826 results on complex logarithms which took nearly 20 years before all were happy with it.
    • He gave his address as the Inner Temple when he submitted his logarithm paper to the Royal Society of London in 1828.
    • Moving to Oxford, he became an incorporated member of Oriel College on 11 November 1830 and was awarded an M.A.
    • Later in 1831 he was called to the English bar as member of the Inner Temple.
    • He was unsuccessful and wrote a letter to James David Forbes (1809-1868), who was at that time professor of Natural Philosophy at the University of Edinburgh.
    • I wish to make my escape from the bar, being fonder of literary and scientific than of professional pursuits.
    • In 1839 Graves was appointed as a Professor of Jurisprudence at University College, London.
    • In the same year he was elected a fellow of the Royal Society of London.
    • Soon after taking up his professorship, Graves was elected as an examiner in laws for the University of London.
    • Working at University College London, Graves was now a colleague of Augustus De Morgan and the two became close friends.
    • The Society for the Diffusion of Useful Knowledge had been set up by the same reformers who founded London University and De Morgan was an enthusiast for the Society.
    • Graves had certainly not given up mathematics but had continued to correspond with Hamilton.
    • In fact his early work on logarithms was undertaken since he believed that it might lead to the discovery of new imaginary numbers.
    • Graves continued to work on the idea but he thought about such things in a rather different way from Hamilton who wrote to him in 1835 (see for example [',' A Rice, Inexplicable? The status of complex numbers in Britain, 1750-1850, in Jesper Lutzen (ed.), Around Caspar Wessel and the Geometric Representation of Complex Numbers: Proceedings of the Wessel Symposium at The Royal Danish Academy of Sciences and Letters, Copenhagen, August 11-15 1998 (Kgl.
    • we belong to opposite poles in algebra since you, like Peacock, seem to consider algebra as a 'system of signs and their combinations', somewhat analogous to syllogisms expressed in letters while I am never satisfied unless I think that I can look beyond or through the signs to the things signified.
    • After Hamilton made his discovery of the quaternions in 1843 the first person he wrote to telling of his discovery was Graves who replied on 26 October complimenting Hamilton on his novel idea and adding (see for example [',' A Rice, Inexplicable? The status of complex numbers in Britain, 1750-1850, in Jesper Lutzen (ed.), Around Caspar Wessel and the Geometric Representation of Complex Numbers: Proceedings of the Wessel Symposium at The Royal Danish Academy of Sciences and Letters, Copenhagen, August 11-15 1998 (Kgl.
    • You can read part of Hamilton's letter at THIS LINK.
    • In this letter Hamilton mentions some extensions of the quaternions which had been discovered by Graves.
    • In fact in volume 3 of the Proceedings of the Royal Irish Academy (1836-1869) we see precisely how Graves explained his discoveries to Hamilton:- .
    • In a letter dated January 18th, 1844, Mr Graves communicated to Sir William Hamilton his theorem respecting sums of eight squares ..
    • In a letter of somewhat earlier date, but evidently written in haste, upon a journey, and dated December 26th, 1843, analogous expressions had been given, containing, however, some errors in the signs, which were soon afterwards corrected as above.
    • That earlier letter also indicated an expectation that a theory of octaves, including a new and extended system of imaginaries, which had thus been suggested to the writer (J T Graves, Esq.) by Sir William R Hamilton's theory of quaternions, might itself be extended so as to form a theory of what Mr Graves at the time proposed to call 2n-ions: but in a letter written shortly afterwards, doubts were expressed respecting the possibility of this additional extension, from octaves to sets of sixteen.
    • As an aside, we note that Degen was the mathematician to whom the young Niels Henrik Abel submitted his "solution" of the quintic equation.
    • Degen asked Abel to give an example of his method and at that stage Abel himself discovered his error.
    • Graves seems to be one of those people who made remarkable discoveries but luck was against him and he has almost been forgotten.
    • For further details of the Eight Squares Identity and Graves-Cayley Numbers, see THIS LINK.
    • At University College, London, Graves continued to seek systems of algebraic triplets even after Hamilton had discovered the quaternions.
    • On 25 April 1845 the first of twelve lectures by Graves on the law of nations were published in the Law Times.
    • He also wrote [',' A Rice, Graves, John Thomas (1806-1870), Oxford Dictionary of National Biography (Oxford University Press, Oxford, 2004).','4]:- .
    • He was also a contributor to Smith's 'Dictionary of Greek and Roman Biography', his articles including very full lives of the jurists Cato, Crassus, Drusus, and Gaius, and one on the legislation of Justinian.
    • On 24 March 1846 Graves married Amelia Tooke, the daughter of William Tooke and Amelia Shaen.
    • William Tooke was a lawyer who become President of the Society of Arts.
    • Before his marriage Graves had resigned his chair of Jurisprudence at University College, London, giving as the reason for his resignation the 'discouragingly low enrolment for his classes'.
    • He stated that his lectures on Roman law, on international law and on general jurisprudence had been poorly attended in contrast with his lectures on practical topics such as the law of equity.
    • In 1847 he was promoted to one of the poor-law inspectors of England and Wales, a position which came into existence because of the Poor Laws that came into force in that year.
    • I met a Rev J T Graves at Dublin, a Fellow of Trinity College, Dublin, mathematician, a man of renown in these parts who has been employed by the Government in enquiring on Endowed Schools and other Educational matters.
    • He is immensely strong on your point of teaching the science of Observation to all men, especially the young of all classes, and he has reported the same to the Government in perhaps the very words you would have used.
    • There is one aspect of Graves' life which we have not mentioned so far and that was his life-long hobby as a collector of ancient mathematical texts.
    • He bequeathed his collection to University College London, three days before he died at his home, Thirlestaine Lodge in Cheltenham at the age of sixty-three.
    • Several articles describe his amazing collection but perhaps the best would be to quote from [',' The Graves Library, Library News, University College London 4 (October 2000).','3] supplied by the library of University College, London:- .
    • The library acquired contains over 10,000 books, 4,600 pamphlets, 51 manuscripts and numerous periodicals, covering mainly mathematics and astronomy.
    • Probably the most important single collection within the Graves material is the Euclid collection, which contains eighty-three of the editions of Euclid's works printed before 1640.
    • The collection includes the 'editio princeps' published by Erhard Ratdolt at Venice in 1482, and amongst the translations are the first into any modern language, the Italian of 1543, the first German translation (1562), the first French (1564), John Day's edition of the first English translation with John Dee's preface of 1570, the first edition in Arabic (1594) and later translations into Turkish, Chinese, Persian, Hebrew, Finnish and many other languages.
    • The Graves collection includes seventy-five of the Library's incunabula and many famous books such as first editions of Copernicus's 'De revolutionibus' of 1543, Newton's 'Principia' and 'Opticks', and Thomas Salusbury's 'Mathematical Collections' of 1661-65.
    • There are first editions of fascinating "association copies" of the works of Priestley, Boyle, Kepler, Galileo, and Napier and important runs of early scientific periodicals.
    • There are also many treasures such as Henry Cavendish's copy of Pascal's 'Traite de l'equilibre des liqueurs' of 1663 and, of special note, a copy of Galileo's 'Il Saggiatore', published in Rome in 1623 and inscribed to Galileo's friend Morandi.
    • Article by: J J O'Connor and E F RobertsonClick on this link to see a list of the Glossary entries for this page .
    • List of References (6 books/articles) .
    • Multiple entries in The Mathematical Gazetteer of the British Isles .
    • Dictionary of National Biography .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/Biographies/Graves_John.html .


History Topics

  1. Chinese overview
    • Overview of Chinese mathematics .
    • Chinese Mathematics .
    • Several factors led to the development of mathematics in China being, for a long period, independent of developments in other civilisations.
    • The geographical nature of the country meant that there were natural boundaries (mountains and seas) which isolated it.
    • On the other hand, when the country was conquered by foreign invaders, they were assimilated into the Chinese culture rather than changing the culture to their own.
    • There are periods of rapid advance, periods when a certain level was maintained, and periods of decline.
    • The first thing to understand about ancient Chinese mathematics is the way in which it differs from Greek mathematics.
    • Unlike Greek mathematics there is no axiomatic development of mathematics.
    • The Chinese concept of mathematical proof is radically different from that of the Greeks, yet one must not in any sense think less of it because of this.
    • Rather one must marvel at the Chinese approach to mathematics and the results to which it led.
    • Chinese mathematics was, like their language, very concise.
    • It was very much problem based, motivated by problems of the calendar, trade, land measurement, architecture, government records and taxes.
    • It is worth noting that counting boards are uniquely Chinese, and do not appear to have been used by any other civilisation.
    • Our knowledge of Chinese mathematics before 100 BC is very sketchy although in 1984 the Suan shu shu Ⓣ dating from around 180 BC was discovered.
    • The next important books of which we have records are a sixteen chapter work Suanshu Ⓣ written by Du Zhong and a twenty-six chapter work Xu Shang suanshu Ⓣ written by Xu Shang.
    • Neither of these texts has survived and little is known of their content.
    • It is an astronomy text, showing how to measure the positions of the heavenly bodies using shadow gauges which are also called gnomons, but it contains important sections on mathematics.
    • It gives a clear statement on the nature of Chinese mathematics in this period (see for example [',' R Calinger, A contextual history of mathematics (New York, 1999).','2]:- .
    • The method of calculation is very simple to explain but has wide application.
    • This is because a person gains knowledge by analogy, that is, after understanding a particular line of argument they can infer various kinds of similar reasoning ..
    • is the mark of an intelligent person.
    • The Zhoubi suanjing Ⓣ contains a statement of the Gougu rule (the Chinese version of Pythagoras's theorem) and applies it to surveying, astronomy, and other topics.
    • Although it is widely accepted that the work also contains a proof of Pythagoras's theorem, Cullen in [',' C Cullen, Astronomy and Mathematics in Ancient China (Cambridge, 1996).','3] disputes this, claiming that the belief is based on a flawed translation given by Needham in [',' J Needham, Science and Civilisation in China 3 (Cambridge, 1959).','13].
    • In fact much Chinese mathematics from this period was produced because of the need to make calculations for constructing the calendar and predicting positions of the heavenly bodies.
    • The Chinese word 'chouren' refers to both mathematicians and astronomers showing the close link between the two areas.
    • One early 'choren' was Luoxia Hong (about 130 BC - about 70 BC) who produced a calendar which was based on a cycle of 19 years.
    • The most famous Chinese mathematics book of all time is the Jiuzhang suanshu or, as it is more commonly called, the Nine Chapters on the Mathematical Art.
    • The book certainly contains contributions to mathematics which had been made over quite a long period, but there is little in the original text to distinguish the precise period of each.
    • Many later developments came through commentaries on this text, one of the first being by Xu Yue (about 160 - about 227) although this one has been lost.
    • Dong and Yao write [',' Y Z Dong and Y Yao, The mathematical thought of Liu Hui (Chinese), Qufu Shifan Daxue Xuebao Ziran Kexue Ban 13 (4) (1987), 99-108.','24]:- .
    • 19">Liu Hui, a great mathematician in the Wei Jin Dynasty, ushered in an era of mathematical theorisation in ancient China, and made great contributions to the domain of mathematics.
    • From the "Jiu Zhang Suan Shu Zhu" and the "Hai Dao Suan Jing" it can be seen that Liu Hui made skilful use of thinking in images as well as in logical and dialectical ways.
    • Liu Hui gave a more mathematical approach than earlier Chinese texts, providing principles on which his calculations are based.
    • His best approximation of π was 3.14159 which he achieved from a regular polygon of 3072 sides.
    • It is clear that he understood iterative processes and the notion of a limit.
    • Liu also wrote Haidao suanjing Ⓣ (see the article on The Ten Classics) which was originally an appendix to his commentary on Chapter 9 of the Nine Chapters on the Mathematical Art.
    • In it Liu uses Pythagoras's theorem to calculate heights of objects and distances to objects which cannot be measured directly.
    • This was to become one of the themes of Chinese mathematics.
    • About fifty years after Liu's remarkable contributions, a major advance was made in astronomy when Yu Xi discovered the precession of the equinoxes.
    • In mathematics it was some time before mathematics progressed beyond the depth achieved by Liu Hui.
    • However, it does contains a problem solved using the Chinese remainder theorem, being the earliest known occurrence of this type of problem.
    • This text by Sun Zi was the first of a number of texts over the following two hundred years which made a number of important contributions.
    • Xiahou Yang (about 400 - about 470) was the supposed author of the Xiahou Yang suanjing Ⓣ which contains representations of numbers in the decimal notation using positive and negative powers of ten.
    • One of the most significant advances was by Zu Chongzhi (429-501) and his son Zu Geng (about 450 - about 520).
    • Zu Chongzhi was an astronomer who made accurate observations which he used to produce a new calendar, the Tam-ing Calendar (Calendar of Great Brightness), which was based on a cycle of 391 years.
    • With his son Zu Geng he computed the formula for the volume of a sphere using Cavalieri's principle (see [',' L Y Lam, and K S Shen, The Chinese concept of Cavalieri’s principle and its applications, Historia Math.
    • The beginnings of Chinese algebra is seen in the work of Wang Xiaotong (about 580 - about 640).
    • He wrote the Jigu suanjing Ⓣ, a text with only 20 problems which later became one of the Ten Classics.
    • His work is seen as a first step towards the "tian yuan" or "coefficient array method" or "method of the celestial unknown" of Li Zhi for computing with polynomials.
    • Certainly Chinese astronomy was not totally independent of developments taking place in the subject in India and similarly mathematics was influenced to some extent by Indian mathematical works, some of which were translated into Chinese.
    • Historians argue today about the extent of the influence on the Chinese development of Indian, Arabic and Islamic mathematics.
    • It is fair to say that their influence was less than it might have been, for the Chinese seemed to have little desire to embrace other approaches to mathematics.
    • Early trigonometry was described in some of the Indian texts which were translated and there was also development of trigonometry in China.
    • From the sixth century mathematics was taught as part of the course for the civil service examinations.
    • Li Chunfeng (602 - 670) was appointed as the editor-in-chief for a collection of mathematical treatises to be used for such a course, many of which we have mentioned above.
    • However Jia Xian (about 1010 - about 1070) made good contributions which are only known through the texts of Yang Hui since his own writings are lost.
    • He improved methods for finding square and cube roots, and extended the method to the numerical solution of polynomial equations computing powers of sums using binomial coefficients constructed with Pascal's triangle.
    • Although Shen Kua (1031 - 1095) made relatively few contributions to mathematics, he did produce remarkable work in many areas and is regarded by many as the first scientist.
    • He was the first of the great thirteenth century Chinese mathematicians.
    • This was a period of major progress during which mathematics reached new heights.
    • The treatise contains remarkable work on the Chinese remainder theorem, gives an equation whose coefficients are variables and, among other results, Heron's formula for the area of a triangle.
    • Li Zhi (also called Li Yeh) (1192-1279) was the next of the great thirteenth century Chinese mathematicians.
    • It contains the "tian yuan" or "coefficient array method" or "method of the celestial unknown" which was a method to work with polynomial equations.
    • The next major figure from this golden age of Chinese mathematics was Yang Hui (about 1238 - about 1298).
    • He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures.
    • He also gave a wonderful account of magic squares and magic circles.
    • Guo Shoujing (1231-1316), although not usually included among the major mathematicians of the thirteen century, nevertheless made important contributions.
    • He also developed a cubic interpolation formula tabulating differences of the accumulated difference as in Newton's forward difference interpolation method.
    • The last of the mathematicians from this golden age was Zhu Shijie (about 1260 - about 1320) who wrote the Suanxue qimeng Ⓣ published in 1299, and the Siyuan yujian Ⓣ published in 1303.
    • He used an extension of the "coefficient array method" or "method of the celestial unknown" to handle polynomials with up to four unknowns.
    • He also gave many results on sums of series.
    • This represents a high point in ancient Chinese mathematics.
    • The decline in Chinese mathematics from the fourteenth century was not by any means dramatic.
    • Wu Jing was an administrator in the province of Zhejing and his arithmetical encyclopaedia contained all the 246 problems of the Nine Chapters.
    • Again Cheng Dawei (1533 - 1606) published the Suanfa tong zong Ⓣ in 1592 which is written in the style of the Nine Chapters on the Mathematical Art but provides an even larger collection of 595 problems.
    • The books we have just listed show mathematical activity, but they did not take forward the methods of polynomial algebra.
    • On the contrary, the deep works of the 13th century ceased to be even understood much less developed further.
    • Xu Guangqi (1562 - 1633) certainly recognised exactly this and offered possible explanations including scholars neglecting practical computational tools and an identification of mathematics with mystical numerology under the Ming dynasty.
    • Other factors must be that the books describing the advanced methods were, in the Chinese tradition, very terse, and without teachers to pass on an understanding it became increasingly difficult for scholars to learn directly from the texts.
    • Xu Guangqi was the first native of China to publish translations of European books in Chinese.
    • Collaborating with Matteo Ricci he translated Western books on mathematics, hydraulics, and geography.
    • Certainly this does not mark the end of the Chinese mathematics tradition, but from the time of Matteo Ricci and other Western missionaries China was greatly influenced by other mathematical traditions.
    • It is impossible in an article of this length to mention many of the numerous contributions from this period on.
    • The most famous member of this family was Mei Wending (1633-1721) and his comment on the golden section is typical of the sensible attitude he took towards Western mathematics (see for example [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','9]):- .
    • After having understood how to make use of the golden section, I began to believe that the different geometrical methods could be understood and that neither the missionaries attitude of considering this simple technique as a divine gift, nor the Chinese attitude of rejecting it as heresy is correct.
    • Mei chose not to take a government post as most mathematicians did, but rather decided to devote himself to mathematics and its teaching.
    • Two of his brothers, Mei Wenmi and Mei Wennai, worked on astronomy and mathematics.
    • Mei Juecheng (1681-1763), who was Mei Wending's grandson, was asked in 1705 by Emperor Kangxi to be editor-in-chief of the major mathematical encyclopaedia Shuli jingyun Ⓣ (1723).
    • Certain people from the eighteenth century onwards did an excellent job in recording the Chinese tradition so that much of it is still accessible to us today.
    • He edited a new edition of the Nine Chapters on the Mathematical Art after copying the complete text as part of this project.
    • Ruan Yuan (1764 - 1849) produced his famous work the Chouren zhuan Ⓣ containing biographies of 275 Chinese and 41 Western "mathematicians".
    • Many biographical details of Chinese mathematicians recorded in this Archive are known through this work.
    • He was a highly productive mathematician who died when at the height of his abilities.
    • It is to the credit of Chinese mathematicians that they did not let their mathematical tradition be replaced by the western tradition.
    • For example Li Shanlan (1811-1882) is important as a translator of Western science texts but he is most famous for his own mathematical contributions.
    • He produced his own versions of logarithms, infinite series, and combinatorics which did not follow the style of western mathematics but his research naturally developed out of the foundations of Chinese mathematics.
    • There were many other efforts to promote Chinese mathematics, and in particular a mathematics journal, the Suanxue bao, was set up in 1899.
    • Western methods should not be adulated and Chinese methods despised.
    • Western mathematicians began lecturing in China during the early years of the twentieth century.
    • Chinese students began to study mathematics abroad and in 1917 Minfu Tah Hu obtained a doctorate from Harvard.
    • China was represented for the first time at the International Congress of Mathematicians in Zurich in 1932.
    • The Chinese Mathematical Society was formed in 1935.
    • List of References (29 books/articles) .
    • Chinese Mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Chinese_overview.html .

  2. References for Nine chapters
    • V I Ilyushchenko, Gauss elimination method (1849 AD) in the ancient Chinese script Mathematics in nine chapters (152 BC), Communications of the Joint Institute for Nuclear Research (Dubna, 1992).
    • J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).
    • S S Bai, A re-examination of a ring area problem in the 'Jiu zhang suanshu' (Chinese), Beijing Shifan Daxue Xuebao 30 (1) (1994), 139-142.
    • E I Berezkina (trs.), Two texts of Liu Hui on geometry (Russian), in Studies in the history of mathematics, No.
    • K Chemla, Different concepts of equations in 'The nine chapters on mathematical procedures' and in the commentary on it by Liu Hui (3rd century), Historia Sci.
    • K Chemla, Relations between procedure and demonstration : Measuring the circle in the 'Nine chapters on mathematical procedures' and their commentary by Liu Hui (3rd century), in History of mathematics and education: ideas and experiences (Essen, 1992) (1996), 69-112.
    • C Cullen, Learning from Liu Hui? A different way to do mathematics, Notices Amer.
    • J W Dauben, The "Pythagorean theorem" and Chinese mathematics : Liu Hui's commentary on the gou-gu theorem in Chapter Nine of the J'iu zhang suan shu', in Amphora (Basel, 1992), 133-155.
    • Z Q Deng, A comparative study of the Jiu zhang suanshu and the Elements (Chinese), J.
    • L S Feng, 'Jiu zhang suanshu' and Hui Liu's theory of similar right triangles (Chinese), in Collected research papers on the history of mathematics, Vol.
    • 1 (Chinese) (Hohhot, 1990), 37-45.
    • D W Fu, Why did Liu Hui fail to derive the volume of a sphere?, Historia Math.
    • S C Guo, Liu Hui's great contributions to mathematics : celebrating the 1720th anniversary of his commentary on the 'Jiu zhang suanshu' (Chinese), Math.
    • S C Guo, Guo Shuchun's edition of the 'Jiuzhang Suanshu' ('Nine chapters on the mathematical art'), Historia Math.
    • W S Horng, How did Liu Hui perceive the concept of infinity : a revisit, Historia Sci.
    • B S Hu, A further study on the positive-negative principle in the Jiu zhang suanshu (Chinese) , in Collected research papers on the history of mathematics Vol.
    • J M Li, A textual criticism on the "art of milu" in ring measurement in Nine chapters on arithmetic (Chinese), J.
    • Z J Liang, From the elimination method in the Jiu zhang suanshu (Arithmetic in nine sections) to automated theorem proving (Chinese), J.
    • D Liu, A comparison of Archimedes' and Liu Hui's studies of circles, in Chinese studies in the history and philosophy of science and technology 179 (Dordrecht, 1996), 279-287.
    • R Mei, Liu Hui's theories of mathematics, in Chinese studies in the history and philosophy of science and technology 179 (Dordrecht, 1996), 243-254.
    • S K Mo, 'Jiuzhang suanshu' ('Nine chapters on the mathematical art') and Liu Hui's commentary (Chinese), Stud.
    • Z C Shang, A comparison of the Elements and Wei Liu's notes on the theory of proportion and its applications in Jiu Zhang Suanshu (Chinese), in Studies on Euclid's Elements (Hohhot, 1992) , 217-233.
    • H H Shou, L G Liu and G J Wang, An offset approximation algorithm based on Liu Hui's circle subdivision method (Chinese), Appl.
    • Chinese Univ.
    • J Song, The historical value of the 'Nine chapters on the mathematical art' in society and the economy, in Chinese studies in the history and philosophy of science and technology 179 (Dordrecht,1996), 261-266.
    • P D Straffin Jr, Liu Hui and the first golden age of Chinese mathematics, Math.
    • D B Wagner, A proof of the Pythagorean theorem by Liu Hui (third century AD), Historia Math.
    • D B Wagner, An early Chinese derivation of the volume of a pyramid : Liu Hui, third century AD, Historia Math.
    • R B Wang, A preliminary exploration on the logical order of the Shang gong chapter of Jiu zhang suanshu - a parallel discussion on the understanding of the operational rule ab+ac=a(b+c) in 'Jiu zhang' (Chinese), in Collected research papers on the history of mathematics Vol.
    • Z W Xi and S L Zhang, On the characteristic of the dialectical thought of the 'Jiu zhang suanshu' and Hui Liu's commentary (Chinese), Qufu Shifan Daxue Xuebao Ziran Kexue Ban 19 (4) (1993), 103-110.
    • S C Yang, "Ratio" and "power" in Hui Liu's commentary on the 'Jiu zhang suan shu' ('Arithmetic in nine chapters') (Chinese), Dongbei Shida Xuebao (4) (1990), 39-43.

  3. Ten classics
    • Chinese Mathematics .
    • Education became important and mathematics was taught at the Imperial Academy.
    • The T'ang dynasty, which followed the Sui dynasty, continued the educational development which had already begun and formalised the teaching of mathematics.
    • The History of the T'ang records (see [',' J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).','1]):- .
    • As a consequence Li Chunfeng together with Liang Shu, an expert in mathematics from the ministry of education, and Wang Zhenru, a teacher from the national university and others were ordered by imperial decree to annotate the ten mathematical texts such as the Wucao suanjing or the Sunzi suanjing.
    • This allows us to date the start of the work by Li Chunfeng and his colleagues fairly accurately.
    • Wucao suanjing (Mathematical Manual of the Five Administrative Departments) .
    • Jigu suanjing (Continuation of Ancient Mathematics) .
    • Shushu jiyi (Notes on Traditions of Arithmetic Methods) .
    • Zhui shu (Method of Interpolation) .
    • Sandeng shu (Art of the Three Degrees; Notation of Large Numbers) .
    • The way that mathematics was taught at the Imperial Academy was as follows.
    • Thirty students were recruited from the lower ranks of society and divided into two classes each of 15 students.
    • These two classes followed a different syllabus, with one class studying more basic practical mathematics while the other was the advanced class studying techniques.
    • Teaching was done by doctors of mathematics and their assistants.
    • The students spent seven years studying mathematics from The Ten Mathematical Classics and then took the civil service examinations.
    • To pass the examinations a score of 6 out of ten had to be achieved.
    • Let us look briefly at the contents of the texts.
    • The text measures the positions of the heavenly bodies using shadow gauges which are also called gnomons.
    • Duke of Zhu: How great is the art of numbers? Tell me something about the application of the gnomon.
    • Shang Gao: Level up one leg of the gnomon and use the other leg as a plumb line.
    • The Zhoubi suanjing contains calculations of the movement of the sun through the year as well as observations of the moon and stars, particularly the pole star.
    • Perhaps the most important mathematics which is included in the Zhoubi suanjing is related to the Gougu rule, which is the Chinese version of the Pythagoras Theorem.
    • The four "corner" triangles each have area ab/2 giving a total area of 2ab for the four added together.
    • Therefore the hypotenuse of the right angled triangle with sides of length a and b has length √( a2 + b2).
    • The author of the Zhoubi suanjing writes that Emperor Yu:- .
    • However, the text of the Zhoubi suanjing also explains that the reason that mathematics can be applied to so many different cases is as a result of the way that mathematical reasoning allows one to pass from particular to general situations.
    • This realisation of the abstract nature of mathematics is important.
    • This is the most important of all the texts included in the Ten Mathematical Classics, but there is no need to discuss it in the article since our archive contains a separate article on The Nine Chapters on the Mathematical Art.
    • This is a small work consisting of nine problems and it was originally written as part of his commentary on Chapter Nine of The Nine Chapters on the Mathematical Art but later removed and made into a separate work by Li Chunfeng and his colleagues during the creation of The Ten Mathematical Classics.
    • A translation of the Haidao suanjing appears in [',' K Shen, J N Crossley and A W-C Lun, The nine chapters on the mathematical art : Companion and commentary (Beijing, 1999).','3] .
    • The Haidao suanjing shows how to use the Gougu theorem (Pythagoras theorem) to calculate heights of objects and distances to objects which cannot be measured directly.
    • When viewed from X at ground level, 123 pu behind P1, the summit S of the island is in line with the top of P1.
    • Similarly when viewed from Y at ground level, 127 pu behind P2, the top of the island is in line with the top of P2.
    • Calculate the height of the island and its distance from P1.
    • Here the summit of the island refers to the top of a hill.
    • Poles are the tips of vertically standing rods.
    • The line of sight passes through the tip of the pole and the summit of the island.
    • Suppose the poles are of height h and the distance between the poles is d.
    • Liu Hui gives the height of the island as h×d/(P2Y-P1X)+h and the distance to it to be P1X×d/(P2Y-P1X).
    • He then gives: height of the island: 1255 pu; distance from P1 to the island: 30750 pu.
    • Other problems in this work are the height of a tree on the side of a mountain, the distance to a square town, the depth of a gorge, the height of a tower on a hill, the width of a river, the depth of a valley with a lake at the bottom, the width of a ford viewed from a hill, and the size of a town seen from a mountain.
    • Historians have given a wide variety of dates for this text but Wang Ling [',' L Wang, The date of the Sunzi suanjing and the Chinese remainder theorem, in Proc.
    • History of Science, 1962 (Paris, 1964), 489-492.','4] seems to have the most convincing argument:- .
    • The Sunzi suanjing mentions the mein as an item of taxation, and the hu tiao system.
    • Of course this dating assumes that the text was written as a whole, while it seems more likely that it was compiled, like many of the texts, from older sources.
    • In that case Wang Ling's dating will only establish when part of the text was written, some possibly being earlier, while other parts probably have been written later.
    • The Sunzi suanjing consists of three chapters, the first describing systems of measuring with considerable detail on using counting rods to multiply, divide, and compute square roots.
    • The second and third chapters consist of problems (28 and 36 respectively) concerning fractions, areas, volumes etc.
    • similar to, but rather easier than, the problems in the Nine Chapters on the Mathematical Art One problem, however, is of special interest, this being Problem 26 in Chapter 3:- .
    • Suppose we have an unknown number of objects.
    • This, of course, is important for it is a problem which is solved using the Chinese remainder theorem.
    • Multiply the number of units left over when counting in threes by 70, add to the product of the number of units left over when counting in fives by 21, and then add the product of the number of units left over when counting in sevens by 15.
    • If the answer is 106 or more then subtract multiples of 105.
    • Wucao suanjing (Mathematical Manual of the Five Administrative Departments) .
    • The main interest in this text is that although many of the 19 formulas given to find the areas of different shapes of fields in the first chapter give approximately the right answer, they are actually incorrect.
    • None of the problems presents anything new.
    • Another work of three chapters with 15, 22 and 38 problems respectively.
    • A slightly strange work which contains a commentary on specific parts of five non-mathematical texts.
    • The commentary does contain mathematics, particularly relating to questions concerning the calendar and large numbers.
    • Jigu suanjing (Continuation of Ancient Mathematics) .
    • We do know the author of this work, namely Wang Xiaotong.
    • It is a strange mixture of practical problems arising in the construction of dykes and canals with fanciful problems which would not arise in practice.
    • Shushu jiyi (Notes on Traditions of Arithmetic Methods) .
    • The author of this text is claimed to be Xu Yue and to have been written at the beginning of the third century.
    • It is a difficult work to understand, in part showing how very large numbers can be constructed using powers of ten.
    • Parts of the text seem to have more religious content than mathematical.
    • Zhui shu (Method of Interpolation) .
    • He was an outstanding mathematician but sadly the text of the Zhui shu has not survived.
    • Zu Chongzhi seems to have been the first Chinese mathematician to compute correctly the volume of a sphere.
    • Sandeng shu (Art of the Three Degrees; Notation of Large Numbers) .
    • List of References (4 books/articles) .
    • Chinese Mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Ten_classics.html .

  4. Weather forecasting
    • The History of Weather Forecasting .
    • It is one of the most common topics of conversation, it has influenced people's lives for thousands of years, and predicting it requires the most powerful computers on the planet: the weather.
    • Most ancient cultures include weather gods, and weather catastrophes have an important role in creation myths of many cultures, for example the Deluge described in the Bible [','R Wengenmayr, Wettervorhersage.
    • Nowadays, we are more independent of weather conditions due to central heating, air conditioners, greenhouses and so forth, but weather forecasts are more accurate than they ever were [','P Bethge, J Blech, M Dworschak, O Stampf, Das gefuhlte Wetter (Spiegel, 23/2000) ','21].
    • Forecasts, both for the next couple of hours and for the next couple of days, are issued daily.
    • Apart from helping people decide when they should invite their neighbours for a barbecue, weather forecasts provide vital information for a wide range of occupational categories such as farmers, pilots, sailors and soldiers.
    • The great success of the weather channel in U.S.
    • But how are weather forecasts created? And why has forecast accuracy improved so dramatically over the last couple of decades? In this article, I will examine the history of weather forecasting, and especially the historical development of the use of mathematics in forecasting, before describing the basic principles of the mathematical methods used in current forecasting models.
    • I will also present some of the current research projects in weather forecasting.
    • Chapter 2 of the article provides an overview of early forecasting attempts, from the first observations made by ancient cultures to the development of government-supported weather services.
    • Chapter 3 reviews the use of mathematics in weather forecasting, starting with the fathers of numerical weather prediction, Vilhelm Bjerknes and Lewis Fry Richardson.
    • Then, the seminal work of Carl-Gustaf Rossby is treated briefly, before the use of computers in weather forecasting, pioneered by Jule Charney (and others), and the application of chaos theory to meteorology by Edward Lorenz are discussed.
    • An overview of some of the current research projects and research areas in weather forecasting is also given.
    • The reader does not need to have much background in applied mathematics or physics, but is expected to know basic physical principles, for instance gravity, centrifugal force, friction and waves.
    • Furthermore, I have assumed that the reader is familiar with differential equations, differentiation of functions of several variables, Fourier series and Gaussian elimination.
    • Explaining these concepts would have gone beyond the scope of this article.
    • However, as the article should provide only an overview of the mathematical methods used in current forecasting models, I have chosen to include only simple equations and explain some mathematical symbols in order to make understanding the methods easier.
    • Also, a number of mathematical concepts are explained in words rather than with equations.
    • The aim of the article is to give the reader a summary of the historical development of both weather forecasting in general as well as the application and the importance of mathematics in forecasting in particular.
    • In addition, it should give an impression of how sophisticated mathematical models have to be in order to accurately simulate and predict the state of the atmosphere and hence the weather.
    • For a very long time, forecasts were based on observations of the skies both during the day and at night.
    • By the end of the nineteenth century, several European countries and the United States had established the first weather services.
    • It is not known when people first started to observe the skies, but at around 650 BC, the Babylonians produced the first short-range weather forecasts, based on their observations of the stars and clouds.
    • The Chinese also recognised weather patterns, and by 300 BC astronomers had developed a calendar which divided the year into 24 festivals, each associated with a different weather phenomenon.
    • Generally, weather was attributed to the vagaries of the gods, as the wide range of weather gods in various cultures, for example the Egyptian sun god Ra and Thor, the Norse god of thunder and lightning, proves.
    • The work of the philosopher and scientist Aristotle (384-322 BC) is especially noteworthy, as it dominated people's views on and their knowledge of the weather for the next 2000 years.
    • In 340 BC, Aristotle wrote his book Meteorologica Ⓣ, where he tried to explain the formation of rain, clouds, wind and storms.
    • Many of his observations were -- in retrospect -- surprisingly accurate.
    • Throughout the Middle Ages and beyond, the Church was the only official institution that was allowed to explain the causes of weather, and Aristotle's Meteorologica was established as Christian dogma [','J D Cox, Stormwatchers.
    • The Turbulent History of Weather Prediction from Franklin’s Kite to El Nino (Hoboken NJ, 2002)','1, p.
    • Besides, weather observations were passed on in the form of rhymes, which are now known as weather lore.
    • Many of these proverbs are based on very good observations and are accurate, as contemporary meteorologists have discovered [','W Wiedlich, Eine kleine Geschichte der Wettervorhersage (General-Anzeiger, 27.11.2007)','26].
    • By the end of the Renaissance, scientists realised that it would be much easier to observe weather changes if they had instruments to measure fundamental quantities in the atmosphere such as temperature, pressure and moisture.
    • Until then, the only tools available were vanes, to determine the wind direction, and early versions of rain gauges.
    • The discoveries of unknown continents and, by extension, of unknown climates and weather phenomena also played a part in the development of modern meteorology [','R Wengenmayr, Wettervorhersage.
    • One of the first to study storms was the scientist and politician Benjamin Franklin (1706-1790).
    • His most famous scientific achievement is his work on electricity and lightning in particular, but in fact he was very interested in weather and studied it throughout most of his life.
    • Many of Franklin's observations paved the way to a better understanding of climate and the atmosphere, but .
    • Franklin was still a natural philosopher at heart, and he was not inclined to clutter his conjectures with a lot of data or mathematics [','J D Cox, Stormwatchers.
    • The Turbulent History of Weather Prediction from Franklin’s Kite to El Nino (Hoboken NJ, 2002)','1, p.
    • All conjectures and theories on weather were based solely on observations and it was not until the early 20th century that mathematics and physics became part of meteorology.
    • One of the first weather observation networks, which operated from 1654 to 1670, was established by the Tuscan nobleman Ferdinand II [','R Wengenmayr, Wettervorhersage.
    • The society did not produce any forecasts; one of the main obstacles for this was that the data had to be sent to the society's main office by boat and post coaches, which took weeks.
    • The invention of the electrical telegraph in 1837 by Samuel Morse facilitated the production of weather forecasts, as data and any other weather observations now could easily and swiftly be transmitted to another country and even to another continent.
    • Observation wards began to appear all over Europe and Northern America, but it was not until the Crimean War (1853-1856) that people realised the benefits of weather forecasts.
    • The fleet of the Ottoman Empire was surprised by a low-pressure system and, as a result, lost several ships.
    • The French Emperor Napoleon III later ordered that the weather for that day should be analysed; he learned that the storm could have been predicted and that warnings could have been transmitted by telegraph [','M Birke, Wettervorhersage & Wetterdienst (University of Regensburg, 2009)','12].
    • Hence, the loss of the ships could have been prevented.
    • These weather maps enabled the scientists to detect and study storm systems and wind patterns as well as comparing the current meteorological situation to past ones, which ultimately led to the production of forecasts.
    • This method, based on the analysis and comparison of many standardised observations taken simultaneously, is called synoptic weather forecasting.
    • The British Meteorological Department issued regular gale warnings from 1861 onwards; the first US storm-warning system began to operate ten years later, dwarfing the European services with its size and funds [',' http://eh.net/encyclopedia/article/craft.weather.forecasting.history, E D Craft','33].
    • In the 1920s, weather maps became much more detailed due to the invention of the radiosonde: a small lightweight box containing measurement equipment and a radio transmitter.
    • Of course, radiosondes and weather balloons have been improved substantially over the last 80 years, but even the early models allowed meteorologists to observe weather conditions in high altitudes.
    • Today, both polar orbiting satellites (at an altitude of 800-900 km; they revolve around the Earth following the degrees of longitude, passing over both poles) and geostationary satellites (at an altitude of 35,800 km [','R Wengenmayr, Wettervorhersage.
    • Both the financial significance of weather forecasts and the necessity of knowing more about atmospheric processes were understood fairly soon after the first gale warnings were published, but very few people realised that mathematics could be used to describe these processes and produce more accurate forecasts than synoptic meteorology ever could.
    • The discovery of chaos theory and not least the development of computers greatly improved the quality of forecasts.
    • The first mathematician who thought of applying mathematics to weather forecasting was the Norwegian Vilhelm Bjerknes (1862-1951).
    • Already at a young age, Bjerknes engaged in mathematics as he assisted his father with his research in hydrodynamics.
    • He then studied mathematics and physics at the University of Christiania (nowadays Oslo).
    • In 1898 he formulated his circulation theorem: in a nutshell, it explains the evolution and the subsequent decay of circulations in fluids.
    • Possibly even more importantly, the theorem also marks "the move of Vilhelm Bjerknes into meteorology" [','A J Thorpe, H Volkert, M ZiemiaDski, The Bjerknes’ Circulation Theorem: A Historical Perspective (Bulletin of the American Meteorological Society, Volume 84, Issue 4, April 2003), 471-480','11, p.
    • Combining his circulation theorem with hydrodynamics and thermodynamics, Bjerknes discovered that, given initial atmospheric conditions, it is possible to compute the future state of the atmosphere using mathematical formulae.
    • The Turbulent History of Weather Prediction from Franklin’s Kite to El Nino (Hoboken NJ, 2002)','1, p.
    • We must apply the equations of theoretical physics not to ideal cases only, but to the actual existing atmospheric conditions as they are revealed by modern observations.
    • transition zones between warm and cold air masses, and described the life cycle of mid-latitude cyclones.
    • Most of Bjerknes's important results were published in On the Dynamics of the Circular Vortex with Applications to the Atmosphere and to Atmospheric Vortex Wave Motion in 1921.
    • Bjerknes' equations were very complicated and not very practical for predicting the weather as they required immense computational power, a fact of which he was aware himself.
    • The first attempt to use mathematics in order to predict the weather was made by the British mathematician Lewis Fry Richardson (1881-1953), who simplified Bjerknes' equations so that solving them became more feasible.
    • Richardson, who had studied a number of sciences at both Newcastle University and the University of Cambridge, worked in the British Meteorological Office from 1913 until 1916.
    • In 1919, he resumed his former job there, but resigned only a year later when the Meteorological Office became part of the Air Ministry.
    • He also worked for the National Peat Industries for some time, and in order to solve differential equations modelling the flow of water in peat, he invented his method for finite differences, which produces highly accurate results.
    • A differential equation with a smooth variable is converted into a function (or an approximation thereof) that relates the changes of the variable and given steps in time and/or space, meaning that the changes are calculated at discrete points rather than at infinitely many points.
    • So in the place of the differential equation you get many equations which can be solved using arithmetic.
    • Finite difference methods are widely used nowadays, but in Richardson's days, other mathematicians considered the method to be "approximate mathematics" [','J C R Hunt, Lewis Fry Richardson and his Contributions to Mathematics, Meteorology, and Models of Conflict (Annual Review for Fluid Mechanics, Volume 30, January 1998), xiii-xxxvi ','8, p.
    • Nevertheless, when Richardson came across problems of the dynamics of the atmosphere at his work at the Meteorological Office, he decided to solve them using his method.
    • By dividing the surface of the Earth into thousands of grid squares, and the atmosphere into several horizontal layers, he obtained a large number of grid boxes, connected to one another by mathematical equations.
    • Fundamentally, Richardson applied Bjerknes' vision of calculating the future state of the atmosphere using observations of its current state to the grid and added to it the idea that there is a connection between the grid boxes.
    • Richardson imagined a "forecast factory" where thousands of human computers would be seated on galleries along the walls of a huge hall.
    • In the centre of the room, there was to be a pulpit with a "conductor" who would make sure that the computers all worked at uniform speed, in time with each other.
    • Richardson also included a research department in charge of refining the models [','R Stewart, Weather Forecasting by Computer (2009)','10; 1, pp.
    • This forecast factory is "remarkably similar to descriptions of modern multiple-processor supercomputers used in weather forecasting today" [','R Stewart, Weather Forecasting by Computer (2009)','10].
    • In Richardson's days, however, all the computations had to be done manually, and he estimated that his factory would need 64,000 human computers to master the mammoth task of calculating the weather in time with the weather actually happening.
    • The Turbulent History of Weather Prediction from Franklin’s Kite to El Nino (Hoboken NJ, 2002)','1, p.
    • In 1916, Richardson decided to join the Friends Ambulance Unit (that had been formed for conscientious objectors) and serve in the First World War, out of curiosity.
    • The Turbulent History of Weather Prediction from Franklin’s Kite to El Nino (Hoboken NJ, 2002)','1, p.
    • Very few understood the importance and the novelty of Richardson's work at the time, as, as he wrote himself, "the scheme is complicated because the atmosphere is complicated" [','J D Cox, Stormwatchers.
    • The Turbulent History of Weather Prediction from Franklin’s Kite to El Nino (Hoboken NJ, 2002)','1, p.
    • This lack of recognition and the affiliation of the Meteorological Office to the Air Ministry led Richardson to abandon meteorology.
    • He spent a considerable amount of the rest of his life studying the mathematics of war.
    • Richardson's theories were not put into practice again until the mid-1940s, when a team of scientists at the Institute for Advanced Study in Princeton developed the world's first computers.
    • Meanwhile, the groundbreaking research of meteorologists such as Jacob Bjerknes (1897-1975) and Carl-Gustaf Rossby (1898-1957) furthered scientists' knowledge of the atmosphere and helped pave the way for the eventual triumph of numerical weather forecasting.
    • Rossby in particular should be credited with this, as he almost single-handedly changed the practices of the U.S.
    • He brought the school of thought of the Bergen scientists across the pond, where he established several meteorological centres.
    • In 1939, when he held a professorship at the Massachusetts Institute of Technology (MIT), he discovered the so-called Rossby waves, which are meanders of large-scale airflows in the atmosphere.
    • They form when an airflow, for example the jet stream, is deflected either to the north or to the south (by a mountain range for example), but because of the conservation of "potential vorticity", the flow returns to its original latitude.
    • Rossby waves are very long, with only three to six oscillations around the entire planet; they play a very important role in the formation of cyclones (i.e.
    • highs) [','S Dunlop, A Dictionary of Weather, second edition (Oxford, 2008)','2, p.
    • where U is the mean westerly wind, ω is the angular velocity of the Earth, φ is the latitude, RE is the radius of the Earth and λ is the wavelength [','http://weatherfaqs.org.uk/node.145','46; 4, p.
    • The middle term on the right-hand side is the so-called Rossby parameter, which describes the variations of the Coriolis parameter with latitude, where the Coriolis parameter fC is given by: .
    • Calculating the wave speed gives the movement of the wave crests from west to east [','R B Stull, Meteorology for Scientists and Engineers, second edition (Pacific Grove CA, 2000)','4, p.
    • In terms of numerical weather prediction, this equation is important as it facilitated extended five-day forecasts [','J D Cox, Stormwatchers.
    • The Turbulent History of Weather Prediction from Franklin’s Kite to El Nino (Hoboken NJ, 2002)','1, p.
    • The work of Vilhelm Bjerknes and Richardson had divided the meteorological world into two camps: one camp defended the old method of comparing the current state of the atmosphere with past observations; the other camp campaigned for the use of physics in weather forecasting.
    • A turning point was the 6th of June 1944: D-Day.
    • For the invasion of Normandy to be a success, the commanders of the Allied forces wanted certain weather conditions (clear enough skies, relatively calm winds).
    • Their requirements for the tide and the moon phase restrained the time span in which to undertake the invasion to early June; the exact day should be determined on the basis of weather forecasts.
    • The commanders had employed advocators of the two opposing schools of thought in weather forecasting: the Norwegian meteorologist Sverre Petterssen, a disciple of the Bergen School, and the American Irving Krick, whose reputation as a forecaster mainly stemmed from his talent for selling himself.
    • The commanders decided to trust Petterssen's forecast, thereby demonstrating a great deal of trust in science [','J D Cox, Stormwatchers.
    • The Turbulent History of Weather Prediction from Franklin’s Kite to El Nino (Hoboken NJ, 2002)','1, pp.
    • D-Day not only changed world history, it also highlighted the importance of weather forecasts [','R Stewart, Weather Forecasting by Computer (2009)','10].
    • Other groundbreaking research results, be it the formulation of chaos theory, be it the invention of computers, significantly changed weather forecasting.
    • The mathematician John von Neumann (1903-1957), one of the fathers of computer science, was the first one to think of using computers to predict the weather.
    • In 1946, he presented his ideas to a group of leading meteorologists, including Carl-Gustaf Rossby and the young, gifted mathematician Jule Charney (1917-1981).
    • Charney became one of the leading scientists in von Neumann's Meteorology Project at the Institute for Advanced Study in Princeton.
    • At first, experienced human forecasters were sceptical about the quality of these forecasts, which admittedly were not as good as forecasts made by humans, but the rapid development of computers and hence their speed dramatically improved forecast quality.
    • In 1948, Charney developed the quasi-geostrophic approximation, which reduces several equations of atmospheric motions to only two equations in two unknown variables [','R S Harwood, Atmospheric Dynamics (Chapter 1: Basics, Chapter 5: Balance of Forces in Synoptic Scale Flow, Chapter 13: Quasi-Geostrophic Equations) (University of Edinburgh, 2005) ','14, chapter 13].
    • (Geostrophic winds are hypothetical winds for which the balance between the pressure-gradient force and the Coriolis effect is exact (see section 3.3) [','S Dunlop, A Dictionary of Weather, second edition (Oxford, 2008)','2, p.
    • single-layer) model of the atmosphere.
    • Further research, both in meteorology and in computer science, finally allowed the application of baroclinic (i.e.
    • Today, the world's leading meteorological centres use the most powerful computers on the planet; the new computer at the British Met Office for example is capable of 125 trillion calculations per second [','http://www.metoffice.gov.uk','42].
    • Meteorologists were thrilled by the possibilities that seemed to open up due the use of computers in weather forecasting, and were convinced that it would only be a matter of time before man could not only accurately predict, but also control the weather.
    • As opposed to other meteorologists at the time, Lorenz did not predict the weather, but investigated how predictable it is, in other words, if there are periodic patterns (the existence of patterns would have supported the belief of some old-school meteorologists that weather forecasting based on the study of past weather events yields accurate results).
    • For this he ran a shortened forecasting model on his computer, and to his great surprise, inputting data that differed from previously entered values only in the fourth decimal place, significantly changed the weather the computer predicted [',' http://www-history.mcs.st-andrews.ac.uk/Biographies/Lorenz_Edward.html, J J O’Connor, E F Robertson','38].
    • But, in the case of the atmosphere (and many other systems as well, chaotic behaviour can be found in every branch of science), you need to enter the exact same data as initial conditions in order to get the same results if you run the model several times.
    • A very simplified example for chaotic behaviour is the trajectory of a paper aeroplane: Imagine you throw a paper aeroplane in a similar manner and in the same direction say ten times.
    • Every time, the trajectory of the plane will be different, because you will never be able to throw the plane in the exact same manner twice: the force you exert or the way you hold the plane in your hand when throwing it will differ ever so slightly from throw to throw, resulting in very different, unpredictable flights.
    • Chaotic behaviour is commonly known as the "butterfly effect", a term coined due to the title of a talk Lorenz gave in 1972: Predictability: Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas? Meteorologists had presumed that small weather changes in some specified places would affect the weather in other places, but after Lorenz's discovery, they had to accept that it did not matter if a butterfly flapped its wings in Brazil, or Bulgaria, or Bangladesh, the result might still be a tornado in Texas (or somewhere else for that matter).
    • This, combined with the fact that observations of the atmosphere are usually slightly erroneous, meant that long-range forecasts would not be possible, as the small errors would build up very quickly and change the outcome considerably.
    • After about ten days it is essentially impossible to forecast weather with any degree of accuracy.
    • Modern forecast models allow for the chaotic nature of the atmosphere by a process called ensemble forecasting, meaning that the model is run several times, each time with slightly different initial conditions.
    • This method can also suggest unlikely weather conditions with quite a high degree of accuracy [','R B Stull, Meteorology for Scientists and Engineers, second edition (Pacific Grove CA, 2000)','4, p.
    • Finding a balance between detailed, accurate long-range forecasts and chaotic behaviour is still a part of modern meteorological research.
    • But before looking at current forecasting models, let us look at the primitive equations that form the basis of every such model.
    • For each weather forecast, meteorologists need to know the values of seven physical quantities: temperature, pressure, density, humidity, and wind velocity, containing three components accounting for the three different wind directions.
    • Some forecasting models also include the content of water and ice in clouds and at the ground.
    • Scientists treat the Earth's atmosphere as if it were a fluid on a rotating sphere in order to describe large-scale atmospheric processes using the fundamental laws of thermodynamics and hydrodynamics, also called the primitive equations.
    • Essentially, they are the equations of motion, one for each of the three wind directions, the continuity equation, describing the conservation of mass, the ideal gas law, and the first law of thermodynamics, describing the conservation of energy.
    • There is also an equation for determining the humidity, which is not always included in the set of the primitive equations (and is not treated here).
    • Kapitel 6: Dynamik der Atmosphare (University of Bern, 2009), 118-123','15, p.
    • Here, only the very basic versions of the primitive equations are described.
    • The equations of motion are based on Newton's second law : force equals the product of mass and acceleration.
    • The movement of an air parcel, i.e.
    • where indicates the change of velocity with respect to the time t.
    • The pressure gradient force (per unit mass) is perpendicular to the isobars (lines on weather charts joining points where the atmospheric pressure is the same [','S Dunlop, A Dictionary of Weather, second edition (Oxford, 2008)','2, p.
    • Depending on the domain in which it is applied, it denotes various quantities, for example the gradient of a scalar field, or the divergence of a vector field.
    • Because of the Earth's rotation, air moving above the planet is deflected sideways, forced to follow a curved path.
    • In the Northern hemisphere, air is deflected to the right of its direction of motion; in the Southern hemisphere, it turns to the left.
    • with being the angular velocity of a rotating system (here of the Earth) and being the velocity of an object relative to the system.
    • in the "air belt" of 1-2 km altitude above the Earth's surface, is exposed to drag against the ground.
    • Thus, the equation of motion can be rewritten as .
    • The three components of wind velocity are written as follows: .
    • Here, u is the zonal wind, parallel to the circles of latitude; v is the meridional wind, parallel to the circles of longitude; and w is the vertical wind component [','U Langematz, Vorlesung 6: Dynamik I (FU Berlin, 2009) www.geo.fu-berlin.de/met/ag/strat/lehre/sose09/Vorlesung_Mittlere_Atmosphaere/','16].
    • For the vertical component of velocity w we then have .
    • Then the equations of motion in spherical coordinates are: .
    • where Fλ and Fφ are components of frictional force.
    • For predicting the motion of winds at discrete points, a so-called Eulerian reference frame is used.
    • In this frame, the time rate of change of any quantity is expressed by .
    • The basic principle underlying the continuity equation is the conservation of mass.
    • Fundamentally, matter can be neither created out of thin air nor can it be destroyed completely; but it can be rearranged.
    • This means that if you subtract the mass flowing out of an air parcel from the mass that flew in, you obtain the change of mass within the parcel [','R B Stull, Meteorology for Scientists and Engineers, second edition (Pacific Grove CA, 2000)','4, p.
    • where ρ0(z) is the base-state density, an exponentially decreasing function of height.
    • Substituting the continuity equation by the above filter condition is called anelastic approximation [','G Doms, U Schattler, A Description of the Nonhydrostatic Regional Model LM.
    • The pressure in an air parcel is found using the equation of state, which relates pressure, temperature and density.
    • But instead of using a different gas constant, a virtual temperature defining this effect can be used.
    • In order to determine the temperature in an air parcel, forecasters make use of the first law of thermodynamics.
    • This law expresses the conservation of energy, or heat, as heat is a form of energy.
    • For example, if you have a bonfire, you need to input energy in the form of wood, and energy is released in the form of heat and light.
    • The first law of thermodynamics also states that the amount of energy added to a system cannot be bigger than the amount of energy released from the system, in other words, the amount of energy supplied is exactly balanced by the work done (within the system).
    • In terms of the atmosphere, the conservation of energy means that the temperature in an air parcel changes only when heat is added or removed.
    • This property is very important for the formation of clouds [','R B Stull, Meteorology for Scientists and Engineers, second edition (Pacific Grove CA, 2000)','4, p.
    • The equation describing the change of temperature T with respect to time t is: .
    • Here, cp is the specific heat of air, with constant pressure, and is the diabatic rate of latent heat release per unit mass.
    • The term latent heat means hidden heat, and describes the amount of energy released or absorbed by a substance -- water, in the case of the atmosphere -- during a change of state of that substance.
    • Jule Charney and his colleagues had simplified them so that the early computers could handle them, but nowadays meteorologists have gone back to use all of Richardson's equations.
    • A six-hour forecast for North America for 11 April 2010, with fine resolution, produced on 11 April 2010 using the North American Mesoscale Model (NAM) of the U.
    • Also, the models include equations accounting for the effects of small-scale processes such as convection, radiation, turbulence and the effects of mountains that cannot be represented explicitly by the forecasting models, as their resolution is not high enough.
    • During data assimilation, real observations are combined with predicted conditions so as to give the best possible estimate of the actual state of the atmosphere.
    • Assuming that hydrostatic balance applies limits the smallest possible grid spacing to about 5 - 10 km [','http://www.dwd.de','29], which is not fine enough a resolution for detailed, accurate forecasts of small-scale weather events like thunderstorms.
    • Again, the primitive equations have to be re-written in terms of ζ [','G Doms, U Schattler, A Description of the Nonhydrostatic Regional Model LM.
    • Discretization means that the atmosphere (or the part of it that you want to study) is represented by a finite number of numerically approximated values.
    • In order to make a forecast for the future (at time step t + Δt), you do not start at the present time step t, but at the previous step t - Δt, and the forecast leaps over the time step t (with Δt denoting the size of the time step, that is the difference between two points in time).
    • One of the main causes for instability are truncation errors, which happen when a variable ψ is represented by a Taylor series, i.e.
    • as an infinite sum of its values at the individual grid points.
    • Due to computational reasons, only the very first terms of the series, which are in fact the most important ones, can be used, but the higher-order terms influence the accuracy of the series.
    • As a second order scheme, the leapfrog scheme is fairly accurate; an even better scheme is the so-called Runge-Kutta method, which is of fourth order (but it will not be described here).
    • However in most of the current models only second order schemes can be applied due to computer capacities.
    • The accuracy of all forecasting models is tested regularly; statistical models have been developed for this.
    • The traditional grid structure is based on dividing the Earth's surface into a large number of squares, such that there is a high air column above each square, as illustrated in figure 6.
    • The atmosphere is then divided into a number of layers, resulting in a three-dimensional grid, in which the primitive equations can be solved for each grid point.
    • Processes in the upper atmosphere influence the weather, though, so the whole of the atmosphere has to be considered in a forecasting model.
    • Over the years, the resolution of the grids has become higher (i.e.
    • the edge length of each square has become smaller).
    • The world's leading weather services such as the British Met Office and the German DWD use three different grids: a global grid spanning the whole planet, a so-called regional model covering Europe (and North America in the case of the Met Office's model), and a local model covering the UK or Germany, respectively.
    • The regional model of the DWD has a resolution of 7 km, and the local model has a resolution of up to 2.8 km (the Met Office's models have coarser resolution as the Met Office does not work with non-hydrostatic equations).
    • Both of these local models, and all of the Met Office models, are based on a rectangular grid, whereas the DWD's global model is based on a triangular grid with a 40 km resolution (see figure 7).
    • The great advantage of the triangular grid is that the primitive equations can be solved in air parcels close to the poles without any problems, as opposed to the rectangular grid, where the longitudes approach each other, resulting in erroneous computations.
    • The position of the grid points in the computational space is defined by .
    • Here, Nα is the number of grid points in the α-direction; λ0 and φ0 are the values of λ and φ in the southwest corner of the model domain [','G Doms, U Schattler, A Description of the Nonhydrostatic Regional Model LM.
    • For this, we need to define approximations to the derivatives at a specified grid point xl in terms of finite differences.
    • The value of a variable ψ at xl is given by ψl; and the finite difference for ψl is given using the values of ψl+1 and ψl-1, i.e.
    • The behaviour of these two terms can be described using Taylor expansion: .
    • Subtracting the second from the first expansion gives the centred finite difference approximation to the first derivative of ψl: .
    • However, the lowest power of the difference of x, Δx, in E gives the order of the approximation.
    • These schemes both have order 1, but there are situations where it is favourable to use these approximations instead of a centred approximation.
    • It is also possible to derive the finite difference approximations to the second and third derivatives of ψl.
    • Not only space, but also time has to be discretized, and time derivatives can also be represented as finite difference approximations, that is in terms of values at discrete time levels [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • The grid point value of the variable ψl at time step tn is denoted by ψnl; its derivative can be expressed as a centred finite difference approximation: .
    • The explicit scheme is much easier to solve than the implicit one, as it is possible to compute the new value of ψl at time n+1 for every grid point, provided the values of ψl are known for every grid point at the current time step n [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • But the choice of the time step is limited in order to keep the scheme stable.
    • The implicit scheme, on the other hand, is absolutely stable, but it results in a system of simultaneous equations, so is more difficult to solve [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • The approximations described above are very simple examples illustrating the general idea of finite differences.
    • When the primitive equations are expressed in terms of finite differences, the equations soon become very long and take some computational effort to solve.
    • These time steps are then subdivided into several small time steps Δτ, over which the terms sψ are integrated [','G Doms, U Schattler, A Description of the Nonhydrostatic Regional Model LM.
    • representing a set of equations that can be solved using Gaussian elimination [','G Doms, U Schattler, A Description of the Nonhydrostatic Regional Model LM.
    • The term f nψ is constant throughout the small time step integrations, but the value of ψn+1 is not known before the last one of these integrations has been completed.
    • The term is the result of a process called averaging; you assume that the mean value of ψn+1 does not vary as fast with respect to both space and time than deviations from the mean would.
    • If you re-write the primitive equations using finite differences, you get, "after considerable algebra" [','G Doms, U Schattler, A Description of the Nonhydrostatic Regional Model LM.
    • 67], a linear tridiagonal system of simultaneous equations which can be written in the general form .
    • However, the application of this method to the primitive equations was crucial to the development of numerical weather forecasting, as it was the only mathematical method that could simplify partial differential equations needed for forecasting for several decades.
    • A further disadvantage of the finite difference method, other than the great number of equations you have to solve, is that it does not reveal anything about the behaviour of the variables between the individual grid points.
    • In 1976, the Australian and Canadian weather services were the first ones to adopt this method, which is now used by a range of weather services across the globe; the European Forecasting Centre ECMWF in Reading, for example, adopted it in 1983 [','F Baer, The Spectral Method: Its Impact on NWP (2004) ','20].
    • One of the advantages of the spectral method is that the primitive equations can be solved in terms of global functions rather than in terms of approximations at specific points as in the finite difference method.
    • For the spectral method, the atmosphere has to be represented in terms of spectral components.
    • In the ECMWF model, the atmosphere is divided into 91 layers (in comparison, the DWD's and the Met Office's global models have 40 layers), with the number of layers in the boundary layer equalling the number of layers in the uppermost 45 km of the atmosphere.
    • The partial differential equations are represented in terms of spherical harmonics, which are truncated at a total wave number of 799.
    • This corresponds to a grid length of roughly 25 km [','F Grazzini, A Persson, User Guide to ECMWF forecast products, version 4.0 (Meteorological Bulletin M3.2, 14.03.2007)','7] (the DWD's and the Met Office's global model has a resolution of 40 km).
    • In essence, in using the spectral method, you assume that an unknown variable ψ can be approximated in terms of a sum of N+1 linearly dependent basis functions ψn(x): .
    • When this series is substituted into an equation of the form Lψ = f (x), where L is a differential operator, you get a so-called residual function: .
    • The residual function is zero when the solution of the equation above is exact, therefore the series coefficients an should be chosen such that the residual function is minimised, i.e.
    • In the majority of cases, polynomial approximations, such as Fourier series or Chebyshev polynomials, are the best choice; but when it comes to weather forecasting, the use of spherical coordinates demands that spherical harmonics are used as expansion functions.
    • This increases the complexity of the problem, and the computational effort required to solve it.
    • A simple example that can be solved in terms of a Fourier series illustrates the idea of the spectral method: One of the processes described by the primitive equations is advection (which is the transport of for instance heat in the atmosphere), and the non-linear advection equation is given by .
    • This can be re-written in terms of the longitude λ: .
    • Having chosen appropriate boundary conditions, the equation can be expanded in terms of a finite Fourier series: .
    • where Fm is a series in terms of the um.
    • As each of the terms on the left-hand side of the equation has been truncated at a different wave number, there will always be a residual function.
    • It is difficult to calculate the non-linear terms of a differential equation in the context of the spectral method, but you can get around this problem by using a so-called transform method.
    • Firstly, the individual components of the non-linear term u and are expressed in terms of spectral coefficients at discrete grid points λi: .
    • Secondly, the advection term, that is the product of these components, is calculated at every grid point in the discretizised space: .
    • This procedure has to be done at every time level, so results in a significant amount of calculation.
    • As has been mentioned above, a dependent variable ψ has to be expanded in terms of spherical harmonics rather than Fourier coefficients when spherical coordinates are used.
    • Spherical harmonics Ynm(λ, φ) are the angular part of the solution to Laplace's equation.
    • The vertical components of velocity transform like scalars, so can be expanded in terms of spherical harmonics straightaway.
    • Expansion in terms of spherical harmonics requires transform methods as well, but these methods are more complicated and computationally more expensive than Fast Fourier Transforms (although the use of transform methods greatly reduces the time required for calculations).
    • Spherical harmonics are two-dimensional, so they are much more difficult to solve than expansions in terms of Fourier series.
    • In general, spectral method algorithms are more difficult to program than their finite difference counterparts; also the domains in which they are used have to be regular in order to keep the high accuracy of this method.
    • However, the spectral method has a number of advantages: for example, there is no pole problem when the method is used.
    • Furthermore, it can handle finite elements of higher orders than the finite difference method can.
    • As a result, the solutions of many problems are very accurate.
    • Moreover, a finite series expansion in terms of linearly independent functions approximates the variation of ψ within a specified element (e.g.
    • a set of grid points) [','R W Riddaway, Numerical Methods, revised March 2001 (by M Hortal) (Meteorological Training Course Lecture Series, 2002) ','17, p.
    • The domain for which the partial differential equations have to be solved is divided into a number of subdomains, and a different polynomial is used to approximate the solution for each subdomain.
    • The fact that only low-order polynomials can be used is reflected in the comparatively low accuracy, but the amount of necessary calculations is much smaller than for finite differences or for spectral methods [','J P Boyd, Chebyshev and Fourier Spectral Methods, second edition (Mineola NY, 2000) ','5, p.
    • On the other hand, there are a number of choices for the basis functions, and depending on which functions are used, the finite element method can give very accurate results when it is applied to irregular grids.
    • Thus, the use of this method is not restricted to triangular and rectangular grids only as is the finite difference method.
    • To get to the point, most of today's research in meteorology is devoted to improving current forecasting models.
    • This involves the use of enhanced observation and measurement techniques as well as refining the mathematical models.
    • Several ways to improve forecast quality are already known in theory, but they cannot be implemented due to a lack of computer power.
    • Since the 1960s, when issuing numerical weather forecasts calculated by computers on a regular basis begun, forecast accuracy has been accompanied by the development of faster computers, and it seems that this will be the case for the foreseeable future.
    • One of the easiest ways to increase the quality of weather predictions is to increase the orders of the numerical approximations to partial differential equations.
    • Most schemes used in current models are of second order, but using third-order schemes would greatly improve forecast accuracy [','http://www.dwd.de','29].
    • Thereby, the initial conditions entered into forecasting models represent the actual state of the atmosphere more accurately, which leads to better forecasts [','http://www.dwd.de','29].
    • More detailed and more accurate mathematical methods as well as increased computer power will allow meteorologists to increase the resolution of their grids.
    • Most models include as many topographical features as possible, but according to the DWD, there is a new grid model using horizontal planes cutting through mountains, permitting a more accurate representation of the equilibrium of the atmospheric forces in the proximity of mountains.
    • The accuracy of a medium-range forecast is measured in terms of the anomaly correlation coefficient (ACC), which has to be above 60% in order for a forecast to be considered accurate.
    • The February forecasts of the ECMWF were consistently above this limit [','http://ecmwf.int/','32].
    • Of course, fundamental research both in mathematics and in physics concerning atmospheric processes and the underlying mathematical and physical laws will continue to be crucial for the development of better forecasting models.
    • A team of international researchers, led by Shaun Lovejoy of McGill University in Montreal, follows a different path than most scientist studying atmospheric processes.
    • Apart from founding numerical weather forecasting, Richardson devoted a considerable amount of time to studying the atmosphere, in particular eddies and whirls.
    • He noticed that atmospheric phenomena, for example clouds, seem to be cascade-like processes, with large-scale structures containing many ever-smaller copies of themselves, like Russian dolls sitting inside each other.
    • In modern mathematics, this kind of phenomenon is called a fractal and its behaviour can be described using power laws (meaning that one quantity changes as another quantity is raised to some power).
    • Richardson did not have the mathematical means to prove his assumptions, and mathematicians have only recently begun to investigate if the atmosphere can be described by a set of power laws.
    • In a report published in 2009, Lovejoy's team provides evidence that rainfall is indeed a collection of fractals.
    • cloudy with a chance of fractals (NewScientist, No 2733, 7 November 2009), 40-43','25].
    • Weather forecasting has come a very long way since the Babylonians and the Greeks started observing the skies, and it was the pioneering work of Vilhelm Bjerknes and Lewis Fry Richardson at the beginning of the 20th century that kicked off the development of modern weather forecasting.
    • But without the invention and subsequent improvement of computers, numerical weather prediction would still be in its infancy.
    • Current one-day forecasts are accurate in 9 out of 10 cases, and three-day forecasts still have a hit rate of 70%.
    • Apart from normal weather forecasts, weather services also issue specialised forecasts for a variety of domains, such as agriculture, aviation and shipping, which help save lives and money.
    • The two different types of forecasting models, one of them based on finite differences, the other one based on the spectral method, are currently competing as to which one of them yields more accurate forecasts for a given computational cost.
    • But at the end of the day, each model has its strengths and weaknesses; so using both models side by side will probably give the best results.
    • Article by: Stefanie Eminger (University of St Andrews .
    • List of References (48 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Weather_forecasts.html .

  5. References for Chinese overview
    • References for Chinese overview .
    • D Bodde, Chinese Thought, Society and Science : The Intellectual and Social Background of Science and Technology in Pre-Modern China (Honolulu, 1991).
    • R Calinger, A contextual history of mathematics (New York, 1999).
    • C Cullen, Astronomy and Mathematics in Ancient China (Cambridge, 1996).
    • L Y Lam and T S Ang, Fleeting footsteps : Tracing the conception of arithmetic and algebra in ancient China (River Edge, NJ, 1992).
    • Chinese mathematics in the thirteenth century (Cambridge, Mass., 1973).
    • Y Li, Materials for the Study of the History of Ancient Chinese Mathematics (Chinese) (Shanghai, 1963).
    • Y Li and S R Du, Chinese mathematics (New York, 1987).
    • J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).
    • Y Mikami, The Development of Mathematics in China and Japa (New York, 1974).
    • Chinese science : explorations of an ancient tradition (Cambridge, Mass., 1973).
    • B Qian, History of Chinese mathematics (Chinese) (Peking, 1981).
    • B Qian (ed.), Ten Mathematical Classics (Chinese) (Beijing, 1963).
    • N Sivin, Cosmos and early Chinese mathematical astronomy (Leiden, 1969).
    • F J Swetz, The Sea Island Mathematical Manual : surveying and mathematics in ancient China (Pennsylvania, 1992).
    • F J Swetz and T I Kao, Was Pythagoras Chinese? (Reston, Virginia, 1977).
    • Li Yen, An outline of Chinese mathematics (Chinese) ( Peking 1958).
    • K Chemla, Reflections on the world-wide history of the rule of false double position, or : How a loop was closed, Centaurus 39 (2) (1997), 97-120.
    • C Cullen, Learning from Liu Hui? A different way to do mathematics, Notices Amer.
    • Y Z Dong and Y Yao, The mathematical thought of Liu Hui (Chinese), Qufu Shifan Daxue Xuebao Ziran Kexue Ban 13 (4) (1987), 99-108.
    • L Y Lam, and K S Shen, The Chinese concept of Cavalieri's principle and its applications, Historia Math.
    • D Liu, 400 years of the history of mathematics in China - an introduction to the major historians of mathematics since 1592, Historia Sci.
    • S K Mo, Jiuzhang suanshu (Nine chapters on the mathematical art) and Liu Hui's commentary (Chinese), Stud.
    • K S Shen, Historical development of the Chinese remainder theorem, Arch.
    • P D Straffin Jr., Liu Hui and the first golden age of Chinese mathematics, Math.

  6. Nine chapters
    • Chinese Mathematics .
    • The Jiuzhang suanshu or Nine Chapters on the Mathematical Art is a practical handbook of mathematics consisting of 246 problems intended to provide methods to be used to solve everyday problems of engineering, surveying, trade, and taxation.
    • It has played a fundamental role in the development of mathematics in China, not dissimilar to the role of Euclid's Elements in the mathematics which developed from the foundations set up by the ancient Greeks.
    • There is one major difference which we must examine right at the start of this article and this is the concept of proof.
    • It is well known what that Euclid, for example, gives rigorous proofs of his results.
    • Failure to see similar rigorous proofs in Chinese works such as the Nine Chapters on the Mathematical Art led to historians believing that the Chinese gave formulas without justification.
    • This however is simply an example of historians well versed in mathematics which is essentially derived from Greek mathematics, thinking that Chinese mathematics was inferior since it was different.
    • Recent work has begun to correct this false impression and understand that there are different understandings of "proof".
    • For example in [',' K Chemla, Relations between procedure and demonstration : Measuring the circle in the ’Nine chapters on mathematical procedures’ and their commentary by Liu Hui (3rd century), in History of mathematics and education: ideas and experiences (Essen, 1992) (1996), 69-112.','8] Chemla shows that Chinese mathematicians certainly understood how to give convincing arguments that their methodology for solving particular problems was correct.
    • Let us now give a short description of each of the nine chapters of the book.
    • This consists of 38 problems on land surveying.
    • It looks first at area problems, then looks at rules for the addition, subtraction, multiplication and division of fractions.
    • The Euclidean algorithm method for finding the greatest common divisor of two numbers is given.
    • The types of shapes for which the area is calculated include triangles, rectangles, circles, trapeziums.
    • This chapter contains 46 problems concerning the exchange of goods, particularly the exchange rates among twenty different types of grains, beans, and seeds.
    • The mathematics involves a study of proportion and percentages and introduces the rule of three for solving proportion problems.
    • Many of the problems seem simple an excuse to give the reader practice at handling difficult calculations with fractions.
    • Here there are 20 problems which again involve proportion, many involving different sums given to or owed by officials of various different ranks.
    • In particular arithmetic and geometric progressions are used in some of the problems.
    • This chapter contains 24 problems and takes its name from the first eleven problems which ask what the length of a field will be if the width is increased but the area kept constant.
    • These first eleven problems involve unit fractions are all of the following type, where n = 2, 3, 4, ..
    • Problems 12 to 18 involve the extraction of square roots, and the remaining problems involve the extraction of cube roots.
    • Notions of limits and infinitesimals appear in this chapter.
    • Liu Hui whose commentary of 263 AD has become part of the text attempts to find the volume of a sphere, gives an approximate formula which he shows to be incorrect, then charmingly writes:- .
    • Here there are 28 problems on the construction of canals, ditches, dykes, etc.
    • Volumes of solids such as prisms, pyramids, tetrahedrons, wedges, cylinders and truncated cones are calculated.
    • Liu Hui, in his commentary, discusses a "method of exhaustion" he has invented to find the correct formula for the volume of a pyramid.
    • Chapter 6: Fair Distribution of Goods.
    • The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit.
    • [Answer: 15/74 of a day] .
    • The 20 problems give a rule of double false position.
    • we try x = i, and instead of c we get c + d.
    • Then we try x = j, and instead of c we obtain c + e.
    • Find the number of people and the total cost of the items.
    • [Answer: There are 7 people and the total cost of the items is 53 coins.] .
    • The two piles of coins weigh the same.
    • It is now found that the pile of mainly gold coins weighs 13 units less than the pile of mainly silver coins.
    • Find the weight of a silver coin and of a gold coin.
    • Here 18 problems which reduce to solving systems of simultaneous linear equations are given.
    • However the method given is basically that of solving the system using the augmented matrix of coefficients.
    • The matrix is then reduced to triangular form, using elementary column operations as is done today in the method of Gaussian elimination, and the answer interpreted for the original problem.
    • The first 13 problems are solved using an application of Pythagoras's theorem, which the Chinese knew as the Gougu rule.
    • Two problems study what are now called Pythagorean triples, while the remainder use the theory of similar triangles.
    • Here is an example of one using similar triangles; it is Problem 20:- .
    • There is a square town of unknown dimensions.
    • There is a gate in the middle of each side.
    • What are the dimensions of the town.
    • From C the tree at A is just visible so the line CA passes through the corner D of the square.
    • [Answer: The side of the town is 250 paces] .
    • Quadratic equations are considered for the first time in Chapter 9, are solved by an analogue of division using ideas from geometry, in fact from the Chinese square-root algorithm, rather than from algebra.
    • Having looked at the content of the work, let us think next about its date.
    • In the past, the tyrant Qin burnt written documents, which led to the destruction of classical knowledge.
    • Later, Zhang Cang, Marquis of Peiping and Geng Shouchang, Vice-President of the Ministry of Agriculture, both became famous through their talent for calculation.
    • Because of the ancient texts had deteriorated, Zhang Cang and his team produced a new version removing the poor parts and filling in the missing parts.
    • Most historians, however, would not believe that the original text of the Nine Chapters on the Mathematical Art was nearly as old as Liu Hui believed.
    • In fact most historians think that the text originated around 200 BC after the burning of the books by Shih Huang Ti.
    • What methods are used to try to date the material? Perhaps the most important is to examine the units of length, volume and weight which appear in the various problems.
    • Standard decimal units of length were established in China around 200 BC and later further subdivisions occurred.
    • That the basic units are used, but not the later subdivisions, leads to a date of shortly after 200 BC.
    • Of course, the dating using units of length is not conclusive.
    • If you pick up a book with mathematics problems given in decimal currency then we could argue as above and say that the book was written after 1970.
    • However new editions of popular textbooks were brought out when the currency changed, so many older books appeared in decimal editions.
    • The Nine Chapters on the Mathematical Art was certainly an important text, so may have had its units of length brought up to date as it evolved.
    • Is there other evidence for dating parts of the Nine Chapters on the Mathematical Art other than units of measurement? Yes, there are.
    • Problems contain references to taxes, methods of distributing goods, towns, and parks which all point to slightly different dates for different parts of the text but 206 BC to 50 AD covering these different dates.
    • In addition to Liu Hui's commentary of 263, there was another important later commentary, namely that of Li Chunfeng whose commentary was written around 640 when he headed a team asked to annotate The Ten Classics.
    • Li Chunfeng corrected and clarified some of Liu Hui's comments, expanding on much of what had been pretty concisely written.
    • has dominated the history of Chinese mathematics.
    • Now although European science does not appear to have reached China in sixteenth century, it has been pointed out that a number of mathematical formulas and rules which were widely used in Europe during that century are essentially identical to formulas written down in the Nine Chapters on the Mathematical Art.
    • This leads to an interesting question which historians have as yet no convincing answer, namely were the European formulas taken directly from those of China.
    • List of References (32 books/articles) .
    • Chinese Mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Nine_chapters.html .

  7. Indian numerals
    • Ancient Indian Mathematics index .
    • It is worth beginning this article with the same quote from Laplace which we give in the article Overview of Indian mathematics.
    • 3">The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India.
    • the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.
    • The purpose of this article is to attempt the difficult task of trying to describe how the Indians developed this ingenious system.
    • We will examine two different aspects of the Indian number systems in this article.
    • Of course it is important to realise that there is still no standard way of writing these numerals.
    • The different fonts on this computer can produce many forms of these numerals which, although recognisable, differ markedly from each other.
    • The second aspect of the Indian number system which we want to investigate here is the place value system which, as Laplace comments in the quote which we gave at the beginning of this article, seems "so simple that its significance and profound importance is no longer appreciated." We should also note the fact, which is important to both aspects, that the Indian number systems are almost exclusively base 10, as opposed to the Babylonian base 60 systems.
    • It was the advent of printing which motivated the standardisation of the symbols.
    • One of the important sources of information which we have about Indian numerals comes from al-Biruni.
    • Before he went there al-Biruni already knew of Indian astronomy and mathematics from Arabic translations of some Sanskrit texts.
    • In India he made a detailed study of Hindu philosophy and he also studied several branches of Indian science and mathematics.
    • Al-Biruni wrote 27 works on India and on different areas of the Indian sciences.
    • In particular his account of Indian astronomy and mathematics is a valuable contribution to the study of the history of Indian science.
    • And just as the shape of the letters that they use for writing is different in different regions of their country, so the numerical symbols vary.
    • Historians trace them all back to the Brahmi numerals which came into being around the middle of the third century BC.
    • Of course different inscriptions differ somewhat in the style of the symbols.
    • Here is one style of the Brahmi numerals.
    • We should now look both forward and backward from the appearance of the Brahmi numerals.
    • Moving forward leads to many different forms of numerals but we shall choose to examine only the path which has led to our present day symbols.
    • First, however, we look at a number of different theories concerning the origin of the Brahmi numerals.
    • There have been quite a number of theories put forward by historians over many years as to the origin of these numerals.
    • In [',' G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).','1] Ifrah lists a number of the hypotheses which have been put forward.
    • The Brahmi numerals came from the Indus valley culture of around 2000 BC.
    • Basically these hypotheses are of two types.
    • One is that the numerals came from an alphabet in a similar way to the Greek numerals which were the initial letters of the names of the numbers.
    • The second type of hypothesis is that they derive from an earlier number system of the same broad type as Roman numerals.
    • For example the Aramaean numerals of hypothesis 2 are based on I (one) and X (four): .
    • Ifrah examines each of the six hypotheses in turn and rejects them, although one would have to say that in some cases it is more due to lack of positive evidence rather than to negative evidence.
    • Ifrah proposes a theory of his own in [',' G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).','1], namely that:- .
    • the first nine Brahmi numerals constituted the vestiges of an old indigenous numerical notation, where the nine numerals were represented by the corresponding number of vertical lines ..
    • To enable the numerals to be written rapidly, in order to save time, these groups of lines evolved in much the same manner as those of old Egyptian Pharonic numerals.
    • Taking into account the kind of material that was written on in India over the centuries (tree bark or palm leaves) and the limitations of the tools used for writing (calamus or brush), the shape of the numerals became more and more complicated with the numerous ligatures, until the numerals no longer bore any resemblance to the original prototypes.
    • However, it would appear that we will never find convincing proof for the origin of the Brahmi numerals.
    • The name literally means the "writing of the gods" and it was the considered the most beautiful of all the forms which evolved.
    • What we [the Arabs] use for numerals is a selection of the best and most regular figures in India.
    • The way in which the Indian numerals were spread to the rest of the world between the 7th to the 16th centuries in examined in detail in [',' R C Gupta, Spread and triumph of Indian numerals, Indian J.
    • In this paper, however, Gupta claims that Indian numerals had reached Southern Europe by the end of the 5th century but his argument is based on the Geometry of Boethius which is now known to be a forgery dating from the first half of the 11th century.
    • We now turn to the second aspect of the Indian number system which we want to examine in this article, namely the fact that it was a place-value system with the numerals standing for different values depending on their position relative to the other numerals.
    • Although our place-value system is a direct descendant of the Indian system, we should note straight away that the Indians were not the first to develop such a system.
    • The Indians were the first to develop a base 10 positional system and, considering the date of the Babylonian system, it came very late indeed.
    • The oldest dated Indian document which contains a number written in the place-value form used today is a legal document dated 346 in the Chhedi calendar which translates to a date in our calendar of 594 AD.
    • This document is a donation charter of Dadda III of Sankheda in the Bharukachcha region.
    • Therefore, despite the doubts, we can be fairly sure that this document provides evidence that a place-value system was in use in India by the end of the 6th century.
    • Many other charters have been found which are dated and use of the place-value system for either the date or some other numbers within the text.
    • a donation charter of Dhiniki dated 794 in the Vikrama calendar which translates to a date in our calendar of 737 AD.
    • an inscription of Devendravarman dated 675 in the Shaka calendar which translates to a date in our calendar of 753 AD.
    • a donation charter of Danidurga dated 675 in the Shaka calendar which translates to a date in our calendar of 737 AD.
    • a donation charter of Shankaragana dated 715 in the Shaka calendar which translates to a date in our calendar of 793 AD.
    • a donation charter of Nagbhata dated 872 in the Vikrama calendar which translates to a date in our calendar of 815 AD.
    • an inscription of Bauka dated 894 in the Vikrama calendar which translates to a date in our calendar of 837 AD.
    • All of these are claimed to be forgeries by some historians but some, or all, may well be genuine.
    • The first inscription which is dated and is not disputed is the inscription at Gwalior dated 933 in the Vikrama calendar which translates to a date in our calendar of 876 AD.
    • Further details of this inscription is given in the article on zero.
    • The problem here is the dating of this manuscript, a topic which is examined in detail in our article on the Bakhshali manuscript.
    • We are left, of course, with asking the question of why the Indians developed such an ingenious number system when the ancient Greeks, for example, did not.
    • A number of theories have been put forward concerning this question.
    • A second hypothesis is that the idea for place-value in Indian number systems came from the Chinese.
    • In particular the Chinese had pseudo-positional number rods which, it is claimed by some, became the basis of the Indian positional system.
    • This view is put forward by, for example, Lay Yong Lam; see for example [',' L Y Lam, Linkages : exploring the similarities between the Chinese rod numeral system and our numeral system, Arch.
    • Lam argues that the Chinese system already contained what he calls the:- .
    • three essential features of our numeral notation system: (i) nine signs and the concept of zero, (ii) a place value system and (iii) a decimal base.
    • A third hypothesis is put forward by Joseph in [',' G G Joseph, The crest of the peacock (London, 1991).','2].
    • Freudenthal is another historian of mathematics who supports the theory that the idea came entirely from within India.
    • To see clearly this early Indian fascination with large numbers, we can take a look at the Lalitavistara which is an account of the life of Gautama Buddha.
    • In Lalitavistara Gautama, when he is a young man, is examined on mathematics.
    • He lists the powers of 10 up to 1053.
    • It is stories such as this, and many similar ones, which convince Joseph that the fascination of the Indians with large numbers must have driven them to invent a system in which such numbers are easily expressed, namely a place-valued notation.
    • He writes in [',' G G Joseph, The crest of the peacock (London, 1991).','2]:- .
    • The early use of such large numbers eventually led to the adoption of a series of names for successive powers of ten.
    • The importance of these number names cannot be exaggerated.
    • The word-numeral system, later replaced by an alphabetic notation, was the logical outcome of proceeding by multiples of ten.
    • The decimal place-value system developed when a decimal scale came to be associated with the value of the places of the numbers arranged left to right or right to left.
    • However, the same story in Lalitavistara convinces Kaplan (see [',' R Kaplan, The nothing that is : a natural history of zero (London, 1999).','3]) that the Indians' ideas of numbers came from the Greeks, for to him the story is an Indian version of Archimedes' Sand-reckoner.
    • All that we know is that the place-value system of the Indians, however it arose, was transmitted to the Arabs and later into Europe to have, in the words of Laplace, profound importance on the development of mathematics.
    • List of References (11 books/articles) .
    • Ancient Indian Mathematics index .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_numerals.html .

  8. Zero
    • A history of Zero .
    • Ancient Indian Mathematics index .
    • One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form.
    • If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it.
    • The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different.
    • Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct.
    • The second use of zero is as a number itself in the form we use it as 0.
    • There are also different aspects of zero within these two uses, namely the concept, the notation, and the name.
    • Neither of the above uses has an easily described history.
    • Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today.
    • There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five".
    • Remarkably, original texts survive from the era of Babylonian mathematics.
    • The Babylonians wrote on tablets of unbaked clay, using cuneiform writing.
    • The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform).
    • Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended).
    • The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used.
    • There is one common feature to this use of different marks to denote an empty position.
    • This is the fact that it never occured at the end of the digits but always between two digits.
    • Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.
    • We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.
    • Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics.
    • How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry.
    • In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines.
    • Here we find the first use of the symbol which we recognise today as the notation for zero, for Greek astronomers began to use the symbol O.
    • Some historians favour the explanation that it is omicron, the first letter of the Greek word for nothing namely "ouden".
    • Other explanations offered include the fact that it stands for "obol", a coin of almost no value, and that it arises when counters were used for counting on a sand board.
    • By this time Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived.
    • Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself.
    • The idea of the zero place (certainly not thought of as a number by Ptolemy who still considered it as a sort of punctuation mark) makes its next appearance in Indian mathematics.
    • Of course that is not to say that the Indian system did not owe something to earlier systems and many historians of mathematics believe that the Indian use of zero evolved from its use by Greek astronomers.
    • As well as some historians who seem to want to play down the contribution of the Indians in a most unreasonable way, there are also those who make claims about the Indian invention of zero which seem to go far too far.
    • For example Mukherjee in [',' R Mukherjee, Discovery of zero and its impact on Indian mathematics (Calcutta, 1991).','6] claims:- .
    • the mathematical conception of zero ..
    • What is certain is that by around 650AD the use of zero as a number came into Indian mathematics.
    • In fact there is evidence of an empty place holder in positional numbers from as early as 200AD in India but some historians dismiss these as later forgeries.
    • The first record of the Indian use of zero which is dated and agreed by all to be genuine was written in 876.
    • The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple.
    • Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.
    • We now come to considering the first appearance of zero as a number.
    • From early times numbers are words which refer to collections of objects.
    • Certainly the idea of number became more and more abstract and this abstraction then makes possible the consideration of zero and negative numbers which do not arise as properties of collections of objects.
    • Of course the problem which arises when one tries to consider zero and negatives as numbers is how they interact in regard to the operations of arithmetic, addition, subtraction, multiplication and division.
    • The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
    • In 830, around 200 years after Brahmagupta wrote his masterpiece, Mahavira wrote Ganita Sara Samgraha which was designed as an updating of Brahmagupta's book.
    • Since this is clearly incorrect my use of the words "seem to lead him into error" might be seen as confusing.
    • Despite the passage of time he is still struggling to explain division by zero.
    • A quantity divided by zero becomes a fraction the denominator of which is zero.
    • In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
    • At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not.
    • The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero.
    • Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.
    • However their use of zero goes back further than this and was in use before they introduced the place-valued number system.
    • You can see a separate article about Mayan mathematics.
    • The brilliant work of the Indian mathematicians was transmitted to the Islamic and Arabic mathematicians further west.
    • It came at an early stage for al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning which describes the Indian place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.
    • Ibn Ezra, in the 12th century, wrote three treatises on numbers which helped to bring the Indian symbols and ideas of decimal fractions to the attention of some of the learned people in Europe.
    • The Book of the Number describes the decimal system for integers with place values from left to right.
    • In 1247 the Chinese mathematician Qin Jiushao wrote Mathematical treatise in nine sections which uses the symbol O for zero.
    • A little later, in 1303, Zhu Shijie wrote Jade mirror of the four elements which again uses the symbol O for zero.
    • Fibonacci was one of the main people to bring these new ideas about the number system to Europe.
    • As the authors of [',' L Pogliani, M Randic and N Trinajstic, Much ado about nothing - an introductive inquiry about zero, Internat.
    • 75">An important link between the Hindu-Arabic number system and the European mathematics is the Italian mathematician Fibonacci.
    • It is significant that Fibonacci is not bold enough to treat 0 in the same way as the other numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 since he speaks of the "sign" zero while the other symbols he speaks of as numbers.
    • Although clearly bringing the Indian numerals to Europe was of major importance we can see that in his treatment of zero he did not reach the sophistication of the Indians Brahmagupta, Mahavira and Bhaskara nor of the Arabic and Islamic mathematicians such as al-Samawal.
    • One might have thought that the progress of the number systems in general, and zero in particular, would have been steady from this time on.
    • He would have found his work in the 1500's so much easier if he had had a zero but it was not part of his mathematics.
    • By the 1600's zero began to come into widespread use but still only after encountering a lot of resistance.
    • Of course there are still signs of the problems caused by zero.
    • Of course they celebrated the passing of only 1999 years since when the calendar was set up no year zero was specified.
    • List of References (14 books/articles) .
    • A History topic on Mayan mathematics .
    • Ancient Indian Mathematics index .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html .

  9. References for Babylonian mathematics
    • References for Babylonian mathematics .
    • A Aaboe, Episodes from the Early History of Mathematics (1964).
    • R Calinger, A conceptual history of mathematics (Upper Straddle River, N.
    • J Friberg, The third millenium roots of Babylonian mathematics.
    • A method for the decipherment, through mathematical and metrological analysis, of proto-Sumerian and proto-Elamite semipictographic inscriptions, Department of Mathematics, University of Goteborg No.
    • G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
    • G G Joseph, The crest of the peacock (London, 1991).
    • S K Adhikari, Babylonian mathematics, Indian J.
    • L Brack-Bernsen and O Schmidt, Bisectable trapezia in Babylonian mathematics, Centaurus 33 (1) (1990), 1-38.
    • E M Bruins, A contribution to the interpretation of Babylonian mathematics; triangles with regular sides, Nederl.
    • E M Bruins, Fermat problems in Babylonian mathematics, Janus 53 (1966), 194-211.
    • E M Bruins, Refinement of approximations in Babylonian mathematics (Russian), Istor.-Mat.
    • J Friberg, Methods and traditions of Babylonian mathematics.
    • S Gandz, A few notes on Egyptian and Babylonian mathematics, in Studies and Essays in the History of Science and Learning Offered in Homage to George Sarton on the Occasion of his Sixtieth Birthday, 31 August 1944 (New York, 1947), 449-462.
    • S Gandz, Studies in Babylonian mathematics.
    • Indeterminate analysis in Babylonian mathematics, Osiris 8 (1948), 12-40.
    • S Gandz, Studies in Babylonian mathematics.
    • Conflicting interpretations of Babylonian mathematics, Isis 31 (1940), 405-425.
    • S Gandz, Studies in Babylonian mathematics.
    • Isoperimetric problems and the origin of the quadratic equations, Isis 32 (1940), 101-115.
    • J Hoyrup, Babylonian mathematics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 21-29.
    • D E Knuth, Ancient Babylonian algorithms, Twenty-fifth anniversary of the Association for Computing Machinery, Comm.
    • H Lewy, Marginal notes on a recent volume of Babylonian mathematical texts, J.
    • J S Liu, A general survey of Babylonian mathematics (Chinese), Sichuan Shifan Daxue Xuebao Ziran Kexue Ban 16 (1) (1993), 80-87.
    • K Muroi, An enigmatic sentence in the old Babylonian table of exponents and logarithms, Historia Sci.
    • K Muroi, Babylonian mathematics - ancient mathematics written in cuneiform writing (Japanese), in Studies on the history of mathematics (Kyoto, 1998), 160-171.
    • K Muroi, Extraction of cube roots in Babylonian mathematics, Centaurus 31 (3-4) (1988), 181-188.
    • K Muroi, Inheritance problems of Babylonian mathematics, Historia Sci.
    • K Muroi, Interest calculation of Babylonian mathematics: new interpretations of VAT 8521 and VAT 8528, Historia Sci.
    • K Muroi, Small canal problems of Babylonian mathematics, Historia Sci.
    • K Muroi, The expressions of zero and of squaring in the Babylonian mathematical text VAT 7537, Historia Sci.
    • K Muroi, Two harvest problems of Babylonian mathematics, Historia Sci.
    • M A Powell Jr, The antecedents of old Babylonian place notation and the early history of Babylonian mathematics, Historia Math.
    • A E Raik, From the early history of algebra.
    • Reciprocals of regular sexagesimal numbers, J.
    • Schools of Oriental Research no.
    • G Sarton, Remarks on the study of Babylonian mathematics, Isis 31 (1940), 398-404.
    • M Wygodski, Mathematics of the ancient Babylonians (Russian), Uspekhi Matem.
    • M Wygodski, The mathematics of the ancient Babylonians (Russian), Uspekhi Matem.

  10. Tait's scrapbook
    • On 22 March 2006 I [EFR] lectured to the Institute of Physics on Peter Guthrie Tait.
    • I gave the lecture at 14 India Street, Edinburgh, the birthplace in 1831 of James Clerk Maxwell, and now owned by the James Clerk Maxwell Foundation.
    • The house where James Clerk Maxwell was born is at 14 India Street, Edinburgh about a fifteen-minute walk from the railway station which is in the centre of Edinburgh.
    • The hall of the house is impressive with two marble pillars giving an instant impression of grandeur.
    • On the wall facing you as you enter are two large portraits, the rightmost one of James Clerk Maxwell, the left most one being a portrait of his school friend P G Tait.
    • Portraits of James Clerk Maxwell's family are on the walls and a Display Cabinet near the windows contains a fine collection of items associated with Maxwell.
    • Documents relating to the purchase of the house, which was built in 1820, are in the Display Cabinet and, especially for the occasion, Tait's school medals are on display.
    • Mr Murray Tait, the great-grandson of P G Tait, presented it to the James Clerk Maxwell Foundation in 2003.
    • It must have been created by members of his family after his death using papers and newspaper cuttings which Tait had preserved.
    • The Scrapbook contains obituaries of Tait (these are the first items), newspaper cuttings which contain anything about him, letters he sent to the newspapers, copies of examinations he set, syllabuses for courses he gave, letters sent to him by Maxwell, Thomson and many others, articles about golf which mention his work in that area, poems by Tait, Maxwell and others etc.
    • There are a host of cuttings which are reviews of the book, speculation as to who the authors were, etc.
    • P G Tait was born in Dalkeith on 28 April 1831, the son of John Tait, secretary to Walter Francis Scott, fifth duke of Buccleuch, and his wife, Mary Ronaldson.
    • There were two important aspects of the move to Edinburgh.
    • One was that the family went to live in Somerset Cottage, the home of John Ronaldson, Mary's brother.
    • The other aspect was that he became friendly with Maxwell around the middle of his time at the Academy.
    • From this time forward I became very intimate with him, and we discussed together, with schoolboy enthusiasm, numerous schoolboy problems, among which I remember particularly the various plane sections of a ring or tore, and the form of a cylindrical mirror which should show one his own image unperverted.
    • Much of the manuscript is written in beautiful calligraphy.
    • Most of the manuscripts are by Tait and signed 'fecit P G Tait' with a date.
    • Tait was top of his class in each one of his six years at Edinburgh Academy but, of course, Maxwell was not in the same class.
    • There were school prizes open to all pupils and in 1846 Tait came third overall but first in mathematics, while in the following year Maxwell came first in mathematics with Tait second.
    • Here is an extract from the announcement of the 1846 prizes (taken from the Scrapbook):- .
    • The Academical Club Prize was this year awarded to the successful Competitor in all the departments of study pursued by the Academy.
    • The branches of study were divided into five departments, viz.
    • Greek - Latin - Mathematics - English and French - Geography, History and Scripture; and printed Examination Papers, containing questions and exercises on each of these, were successively put into the hands of the Competitors, who returned written answers, without leaving the school-room, and without any assistance of any kind.
    • In Mathematics Tait (fifth class) came top, Lewis Campbell (sixth class) came second, Maxwell (sixth class) came third.
    • It is indicated that these three did significantly better than six others who also gained distinction in mathematics.
    • Tait also gained distinction in Latin (Maxwell didn't), Lewis Campbell came second in Greek despite going on to have a distinguished career as a professor of Greek, Tait and Maxwell both achieved distinctions in English and French and in Geography, History and Scripture.
    • In November 1847, Tait entered the University of Edinburgh.
    • Maxwell entered Edinburgh University at the same time at Tait and together they attended the second mathematics class taught by Kelland and the natural philosophy (physics) class taught by James David Forbes.
    • In January 1852, at the age of twenty, he graduated as senior Wrangler in the Mathematical Tripos.
    • This means that he was placed first among the First Class degrees in mathematics awarded by Cambridge in that year.
    • The full list of results in the Mathematical Tripos of that year are in the Scrapbook:- .
    • List of Honors .
    • Bachelor of Arts' Commencement .
    • In this he met with limited success: only one student employed him as a coach but that student was more successful than any of Hopkins' students.
    • Of the Mathematical Tripos he also commented:- .
    • coaches spend their lives in discovering which pages of a textbook a man ought to read.
    • Also during these years he collaborated with William John Steele on writing A Treatise on Dynamics of a Particle intended as a text for Cambridge students.
    • Steele was a friend who after being a pupil of Professor William Thomson at Glasgow University was at Peterhouse and graduated second to Tait in the Tripos.
    • In September 1854 Tait took up the professorship of mathematics at Queen's College Belfast.
    • He wrote later of Andrews:- .
    • I have always regarded it as one of the most important determining factors in my own life (private as well as scientific) and one for which I cannot be sufficiently thankful, that my appointment to the Queen's College at the age of twenty-three brought me for six years into almost daily association with such a friend.
    • Another important friendship that evolved from this time was with his engineering colleague James Thomson (brother of William, later Lord Kelvin).
    • it was only in August last that I suddenly bethought me of certain formulas I had admired years ago on page 610 of your Lectures - and I thought (and still think) likely to serve my purpose exactly.
    • The title (in German) I forget - but a manuscript translation of my own which I now have beside me is headed "Vortex motion" ..
    • Two of his friends at Peterhouse were sons of the Rev James Porter and through them Tait met their sister, Margaret Archer Porter, who he married in Belfast on 13 October 1857.
    • The Chair of Natural Philosophy at the University of Edinburgh became vacant in 1859, J D Forbes having moved to the University of St Andrews to become Principal.
    • there is another quality which is desirable in a Professor in a University like ours and that is the power of oral exposition proceeding on the supposition of imperfect knowledge or even total ignorance on the part of pupils.
    • The first extract is from an article about Tait in the student newspaper of the University of Edinburgh:- .
    • Did one wish to give a stranger a good impression of the style and quality of the lectures which are delivered to Edinburgh students, without a doubt he would take him some forenoon to hear Professor Tait.
    • Two of us a few days ago freshened old memories by a visit to the Natural Philosophy Classroom.
    • We were not long seated before the door leading to the Physical Laboratory opened, and the tall figure of the Professor slipped round it.
    • The lecturer has lost nothing of his ancient power of graphic illustration and charming style.
    • The conclusions are obtained from various trains of reasoning, and apt reference to everyday commonplaces clinches the argument, while the subtle humour of the man is every now and then revealed, and serves to keep his audience in good fettle.
    • Professor Tait was the last of the old school amongst the Edinburgh professoriate, and he always insisted on maintaining its traditional vogue, no matter how quaint these might now be.
    • He was accustomed, therefore, to make his daily entrance to his lecture theatre a procedure of some ceremony.
    • The Professor himself, invariably appeared clad in his academic robes; first of all, an upright majestic figure was seen in the background, then a trip over the doorway, as he entered the classroom with a magnificent and courtly bow to the class.
    • This, of course, always met with a rapturous applause from the class, which as inevitably brought a smile to the old man's face as he stalked to his desk, and at once began to lecture.
    • It was not the force of discipline, because Tait seemed to ignore that, but the sheer force of interest which the Professor compelled in his hearers as he explained the forces physical which he loved so well.
    • As a lecturer his reputation was so great that he drew students to study his one subject alone from all parts of the civilised world.
    • Tait was proud of the University of Edinburgh as is clear in his address to graduates in 1888 (taken from the Scrapbook):- .
    • The University of Edinburgh never stood higher, in the estimation of those at least whose judgement is of any value, than on the occasion of its Tercentenary four short years ago.
    • To speak of my own department alone: - what University, home or foreign, has been fortunate enough to see assembled at its celebrations such a collection of the very foremost of the world's mathematicians and physicists as then graced this hall? To name only a very few, we had from the continent Cremona, Helmholtz, Hermite, and Mendeleeff; with Cayley, Salmon, Stokes, Sylvester, and Thomson from our own islands.
    • I have always held, I think in accordance with all the highest authority, that the great object of a University is to teach what is alike surely known and of value when known, and to add to the utmost of our power to the store of really useful human knowledge.
    • In comparison with these great objects, the conferring of degrees is a secondary and relatively unimportant matter.
    • Professor Tait's Report of the Class in Experimental Physics .
    • I undertook, with considerable hesitation, to give a course of lectures on Experimental Physics, which should be (though elementary) strictly scientific, and not (in the common and degraded sense of the word) popular.
    • In other words, I determined that, if I tried the experiment at all, I should do it with the sole view of imparting accurate information; and all mere sensational displays being sedulously avoided.
    • I intimated in my opening lecture that I was perfectly ignorant of what might reasonably be expected of my class - so far at least as regarded their fitness by preparation, not by natural capacity, for attacking the subject.
    • I was greatly pleased to find that the want of preliminary training did not interfere, to any serious extent, with their progress.
    • I look upon the experiment as a very successful one indeed, and have no longer any fears of the effects of defective previous training.
    • I have never had, nor (considering the small number of lectures into which so much had to be compressed) one in which the progress made was more marked.
    • seems to threaten in a direct manner one of the most valuable of a Scotsman's birthrights.
    • The Scottish Universities have hitherto made it their proudest boast that they are the property of the Scottish people, without distinction of rank, age or sect.
    • Is he now to be deprived of this right, or are obstacles to be gratuitously put in the way of enjoying it? I said "gratuitously", I should rather have said "wantonly".
    • Experience has led me to the deliberate conclusion that the less a man knows of Natural Philosophy when he enters my Ordinary Class, the better his progress in that fascinating subject.
    • Again many men who have exceptionally high qualifications for the study of experimental science, men who may, if properly guided, render invaluable service to science and to their country, are hopelessly incapable of mastering elementary mathematics, or even the trivial pedantries of grammar.
    • Are these men to be stopped on the very threshold of the career for which nature has specially qualified them? To them a degree is not an object; they come to the University to obtain from it the information they desire, and which it is the primary duty of a University to give to all comers.
    • The Scrapbook contains newspaper cuttings of letters which Tait had sent on a wide variety of topics.
    • In the following exchange he is complaining of sewage problems in St Andrews in 1878.
    • SIR, - In the Fifeshire Journal of today I observe a long letter from Professor Tait complaining of a sickening smell arising from some drain or sewer at Pilmour Links, and I had the curiosity to visit the spot and judge for myself how matters really stood; but to my surprise I could find no trace whatever of the sickening smell complained of.
    • SIR, -One who can think it possible to mistake the smell of coal gas for the stench of sewage is scarcely qualified to speak in the name of the Drainage Committee.
    • The drainage of St Andrews is about as bad as it could be; and, when this has been pointed out, it is vain to try to trail a red-herring across the scent, as you correspondent "Civis" has done.
    • Instant and thorough action is necessary if the prosperity of the town is to be maintained.
    • The Scrapbook contains information about Tait's solution of the 15 puzzle.
    • Tait had solved the puzzle and submitted a paper on the solution when he saw two articles in the American Journal of Mathematics in 1879, one by W W Johnson and one by W E Story.
    • Johnson proved that an odd permutation of the pieces of the puzzle is impossible to obtain, Story proved that every even permutation of the squares is possible.
    • He writes a permutation as the product of disjoint cycles, then notes that a cycle of odd length can be written as the product of an even number of transpositions.
    • Similarly Chinese would signify the Aryan rotated right-handedly through a quadrant, and Mongol the Semitic rotated left-handedly through a quadrant.
    • Now it is easily seen that Aryan is changed into Semitic, and Chinese into Mongol, or vice versa, by an odd number of interchanges.
    • Similarly Aryan and Mongol, and Semitic and Chinese, differ by an even number of interchanges.
    • The former can be changed into Mongol, the latter into Chinese.
    • It is tempting to think of Tait as a minor figure compared with the other two giants of Scottish science, but this is far from the truth.
    • Maxwell, who we now rate as the most important of the three, was the most modest and least forceful.
    • We note Maxwell's admiration for the work of Tait expressed in a letter of 1871 to Thomson:- .
    • You should let the world know that the true source of mathematical methods as applicable to physics is to be found in the Proceedings of the Royal Society of Edinburgh.
    • The volume- surface- and line-integrals of vectors and quaternions and their properties as in the course of being worked out by T' (Tait) is worth all that is going on in other seats of learning.
    • It has been said that everything that William Thomson did in the first half of his career was brilliant, and everything he did in the second half of his career was wrong.
    • However modern physics has shown that Thomson's intuition was rather incredible since string theory bases the understanding of fundamental particles on knot theory.
    • Geniuses are sometimes wrong, but it is always worth looking at their failures for they often contain gems of wisdom that should not be lost.
    • The Scrapbook contains lots of examples of the correspondence between the three friends.
    • Here is an example of a letter from Maxwell to Tait:- .
    • How did Butcher discover the power of the dot?..
    • If you make the density of electricity zero and it velocity infinity in a current keeping the product what it ought to be, you may call "Thermoelectric Power" "Entropy of Electricity.
    • Also the momentum of the current vanishes as compared with its energy.
    • But it is much more likely that electricity is very dense and very slow so the less said of Entropy the better.
    • If the Earl of March were created a May-Duke determine the probability of his becoming a Winter-King and if he were elected Emperor would it be right to address him as Semper Augustus.
    • Of course James Clerk Maxwell was jcm ( and therefore became dp/dt).
    • His son Freddie Tait went on to be the double winner of the amateur golf championship.
    • One of Professor Tait's most original and interesting investigations was carried on with the assistance of the ever ready Lindsay, on the trajectory and velocity of a driven golf ball.
    • The many experiments extended over several months, during which the Professor used to "bunker" himself intentionally and drive the ball against a wall of sand in order to obtain a measure of the force employed.
    • The activity was at last concluded, however, and the results embodied in a paper which was submitted to the Royal Society of Edinburgh.
    • Scientifically the statement was accepted, but, unfortunately for its permanent accuracy, the ever jolly "Freddy" proved his father to be quite in the wrong by driving his ball some five yards farther within a fortnight of the announcement of the "maximum possible drive!".
    • Here are a small number of further extracts from the Scrapbook.
    • Isabelle must feel herself now half a Scotch girl - I beg pardon; I should have said Scottish; at least Tait instructed me that I might speak of scotching a snake, but I was by no means to apply such a term to the inhabitants of that part of the island that lies north of the Tweed.
    • Here is just a small flavour of this aspect of Tait contained in an extract from the Scrapbook:- .
    • Clausius ingeniously endeavours to put me out of court by calling me 'der englische Mathematiker, Tait.' Of course, if I were merely a mathematician, and had never made physical investigations, I should have no title to write on such matters at all.
    • Still, I have preferred to retain the discussion of these so-called claim, inasmuch as I may thus be spared the necessity for farther discussion of a subject on which my opponents will at least not allow that they are and have been hopelessly in error.
    • To call a Scotsman "English" is, of course, one of the greatest insults possible! .
    • Finally let us record that the Scrapbook contains an invitation to Tait to attend the degree ceremony where he is to be awarded an honorary degree by the University of Edinbirgh.
    • The University of Edinburgh conferred "the honorary degree of Doctor of Laws on Professor P.
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Tait_scrapbook.html .

  11. Chinese problems
    • Chinese problems .
    • Chinese Mathematics .
    • We give here a collection of Chinese problems which are extracted from various articles in our archive on Chinese mathematics or Chinese mathematicians.
    • Many of the problems have answers given in the corresponding article, and some have a description of the method.
    • The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit.
    • Now 1 cubic cun of jade weighs 7 liang, and 1 cubic cun of rock weighs 6 liang.
    • Now there is a cube of side 3 cun consisting of a mixture of jade and rock which weighs 11 jin.
    • Tell: what are the weights of jade and rock in the cube.
    • How many are there of each? .
    • Find the number of people and the total cost of the items.
    • The two piles of coins weigh the same.
    • It is now found that the pile of mainly gold coins weighs 13 units less than the pile of mainly silver coins.
    • Find the weight of a silver coin and of a gold coin.
    • There is a square town of unknown dimensions.
    • There is a gate in the middle of each side.
    • What are the dimensions of the town.
    • Suppose we have an unknown number of objects.
    • Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points.
    • B walks a distance of 256 pu eastwards.
    • Then A walks a distance of 480 pu south before he can see B.
    • Find the diameter of the town.
    • Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points.
    • B leaves the east gate and walks straight ahead a distance of 16 pu, when he just sees A.
    • Find the diameter of the town.
    • Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points.
    • 135 pu directly out of the south gate is a tree.
    • If one walks 15 pu out of the north gate and then turns east for a distance of 208 pu, the tree comes into sight.
    • Find the diameter of the town.
    • Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points.
    • A tree lies three li north of the northern gate.
    • Find the circumference and the diameter of the city wall.
    • The land area is 13 mou and 71/2 tenths of a mou.
    • Find the length of the side of the farm and the diameter of the pond.
    • Now a pile of rice is against the wall with a base circumference 60 chi and an altitude of 12 chi.
    • What is the volume? Another pile is at an inner corner, with a base circumference of 30 chi and an altitude of 12 chi.
    • What is the volume? Another pile is at an outer corner, with base circumference of 90 chi and an altitude of 12 chi.
    • Given the diameter of the field and the breadth of the river find the area of the non-flooded part of the field.
    • In the right-angled triangle with sides of length a, b and c with a > b > c, we know that a + b = 81 ken and a + c = 72 ken.
    • The sum of the base and height of the triangle is 17 bu.
    • What is the sum of the base and hypotenuse? .
    • What are the values of the three sides.
    • There are three persons, A, B, and C each with a number of coins.
    • A says "If I take 2/3 of B's coins and 1/3 of C's coins then I hold 100".
    • B says If I take 2/3 of A's coins and 1/2 of C's coins then I hold 100 coins".
    • C says "If I take 2/3 of A's coins and 2/3 of B's coins, then I hold 100 coins".
    • If a Wenzhou orange costs 7 coins, a green orange 3 coins, and 3 golden oranges cost 1 coin, how many oranges of the three kinds will be bought? .
    • A number of pheasants and rabbits are placed together in the same cage.
    • Find the number of pheasants and rabbits.
    • Given the relations 2yz = z2 + xz and 2x + 4y + 4z = x(y2 - z + x) between the sides of a right angled triangle x, y, z where z is the hypotenuse, find d = 2x + 2y.
    • If the cube law is applied to the rate of recruiting soldiers and it is found that on the first day 3 cubed are recruited, 4 cubed on the second day, and on each succeeding day the cube of a number one greater than the previous day are recruited, how many soldiers in total will have been recruited after 15 days? How many after n days? .
    • Let d be the diameter of the circle inscribed in a right triangle (you should use the relation d = x + y - z where x, y, z are as defined below).
    • Let x, y be the lengths of the two legs and z the length of the hypotenuse of the triangle.
    • Chinese Mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Chinese_problems.html .

  12. Fractal Geometry
    • A History of Fractal Geometry .
    • Any mathematical concept now well-known to school children has gone through decades, if not centuries of refinement.
    • A typical student will, at various points in her mathematical career -- however long or brief that may be -- encounter the concepts of dimension, complex numbers, and "geometry".
    • If the field of mathematics does not particularly interest her, this student might see these concepts as distinct and unrelated and, in particular, she might make the mistake of thinking that the Euclidean geometry taught to her in school encompasses the whole of the field of geometry.
    • However, if she were to pursue mathematics at the university level, she might discover an exciting and relatively new field of study that links the aforementioned ideas in addition to many others: fractal geometry.
    • While the lion's share of the credit for the development of fractal geometry goes to Benoit Mandelbrot, many other mathematicians in the century preceding him had laid the foundations for his work.
    • Moreover, Mandelbrot owes a great deal of his advancements to his ability to use computer technology -- an advantage that his predecessors distinctly lacked; however, this in no way detracts from his visionary achievements.
    • Nevertheless, while acknowledging and understanding the accomplishments of Mandelbrot, it undoubtedly helps to have some familiarity with the relevant works of Karl Weierstrass, Georg Cantor, Felix Hausdorff, Gaston Julia, Pierre Fatou and Paul Levy -- not only to make Mandelbrot's work clearer -- but to see its connections to other branches of mathematics.
    • Equally, while most authors will not fail to include at least brief discussion of Mandelbrot's rather interesting and slightly unconventional (for a modern mathematician) life in their texts on fractals, it seems only fair to give some, if not equal, consideration to his predecessors.
    • Until the 19th century, mathematics had concerned itself only with functions that produced differentiable curves.
    • Indeed, the conventional wisdom of the day said that any function with an analytic formula (i.e.
    • sum of a convergent power series) would certainly produce such a curve.
    • ','3] However, on July 18, 1872, Karl Weierstrass presented a paper at the Royal Prussian Academy of Sciences showing that for a a positive integer and 0 < b < 1 .
    • Using the limit definition of a derivative, he showed that the difference quotient of the function .
    • gets arbitrarily large as the index of summation increases.
    • as an example of a non-differentiable analytic function, but never published a proof, nor could anyone replicate it.
    • A E Gerald (Addison -Wesley, 1993).','14] Thus, Weierstrass's proof stands as the first rigorously proven example of a function that is analytic, but not differentiable.
    • While Weierstrass, and indeed, much of the mathematical establishment of the time eschewed the use of graphs in favour of symbolic manipulation in order to prove results, future mathematicians such as Helge von Koch and Mandelbrot himself found it useful to represent their results graphically.
    • [','H von Koch, Dictionary of Scientific Biography.
    • While these are both approximations, one can see that these functions lack the smoothness of parabolas or of the sine and cosine functions.
    • These functions resisted traditional analysis and were -- though not due to their appearance, which was beyond the ability of mathematicians of the day to represent -- labelled "monsters" by Charles Hermite and were largely ignored by the contemporary mathematical community.
    • In 1883 Georg Cantor, who attended lectures by Weierstrass during his time as a student at the University of Berlin [','J J O’Connor, J.J., and E F Robertson.
    • ','3] introduced a new function, ψ , for which ψ' = 0 except on the set of points, {z}.
    • The Cantor set has a Lebesgue measure of zero; however, it is also uncountably infinite.
    • ','3] What is more, it has the property of being self-similar, meaning that if one magnifies a section of the set, one obtains the whole set again.
    • Looking at Figure 4, one can easily see that each horizontal line is one third the size of the horizontal line directly above it.
    • In fact, self-similarity is a feature of fractals, and the Cantor set is an early example of a fractal, though self-similarity was not defined until 1905 (by Cesaro, who was analysing the paper by Helge von Koch discussed below) and fractals were not defined until Mandelbrot in 1975, [','R M Crownover, Introduction to Fractals and Chaos (London, 1995).','2] thus Cantor would not have thought of it in those terms.
    • it seems to me that his [Weierstrass's] example is not satisfactory from the geometrical point of view since the function is defined by an analytic expression that hides the geometrical nature of the corresponding curve and so from this point of view one does not see why the curve has no tangent.
    • Von Koch's curve, like the Cantor set, has the property of self-similarity.
    • He merely aimed to provide an alternative way of proving that functions that were non-differentiable (i.e.
    • In doing so, von Koch expressed a link between these non-differentiable "monsters" of analysis and geometry.
    • Many of his other results were derived from those of Henri Poincare, from whom he knew it was possible to obtain "pathological" results -- i.e.
    • these so-called "monsters" -- but never really explored them, outside of the aforementioned essay.
    • [','H von Koch, Dictionary of Scientific Biography.
    • 1972 ','5] Poincare, it should be noted, studied non-linear dynamics in the later 19th century, which eventually led to chaos theory, [','R M Crownover, Introduction to Fractals and Chaos (London, 1995).','2] a field closely related to fractal geometry, though beyond the scope of this paper.
    • It is therefore fitting that a mathematician whose work followed that of Poincare so closely would turn out to be one of the forefathers of a field that is closely related to the area of study for which Poincare himself helped lay the foundations.
    • An absolutely key concept in the study of fractals, aside from the aforementioned self-similarity and non-differentiability, is that of Hausdorff dimension, a concept introduced by Felix Hausdorff in March of 1918.
    • Hausdorff's results from the same paper were important to the field of topology, as well; [','G A Edgar, ed.
    • ','3] however that his definition of dimension extended the previous definition to allow for sets to have a dimension that is an arbitrary, non-zero value [','F Hausdorff, Dictionary of Scientific Biography.
    • 1972 ','4] (unlike topological dimension) ended up being integral to the definition of a fractal, as Mandelbrot defined fractals "a set having Hausdorff dimension strictly greater than its topological dimension." [','R M Crownover, Introduction to Fractals and Chaos (London, 1995).','2] .
    • As soon as Hausdorff introduced this new, expanded definition of dimension, it was the subject of investigation -- in particular by Abraham Samilovitch Besicovitch, who, from 1934 to early 1937 wrote no less than three papers referencing Hausdorff's work.
    • He was forced to give up his post as a professor at the University of Bonn in 1935, and even though he continued to work on set theory and topology, his work could only be published outside of Germany.
    • [','F Hausdorff, Dictionary of Scientific Biography.
    • The Hausdorff dimension, d, of a self-similar set -- its connection to fractal geometry, though, as previously stated, there are many other applications of Hausdorff dimension -- which is scaled down by ratios r1 , r2 , ..
    • the first iteration of the set is the whole set, scaled down by a factor of r1) satisfies the following two equations [','R M Crownover, Introduction to Fractals and Chaos (London, 1995).','2]: .
    • These equations, however, do not appear in Hausdorff's paper, as they relate directly to fractals (and calculating the dimension of a fractal), which were ideas that would have been unknown to Hausdorff.
    • They studied mappings of the complex plane and iterative functions.
    • Their work with iterative functions led to the ideas of attractors, points in space which attract other points to them; and repellors, points in space that repel other points, usually to another attractor.
    • The boundaries of the various basins of attraction turned out to be very complicated and are known today as Julia sets, [','N Lesmoir-Gordon, W Rood, and R Edney.
    • Introducing Fractal Geometry (Cambridge, 2000).','7] an example of which can be seen in Figure 6.
    • A more analytic definition of a Julia set for a function, f (z), is [','R M Crownover, Introduction to Fractals and Chaos (London, 1995).','2] .
    • Namely, "the Julia set of f is the boundary of the set of points z ∈ C that escape to infinity under repeated iteration by f (z)." [','R M Crownover, Introduction to Fractals and Chaos (London, 1995).','2] .
    • Because Fatou and Julia (and, by extension, their work) predated computers, they were unable to generate pictures such as the one on the right, which is the graph of millions of iterations of a function.
    • Introducing Fractal Geometry (Cambridge, 2000).','7] Julia published a 199-page paper in 1918 called Memoire sur l'iteration des fonctions rationelles Ⓣ, which discussed much of his work on iterative functions and describing the Julia set.
    • With this paper, Julia won the Grand Prix of the Academie des Sciences and became extremely famous in mathematical circles throughout the 1920s.
    • Fatou, on the other hand, did not achieve the same level of fame as Julia, even contemporarily, despite discovering very similar results -- though in a different manner -- and also submitting them to be published.
    • He submitted an announcement of his results to Comptes Rendus, while Julia had chosen to send his opus to the Journal de Mathematiques Pures et Appliquees.
    • Julia, protective of his work, sent letters to Comptes Rendus asking them to investigate whose results had priority.
    • On rare occasions, they can be "dendrites" (Figure 8), where they are "made up completely of continuously sub-branching lines, which are only just connected since the removal of any point from them would split them in two," [','N Lesmoir-Gordon, W Rood, and R Edney.
    • The method for deciding whether or not a set is connected is to calculate out the orbit of the starting point.
    • In 1938, the year after Besicovitch's last paper on Hausdorff dimension, Paul Levy produced a comprehensive treatment on the property of self-similarity.
    • He showed that the von Koch curve was just one of many examples of a self-similar curve, though von Koch himself had stated that his curve could be generalized.
    • The curves generated by Levy (see Figure 9 for an example -- the green and blue sets are two smaller copies of the larger set) were iterative and connected and, with enough iterations, covers (or tiles) the plane.
    • Levy's curves, however, are not fractals, as they have both a Hausdorff and a topological dimension of two.
    • Little did anyone at this time suspect that there was someone, albeit still a very young person, who would unite the works of Levy and Hausdorff.
    • One of Mandelbrot's uncles, Szolem Mandelbrojt, was a pure mathematician, who took an interest in the young Mandelbrot and tried to steer him towards mathematics.
    • In fact, in 1945, Mandelbrojt showed his nephew the works of Fatou and Julia, though the young Mandelbrot initially did not take much of an interest.
    • Mandelbrot, like Helge von Koch before him, preferred visual representations of mathematical problems, as opposed to the symbolic, [','N Lesmoir-Gordon, W Rood, and R Edney.
    • Introducing Fractal Geometry (Cambridge, 2000).','7] though this may also stem from his lack of formal education, due to World War II.
    • ','13] Unfortunately, this would bring him into direct conflict with the teaching style of "Bourbaki", a group of mathematicians whose belief in solving problems analytically (as opposed to visually) dominated the teaching of mathematics in France at the time.
    • He did very well in the mathematics section, where he could employ his ability to solve problems through visualisation to answer questions.
    • Introducing Fractal Geometry (Cambridge, 2000).','7] and after a one-day career at the Ecole Normale, Mandelbrot started at the Ecole Polytechnique, where he met another of his mentors, Paul Levy, [','---.
    • The company gave him a free hand in choosing a topic of study, which allowed him to explore and develop concepts using his own methods, without having to worry about the reaction of the academic community.
    • In 1967, while still there, Mandelbrot wrote his landmark essay, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension [','B Mandelbrot, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.
    • Science, New Series 156 3775 (May 5, 1967): 636-638.','8], in which he linked the idea of previous mathematicians to the real world -- namely coastlines, which he claimed were "statistically self-similar".
    • He argued that [','B Mandelbrot, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension.
    • Self-similarity methods are a potent tool in the study of chance phenomena, including geostatics, as well as economics and physics.
    • After this essay and with the aid of computers, Mandelbrot returned to the work of Julia and Fatou.
    • With the ability to see, for the first time, what these sets looked like in their limits, Mandelbrot came up with the idea of mapping the values of c ∈ C for which the Julia set for the function fc (z) = z2 + c is connected.
    • When one zooms in on some part of the edge, one notices that the Mandelbrot set is, indeed, self-similar.
    • Furthermore, if one zooms in even further on various sections of the edge, one obtains different Julia sets.
    • In fact, it is "asymptotically similar to Julia sets near any point on its boundary," as proved in a theorem by the Chinese mathematician Tan Lei.
    • Mandelbrot has managed not only to invent the discipline of fractal geometry, but has also popularized it through its applications to other areas of science.
    • The rare scholars who are nomads-by-choice are essential to the intellectual welfare of the settled disciplines.
    • As he hinted in How Long Is the Coast of Britain? fractal geometry comes in useful in representing natural phenomena; things such as coastlines, the silhouette of a tree, or the shape of snowflakes -- things are not easily represented using traditional Euclidean geometry.
    • Equally, no simple shape from Euclidean geometry comes to mind when contemplating things such as the path of a river.
    • Furthermore, fractal geometry and chaos theory have important connections to physics, medicine, and the study of population dynamics.
    • Introducing Fractal Geometry (Cambridge, 2000).','7] However, even if the field lacked these links, it would be hard for those so inclined to resist the aesthetic appeal of most fractals.
    • Mandelbrot's non-traditional approach led him to invent an amazing and useful new form of mathematics.
    • He also benefitted from access to computers, which allowed him not only to build upon the works of others in a new way -- one which had definitely not been done before -- but to use his preferred method of solving problems -- namely visualisation.
    • Furthermore, his invention also makes a case for the importance of the study of pure mathematics: until Mandelbrot came along and united the eclectic ideas of Hausdorff, Julia, et al, they represented very abstract mathematical ideas from varying branches of (pure) mathematics.
    • However, through fractal geometry, many of these seemingly abstract ideas (from mathematicians who are relatively unknown outside of their own spheres of research) develop applications that other scientists and even non-scientists can appreciate.
    • Thus, the work that eventually led to fractals and their applications are an excellent counterexample to the arguments of anyone who would dare to denigrate the study of pure mathematics.
    • Article by: Holly Trochet (University of St Andrews) .
    • List of References (14 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/fractals.html .

  13. Chinese numerals
    • Chinese numerals .
    • Chinese Mathematics .
    • In 1899 a major discovery was made at the archaeological site at the village of Xiao dun in the An-yang district of Henan province.
    • Thousands of bones and tortoise shells were discovered there which had been inscribed with ancient Chinese characters.
    • The site had been the capital of the kings of the Late Shang dynasty (this Late Shang is also called the Yin) from the 14th century BC.
    • The last twelve of the Shang kings ruled here until about 1045 BC and the bones and tortoise shells discovered there had been used as part of religious ceremonies.
    • Questions were inscribed on one side of a tortoise shell, the other side of the shell was then subjected to the heat of a fire, and the cracks which appeared were interpreted as the answers to the questions coming from ancient ancestors.
    • The importance of these finds, as far as learning about the ancient Chinese number system, was that many of the inscriptions contained numerical information about men lost in battle, prisoners taken in battle, the number of sacrifices made, the number of animals killed on hunts, the number of days or months, etc.
    • Here is a selection of the symbols that were used.
    • There was also a symbol for 10000 which we have not included in the illustration above but it took the form of a scorpion.
    • The additive nature of the system was that symbols were juxtaposed to indicate addition, so that 4359 was represented by the symbol for 4000 followed by the symbol for 300, followed by the symbol of 50 followed by the symbol for 9.
    • Because we have not illustrated many numbers above here is one further example of a Chinese oracular number.
    • There are a number of fascinating questions which we can consider about this number system.
    • Although the representation of the numbers 1, 2, 3, 4 needs little explanation, the question as to why particular symbols are used for the other digits is far less obvious.
    • By this we mean that since the number nine looks like a fish hook, then perhaps the sound of the word for 'nine' in ancient Chinese was close to the sound of the word for 'fish hook'.
    • Again the symbol for 1000 is a 'man' so perhaps the word for 'thousand' in ancient Chinese was close to the sound of the word for 'man'.
    • A second theory about the symbols comes from the fact that numbers, and in fact all writing in this Late Shang period, were only used as part of religious ceremonies.
    • We have explained above how the inscriptions were used by soothsayers, who were the priests of the time, in their ceremonies.
    • This theory suggests that the number symbols are of religious significance.
    • Of course it is possible that some of the symbols are explained by the first of these theories, while others are explained by the second.
    • Again symbols such as the scorpion may simply have been used since swarms of scorpions meant "a large number' to people at that time.
    • Perhaps the symbol for 100 represents a toe (it does look like one), and one might explain this if people at the time counted up to ten on their fingers, then 100 for each toe, and then 1000 for the 'man' having counted 'all' parts of the body.
    • However a second form of Chinese numerals began to be used from the 4th century BC when counting boards came into use.
    • A counting board consisted of a checker board with rows and columns.
    • The most significant property of representing numbers this way on the counting board was that it was a natural place valued system.
    • The Chinese adopted a clever way to avoid this problem.
    • They used both forms of the numbers given in the above illustration.
    • The alternating forms of the numbers again helped to show that there was indeed a space.
    • For example Sun Zi, in the first chapter of the Sunzi suanjing Ⓣ, gives instructions on using counting rods to multiply, divide, and compute square roots.
    • What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10.
    • This illustrates the significance of using counting board numerals.
    • Now the Chinese counting board numbers were not just used on a counting board, although this is clearly their origin.
    • They were used in written texts, particularly mathematical texts, and the power of the place valued notation led to the Chinese making significant advances.
    • In particular the "tian yuan" or "coefficient array method" or "method of the celestial unknown" developed out of the counting board representation of numbers.
    • This was a notation for an equation and Li Zhi gives the earliest source of the method, although it must have been invented before his time.
    • Certainly this, like the counting board, seems to have been a Chinese invention.
    • In many ways it was similar to the counting board, except instead of using rods to represent numbers, they were represented by beads sliding on a wire.
    • Arithmetical rules for the abacus were analogous to those of the counting board (even square roots and cube roots of numbers could be calculated) but it appears that the abacus was used almost exclusively by merchants who only used the operations of addition and subtraction.
    • Here is an illustration of an abacus showing the number 46802.
    • For numbers up to 4 slide the required number of beads in the lower part up to the middle bar.
    • List of References (6 books/articles) .
    • Chinese Mathematics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Chinese_numerals.html .

  14. Mathematical games
    • Alphabetical list of History Topics .
    • The whole history of mathematics is interwoven with mathematical games which have led to the study of many areas of mathematics.
    • Number games, geometrical puzzles, network problems and combinatorial problems are among the best known types of puzzles.
    • The Rhind papyrus shows that early Egyptian mathematics was largely based on puzzle type problems.
    • For example the papyrus, written in around 1850 BC, contains a rather familiar type of puzzle.
    • You can see a picture of the Rhind papyrus at THIS LINK.
    • Each mouse had eaten seven ears of grain.
    • Each ear of grain would have produced seven hekats of wheat.
    • What is the total of all of these? .
    • Similar problems appear in Fibonacci's Liber Abaci Ⓣ written in 1202 and the familiar St Ives Riddle of the 18th Century based on the same idea (and on the number 7).
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    • Greek mathematics produced many classic puzzles.
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    • If thou art diligent and wise, O Stranger, compute the number of cattle of the Sun..
    • In some interpretations of the problem the number of cattle turns out to be a number with 206545 digits! .
    • Archimedes also invented a division of a square into 14 pieces leading to a game similar to Tangrams involving making figures from the 14 pieces.
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    • Tangrams are of Chinese origin and require little mathematical skill.
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    • You can see a picture of the Alice tangrams at THIS LINK.
    • Fibonacci, already mentioned above, is famed for his invention of the sequence 1, 1, 2, 3, 5, 8, 13, ..
    • where each number is the sum of the previous two.
    • In fact a wealth of mathematics has arisen from this sequence and today a Journal is devoted to topics related to the sequence.
    • A certain man put a pair of rabbits in a place surrounded on all sides by a wall.
    • How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begins a new pair which from the second month on becomes productive? .
    • Fibonacci writes out the first 13 terms of the sequence but does not give the recurrence relation which generates it.
    • One of the earliest mentions of Chess in puzzles is by the Arabic mathematician Ibn Kallikan who, in 1256, poses the problem of the grains of wheat, 1 on the first square of the chess board, 2 on the second, 4 on the third, 8 on the fourth etc.
    • One of the earliest problem involving chess pieces is due to Guarini di Forli who in 1512 asked how two white and two black knights could be interchanged if they are placed at the corners of a 3 × 3 board (using normal knight's moves).
    • ., n2 to fill the squares of an n × n board so that each row, each column and both main diagonals sum to the same number.
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    • They are claimed to go back as far as 2200 BC when the Chinese called them lo-shu.
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    • Durer's famous engraving of Melancholia made in 1514 includes a picture of a magic square.
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    • You can see a picture of Melancholia at THIS LINK and of the magic square in it at THIS LINK.
    • The number of magic squares of a given order is still an unsolved problem.
    • There are 880 squares of size 4 and 275305224 squares of size 5, but the number of larger squares is still unknown.
    • Euler studied this type of square known as a pandiagonal square.
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    • No pandiagonal square of order 2(2n + 1) can exist but they do for all other orders.
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    • For n = 4 there are 880 magic squares of which 48 are pandiagonal.
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    • Other early inventors of games included Recorde and Cardan.
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    • Cardan invented a game consisting of a number of rings on a bar.
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    • You can see a picture of Cardan's rings at THIS LINK.
    • It appears in the 1550 edition of his book De Subtililate Ⓣ.
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    • This problem is similar to the Towers of Hanoi described below.
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    • In fact Lucas (the inventor of the Towers of Hanoi) gives a pretty solution to Cardan's Ring Puzzle using binary arithmetic.
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    • Tartaglia, who with Cardan jointly discovered the algebraic solution of the cubic, was another famous inventor of mathematical recreations.
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    • He invented many arithmetical problems, and contributed to problems with weighing masses with the smallest number of weights and Ferry Boat type problems which now have solutions using graph theory.
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    • Bachet was famed as a poet, translator and early mathematician of the French Academy.
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    • He is best known for his translation of 1621 of Diophantus's Arithmetica.
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    • Bachet, however, is also famed as a collector of mathematical puzzles which he published in 1612 Problemes plaisans et delectables qui font par les nombres Ⓣ.
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    • It contains many of the problems referred to above, river crossing problems, weighing problems, number tricks, magic squares etc.
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    • Here is an example of one of Bachet's weighing problems .
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    • What is the least number of weights that can be used on a scale pan to weigh any integral number of pounds from 1 to 40 inclusive, if the weights can be placed in either of the scale pans? .
    • In addition to magic square problems and number problems he considered the Knight's Tour of the chess board, the Thirty Six Officers problem and the Seven Bridges of Konigsberg.
    • De Moivre and Montmort had looked at it and solved the problem in the early years of the 18th Century after the question had been posed by Taylor.
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    • Ozanam and Montucla quote the solutions of both De Moivre and Montmort.
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    • Euler, in 1759 following a suggestion of L Bertrand of Geneva, was the first to make a serious mathematical analysis of it, introducing concepts which were to become important in graph theory.
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    • Lagrange also contributed to the understanding of the Knight's Tour problem, as did Vandermonde.
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    • The Seven Bridges of Konigsberg heralds the beginning of graph theory and topology.
    • You can see a picture of the Konigsberg bridges at THIS LINK.
    • The Thirty Six Officers Problem, posed by Euler in 1779, asks if it is possible to arrange 6 regiments consisting of 6 officers each of different ranks in a 6 × 6 square so that no rank or regiment will be repeated in any row or column.
    • There is a unique solution (up to symmetry) to the 6 × 6 problem and the puzzle, in the form of a wooden board with 36 holes into which pins were placed, was sold on the streets of London for one penny.
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    • In 1857 Hamilton described his Icosian game at a meeting of the British Association in Dublin.
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    • Jacques and Sons, makers of high quality chess sets, for £25 and patented in London in 1859.
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    • You can see a picture of the Icosian game at THIS LINK.
    • The problem, posed in 1850, asks how 15 school girls can walk in 5 rows of 3 each for 7 days so that no girl walks with any other girl in the same triplet more than once.
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    • It is important in the modern theory of combinatorics.
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    • Around this time two professional inventors of mathematical puzzles, Sam Loyd and Henry Ernest Dudeney, were entertaining the world with a large number of mathematical games and recreations.
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    • It illustrates important properties of permutations.
    • He invented a number of puzzles, some of which are very hard, which he published in the first number of the American Chess Journal.
    • Edouard Lucas invented the Towers of Hanoi in 1883.
    • The game of pentominoes is of more recent invention.
    • The problem of tiling an 8 × 8 square with a square hole in the centre was solved in 1935.
    • It is still an unsolved problem how many distinct polyominoes of each order there are.
    • There is a 3-dimensional version of pentominoes where cubes are used as the basic elements instead of squares.
    • This consists of 7 pieces, 6 pieces consisting of 4 small cubes and one of 3 small cubes.
    • The aim of this game is to assemble a 3 × 3 × 3 cube.
    • You can see a picture of the Soma Cubes at THIS LINK.
    • There are 30 cubes which have all possible permutations of precisely 6 colours for their faces.
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    • (Can you prove there are exactly 30 such cubes?) Choose a cube at random and then choose 8 other cubes to make a 2 × 2 × 2 cube with the same arrangement of colours for it's faces as the first chosen cube.
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    • Each face of the 2 × 2 × 2 cube has to be a single colour and the interior faces have to match in colour.
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    • Raymond Smullyan, a mathematical logician, composed a number of chess problems of a very different type from those usually composed.
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    • They are know known as problems of retrograde analysis and their object is to deduce the past history of a game rather than the future of a game which is the conventional problem.
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    • Problems of retrograde analysis are problems in mathematical logic.
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    • One of the most important of the modern professional puzzle inventors and collectors is Martin Gardner who wrote an extremely good column in Scientific American for about 30 years, stopping about four years ago.
    • He published some of Smullyan's retrograde analysis chees problems in 1973.
    • Of course the advent of personal computers has made both the writing and playing of mathematical games for computers an important new direction.
    • The game Gardner reported on was 'Spirolaterals' devised by Frank Olds with only 3 or 4 lines of code.
    • You can see some examples of spirolaterals at THIS LINK.
    • The most famous of recent puzzles in the of Rubik's cube invented by the Hungarian Erno Rubik.
    • By 1982 10 million cubes had been sold in Hungary, more than the population of the country.
    • The cube consists of 3 × 3 × 3 smaller cubes which, in the initial configuration, are coloured so that the 6 faces of the large cube are coloured in 6 distinct colours.
    • There are 43,252,003,274,489,856,000 different arrangements of the small cubes, only one of these arrangements being the initial position.
    • Solving the cube shows the importance of conjugates and commutators in a group.
    • List of References (13 books/articles) .
    • Solution of Archimedes cattle problem .
    • The Towers of Hanoi .
    • Examples of spirolaterals .
    • David Joyner (A history of puzzles) .
    • Alphabetical list of History Topics .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Mathematical_games.html .

  15. Matrices and determinants
    • The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC.
    • However it was not until near the end of the 17th Century that the ideas reappeared and development really got underway.
    • It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations.
    • The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive.
    • One produces grain at the rate of 2/3 of a bushel per square yard while the other produces grain at the rate of 1/2 a bushel per square yard.
    • If the total yield is 1100 bushels, what is the size of each field.
    • The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians.
    • Indeed it is fair to say that the text Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods.
    • There are three types of corn, of which three bundles of the first, two of the second, and one of the third make 39 measures.
    • Two of the first, three of the second and one of the third make 34 measures.
    • And one of the first, two of the second and three of the third make 26 measures.
    • How many measures of corn are contained of one bundle of each type? .
    • He sets up the coefficients of the system of three linear equations in three unknowns as a table on a 'counting board'.
    • Our late 20th Century methods would have us write the linear equations as the rows of the matrix rather than the columns but of course the method is identical.
    • from which the solution can be found for the third type of corn, then for the second, then the first by back substitution.
    • Cardan, in Ars Magna (1545), gives a rule for solving a system of two linear equations which he calls regula de modo and which [',' E Knobloch, Determinants, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 766-774.','7] calls mother of rules ! This rule gives what essentially is Cramer's rule for solving a 2 × 2 system although Cardan does not make the final step.
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    • Cardan therefore does not reach the definition of a determinant but, with the advantage of hindsight, we can see that his method does lead to the definition.
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    • Many standard results of elementary matrix theory first appeared long before matrices were the object of mathematical investigation.
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    • For example de Witt in Elements of curves, published as a part of the commentaries on the 1660 Latin version of Descartes' Geometrie , showed how a transformation of the axes reduces a given equation for a conic to canonical form.
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    • The idea of a determinant appeared in Japan before it appearedin Europe.
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    • In 1683 Seki wrote Method of solving the dissimulated problems which contains matrix methods written as tables in exactly the way the Chinese methods described above were constructed.
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    • Using his 'determinants' Seki was able to find determinants of 2 × 2, 3 × 3, 4 × 4 and 5 × 5 matrices and applied them to solving equations but not systems of linear equations.
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    • The first appearance of a determinant in Europe was ten years later.
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    • He explained that the system of equations .
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    • His unpublished manuscripts contain more than 50 different ways of writing coefficient systems which he worked on during a period of 50 years beginning in 1678.
    • Leibniz used the word 'resultant' for certain combinatorial sums of terms of a determinant.
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    • As well as studying coefficient systems of equations which led him to determinants, Leibniz also studied coefficient systems of quadratic forms which led naturally towards matrix theory.
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    • In the 1730's Maclaurin wrote Treatise of algebra although it was not published until 1748, two years after his death.
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    • Cramer gave the general rule for n × n systems in a paper Introduction to the analysis of algebraic curves (1750).
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    • It arose out of a desire to find the equation of a plane curve passing through a number of given points.
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    • One finds the value of each unknown by forming n fractions of which the common denominator has as many terms as there are permutations of n things.
    • Cramer does go on to explain precisely how one calculates these terms as products of certain coefficients in the equations and how one determines the sign.
    • He also says how the n numerators of the fractions can be found by replacing certain coefficients in this calculation by constant terms of the system.
    • In 1764 Bezout gave methods of calculating determinants as did Vandermonde in 1771.
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    • In 1772 Laplace claimed that the methods introduced by Cramer and Bezout were impractical and, in a paper where he studied the orbits of the inner planets, he discussed the solution of systems of linear equations without actually calculating it, by using determinants.
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    • Rather surprisingly Laplace used the word 'resultant' for what we now call the determinant: surprising since it is the same word as used by Leibniz yet Laplace must have been unaware of Leibniz's work.
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    • Laplace gave the expansion of a determinant which is now named after him.
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    • Lagrange, in a paper of 1773, studied identities for 3 × 3 functional determinants.
    • However this comment is made with hindsight since Lagrange himself saw no connection between his work and that of Laplace and Vandermonde.
    • This 1773 paper on mechanics, however, contains what we now think of as the volume interpretation of a determinant for the first time.
    • He used the term because the determinant determines the properties of the quadratic form.
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    • However the concept is not the same as that of our determinant.
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    • In the same work Gauss lays out the coefficients of his quadratic forms in rectangular arrays.
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    • He describes matrix multiplication (which he thinks of as composition so he has not yet reached the concept of matrix algebra) and the inverse of a matrix in the particular context of the arrays of coefficients of quadratic forms.
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    • Gaussian elimination, which first appeared in the text Nine Chapters on the Mathematical Art written in 200 BC, was used by Gauss in his work which studied the orbit of the asteroid Pallas.
    • Using observations of Pallas taken between 1803 and 1809, Gauss obtained a system of six linear equations in six unknowns.
    • Cauchy's work is the most complete of the early works on determinants.
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    • He reproved the earlier results and gave new results of his own on minors and adjoints.
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    • In the 1812 paper the multiplication theorem for determinants is proved for the first time although, at the same meeting of the Institut de France, Binet also read a paper which contained a proof of the multiplication theorem but it was less satisfactory than that given by Cauchy.
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    • In 1826 Cauchy, in the context of quadratic forms in n variables, used the term 'tableau' for the matrix of coefficients.
    • He found the eigenvalues and gave results on diagonalisation of a matrix in the context of converting a form to the sum of squares.
    • Cauchy also introduced the idea of similar matrices (but not the term) and showed that if two matrices are similar they have the same characteristic equation.
    • He also, again in the context of quadratic forms, proved that every real symmetric matrix is diagonalisable.
    • Jacques Sturm gave a generalisation of the eigenvalue problem in the context of solving systems of ordinary differential equations.
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    • In fact the concept of an eigenvalue appeared 80 years earlier, again in work on systems of linear differential equations, by D'Alembert studying the motion of a string with masses attached to it at various points.
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    • It should be stressed that neither Cauchy nor Jacques Sturm realised the generality of the ideas they were introducing and saw them only in the specific contexts in which they were working.
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    • Jacobi from around 1830 and then Kronecker and Weierstrass in the 1850's and 1860's also looked at matrix results but again in a special context, this time the notion of a linear transformation.
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    • These were important in that for the first time the definition of the determinant was made in an algorithmic way and the entries in the determinant were not specified so his results applied equally well to cases were the entries were numbers or to where they were functions.
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    • These three papers by Jacobi made the idea of a determinant widely known.
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    • Cayley, also writing in 1841, published the first English contribution to the theory of determinants.
    • In this paper he used two vertical lines on either side of the array to denote the determinant, a notation which has now become standard.
    • Eisenstein in 1844 denoted linear substitutions by a single letter and showed how to add and multiply them like ordinary numbers except for the lack of commutativity.
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    • It is fair to say that Eisenstein was the first to think of linear substitutions as forming an algebra as can be seen in this quote from his 1844 paper:- .
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    • An algorithm for calculation can be based on this, it consists of applying the usual rules for the operations of multiplication, division, and exponentiation to symbolic equations between linear systems, correct symbolic equations are always obtained, the sole consideration being that the order of the factors may not be altered.
    • Sylvester defined a matrix to be an oblong arrangement of terms and saw it as something which led to various determinants from square arrays contained within it.
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    • After leaving America and returning to England in 1851, Sylvester became a lawyer and met Cayley, a fellow lawyer who shared his interest in mathematics.
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    • Cayley quickly saw the significance of the matrix concept and by 1853 Cayley had published a note giving, for the first time, the inverse of a matrix.
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    • Cayley in 1858 published Memoir on the theory of matrices which is remarkable for containing the first abstract definition of a matrix.
    • He shows that the coefficient arrays studied earlier for quadratic forms and for linear transformations are special cases of his general concept.
    • He gave an explicit construction of the inverse of a matrix in terms of the determinant of the matrix.
    • Cayley also proved that, in the case of 2 × 2 matrices, that a matrix satisfies its own characteristic equation.
    • I have not thought it necessary to undertake the labour of a formal proof of the theorem in the general case of a matrix of any degree.
    • In fact he also proved a special case of the theorem, the 4 × 4 case, in the course of his investigations into quaternions.
    • It appears in the context of a canonical form for linear substitutions over the finite field of order a prime.
    • Frobenius, in 1878, wrote an important work on matrices On linear substitutions and bilinear forms although he seemed unaware of Cayley's work.
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    • Frobenius in this paper deals with coefficients of forms and does not use the term matrix.
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    • However he proved important results on canonical matrices as representatives of equivalence classes of matrices.
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    • He cites Kronecker and Weierstrass as having considered special cases of his results in 1874 and 1868 respectively.
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    • This 1878 paper by Frobenius also contains the definition of the rank of a matrix which he used in his work on canonical forms and the definition of orthogonal matrices.
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    • The nullity of a square matrix was defined by Sylvester in 1884.
    • He defined the nullity of A, n(A), to be the largest i such that every minor of A of order n-i+1 is zero.
    • Sylvester was interested in invariants of matrices, that is properties which are not changed by certain transformations.
    • In 1896 Frobenius became aware of Cayley's 1858 Memoir on the theory of matrices and after this started to use the term matrix.
    • An axiomatic definition of a determinant was used by Weierstrass in his lectures and, after his death, it was published in 1903 in the note On determinant theory.
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    • With these two publications the modern theory of determinants was in place but matrix theory took slightly longer to become a fully accepted theory.
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    • An important early text which brought matrices into their proper place within mathematics was Introduction to higher algebra by Bocher in 1907.
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    • Turnbull and Aitken wrote influential texts in the 1930's and Mirsky's An introduction to linear algebra in 1955 saw matrix theory reach its present major role in as one of the most important undergraduate mathematics topic.
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    • List of References (13 books/articles) .
    • The URL of this page is: .
    • https://www-history.mcs.st-andrews.ac.uk/HistTopics/Matrices_and_determinants.html .

  16. Debating topics
    • Debating topics on mathematics .
    • Alphabetical list of History Topics .
    • The first ideas we present are simply to make people think about numbers, and in particular to encourage the use of the history archive to find birth dates and death dates before making calculations.
    • Each year in the Chinese calendar is named after one of 12 animals.
    • The year 1985 was the Year of the Ox.
    • How many in the 20th century were "Years of the Ox"? .
    • In one sense the answer might be that we still use the system set up by the Babylonians and their number system had a base of 60.
    • Perhaps it was chosen because it had lots of factors.
    • But surely ancient civilisations do not choose the base of their number system.
    • We could debate the nature of number.
    • One is tempted to say "Of course it is." But the ancient Greeks did not consider 1 to be a number.
    • Is the square root of 2 a number? .
    • What is wrong with having the hypotenuse of a right angled triangle, whose shorter sides are each of one unit, not corresponding to a number? Why should there be a number corresponding to the length of every line we draw? .
    • If π is the ratio of the circumference of a circle to its diameter, then why is its area π times the radius squared.
    • Did other civilisations develop the same type of number system? .
    • If the place value number system was a better system than that of the Romans then why was there so much resistance to using it? .
    • How did the Chinese represent equations? .
    • What does it mean to say that equations of degree 5 cannot be solved? .
    • Why did the ancient Chinese not worry about solving cubic, quartic, quintic equations? .
    • Do we need to introduce negative numbers