Search Results for Greek*
Biographies
- Aristarchus biography
- For example Heath begins Volume II of his history of Greek mathematics with the words [A history of Greek mathematics I, II (Oxford, 1931).',5)">5]:-
.
- The Greeks knew better; they called him 'Aristarchus the mathematician'.
- The ancient Copernicus : Reprint of the 1913 original (New York, 1981).',4)">4], or [A history of Greek mathematics I, II (Oxford, 1931).',5)">5]):-
.
- We know of no earlier hypothesis of this type but in fact the theory was not accepted by the Greeks so apparently never had any popularity.
- The ancient Copernicus : Reprint of the 1913 original (New York, 1981).',4)">4], or [A history of Greek mathematics I, II (Oxford, 1931).',5)">5], or see [Dictionary of Scientific Biography (New York 1970-1990).
- In fact the way that Aristarchus expresses his proportions is, according to Heath, similar to other expressions which occur in Greek writings and indicated that Aristarchus considered that the radius of the sphere of the fixed stars was infinitely large compared with the orbit of the earth.
- a difficult passage in an anonymous commentary written in Greek during the 2nd century AD on Book 20 of Homer's Odyssey.
- [His] thesis concerning the times when solar eclipses may occur rests on an analysis of Greek and Egyptian calendrical conventions, rather than on an appeal to observation of solar eclipses.
- History Topics: Greek Astronomy
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- History Topics: Greek number systems
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- Banu Musa biography
- The Banu Musa brothers were among the first group of mathematicians to begin to carry forward the mathematical developments begun by the ancient Greeks.
- The first steps were being taken to allow Greek mathematics to spread through the Islam empire.
- Al-Ma'mun had continued the patronage of learning started by his father and had founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated.
- The Banu Musa were among the first Arabic scientists to study the Greek mathematical works and to lay the foundation of the Arabic school of mathematics.
- They may be called disciples of Greek mathematics, yet they deviated from classical Greek mathematics in ways that were very important to the development of some mathematical concepts.
- (Moscow, 1966), 131-139.',2)">2] suggests, this may have been due to a lack of understanding of the finer points of Greek geometric thinking.
- The Greeks had not thought of areas and volumes as numbers, but had only compared ratios of areas etc.
- The Banu Musa's concept of number is broader than that of the Greeks.
- Banu Musa (print-only)
- The Banu Musa brothers were among the first group of mathematicians to begin to carry forward the mathematical developments begun by the ancient Greeks.
- The first steps were being taken to allow Greek mathematics to spread through the Islam empire.
- Al-Ma'mun had continued the patronage of learning started by his father and had founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated.
- The Banu Musa were among the first Arabic scientists to study the Greek mathematical works and to lay the foundation of the Arabic school of mathematics.
- They may be called disciples of Greek mathematics, yet they deviated from classical Greek mathematics in ways that were very important to the development of some mathematical concepts.
- This in itself may not have been a step forward for, as the author of [2] suggests, this may have been due to a lack of understanding of the finer points of Greek geometric thinking.
- The Greeks had not thought of areas and volumes as numbers, but had only compared ratios of areas etc.
- The Banu Musa's concept of number is broader than that of the Greeks.
- Banu Musa biography
- The Banu Musa brothers were among the first group of mathematicians to begin to carry forward the mathematical developments begun by the ancient Greeks.
- The first steps were being taken to allow Greek mathematics to spread through the Islam empire.
- Al-Ma'mun had continued the patronage of learning started by his father and had founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated.
- The Banu Musa were among the first Arabic scientists to study the Greek mathematical works and to lay the foundation of the Arabic school of mathematics.
- They may be called disciples of Greek mathematics, yet they deviated from classical Greek mathematics in ways that were very important to the development of some mathematical concepts.
- (Moscow, 1966), 131-139.',2)">2] suggests, this may have been due to a lack of understanding of the finer points of Greek geometric thinking.
- The Greeks had not thought of areas and volumes as numbers, but had only compared ratios of areas etc.
- The Banu Musa's concept of number is broader than that of the Greeks.
- Yavanesvara biography
- The Indian methods of computing horoscopes all date back to the translation of a Greek astrology text into Sanskrit prose by Yavanesvara in 149 AD.
- Yavanesvara (or Yavanaraja) literally means "Lord of the Greeks" and it was a name given to many officials in western India during the period 130 AD - 390 AD.
- During this period the Ksatrapas ruled Gujarat (or Madhya Pradesh) and these "Lord of the Greeks" officials acted for the Greek merchants living in the area.
- The particular "Lord of the Greeks" official Yavanesvara who we are interested in here worked under Rudradaman.
- The Greek astrology text in question was written in Alexandria some time round about 120 BC.
- Yavanesvara did far more than just translate the Greek text for such a translation would have had little relevance to the Indians.
- Instead of the Greek gods who appear in the original, Yavanesvara used Hindu images.
- In order to do this Yavanesvara put into his work an explanation of the Greek version of the Babylonian theory of the motions of the planets.
- Thales biography
- Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer.
- Proclus, the last major Greek philosopher, who lived around 450 AD, wrote:-
.
- Plutarch, writing of these Seven Sages, says that (see [A History of Greek Mathematics I (Oxford, 1921).',8)">8]):-
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- Diogenes Laertius writing in the second century AD quotes Hieronymus, a pupil of Aristotle [Lives of eminent philosophers (New York, 1925).',6)">6] (or see [A History of Greek Mathematics I (Oxford, 1921).',8)">8]):-
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- A similar statement is made by Pliny (see [A History of Greek Mathematics I (Oxford, 1921).',8)">8]):-
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- Thales is said to have travelled in Egypt, and to have thence brought to the Greeks the science of geometry.
- What Egyptians knew of geometry was mainly rules of thumb, and there is no reason to believe that Thales arrived at deductive proofs, such as later Greeks discovered.
- Proclus writes (see [A History of Greek Mathematics I (Oxford, 1921).',8)">8]):-
.
- Heath in [A History of Greek Mathematics I (Oxford, 1921).',8)">8] gives three different methods which Thales might have used to calculate the distance to a ship at sea.
- History Topics: Greek Astronomy
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- Thales biography
- Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer.
- Proclus, the last major Greek philosopher, who lived around 450 AD, wrote:-
.
- Plutarch, writing of these Seven Sages, says that (see [A History of Greek Mathematics I (Oxford, 1921).',8)">8]):-
.
- Diogenes Laertius writing in the second century AD quotes Hieronymus, a pupil of Aristotle [Lives of eminent philosophers (New York, 1925).',6)">6] (or see [A History of Greek Mathematics I (Oxford, 1921).',8)">8]):-
.
- A similar statement is made by Pliny (see [A History of Greek Mathematics I (Oxford, 1921).',8)">8]):-
.
- Thales is said to have travelled in Egypt, and to have thence brought to the Greeks the science of geometry.
- What Egyptians knew of geometry was mainly rules of thumb, and there is no reason to believe that Thales arrived at deductive proofs, such as later Greeks discovered.
- Proclus writes (see [A History of Greek Mathematics I (Oxford, 1921).',8)">8]):-
.
- Heath in [A History of Greek Mathematics I (Oxford, 1921).',8)">8] gives three different methods which Thales might have used to calculate the distance to a ship at sea.
- History Topics: Greek Astronomy
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- Thabit biography
- The sect, with strong Greek connections, had in earlier times adopted Greek culture, and it was common for members to speak Greek although after the conquest of the Sabians by Islam, they became Arabic speakers.
- This language was Thabit ibn Qurra's native language, but he was fluent in both Greek and Arabic.
- At this time there were many patrons who employed talented scientists to translate Greek text into Arabic and Thabit, with his great skills in languages as well as great mathematical skills, translated and revised many of the important Greek works.
- In fact many Greek texts survive today only because of this industry in bringing Greek learning to the Arab world.
- However we must not think that the mathematicians such as Thabit were mere preservers of Greek knowledge.
- The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied.
- History Topics: How do we know about Greek mathematics?
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- Caratheodory biography
- Constantin Caratheodory's father, Stephanos Caratheodory, was an Ottoman Greek who had studied law in Berlin and then served as secretary in the Ottoman embassies in Berlin, Stockholm and Vienna.
- Stephanos had married Despina Petrocochino, who came from a Greek family of businessmen who had settled in Marseille.
- Constantin was already bilingual in French and Greek by this time.
- This put Caratheodory in a difficult position since he sided with the Greeks, yet his father served the government of the Ottoman Empire.
- After five years Gottingen he was appointed to the University of Berlin in 1918 but after he had been there for a year, at the request of the Greek government, he ended his contract with Berlin on 31 December 1919 and travelled to Greece to undertake a new venture.
- The Greek government had asked Caratheodory to establish a second university in Smyrna.
- On 14 July the Greek government published a bill setting up a Greek University in Smyrna and soon others were appointed at assist Caratheodory.
- Caratheodory was able to save the university library, which he had worked so hard to establish, and most of the equipment which he had been purchased for the science departments, and escaped to Athens on a Greek battleship.
- In particular he continued to work on reorganising the Greek universities, particularly during 1930-32, with the aim of integrating Greece academically into Europe.
- Apollonius biography
- Born: about 262 BC in Perga, Pamphylia, Greek Ionia (now Murtina, Antalya, Turkey)
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- Apollonius of Perga should not be confused with other Greek scholars called Apollonius, for it was a common name.
- ',1)">1] details of others with the name of Apollonius are given: Apollonius of Rhodes, born about 295 BC, a Greek poet and grammarian, a pupil of Callimachus who was a teacher of Eratosthenes; Apollonius of Tralles, 2nd century BC, a Greek sculptor; Apollonius the Athenian, 1st century BC, a sculptor; Apollonius of Tyana, 1st century AD, a member of the society founded by Pythagoras; Apollonius Dyscolus, 2nd century AD, a Greek grammarian who was reputedly the founder of the systematic study of grammar; and Apollonius of Tyre who is a literary character.
- Pergamum, today the town of Bergama in the province of Izmir in Turkey, was an ancient Greek city in Mysia.
- Conics was written in eight books but only the first four have survived in Greek.
- What militates against its being read in its original form is the great extent of the exposition (it contains 387 separate propositions), due partly to the Greek habit of proving particular cases of a general proposition separately from the proposition itself, but more to the cumbersomeness of the enunciations of complicated propositions in general terms (without the help of letters to denote particular points) and to the elaborateness of the Euclidean form, to which Apollonius adheres throughout.
- Apollonius was also an important founder of Greek mathematical astronomy, which used geometrical models to explain planetary theory.
- History Topics: Greek Astronomy
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- History Topics: How do we know about Greek mathematicians?
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- History Topics: Greek number systems
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- Eratosthenes biography
- Heath writes [A History of Greek Mathematics (2 vols.) (Oxford, 1921).',4)">4]:-
.
- In particular he described there the history of the problem of duplicating the cube (see Heath [A History of Greek Mathematics (2 vols.) (Oxford, 1921).',4)">4]):-
.
- when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an alter double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an alter of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.
- Eratosthenes erected a column at Alexandria with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [A History of Greek Mathematics (2 vols.) (Oxford, 1921).',4)">4]:-
.
- Heath [A History of Greek Mathematics (2 vols.) (Oxford, 1921).',4)">4] argues that Eratosthenes used 24° and that 11/83 of 180° was a refinement due to Ptolemy.
- Eratosthenes writings include the poem Hermes, inspired by astronomy, as well as literary works on the theatre and on ethics which was a favourite topic of the Greeks.
- History Topics: Greek Astronomy
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- Pappus biography
- Pappus of Alexandria is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry.
- There is a reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Pappus was a contemporary of Theon of Alexandria (see for example [Dictionary of Scientific Biography (New York 1970-1990).
- Heath in [A History of Greek Mathematics II (Oxford, 1921).',4)">4] is completely convinced saying that [A History of Greek Mathematics II (Oxford, 1921).',4)">4]:-
.
- Heath in [A History of Greek Mathematics II (Oxford, 1921).',4)">4] describes the Mathematical Collection as follows:-
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- Obviously written with the object of reviving the classical Greek geometry, it covers practically the whole field.
- It is, however, a handbook or guide to Greek geometry rather than an encyclopaedia; it was intended, that is, to be read with the original works (where still extant) rather than to enable them to be dispensed with.
- He concludes his discussion of honeycombs and introduces the aims of his work as follows (see for example [Selections illustrating the history of Greek mathematics II (London, 1941).',3)">3] or [A History of Greek Mathematics II (Oxford, 1921).',4)">4]):-
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- In Book VII Pappus writes about the Treasury of Analysis (see for example [Selections illustrating the history of Greek mathematics II (London, 1941).',3)">3]):-
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- Pappus then goes on to explain the different approaches of analysis and synthesis [Selections illustrating the history of Greek mathematics II (London, 1941).',3)">3]:-
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- We quote Pappus's own description of the subject (see for example [Selections illustrating the history of Greek mathematics II (London, 1941).',3)">3]):-
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- He writes well, shows great clarity of thought and the Mathematical Collection is a work of very great historical importance in the study of Greek geometry.
- Of course Pappus did not write "Almagest" but the Greek title of the work.
- History Topics: Greek number systems
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- Pappus biography
- Pappus of Alexandria is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry.
- There is a reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Pappus was a contemporary of Theon of Alexandria (see for example [Dictionary of Scientific Biography (New York 1970-1990).
- Heath in [A History of Greek Mathematics II (Oxford, 1921).',4)">4] is completely convinced saying that [A History of Greek Mathematics II (Oxford, 1921).',4)">4]:-
.
- Heath in [A History of Greek Mathematics II (Oxford, 1921).',4)">4] describes the Mathematical Collection as follows:-
.
- Obviously written with the object of reviving the classical Greek geometry, it covers practically the whole field.
- It is, however, a handbook or guide to Greek geometry rather than an encyclopaedia; it was intended, that is, to be read with the original works (where still extant) rather than to enable them to be dispensed with.
- He concludes his discussion of honeycombs and introduces the aims of his work as follows (see for example [Selections illustrating the history of Greek mathematics II (London, 1941).',3)">3] or [A History of Greek Mathematics II (Oxford, 1921).',4)">4]):-
.
- In Book VII Pappus writes about the Treasury of Analysis (see for example [Selections illustrating the history of Greek mathematics II (London, 1941).',3)">3]):-
.
- Pappus then goes on to explain the different approaches of analysis and synthesis [Selections illustrating the history of Greek mathematics II (London, 1941).',3)">3]:-
.
- We quote Pappus's own description of the subject (see for example [Selections illustrating the history of Greek mathematics II (London, 1941).',3)">3]):-
.
- He writes well, shows great clarity of thought and the Mathematical Collection is a work of very great historical importance in the study of Greek geometry.
- Of course Pappus did not write "Almagest" but the Greek title of the work.
- History Topics: Greek number systems
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- Heath biography
- He was a specialist in the history of Greek mathematics, writing articles on 'Pappus' and 'Porisms' for Encyclopaedia Britannica while still an undergraduate.
- Cayley recommended its publication by Cambridge University Press and Diophantus of Alexandria: a study in the history of Greek algebra appeared in 1885.
- In 1896 he published Apollonius of Perga which presented the important text on conic sections using modern notation, and it contained an important preface giving details of previous Greek work on conic sections.
- Between these two editions, in 1920, he published his version of Book I of the Elements written in Greek.
- This was intended to encourage schools to teach directly from Book I, but it does make us realise how things have changed in British schools today - how many of today's schoolchildren could use a book written in Greek as their mathematics textbook!
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- Greek astronomical work also attracted Heath's attention and in 1913 he published a translation of Aristarchus's On the sizes and distances of the sun and moon again with an important preface, this time giving a thorough account of Greek astronomy.
- Perhaps his most famous work, however, is History of Greek Mathematics which appeared in 1921.
- A single volume version on Greek mathematics, condensing the material from his earlier work, appeared in 1931 under the title A manual of Greek mathematics and the following year he produced a companion volume Greek astronomy.
- Following the publication of History of Greek Mathematics he was elected president of the Mathematical Association for 1922-23.
- History Topics: How do we know about Greek mathematicians?
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- History Topics: How do we know about Greek mathematics?
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- Euclid biography
- Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [Dictionary of Scientific Biography (New York 1970-1990).
- ',1)">1] or [A history of Greek mathematics 1 (Oxford, 1931).',9)">9] or many other sources):-
.
- None of Euclid's works have a preface, at least none has come down to us so it is highly unlikely that any ever existed, so we cannot see any of his character, as we can of some other Greek mathematicians, from the nature of their prefaces.
- Another story told by Stobaeus [A history of Greek mathematics 1 (Oxford, 1931).',9)">9] is the following:-
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- Heath says [A history of Greek mathematics 1 (Oxford, 1931).',9)">9]:-
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- Greek mathematics can boast no finer discovery than this theory, which put on a sound footing so much of geometry as depended on the use of proportion.
- As Heath writes in [A history of Greek mathematics 1 (Oxford, 1931).',9)">9]:-
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- Even in Greek times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language here and there, mostly with a view to greater clearness and consistency..
- Heath [A history of Greek mathematics 1 (Oxford, 1931).',9)">9] discusses many of the editions and describes the likely changes to the text over the years.
- Euclid also wrote the following books which have survived: Data (with 94 propositions), which looks at what properties of figures can be deduced when other properties are given; On Divisions which looks at constructions to divide a figure into two parts with areas of given ratio; Optics which is the first Greek work on perspective; and Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set.
- (This was the first Latin translation directly from the Greek.)
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- History Topics: Greek Astronomy
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- History Topics: How do we know about Greek mathematicians?
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- History Topics: How do we know about Greek mathematics?
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- Euclid biography
- Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [Dictionary of Scientific Biography (New York 1970-1990).
- ',1)">1] or [A history of Greek mathematics 1 (Oxford, 1931).',9)">9] or many other sources):-
.
- None of Euclid's works have a preface, at least none has come down to us so it is highly unlikely that any ever existed, so we cannot see any of his character, as we can of some other Greek mathematicians, from the nature of their prefaces.
- Another story told by Stobaeus [A history of Greek mathematics 1 (Oxford, 1931).',9)">9] is the following:-
.
- Heath says [A history of Greek mathematics 1 (Oxford, 1931).',9)">9]:-
.
- Greek mathematics can boast no finer discovery than this theory, which put on a sound footing so much of geometry as depended on the use of proportion.
- As Heath writes in [A history of Greek mathematics 1 (Oxford, 1931).',9)">9]:-
.
- Even in Greek times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language here and there, mostly with a view to greater clearness and consistency..
- Heath [A history of Greek mathematics 1 (Oxford, 1931).',9)">9] discusses many of the editions and describes the likely changes to the text over the years.
- Euclid also wrote the following books which have survived: Data (with 94 propositions), which looks at what properties of figures can be deduced when other properties are given; On Divisions which looks at constructions to divide a figure into two parts with areas of given ratio; Optics which is the first Greek work on perspective; and Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set.
- (This was the first Latin translation directly from the Greek.)
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- History Topics: Greek Astronomy
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- History Topics: How do we know about Greek mathematicians?
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- History Topics: How do we know about Greek mathematics?
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- Theon biography
- There is a reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Theon of Alexandria lived under the Emperor Theodosius I (who reigned from 379 to 395).
- Theon's version of Euclid's Elements (with textual changes and some additions) is thought to have been written with the assistance of his daughter Hypatia and was the only Greek text of the Elements known, until an earlier one was discovered in the Vatican in the late 19th century.
- Heath writes of Theon's edition of the Elements [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
.
- while making only inconsiderable additions to the content of the "Elements", he endeavoured to remove difficulties that might be felt by learners in studying the book, as a modern editor might do in editing a classical text-book for use in schools; and there is no doubt that his edition was approved by his pupils at Alexandria for whom it was written, as well as by later Greeks who used it almost exclusively..
- As to Theon's commentary on Ptolemy's Syntaxis Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
.
- This commentary is not calculated to give us a very high opinion of Theon's mathematical calibre, but it is valuable for several historical notices that it gives, and we are indebted to it for a useful account of the Greek method of operating with sexagesimal fractions, which is illustrated with examples of multiplication, division, and the extraction of the square root of a non-number by way of approximation.
- History Topics: How do we know about Greek mathematics?
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- Diophantus biography
- Heath [Diophantus of Alexandria: A Study in the History of Greek Algebra (New York, 1964).',3)">3] quotes from a letter by Michael Psellus who lived in the last half of the 11th century.
- Psellus wrote (Heath's translation in [Diophantus of Alexandria: A Study in the History of Greek Algebra (New York, 1964).',3)">3]):-
.
- Knorr gives a different translation of the same passage (showing how difficult the study of Greek mathematics is for anyone who is not an expert in classical Greek) which has a remarkably different meaning:-
.
- The most details we have of Diophantus's life (and these may be totally fictitious) come from the Greek Anthology, compiled by Metrodorus around 500 AD.
- Heath writes in [A history of Greek mathematics 2 (Oxford, 1931).',4)">4] in 1920:-
.
- 28 (1975), 3-30.',20)">20] Rashed compares the four books in this Arabic translation with the known six Greek books and claims that this text is a translation of the lost books of Diophantus.
- The reviewer, familiar with the Arabic text of this manuscript, does not doubt that this manuscript is the translation from the Greek text written in Alexandria but the great difference between the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of the theory of numbers and these books containing only algebraic material make it very probable that this text was written not by Diophantus but by some one of his commentators (perhaps Hypatia?).
- It is time to take a look at this most outstanding work on algebra in Greek mathematics.
- Among such results are [A history of Greek mathematics 2 (Oxford, 1931).',4)">4]:-
.
- No one has yet translated from the Greek into Latin the thirteen Books of Diophantus, in which the very flower of the whole of arithmetic lies hid..
- Diophantus biography
- Heath [Diophantus of Alexandria: A Study in the History of Greek Algebra (New York, 1964).',3)">3] quotes from a letter by Michael Psellus who lived in the last half of the 11th century.
- Psellus wrote (Heath's translation in [Diophantus of Alexandria: A Study in the History of Greek Algebra (New York, 1964).',3)">3]):-
.
- Knorr gives a different translation of the same passage (showing how difficult the study of Greek mathematics is for anyone who is not an expert in classical Greek) which has a remarkably different meaning:-
.
- The most details we have of Diophantus's life (and these may be totally fictitious) come from the Greek Anthology, compiled by Metrodorus around 500 AD.
- Heath writes in [A history of Greek mathematics 2 (Oxford, 1931).',4)">4] in 1920:-
.
- 28 (1975), 3-30.',20)">20] Rashed compares the four books in this Arabic translation with the known six Greek books and claims that this text is a translation of the lost books of Diophantus.
- The reviewer, familiar with the Arabic text of this manuscript, does not doubt that this manuscript is the translation from the Greek text written in Alexandria but the great difference between the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of the theory of numbers and these books containing only algebraic material make it very probable that this text was written not by Diophantus but by some one of his commentators (perhaps Hypatia?).
- It is time to take a look at this most outstanding work on algebra in Greek mathematics.
- Among such results are [A history of Greek mathematics 2 (Oxford, 1931).',4)">4]:-
.
- No one has yet translated from the Greek into Latin the thirteen Books of Diophantus, in which the very flower of the whole of arithmetic lies hid..
- Barrow biography
- At Felstead Barrow learnt Greek, Latin, Hebrew and logic in preparation for University.
- Duport, the Regius Professor of Greek at Cambridge, tutored Barrow without taking any fees, both due to Barrow's talent and that both were royalists.
- Under Duport, Barrow studied Greek, Latin, Hebrew, French, Spanish, Italian, literature, chronology, geography and theology.
- In 1654 he defended the University in a speech in which he spoke of the importance of learning Greek, Latin and literature for the purpose of acquiring a firm basis for learning.
- When the professorship in Greek became available it was expected that Barrow would be appointed to replace Duport who had been forced to leave the chair on political grounds.
- While in Constantinople Barrow spent much time studying divinity and in particular the local Greek Church.
- The Professor of Greek voluntarily resigned in recognition that there were others who were better suited to the position - one of these people being Barrow himself.
- He could also hold this chair while continuing as Professor of Greek at Cambridge.
- Barrow was an obvious choice for this position and he relinquished the Greek chair for the mathematics because, he explained, of his greater interest in mathematics than Greek, because less work was involved, and that it had always been his intention to hold the Greek chair temporarily.
- Posidonius biography
- Although he was born in Apameia in Syria, Posidonius was from a Greek family and he was brought up in the Greek tradition.
- While there Posidonius became friends with Pompey the Great who had been educated in the Greek tradition.
- Posidonius made some minor contributions to pure mathematics where he is [A history of Greek mathematics I, II (Oxford, 1931).',2)">2]:-
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- The work is in two volumes and as Heath comments [A history of Greek mathematics I, II (Oxford, 1931).',2)">2]:-
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- These works give an account of the Roman civil wars and the contacts by the Greeks and the Romans with other peoples such as the Celts, Germans, and peoples of Spain and Gaul.
- Allman biography
- He also wrote a number of articles in the ninth edition of the 'Encyclopaedia Britannica' on Greek mathematicians.
- Among the articles Allman contributed to the ninth edition of the Encyclopaedia Britannica were those on Thales, Pythagoras, Ptolemy and other Greek philosophers.
- However, Allman's most significant contribution was Greek geometry from Thales to Euclid published in Dublin in 1889.
- In studying the development of Greek Science, two periods must be carefully distinguished.
- The founder of Greek philosophy - Thales and Pythagoras - were also the founders of Greek science, and from the time of Thales to that of Euclid and the foundation of the museum of Alexandria, the development of science was, for the most part, the work of the Greek philosophers.
- Allman developed methods of deduction of historical information which have proved fundamental in discovering information of the type described in our articles How do we know about Greek mathematics? and How do we know about Greek mathematicians? in this archive.For example in [Centaurus 18 (1973/74), 1-5.',3)">3] Neuenschwander uses a philological technique developed by Allman in his studies of Hippocrates of Chios in order to show that Eutocius was by no means quoting directly from Eudemus's History of geometry.
- Menaechmus biography
- Menaechmus is mentioned by Proclus who tells us that he was a pupil of Eudoxus in the following quote (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]):-
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- There is another reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Menaechmus was (see for example [Dictionary of Scientific Biography (New York 1970-1990).
- Some have inferred from this (see for example [Greek Geometry from Thales to Euclid (Dublin-London, 1889), 153-179.',4)">4]) that Menaechmus acted as a tutor to Alexander the Great, and indeed this is not impossible to imagine since as Allman suggests Aristotle may have provided the link between the two.
- There is also an implication in the writings of Proclus that Menaechmus was the head of a School and this is argued convincingly by Allman in [Greek Geometry from Thales to Euclid (Dublin-London, 1889), 153-179.',4)">4].
- ',1)">1], [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3] and [Greek Geometry from Thales to Euclid (Dublin-London, 1889), 153-179.',4)">4] all consider a problem associated with these solutions.
- Allman [Greek Geometry from Thales to Euclid (Dublin-London, 1889), 153-179.',4)">4] suggests that Menaechmus might have drawn the curves by finding many points on them and that this might be considered as a mechanical device.
- Heath [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3] writes:-
.
- Proclus writes about Menaechmus saying that he studied the structure of mathematics [Greek Geometry from Thales to Euclid (Dublin-London, 1889), 153-179.',4)">4]:-
.
- Ptolemy biography
- One of the most influential Greek astronomers and geographers of his time, Ptolemy propounded the geocentric theory in a form that prevailed for 1400 years.
- However, of all the ancient Greek mathematicians, it is fair to say that his work has generated more discussion and argument than any other.
- His name, Claudius Ptolemy, is of course a mixture of the Greek Egyptian 'Ptolemy' and the Roman 'Claudius'.
- This would indicate that he was descended from a Greek family living in Egypt and that he was a citizen of Rome, which would be as a result of a Roman emperor giving that 'reward' to one of Ptolemy's ancestors.
- Its original Greek title translates as The Mathematical Compilation but this title was soon replaced by another Greek title which means The Greatest Compilation.
- Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.
- To all appearances, one will have to credit Ptolemy with giving an essentially richer picture of the Greek firmament after his eminent predecessors.
- History Topics: Greek Astronomy
.
- Archimedes biography
- Chasles said that Archimedes' work on integration (see [A history of Greek mathematics II (Oxford, 1931).',7)">7]):-
.
- We have used the chronological order suggested by Heath in [A history of Greek mathematics II (Oxford, 1931).',7)">7] in listing these works above, except for The Method which Heath has placed immediately before On the sphere and cylinder.
- In the Method, Archimedes described the way in which he discovered many of his geometrical results (see [A history of Greek mathematics II (Oxford, 1931).',7)">7]):-
.
- Heath adds his opinion of the quality of Archimedes' work [A history of Greek mathematics II (Oxford, 1931).',7)">7]:-
.
- History Topics: Greek Astronomy
.
- History Topics: How do we know about Greek mathematicians?
.
- History Topics: How do we know about Greek mathematics?
.
- History Topics: Greek number systems
.
- Geminus biography
- It may be surprising that Geminus's name seems to be Latin rather than Greek but as Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]:-
.
- The occurrence of a Latin name in a centre of Greek culture need not surprise us, since Romans settled in such centres in large numbers during the last century BC.
- Geminus, however, in spite of his name, was thoroughly Greek.
- The authors claim this to be an important contribution to Greek astronomy introducing the use of mean motion.
- Geminus tells us that Pythagoras applied it to [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]:-
.
- Proclus quotes from Geminus (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]), saying that in the case of the parallel postulate:-
.
- History Topics: How do we know about Greek mathematicians?
.
- Hunayn biography
- However, after falling out with this teacher, Hunayn left Baghdad and, probably during a period in Alexandria, became an expert in the Greek language.
- The first steps began to be taken which would allow Greek knowledge to spread through the Islamic empire, a process in which Hunayn was to play a major role.
- He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated.
- It should not be thought that the Arabs who were translating these Greek texts simply sat down with a pile of Greek manuscripts and translated them.
- It is thought that Hunayn, being more skilled in the Greek language than any of the other scholars in Baghdad, was on this expedition.
- Hunayn is important for the many excellent translations of Greek texts which he made into Arabic.
- History Topics: How do we know about Greek mathematics?
.
- Van der Waerden biography
- This is the first book which bases a full discussion of Greek mathematics on a solid discussion of pre-Greek mathematics.
- At Gottingen - the first time I was there - I attended the lectures of Neugebauer, who gave a course on Greek mathematics.
- The papers which appeared in the years 1986-88 include: Francesco Severi and the foundations of algebraic geometry (1986), On Greek and Hindu trigonometry (1987), The heliocentric system in Greek, Persian and Hindu astronomy (1987), The astronomical system of the Persian tables (1988), On the Romaka-Siddhanta (1988), Reconstruction of a Greek table of chords (1988), and The motion of Venus in Greek, Egyptian and Indian texts (1988).
- History Topics: Greek Astronomy
.
- Diophantus (print-only)
- Knorr gives a different translation of the same passage (showing how difficult the study of Greek mathematics is for anyone who is not an expert in classical Greek) which has a remarkably different meaning:-
.
- The most details we have of Diophantus's life (and these may be totally fictitious) come from the Greek Anthology, compiled by Metrodorus around 500 AD.
- In [19] and [20] Rashed compares the four books in this Arabic translation with the known six Greek books and claims that this text is a translation of the lost books of Diophantus.
- The reviewer, familiar with the Arabic text of this manuscript, does not doubt that this manuscript is the translation from the Greek text written in Alexandria but the great difference between the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of the theory of numbers and these books containing only algebraic material make it very probable that this text was written not by Diophantus but by some one of his commentators (perhaps Hypatia?).
- It is time to take a look at this most outstanding work on algebra in Greek mathematics.
- No one has yet translated from the Greek into Latin the thirteen Books of Diophantus, in which the very flower of the whole of arithmetic lies hid..
- Thompson D'Arcy biography
- D'Arcy Thompson senior was appointed Professor of Greek at Queen's College (now University College) Galway when D'Arcy was 3 years old.
- There he won the prize for Classics, Greek Testament, Mathematics and Modern Languages in his final year at school.
- He was a Greek scholar, a naturalist and a mathematician.
- His understanding of mathematics was of the modern subject but based on the firm foundations of an understanding of Greek mathematics.
- D'Arcy related such things to the Greek work on approximating π, √2 and Euclid's Elements.
- His most important publications in this area were A glossary of Greek birds (1895), a translation of Aristotle's Historia Animalium (1910), and A glossary of Greek fishes (1945).
- D'Arcy Thompson on Greek irrationals
.
- Heron biography
- Columella, in a text written in about 62 AD [A history of Greek mathematics I, II (Oxford, 1931).',5)">5]:-
.
- Pappus writes (see for example [Selections illustrating the history of Greek mathematics II (London, 1941).',8)">8]):-
.
- Heron gives this in the following form (see for example [A history of Greek mathematics I, II (Oxford, 1931).',5)">5]):-
.
- 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 [A history of Greek mathematics I, II (Oxford, 1931).',5)">5]):-
.
- 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.
- Finally Heath writes in [A history of Greek mathematics I, II (Oxford, 1931).',5)">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.
- History Topics: Greek Astronomy
.
- Johnson Woolsey biography
- from Johns Hopkins in 1896 for his thesis Musical Pitch and the Measurement of Intervals Among the Ancient Greeks.
- I am glad to say he is his mother's boy, but he has mathematics enough in him to have written his thesis on the mathematical theory of music as it appears in the old Greek philosophers, Musical Pitch and the Measurement of Intervals among the Ancient Greeks.
- In fact Charles Johnson became an Instructor in Greek at Yale University publishing his thesis and the book The motion of the voice: In the theory of ancient music (1899).
- Thales (print-only)
- Thales seems to be the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer.
- Proclus, the last major Greek philosopher, who lived around 450 AD, wrote:-
.
- Thales is said to have travelled in Egypt, and to have thence brought to the Greeks the science of geometry.
- What Egyptians knew of geometry was mainly rules of thumb, and there is no reason to believe that Thales arrived at deductive proofs, such as later Greeks discovered.
- Diocles biography
- Although Diocles' text was largely ignored by later Greeks, it had considerable influence on the Arab mathematicians, in particular on al-Haytham.
- : the whole of the introduction confirms the impression we derive from other contemporary sources, that mathematics during the Hellenistic period was pursued, not in schools established in cultural centres, but by individuals all over the Greek world, who were in lively communication with each other both by correspondence and in their travels.
- No writing of Diocles was known to Heath in 1921 when he wrote [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3], but Toomer translated and published the newly found Arabic translation of the lost treatise On burning mirrors by Diocles in 1976.
- History Topics: How do we know about Greek mathematicians?
.
- Tannery Paul biography
- Rose Illus.) 80 (1942), 99-103.',5)">5] describes three letters he exchanged with Zeuthen dealing with questions in the Greek theory of conic sections and the significance of certain constructions by means of "neusis."
.
- This was granted and, although he did not remain there for very long, it proved a period in which his work on the history of Greek geometry flourished.
- His main contributions were to the history of Greek mathematics and to the philosophy of mathematics.
- He published a history of Greek science in 1887, a history of Greek geometry in the same year, and a history of ancient astronomy in 1893.
- Tannery became so skilled in using Greek numerals in his historical work that he believed that they had certain advantages over our present system.
- History Topics: How do we know about Greek mathematicians?
.
- Eudoxus biography
- Heath [A History of Greek Mathematics I (Oxford, 1921).',3)">3] writes of Eudoxus as a student in Athens:-
.
- The definition states (in Heath's translation [A History of Greek Mathematics I (Oxford, 1921).',3)">3]):-
.
- This appears as Euclid's Elements Book V Definition 5 which is, in Heath's translation [A History of Greek Mathematics I (Oxford, 1921).',3)">3]:-
.
- Heath [A History of Greek Mathematics I (Oxford, 1921).',3)">3] writes that Eudoxus's definition of equal ratios:-
.
- Heath, however, doubts Tannery's suggestions [A History of Greek Mathematics I (Oxford, 1921).',3)">3]:-
.
- As Heath writes [A History of Greek Mathematics I (Oxford, 1921).',3)">3]:-
.
- History Topics: Greek Astronomy
.
- Ball biography
- It taught Classical Greek, Latin, French, German, Mathematics, Chemistry, and English.
- I never remember hearing him take an active part in philosophical discussion, or comment on philosophical books: and it is difficult to believe that the account of the Greek period in his History of Mathematics (long and valuable though it is) was the work of a mind fundamentally interested in philosophical questions.
- The Senior Wrangler was George Constantine Calliphronas (1852-1913) whose Greek father was the Rev Demetrius Panaghis Calliphronas, vicar of Walpole St Andrew, Wisbech.
- The three famous problems of antiquity - the duplication of the cube, the trisection of the angle, and the quadrature of the circle - he introduces by legends and discusses as isolated puzzles: this is hardly adequate, for to the Greeks the three problems were involved in a serious and far-reaching logical programme of research into the nature of things.
- Maurolico biography
- Francesco Maurolico's name is Greek and is transcribed in a variety of different ways in addition to 'Francesco Maurolico' which is the most common.
- His family were originally Greek but had fled to Messina, in Sicily, to escape from Turkish invasions of their homeland, so their first language was Greek.
- Francesco's father, Antonio Maurolico, had fled from Constantinople to Messina where he was tutored by the Greek scholar and grammarian Constantine Lascaris who had also fled from Constantinople, settling in Messina in 1466.
- Much of Francesco's education came from his mother, described as a wise and noble women, and his father, who taught him Greek, mathematics and astronomy.
- Maurolico wrote important books on Greek mathematics, restored many ancient works from scant information and translated many ancient texts such as those by Theodosius, Menelaus, Autolycus, Euclid, Apollonius and Archimedes.
- It includes humanistic works, two libelli of carmina and epigrams, and his Latin verse translation from the Greek, Poemata Phocylidis et Pythagorae Moralia, as well as six books of Diodorus Siculus and six books of the elements of grammar.
- Anaxagoras biography
- Anaxagoras of Clazomenae was described by Proclus, the last major Greek philosopher, who lived around 450 AD as (see for example [A history of Greek mathematics 1 (Oxford, 1931).',4)">4]):-
.
- As Heath writes in [A history of Greek mathematics 1 (Oxford, 1931).',4)">4]:-
.
- The rotation [A history of Greek mathematics 1 (Oxford, 1931).',4)">4]:-
.
- This is the first record of this problem being studied and this problem, and other similar problems, were to play a major role in the development of Greek mathematics.
- This Greek city on the Asiatic shore of the Hellespont was the place for the worship of Priapus, a god of procreation and fertility.
- The best that we can hope to learn of Anaxagoras's personality is from the story that when once asked what as the point of being born he replied [A history of Greek mathematics 1 (Oxford, 1931).',4)">4]:-
.
- Aristotle biography
- Proxenus taught Aristotle Greek, rhetoric, and poetry which complemented the biological teachings that Nicomachus had given Aristotle as part of training his son in medicine.
- Since in latter life Aristotle wrote fine Greek prose, this too must have been part of his early education.
- (i) Every Greek is a person.
- (iii) Every Greek is mortal.
- Heath [A history of Greek mathematics 1 (Oxford, 1931).',15)">15] explains Aristotle's idea that 'continuous':-
.
- Aristotle writes in Physics (see for example [A history of Greek mathematics 1 (Oxford, 1931).',15)">15]):-
.
- History Topics: Greek Astronomy
.
- Eudoxus biography
- Heath [A History of Greek Mathematics I (Oxford, 1921).',3)">3] writes of Eudoxus as a student in Athens:-
.
- The definition states (in Heath's translation [A History of Greek Mathematics I (Oxford, 1921).',3)">3]):-
.
- This appears as Euclid's Elements Book V Definition 5 which is, in Heath's translation [A History of Greek Mathematics I (Oxford, 1921).',3)">3]:-
.
- Heath [A History of Greek Mathematics I (Oxford, 1921).',3)">3] writes that Eudoxus's definition of equal ratios:-
.
- Heath, however, doubts Tannery's suggestions [A History of Greek Mathematics I (Oxford, 1921).',3)">3]:-
.
- As Heath writes [A History of Greek Mathematics I (Oxford, 1921).',3)">3]:-
.
- History Topics: Greek Astronomy
.
- Zeno of Elea biography
- The Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy.
- Zeno's book of forty paradoxes was, according to Plato [A history of Greek mathematics 1 (Oxford, 1931).',8)">8]:-
.
- For the dichotomy, Aristotle describes Zeno's argument (in Heath's translation [A history of Greek mathematics 1 (Oxford, 1931).',8)">8]):-
.
- Such a paradox is 'The Arrow' and again we give Aristotle's description of Zeno's argument (in Heath's translation [A history of Greek mathematics 1 (Oxford, 1931).',8)">8]):-
.
- As Heath says [A history of Greek mathematics 1 (Oxford, 1931).',8)">8]:-
.
- It is difficult to tell precisely what effect the paradoxes of Zeno had on the development of Greek mathematics.
- Heath however seems to detect a greater influence [A history of Greek mathematics 1 (Oxford, 1931).',8)">8]:-
.
- Hypatia biography
- Hypatia, Heath writes, [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
.
- Heath writes of Theon and Hypatia's edition of the Elements [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
.
- while making only inconsiderable additions to the content of the "Elements", he endeavoured to remove difficulties that might be felt by learners in studying the book, as a modern editor might do in editing a classical text-book for use in schools; and there is no doubt that his edition was approved by his pupils at Alexandria for whom it was written, as well as by later Greeks who used it almost exclusively..
- History Topics: How do we know about Greek mathematics?
.
- Barocius biography
- The island was Greek up to the time of the Fourth Crusade in 1204 when the island was given to the leader of the Crusade after Constantinople was destroyed.
- They treated the Greek inhabitants cruelly, building fortified towns and castles as much to protect themselves from the local Cretan inhabitants as from outside invaders.
- On 28 May 1453 Constantinople was taken by the Turks and one effect was that Byzantine scholars, who had been working on recovering ancient Greek texts and making them more widely available, moved to Iraklion in Crete.
- This city then became the centre for copying manuscripts and was the route for ancient Greek learning to enter Italy.
- Scholars born in Crete and familiar with ancient Greek scholarship went to the universities in Italy adding impetus to the Renaissance.
- Francesco attended school in Padua where he learnt Greek and Latin, then he studied at the University of Padua where mathematics was part of his course.
- Barozzi was part of the movement to revive science by studying Greek texts and we explained above how that movement came to involve Crete.
- Proclus biography
- Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
.
- At the same time he was a believer in all sorts of myths and mysteries, and a devout worshipper of divinities both Greek and Oriental.
- Proclus wrote Commentary on Euclid which is our principal source about the early history of Greek geometry.
- This work is not coloured by his religious beliefs and Martin, writing in the middle of the 19th century, says (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]):-
.
- Heath, describing Proclus's Commentary on Euclid writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
.
- Proclus was not a creative mathematician; but he was an acute expositor and critic, with a thorough grasp of mathematical method and a detailed knowledge of the thousand years of Greek mathematics from Thales to his own time.
- History Topics: How do we know about Greek mathematicians?
.
- Theaetetus biography
- This allows us to give a fairly accurate date for Theaetetus's birth (although some have claimed that the Greek word could describe a man of up to 21 years old).
- There are two references to a 'Theaetetus' in the Suda Lexicon (a work of a 10th century Greek lexicographer).
- Bulmer-Thomas in [Dictionary of Scientific Biography (New York 1970-1990).',1)">1], however, thinks that Allman's explanation in [Greek geometry from Thales to Euclid ((London-Dublin, 1889), 206-215.',5)">5] is the most likely.
- In Heath's translation, see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3], (we repeat in a slightly different form part of the above quotation by Pappus) the theory of irrationals:-
.
- ., √17 were irrational (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]):-
.
- A comment (thought to be due to Geminus) states [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]:-
.
- Democritus biography
- As Heath writes in [A History of Greek Mathematics I (Oxford, 1921).',7)">7]:-
.
- Heath [A History of Greek Mathematics I (Oxford, 1921).',7)">7] writes:-
.
- There is another intriguing piece of information about Democritus which is given by Plutarch in his Common notions against the Stoics where he reports on a dilemma proposed by Democritus as reported by the Stoic Chrysippus (see [A History of Greek Mathematics I (Oxford, 1921).',7)">7], [Sudhoffs Arch.
- Firstly notice, as Heath points out in [A History of Greek Mathematics I (Oxford, 1921).',7)">7], that Democritus has the idea of a solid being the sum of infinitely many parallel planes and he may have used this idea to find the volumes of the cone and pyramid as reported by Archimedes.
- There is much discussion in [A History of Greek Mathematics I (Oxford, 1921).',7)">7], [Isis 63 (217) (1972), 205-220.
- History Topics: Greek Astronomy
.
- Archytas biography
- Archytas of Tarentum was a mathematician, statesman and philosopher who lived in Tarentum in Magna Graecia, an area of southern Italy which was under Greek control in the fifth century BC.
- Archytas led the Pythagoreans in Tarentum and tried to unite the Greek towns in the area to form an alliance against their non-Greek neighbours.
- Heath writes in [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
.
- Another mechanical device was a rattle for children which was useful, in Aristotle's words (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]):-
.
- Simplicius, in his Physics, quotes Archytas's view that the universe is infinite (in Heath's translation [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]):-
.
- Regiomontanus (print-only)
- Cardinal Bessarion was a scholar and native Greek speaker who had a mission to promote classical Greek works in Europe.
- One astrolabe in the group is of particular historical significance because it was presented at Rome in 1462, with a dedicatory inscription, to Cardinal Bessarion, titular Latin patriarch of Constantinople from 1463, and one of the illustrious Greek scholars who contributed to the great revival of letters in the fifteenth century.
- During this time he was able to read other important Greek manuscripts after improving his knowledge of the language with instruction from the native Greek speaker Bessarion.
- He wrote to the mathematician Giovanni Bianchini on 11 February 1464 saying that if he could find a complete copy he would translate the Greek text.
- Hipparchus biography
- Reasonably enough Hipparchus is often referred to as Hipparchus of Nicaea or Hipparchus of Bithynia and he is listed among the famous men of Bithynia by Strabo, the Greek geographer and historian who lived from about 64 BC to about 24 AD.
- It is not too unusual to have few details of the life of a Greek mathematician, but with Hipparchus the position is a little unusual for, despite Hipparchus being a mathematician and astronomer of major importance, we have disappointingly few definite details of his work.
- Heath writes in [A history of Greek mathematics I, II (Oxford, 1931).',6)">6]:-
.
- A corollary of this is that, before Hipparchus, astronomical tables based on Greek geometrical methods did not exist.
- If this is so, Hipparchus was not only the founder of trigonometry but also the man who transformed Greek astronomy from a purely theoretical into a practical predictive science.
- History Topics: Greek Astronomy
.
- Regiomontanus biography
- Cardinal Bessarion was a scholar and native Greek speaker who had a mission to promote classical Greek works in Europe.
- One astrolabe in the group is of particular historical significance because it was presented at Rome in 1462, with a dedicatory inscription, to Cardinal Bessarion, titular Latin patriarch of Constantinople from 1463, and one of the illustrious Greek scholars who contributed to the great revival of letters in the fifteenth century.
- During this time he was able to read other important Greek manuscripts after improving his knowledge of the language with instruction from the native Greek speaker Bessarion.
- He wrote to the mathematician Giovanni Bianchini on 11 February 1464 saying that if he could find a complete copy he would translate the Greek text.
- Regiomontanus biography
- Cardinal Bessarion was a scholar and native Greek speaker who had a mission to promote classical Greek works in Europe.
- One astrolabe in the group is of particular historical significance because it was presented at Rome in 1462, with a dedicatory inscription, to Cardinal Bessarion, titular Latin patriarch of Constantinople from 1463, and one of the illustrious Greek scholars who contributed to the great revival of letters in the fifteenth century.
- During this time he was able to read other important Greek manuscripts after improving his knowledge of the language with instruction from the native Greek speaker Bessarion.
- He wrote to the mathematician Giovanni Bianchini on 11 February 1464 saying that if he could find a complete copy he would translate the Greek text.
- Saint-Vincent biography
- From 1613 he began his career as a teacher, first at Brussels where he taught Greek.
- Then he continued to teach Greek in a number of Jesuit Colleges - in Bois-le-Duc (now 's-Hertogenbosch in the Netherlands) in 1614, and Coutrai (now Kortrijk in Belgium) in 1615.
- In contrast with classical Greek mathematics, St Vincent thus accepts, for the first time in the history of mathematics, the existence of a limit.
- There was uneasiness in the learned world because no one in that world still believed that under the specific Greek rules the quadrature of a circle could possibly be effected, and few relished the thought of trying to locate an error, or errors, in 1200 pages of text.
- He also applies his summation of series to the classical Greek problem of Zeno, namely Achilles and the tortoise.
- In Books III, IV, V and VI Saint-Vincent treats conic sections, the circle, ellipse, parabola, and hyperbola, using classical Greek geometric methods.
- Pappus (print-only)
- Pappus of Alexandria is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry.
- There is a reference in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Pappus was a contemporary of Theon of Alexandria (see for example [1]):-
.
- Obviously written with the object of reviving the classical Greek geometry, it covers practically the whole field.
- It is, however, a handbook or guide to Greek geometry rather than an encyclopaedia; it was intended, that is, to be read with the original works (where still extant) rather than to enable them to be dispensed with.
- He writes well, shows great clarity of thought and the Mathematical Collection is a work of very great historical importance in the study of Greek geometry.
- Of course Pappus did not write "Almagest" but the Greek title of the work.
- Al-Kindi biography
- Al-Ma'mun was a patron of learning and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated.
- The main task that al-Kindi and his colleagues undertook in the House of Wisdom involved the translation of Greek scientific manuscripts.
- Perhaps, rather surprisingly for a man of such learning whose was employed to translate Greek texts, al-Kindi does not appear to have been fluent enough in Greek to do the translation himself.
- Rather he polished the translations made by others and wrote commentaries on many Greek works.
- For example in his work on optics he is critical of a Greek description by Anthemius of how a mirror was used to set a ship on fire during a battle.
- Nicomachus biography
- The work contains the first multiplication table in a Greek text.
- It is also remarkable in that it contains Arabic numerals, not Greek ones.
- Heath tries to explain the apparent contradiction in [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4], suggesting that:-
.
- However Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
.
- Extract from a Greek multiplication table printed in 1538.
- Hypsicles biography
- ',1)">1] or [A History of Greek Mathematics I (Oxford, 1921).',2)">2]):-
.
- n [A History of Greek Mathematics I (Oxford, 1921).',2)">2 + (n - 1) (m - 2)]/2.
- ',1)">1] or [A History of Greek Mathematics I (Oxford, 1921).',2)">2]):-
.
- Hypsicles considers two problems in this work [A History of Greek Mathematics I (Oxford, 1921).',2)">2]:-.
- Heath writes [A History of Greek Mathematics I (Oxford, 1921).',2)">2]:-
.
- Fowler David biography
- I corresponded with him, particularly regarding the history of ancient Greek mathematics, and in one letter he explained to me how he first became interested in history.
- He began to put forward his own ideas about Greek mathematics publishing papers such as Ratio in early Greek mathematics (1979), Book II of Euclid's Elements and a pre-Eudoxan theory of ratio (1980), Anthyphairetic ratio and Eudoxan proportion (1981), and A generalization of the golden section (1982).
- It is often asserted that the discovery of the phenomenon of incommensurability led to a situation in which the early Greek mathematicians were unable to set the theory of ratio or proportion on firm foundations, within the means at their disposal, until the development by Eudoxus, in the middle of the fourth century B.C., of the proportion theory of Book V.
- The book comprises three parts: Interpretations (of the received Greek mathematics), Evidence (from the manuscripts through which it was transmitted), and Later Developments (of, among other items, continued fractions).
- Oenopides biography
- However, in contrast to these claims, Heath writes [A history of Greek mathematics I, II (Oxford, 1931).',2)">2]:-
.
- Heath writes [A history of Greek mathematics I, II (Oxford, 1931).',2)">2]:-
.
- He writes [A history of Greek mathematics I, II (Oxford, 1931).',2)">2]:-
.
- [Oenopides] may have been the first to lay down the restriction of the means permissible in constructions with ruler and compasses which became a canon of Greek geometry for all plane constructions..
- History Topics: Greek Astronomy
.
- Euclid (print-only)
- Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [1] or [9] or many other sources):-
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- None of Euclid's works have a preface, at least none has come down to us so it is highly unlikely that any ever existed, so we cannot see any of his character, as we can of some other Greek mathematicians, from the nature of their prefaces.
- Greek mathematics can boast no finer discovery than this theory, which put on a sound footing so much of geometry as depended on the use of proportion.
- Even in Greek times the most accomplished mathematicians occupied themselves with it: Heron, Pappus, Porphyry, Proclus and Simplicius wrote commentaries; Theon of Alexandria re-edited it, altering the language here and there, mostly with a view to greater clearness and consistency..
- Euclid also wrote the following books which have survived: Data (with 94 propositions), which looks at what properties of figures can be deduced when other properties are given; On Divisions which looks at constructions to divide a figure into two parts with areas of given ratio; Optics which is the first Greek work on perspective; and Phaenomena which is an elementary introduction to mathematical astronomy and gives results on the times stars in certain positions will rise and set.
- Hippocrates biography
- Heath [A History of Greek Mathematics I (Oxford, 1921), 182-202.',6)">6] recounts two versions of this story:-
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- Proclus, the last major Greek philosopher, who lived around 450 AD wrote:-
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- See [A History of Greek Mathematics I (Oxford, 1921), 182-202.',6)">6] both for the translation which we give and for a discussion of which parts are due to Eudemus:-
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- Again following Heath's translation in [A History of Greek Mathematics I (Oxford, 1921), 182-202.',6)">6]:-
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- As Heath writes in [A History of Greek Mathematics I (Oxford, 1921), 182-202.',6)">6]:-
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- Adelard biography
- one of the translators who made the first wholesale conversion of Arabo-Greek learning from Arabic into Latin.
- Adelard seems to have taken as his source one of al-Hajjaj's Arabic translations from Greek.
- We should make some further comments on his translation of al-Khwarizmi's tables which became the first Latin astronomical tables of the Arabic type with their Greek influences and Indian symbols.
- The remaining two books of the five which compose the treatise cover geometry, which is completely Greek in style, music, and astronomy.
- History Topics: How do we know about Greek mathematics?
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- Pythagoras biography
- Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings.
- Proclus, the last major Greek philosopher, who lived around 450 AD wrote (see [A history of Greek mathematics 1 (Oxford, 1931).',7)">7]):-
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- Heath [A history of Greek mathematics 1 (Oxford, 1931).',7)">7] gives a list of theorems attributed to Pythagoras, or rather more generally to the Pythagoreans.
- History Topics: Greek Astronomy
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- Eutocius biography
- It received its Greek name after it was conquered by Alexander the Great in 332 BC and the city had fine public buildings built by Herod the Great.
- Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
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- Eutocius's commentary on Apollonius's "Conics" is extant for the first four Books, and it is probably owing to their having been commented on by Eutocius, as well as to their being more elementary than the rest, that these four Books alone survive in Greek.
- Heath lists some of these important pieces of information [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
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- History Topics: How do we know about Greek mathematicians?
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- Hippocrates biography
- Heath [A History of Greek Mathematics I (Oxford, 1921), 182-202.',6)">6] recounts two versions of this story:-
.
- Proclus, the last major Greek philosopher, who lived around 450 AD wrote:-
.
- See [A History of Greek Mathematics I (Oxford, 1921), 182-202.',6)">6] both for the translation which we give and for a discussion of which parts are due to Eudemus:-
.
- Again following Heath's translation in [A History of Greek Mathematics I (Oxford, 1921), 182-202.',6)">6]:-
.
- As Heath writes in [A History of Greek Mathematics I (Oxford, 1921), 182-202.',6)">6]:-
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- Gherard biography
- Some of these translations were of Arabic works while others were of Greek works which had been translated into Arabic.
- Often however, the works were a mixture in the sense that they were Arabic commentaries on Greek works.
- Gherard is mentioned in the archive as the translator of (i) works by the Banu Musa brothers, (ii) the Tabulae Jahen (to give them the Latin name as translated by Gherard) of al-Jayyani, (iii) al-Nayrizi's commentary on Euclid's Elements which themselves were based on al-Hajjaj's Arabic translation of the Elements from the Greek, (iv) work by Thabit ibn Qurra, (v) work by Abu Kamil, and (vi) Ahmed ibn Yusuf's work on ratio and proportion.
- The tremendous upsurge of interest in Arabic and Greek science and philosophy in medieval universities from the start of the thirteenth century owes its stimulation in greater part to the work of Gerard of Cremona.
- History Topics: How do we know about Greek mathematics?
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- Delambre biography
- Delambre's interests moved from the study of Greek and Greek literature to Greek science, and he read much on the topic.
- Soon his interest in Greek astronomy led him to find out about modern astronomy and in about 1780 he read Lalande's Traite d'astronomie.
- History Topics: Greek Astronomy
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- Roberval biography
- The parish priest was actually the chaplain to the queen, Marie de Medici, and he not only instructed Gilles Personne in mathematics but also in Latin and probably Greek.
- He compared the lengths of curves, a topic not considered since the times of the ancient Greeks, equating the spiral and parabola in their ordinary forms.
- He had some close friends such as Abbe Gallois who was the editor of Le Journal des Scavans from 1665 to 1674, secretary of the Academy in 1668 and 1669 and later professor of Mathematics and Greek at the College de France.
- Borgi biography
- broke away completely from the Greek theory of numbers ..
- Borgi began his book by saying that he was not concerned with the Greeks' special numbers such as perfect numbers, abundant numbers, etc.
- However, he continued the Greek tradition of not considering 1 to be a number.
- Autolycus biography
- As Heath writes in [A history of Greek mathematics I, II (Oxford, 1931).',3)">3]:-
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- That a remark of this kind should be genuine in any Greek mathematical treatise, Euclidean or not, seems to me utterly implausible; I would assume the obvious, i.e.
- Another important fact regarding Autolycus is that two of his books have survived in the original Greek and we believe that they are the earliest two mathematics works to have survived.
- History Topics: Greek Astronomy
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- Theodorus biography
- Theodorus, in addition to his work in mathematics, was [A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212.',5)">5]:-
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- Theodorus is remembered by mathematicians for his contribution to the development of irrational numbers and it is this aspect of his work which Plato describes (see for example [A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212.',5)">5]):-
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- Heath [A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212.',5)">5] illustrates the use of this result to show that √5 is irrational.
- Heath [A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212.',5)">5] gives a geometric version of this, starting with a right-angled triangle with sides 1, 2 and √5 which may be close to the method that Theodorus used.
- Chrysippus biography
- Another piece of information, which again is not surprising, is that Chrysippus wrote Greek with very poor style.
- Despite his Greek prose being awkward, he was a prolific writer who is said to have written 705 rolls of papyri, none of which are remains today.
- Aristotle, in Metaphysics writes (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',5)">5]):-
.
- However Heath [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',5)">5] does not believe, as Chrysippus does, that Democritus regards mathematical lines as having an atomic structure.
- Lonie biography
- 1836-37 Junior Latin, Junior Greek
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- 1837-38 Greek Provectior, Mathematics 1, Logic
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- 1838-39 Greek Provectior, Latin Provectior, Ethics, Mathematics 2
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- 1839-40 Latin Provectior, Greek Provectior, Mathematics 3, Physics, Philosophy of the Senses
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- Boole (print-only)
- Having learnt Latin from a tutor, George went on to teach himself Greek.
- By the age of 14 he had become so skilled in Greek that it provoked an argument.
- He translated a poem by the Greek poet Meleager which his father was so proud of that he had it published.
- By this time he had already met Mary Everest (a niece of Sir George Everest, after whom the mountain is named) whose uncle was the professor of Greek at Cork and a friend of Boole.
- Hippias biography
- Heath tells us something of this character when he writes in [A History of Greek Mathematics I (Oxford, 1921).',3)">3]:-
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- A rather nice story, which says more of the Spartans than it does of Hippias, is that it was reported that he received no payment for the lectures he gave in Sparta since [A History of Greek Mathematics I (Oxford, 1921).',3)">3]:-
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- Heath [A History of Greek Mathematics I (Oxford, 1921).',3)">3] writes:-
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- Pappus reports that Sporus writes (see [A History of Greek Mathematics I (Oxford, 1921).',3)">3]):-
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- Heraclides biography
- His father was named Euthyphron, a wealthy man of high status from Heraclea Pontica, who was descended from one of the original founders of this Greek city on the south coast of the Black Sea.
- For example Heath [A history of Greek mathematics I, II (Oxford, 1931).',2)">2] writes:-
.
- This theory, first proposed by Tycho Brahe at the end of the 16th century, was never as far as we know put forward by a Greek astronomer.
- This nickname is a Greek pun based on the word Pompikos (meaning stately and magnificent) replacing Pontikos (meaning from Pontus).
- Al-Khwarizmi biography
- He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated.
- Their tasks there involved the translation of Greek scientific manuscripts and they also studied, and wrote on, algebra, geometry and astronomy.
- This section on mensuration certainly has more in common with Hindu and Hebrew texts than it does with any Greek work.
- as opposed to most later Islamic astronomical handbooks, which utilised the Greek planetary models laid out in Ptolemy's "Almagest"..
- Boole biography
- Having learnt Latin from a tutor, George went on to teach himself Greek.
- By the age of 14 he had become so skilled in Greek that it provoked an argument.
- He translated a poem by the Greek poet Meleager which his father was so proud of that he had it published.
- By this time he had already met Mary Everest (a niece of Sir George Everest, after whom the mountain is named) whose uncle was the professor of Greek at Cork and a friend of Boole.
- Seidenberg biography
- 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 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.
- These, he claims, originated from a common source prior to Greek, Babylonian, Chinese, and Vedic mathematics.
- Empedocles biography
- The name Acragas is Greek, while the Latin name for the town was Agrigentum.
- It was, in Empedocles time, a rich city containing the finest Greek culture.
- He travelled throughout the Greek world participating fully in the extraordinary desire for learning and understanding which gripped that part of the world.
- The reason for his four element theory was to argue a modification of the belief of the Eleatic School, one of the leading pre-Socratic schools of Greek philosophy, which had been founded by Parmenides in Elea in southern Italy.
- Boole biography
- Having learnt Latin from a tutor, George went on to teach himself Greek.
- By the age of 14 he had become so skilled in Greek that it provoked an argument.
- He translated a poem by the Greek poet Meleager which his father was so proud of that he had it published.
- By this time he had already met Mary Everest (a niece of Sir George Everest, after whom the mountain is named) whose uncle was the professor of Greek at Cork and a friend of Boole.
- Golius biography
- However, during his studies he became fascinated reading the texts of the ancient Greek mathematicians, particularly the Conics by Apollonius of Perga.
- The masterpiece Conics was written in eight books but only the first four have survived in Greek.
- He also taught his students about the ancient Greek mathematics, usually using the Arabic versions of the Greek texts.
- De Valera biography
- The results he obtained were a pass with honours in all the subjects he had taken, namely: Greek, Latin, English, French, Arithmetic, Euclid and Algebra.
- This irked his Greek teacher, John Maguire, CSSp, to such an extent that he ordered de Valera to write a Greek sentence in his mathematics copybook.
- When de Valera protested saying he had a separate copy for Greek, Maguire rejoined [Dev and his Alma Mater (Paraclete Press, Dublin, 1984).',6)">6]:-
.
- Smith David biography
- Mary Elizabeth Smith taught her son David Latin, Greek and other topics so that he had a superb education in addition to his elementary schooling in Cortland [11]:-
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- he could speak both Latin and Greek as a boy, and much preferred studying the Classics to weeding the garden or other chores assigned him by his father.
- He studied Latin, Greek, Hebrew and took courses such as elocution and rhetoric [11]:-
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- He was able to obtain about 150 manuscripts during these months in Persian, Arabic and Hebrew which were either translations by the Arabic scholars of Greek texts or were works by mathematicians such as Omar Khayyam.
- Dehn biography
- He would declaim in Greek, some passages from the classics, beer stein in hand.
- Dehn taught Mathematics, Philosophy, Greek, and Italian.
- He would declaim in Greek, some passages from the classics, beer stein in hand.
- Antiphon biography
- Aristotle writes in his Physics (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]):-
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- He wrote (translation by Heath given in [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]):-
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- However, according to Heath, this was not what Antiphon claimed [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
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- Philip Flora biography
- Flora Philip's main interests were mathematics and Greek and she was the first woman member of the Edinburgh Mathematical Society, joining the Society in December 1886 three years after the Society was founded.
- She also belonged to Professor Blackie's Greek Club, which met in rotation in the homes of its members, and in her honour Professor Blackie composed an ode in the Greek language.
- Conon biography
- Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
.
- Heath writes in [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4] that Conon was :-
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- History Topics: How do we know about Greek mathematicians?
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- Varahamihira biography
- One treatise which Varahamihira summarises was the Romaka-Siddhanta which itself was based on the epicycle theory of the motions of the Sun and the Moon given by the Greeks in the 1st century AD.
- Other works which Varahamihira summarises are also based on the Greek epicycle theory of the motions of the heavenly bodies.
- Knorr biography
- Owen, born in Wales and educated in England, was an expert on the history of Ancient Greek philosophy.
- He spent 1973-74 and 1974-75 at the University of Cambridge in England where he worked on further research which culminated in the publication of the book The Evolution of the Euclidean Elements : A Study of the Theory of incommensurable Magnitudes and its significance for early Greek Geometry.
- During this time he worked on his next book Ancient Sources of the Medieval Tradition of Mechanics: Greek, Arabic, and Latin studies of the balance (1982).
- Banu Musa al-Hasan biography
- Rashed claims, as we have suggested above, that while Archimedes' texts were being translated into Arabic for the first time, the Banu Musa (perhaps al-Hasan in particular) was trying to give new proofs of the Greek results as well as trying to prove results going beyond what the Greeks had achieved.
- Filep biography
- He published many other articles on the history of mathematics such as Lajos David (1881-1962), historian of Hungarian mathematics (1981), Great female figures of Hungarian mathematics in 19th-20th centuries (1983), The development, and the developing of, the concept of a fraction (2001), The genesis of Eudoxus's infinity lemma and proportion theory (2001), From Fejer's disciples to Erdős's epsilons - change over from analysis to combinatorics in Hungarian mathematics (2002), and Irrationality and approximation of √2 and √3 in Greek mathematics (2004).
- He summarised his 2003 paper Proportion theory in Greek mathematics as follows:-
.
- The concepts of ratio, equal ratio, and proportion played an important role in ancient Greek mathematics.
- Wallis biography
- However he spent 1631-32 at Martin Holbeach's school in Felsted, Essex, where he became proficient in Latin, Greek and Hebrew.
- Wallis made other contributions to the history of mathematics by restoring some ancient Greek texts such as Ptolemy's Harmonics, Aristarchus's On the magnitudes and distances of the sun and moon and Archimedes' Sand-reckoner.
- He was withal a good divine, and no mean critic in the Greek and Latin tongues.
- Albertus biography
- He was sent to the Dominican convent of Saint-Jacques at the University of Paris in about 1241 where he read the new translations, with commentaries, of the Arabic and Greek texts of Aristotle.
- This was a period when the writings of Arabic scholars, and through them the texts of ancient Greek philosophers, was becoming known throughout Christian Europe and it was having to come to terms with this new knowledge.
- Thomas Aquinas died in 1274 (actually on his way to the Council in Lyon) and three years later certain factions within the Church tried to condemn his teachings on the grounds that he was too favourably disposed to non-Christian philosophers, both Arabic and Greek.
- Young Thomas biography
- By the time he left this school in 1786 he was knowledgeable in many languages, including ancient Greek, Latin and Hebrew, as well as French and Italian.
- It was a two volume work based on the lectures he had given at the Royal Institution but also contained papers he had not delivered during his lectures there together with an incredible catalogue of scientific literature from Greek times up to the date of writing.
- The Rosetta Stone, containing texts in three different languages, Greek, demotic script and hieroglyphic script, had gone on display in London in 1802 but at that time he was too involved with his Royal Institution lectures, then in publishing these lectures, and after that his concentration on a medical career diverted him from what, in many ways, was the ideal task for someone of his skills and interests.
- Panini biography
- The first is an attempt to see whether there is evidence of Greek influence.
- Would it be possible to find evidence which would mean that the text had to have been written after the conquests of Alexander the Great? There is a little evidence of Greek influence, but there was Greek influence on this north east part of the Indian subcontinent before the time of Alexander.
- Aaboe biography
- He attended the Ostre Borgerdyd Skole in Copenhagen, a famous school where J L Heiberg, who produced masterly editions of Greek mathematical and astronomical classics, had been headmaster from 1884 to 1896.
- In 1963 he published On a Greek qualitative planetary model of the epicyclic variety.
- In a Greek papyrus of the second century A.D.
- Muir biography
- He attended an independent school in Wishaw, about half way between Biggar and Glasgow, and there his favourite subject was Greek.
- Thomas entered the University of Glasgow intending to study his favourite subject of Greek.
- Indeed he started on a classics course, showing outstanding ability in Greek, but Thomson (later Lord Kelvin) recognised his genius for mathematics and persuaded him to study that subject.
- Nicomedes biography
- As indicated in this quote Pappus also wrote about Nicomedes, in particular he wrote about his solution to the problem of trisecting an angle (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]):-
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- Pappus tells us (see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]):-
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- Eutocius tells us that Nicomedes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
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- Bell biography
- My interest in mathematics began with two school prizes, one in Greek, the other for physical laboratory, both richly bound in full calf.
- The Greek prize was Clerk Maxwell's classic on electricity and magnetism, the other, Homer's Odyssey.
- My cousin got the prize for Greek, I got the other.
- Snell biography
- He had taught Greek, Latin, Hebrew and the liberal arts in a high school earlier in his career and he had studied medicine and Aristotle's works.
- His schooling was from his father who taught him Latin, Greek and philosophy.
- He published two of these under a Greek title which may be translated as The Revived Geometry of Cutting off of a Ratio and Cutting off of an Area (1607).
- Anthemius biography
- Heath [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2] gives one of his problems which leads to the ellipse construction:-
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- Heath [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2] gives Anthemius's solution:-
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- Anthemius studied the focal properties of the parabola and proves that [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
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- Lafforgue biography
- My specific interest in the topic of education began a few years ago when I signed a petition defending Greek and Latin as academic subjects, as they were in grave danger.
- This reading shook me profoundly - Latin and Greek are just the tip of the iceberg! In France, even the teaching of the French language itself was at risk.
- On 15 May 2004, Lafforgue gave the address A mathematician and the classics to a conference organised to support the teaching of Latin and Greek in secondary schools.
- Bessel-Hagen biography
- He not only had a wide-ranging knowledge of various areas of mathematics, but also was well versed in classical Greek, Greek philosophy, and the history of Greek mathematics.
- Copernicus biography
- At Bologna University Copernicus studied Greek, mathematics and astronomy in addition to his official course of canon law.
- In 1509 Copernicus published a work, which was properly printed, giving Latin translations of Greek poetry by the obscure poet Theophylactus Simocattes.
- History Topics: Greek Astronomy
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- Berkeley biography
- librarian, junior dean, Greek lecturer, divinity lecturer, senior proctor, and Hebrew lecturer.
- De Moivre, Taylor, Maclaurin, Lagrange, Jacob Bernoulli and Johann Bernoulli all made attempts to bring the rigorous arguments of the Greeks into the calculus.
- Watson William biography
- He excelled at the school and, after coming top in Latin, Greek and Mathematics in every class he took, he completed his school education in 1902 being the dux and gold medallist.
- He passed the Leaving Certificate examinations in Higher French in June 1901, then in Higher Latin, Greek, Mathematics, English, and Dynamics in June 1902.
- In his first year he took the Ordinary classes in Latin and Greek but he certainly did not change course because of poor results since he achieved 75% in both coures.
- Laszlo biography
- He published many other articles on the history of mathematics such as Lajos David (1881-1962), historian of Hungarian mathematics (1981), Great female figures of Hungarian mathematics in 19th-20th centuries (1983), The development, and the developing of, the concept of a fraction (2001), The genesis of Eudoxus's infinity lemma and proportion theory (2001), From Fejer's disciples to Erdős's epsilons - change over from analysis to combinatorics in Hungarian mathematics (2002), and Irrationality and approximation of √2 and √3 in Greek mathematics (2004).
- He summarised his 2003 paper Proportion theory in Greek mathematics as follows:-
.
- The concepts of ratio, equal ratio, and proportion played an important role in ancient Greek mathematics.
- Aristaeus biography
- 'Solid loci' is the Greek name for conic sections so it is rather confusing that there is another reference by a later writer to a work by Aristaeus called Five Books concerning Conic Sections.
- Heath makes a guess at the possible contents of the Solid Loci and writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]:-
.
- Hypsicles tells us that, in this work, Aristaeus proved that [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]:-
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- Boethius biography
- Boethius was extremely well educated, being fluent in Greek and very familiar with the works of the Greek philosophers.
- His aim was to show the ways in which these two most important Greek philosophers agreed with each other.
- Peurbach biography
- He was one of the most learned scholars of his time, and spread knowledge of Greek language and learning with a personal library that included a large collection of Greek manuscripts.
- The idea which he wanted to sell to Peurbach was to produce a better translation of the Almagest from the Greek.
- Commandino biography
- There is little information about Commandino's youth and all we know of his early education is that he studied Latin and Greek at Fano under the humanist G Torelli.
- Already when he lived in Rome he had begun the task of editing Ptolemy's Planisphere and from that point on he spent the rest of his life publishing translations (mostly Greek into Latin), with commentaries, of the classic texts of Archimedes, Ptolemy, Euclid, Aristarchus, Pappus, Apollonius, Eutocius, Heron and Serenus.
- In [Torquato Tasso and the University (Italian), Ferrara, 1995 (Florence, 1997), 119-141.',9)">9] Napolitani discusses the achievements of Commandino who he rightly suggests had the greatest influence of anybody in ensuring that the classic Greek mathematical texts survived by publishing his editions of them.
- Scot biography
- Human dissections had revealed many anatomical errors by the ancient Greek physician Galen.
- However, Muslim scholarship had retained a better collective memory of the works of the ancient Greeks and other cultures, and Michael Scot played a connecting role in bringing this knowledge into Europe.
- Dionysodorus biography
- Strabo, the Greek geographer and historian (about 64 BC - about 24 AD), describes a mathematician named Dionysodorus who was born in Amisene, Pontus in northeastern Anatolia on the Black Sea.
- One papyrus states [A History of Greek Mathematics II (Oxford, 1921).',3)">3]:-
.
- ',1)">1] and [A History of Greek Mathematics II (Oxford, 1921).',3)">3]).
- Wirtinger biography
- Particularly when he was in the later stages of his career he would hold seminars in the history of mathematics, particularly on the ancient Greek mathematicians.
- His knowledge of ancient Greek allowed him to tackle these topics reading works in the original Greek.
- Simson biography
- It was while he was in England that Edmond Halley suggested to him that he might devote his considerable talents to the restoration of the work of the early Greek geometers, such as Euclid and Apollonius of Perga.
- That Simson's work was not restricted to Greek geometry is illustrated by Tweddle's paper [Arch.
- He was fond of singing Greek odes set to contemporary music.
- Viviani biography
- Throughout his life, one of Viviani's main interests was in ancient Greek mathematics.
- (Solid Loci is the Greek term for conic sections.) Pappus, however, indicated propositions from the work and Viviani reconstructed the original from these references by Pappus.
- Another restoration of a Greek text by Viviani is interesting for a number of reasons.
- Schrodinger biography
- His study of Greek science and philosophy is summarised in Nature and the Greeks (1954) which he wrote while in Dublin.
- Theodosius biography
- The reason for this comes from an error in the Suda Lexicon (a work of a 10th century Greek lexicographer) which states that Theodosius was a (see for example [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]):-
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- Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
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- Two other works by Theodosius have survived in the original Greek.
- Tunstall biography
- Certainly here he gained two degrees and achieved an outstanding reputation as a scholar of great proficiency in Greek, Latin, and mathematics.
- besides a knowledge of Latin and Greek second to none among his countrymen, he has also a seasoned judgment and exquisite taste and, more than that, unheard-of modesty and, last but not least, a lively manner which is amusing with no loss of serious worth.
- Tunstall also has the distinction of having the Grynaeus's edition of 1533, being the first printed edition, of Euclid's Elements in Greek dedicated to him:-
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- Copernicus biography
- At Bologna University Copernicus studied Greek, mathematics and astronomy in addition to his official course of canon law.
- In 1509 Copernicus published a work, which was properly printed, giving Latin translations of Greek poetry by the obscure poet Theophylactus Simocattes.
- History Topics: Greek Astronomy
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- Valerio biography
- His father was Giovanni Valeri who came from Ferrara, and his mother was Giovanna Rodomano who was of Greek extraction.
- At first he taught rhetoric and Greek at the Collegio Greco.
- In addition to these teaching positions, Valerio was also corrector of Greek at the Vatican library for many years.
- Hobbes biography
- Hobbes showed his brilliance at this school and was an outstanding Greek and Latin scholar by the time he left this school at age fourteen, having already translated Euripides' Medea from Greek into Latin iambics.
- On his return Hobbes took up studying Greek and Latin again.
- Booth biography
- This classical school specialised in teaching Greek and Latin.
- He continued to study at Trinity College and was awarded Bishop Berkeley's gold medal for Greek in 1834.
- The subjects of the examination were pure and applied mathematics, experimental physics, mental and moral philosophy, Greek language and literature, Latin language and literature, and Hebrew and cognate languages.
- Mansion biography
- Mansion was particularly grateful to three teachers at the College for the positive influence they had on him, namely J Poumay, who taught him French and German, G Smiet, who taught him mathematics, and J Kunders , who taught him Latin and Greek.
- He wrote on the history of Greek mathematics and on many mathematicians including: Hermite, Abel, de la Vallee Poussin, Saccheri, Lobachevsky, de Tilly, Poincare, Copernicus, Galileo, Kepler, Descartes, Huygens, Leibniz, Newton, d'Alembert, Euler, Laplace, Ampere, Faraday, Quetelet, Lord Kelvin, and Helmholtz.
- He also wrote on the history of physics and on Greek astronomy in Note sur le caractere geometrique de l'ancienne astronomie (1899).
- Schmidt Otto biography
- His passion for learning is illustrated by the fact that he approached the headmaster of the school in Odessa wanting to learn ancient Greek (he was already fluent in Latin).
- Although nobody else in the school was interested in studying Greek, the headmaster found a teacher to teach Schmidt in a class of one.
- Kline biography
- The conversion of mathematics by Greek philosophers into an abstract, deductive system of thought, the Greek and modern doctrine that nature is mathematically designed, the use of mathematics by Hipparchus and Ptolemy and later by Copernicus and Kepler to erect the most impressive astronomical theories, the development of a mathematical system of perspective by Renaissance painters who sought to achieve realism, the deduction by Galileo, Newton, and others of universal scientific laws which "united heaven and Earth", the reorganization of philosophy, religion, literature, and the social sciences in the Age of Reason, the rise of a statistical view of natural laws consequent upon the success of statistical procedures in the physical and social sciences, the effect of the creation of non-Euclidean geometry upon the belief in truth and on the common understanding of the nature of mathematics, and mathematics as an art are some of the illustrations of the cultural influences of mathematics.
- Williamson biography
- He sat the Leaving Certificate examinations passing Mathematics, Latin, Greek, French, and English at Higher grade.
- As an undergraduate at Edinburgh, Williamson studied Ordinary Mathematics, Natural Philosophy and Greek in session 1919-20.
- De Rham biography
- The Gymnase classique in Lausanne specialised in Greek and Latin and taught little in the way of mathematics.
- And the lessons on Greek by Jean Franel were exciting.
- Stifel biography
- Stifel lived in Luther's own house for a while and the two became close friends; also at this time he became friendly with Philipp Melanchthon, the Professor of Greek in Wittenberg and one of Luther's first supporters.
- Unable to read Greek, he studied Euclid's Elements in the Latin translation by Campanus of Novara.
- Zeuthen biography
- He was also an expert on the history of medieval mathematics and produced important studies of Greek mathematics.
- He suggested that the end of Theodorus's proof somehow involved the continued fractions for 17 and 19, a conjecture which is very much in line with modern ideas about Greek mathematics.
- Saunderson biography
- He attended the free school in the nearby small town of Penniston where he learnt Latin, Greek, French, and mathematics.
- Not only did he quickly master Euclid's Elements, which he read in the original Greek, but he also became an accomplished musician.
- Ledermann biography
- There he learnt classics, studying Latin for nine years and Greek for six years.
- Although I was fond of the classics, especially Greek with its wonderful literature, I was fascinated by mathematics immediately after my first lesson at the age of eleven, and I decided there and then to make mathematics my career.
- Jacobi biography
- He received the highest awards for Latin, Greek and history but it was the study of mathematics which he took furthest.
- By the end of academic year 1823-24 Jacobi had passed the examinations necessary for him to be able to teach mathematics, Greek, and Latin in secondary schools.
- Milne Archibald biography
- He took the Edinburgh University Preliminary Examination, passing Latin and Greek at the Lower level and English, Mathematics, and Dynamics at the Higher level in October 1894, and then Higher Latin in April 1895.
- It was doubtless true that the day had passed when Latin and Greek quotations were used in speeches in the House of Commons, but he hoped that they would never come to the day when they might doubt the words of Bacon:- "Reading maketh a full man, conference a ready man, and writing an exact man."
.
- Recorde biography
- It was a course of study which he wanted to be available to everyone, not just the few educated men who could read Latin or Greek.
- In order to do this he had to introduce many new English words to be the equivalent of the Latin or Greek terms in use at that time.
- Al-Maghribi biography
- Another important aspect of Muhyi l'din's work was the critical commentaries which he produced on some of the classic Greek works such as Euclid's Elements, Apollonius's Conics, Theodosius's Spherics, and Menelaus's Spherics.
- Later Book XV was written in Arabic by an unknown author, perhaps using Greek works which are now lost.
- Chauvenet biography
- Although William was extremely good at mathematics and this was the natural subject for him to study at university, he had to also be knowledgeable in Latin and Greek in order to be accepted onto a degree course.
- He did not find this too much trouble and after one year of study he was proficient at Latin and Greek when he entered Yale University in 1836.
- Monte biography
- Guidobaldo's book Liber mechanicorum (1577) was regarded as the greatest work on statics since Greek times.
- It was a return to classical Greek rigour deliberately rejecting the approach of Jordanus, Tartaglia and Cardan.
- Meiklejohn biography
- He sat the Preliminary Examinations of Thurso School Board obtaining passes in Higher Mathematics, Latin, and English in 1892, also passing Lower Greek in the same year.
- He went to the University of Edinburgh to study classics and in his first session studied Ordinary Latin and Greek.
- Killing biography
- The first subjects to attract Killing at the Gymnasium were the classical languages of Greek, Latin and Hebrew.
- After completing his doctorate Killing trained to become a Gymnasium teacher of mathematics and physics, also qualifying to teach Greek and Latin at a lower level.
- Qadi Zada biography
- For example Montucla says that he was a Greek convert to Islam which Dilgan suggests may come from a misunderstanding of the name al-Rumi [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:-
.
- for the peoples who lived in Asia Minor were called Rum, meaning Roman (not Greek), because Asia Minor was once Roman.
- Dunbar biography
- He sat examinations in the Scottish Leaving Certificate in 1905 and 1906 and was awarded Higher passes in English, Mathematics, Latin, and Greek, with Lower passes in Dynamics, and French.
- He first matriculated in 1907 and his course of study included Latin, Greek, Mathematics, Natural Philosophy, Intermediate Honours Mathematics, Thermodynamics, Intermediate Physics, Advanced Natural Philosophy, Advanced Mathematics, and Function Theory.
- Green biography
- He probably learnt a little of Latin, Greek and French at school but it is hard to see how even a bright eight year old boy in a good school could learn more than the briefest of introductions to these subjects.
- The mathematics examinations did not prove hard for Green, but the other topics such as Latin and Greek proved much harder for someone with only four terms of school education.
- Dinostratus biography
- ',1)">1] or [A History of Greek Mathematics I (Oxford, 1921).',3)">3]):-
.
- ',1)">1] or [A History of Greek Mathematics I (Oxford, 1921).',3)">3]):-
.
- Klugel biography
- Klugel believed that the ancient Greeks had not followed the path of seeking to generalise and had, as a consequence, not discovered negative numbers.
- The English mathematicians were, claimed Klugel, following the same road as the ancient Greeks and trying to avoid negative numbers:-
.
- Ceva Tommaso biography
- Caravaggio, who was born in Milan, was professor of mathematics at a college in Milan where he taught both mathematics and Greek literature.
- In particular the academy promoted poetry written in the classical Greek or Roman style, so Ceva's poetic style fitted extremely well and his poetry was rated very highly by the Academy.
- Domninus biography
- Heath [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2] writes of the Manual of Introductory Arithmetic :-
.
- He writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
.
- Grieve biography
- Barrie was educated at Breadalbane Academy and passed English, Mathematics, Latin, Greek, and French at the Higher grade in the Leaving Certificate examinations.
- During his first year of study he took Latin and Greek at Ordinary level, then in the following session, 1904-05, he studied Mathematics, Natural Philosophy, and Chemistry at Ordinary level.
- Lorgna biography
- Lorgna learnt both practical and theoretical aspects of hydraulics and general engineering and also, at this time, he acquired an thorough knowledge of the Croatian and French languages, as well as studying classical Latin and Greek.
- Spallanzani served as professor of logic, metaphysics, and Greek, then as professor of physics at the University of Modena before gaining worldwide recognition as a physiologist while holding a chair at the University of Pavia.
- Tweedie D J biography
- Tweedie obtained passes at Higher level in Latin, Greek, and Mathematics.
- He seems to have set out on a course studying classics for he took courses in English, Latin and Greek during his first three years of university study.
- Whitehead biography
- Whitehead's father taught him Latin from the age of ten and Greek from the age of twelve.
- There was little choice of subjects and all the boys studied as their major subjects Latin, Greek and English, with the minor subjects of mathematics, physical sciences, history, geography and modern languages receiving less attention.
- Cusa biography
- Nicholas studied Latin, Greek, Hebrew, and, in later years, Arabic, though he was not a lover of rhetoric and poetry.
- Also while in Constantinople, he discovered some important Greek manuscripts.
- Burgess biography
- In his first year of study Burgess took courses in Latin, Greek and Mathematics.
- In session 1891-92 he studied Latin, Greek, Mathematics, and Natural Philosophy.
- MacMillan Chrystal biography
- After studying at the University of Berlin over the summer of 1896, Macmillan returned to Edinburgh and matriculated in the Faculty of Arts in October 1896 after passing the Preliminary Examination in Greek.
- She then took courses on Moral Philosophy and Mental Philosophy in session 1896-97, Political Economy and Greek in session 1897-98, Moral Philosophy in 1898-99, and Mental Philosophy in session 1899-1900.
- Weil biography
- He had taught himself classical Greek, read Homer and Plato in Greek, and had also taught himself Latin.
- Van Ceulen biography
- This made his mathematical studies much harder since he could not read Latin or Greek so had to rely on friends to make translations of important texts for him.
- Van Ceulen had a problem since he could not read Greek, but Jan Cornets de Groot, the burgomaster of Delft and father of the jurist, scholar, statesman and diplomat, Hugo Grotius, translated Archimedes' approximation to π for Van Ceulen.
- Marinus biography
- He became a convert to the Greek way of life and joined the Academy in Athens where he was a pupil of Proclus who was head of the Academy.
- Marinus [A history of Greek mathematics I, II (Oxford, 1931).',2)">2]:-
.
- Guldin biography
- They were well known to scholars of that time as constituting the most important geometric work of the late Greek period.
- The argument really centres around the fact that Guldin is a classical geometer following the methods of the ancient Greek mathematicians.
- Fincke biography
- Thomas's father was described as a "learned merchant" who had been a student of Martin Luther and Philipp Melanchthon, the Professor of Greek in Wittenberg and one of Luther's first supporters.
- He received a good broad education at this school, being taught Luther's religious ideas, mainly from textbooks written by Philipp Melanchthon, and taught the ancient languages of Greek, Latin and Hebrew.
- Robins biography
- Pemberton soon had Robins reading, in English translations, the classic Greek texts on geometry by Apollonius, Archimedes and by Pappus.
- In addition to the Greek texts he read works by Fermat, Huygens, de Witt, Sluze, James Gregory, Barrow, Newton, Taylor and Cotes.
- Sluze biography
- After this de Sluze remained in Rome where he greatly enjoyed the scholarly environment and he studied a large number of subjects including many languages such as Greek, Hebrew, Arabic and Syriac, as well as mathematics and astronomy.
- His skill in languages was such that pope Innocent X often asked him to translate various letters sent to the pope by Greek and Armenian bishops and other prelates in the East.
- Copernicus (print-only)
- At Bologna University Copernicus studied Greek, mathematics and astronomy in addition to his official course of canon law.
- In 1509 Copernicus published a work, which was properly printed, giving Latin translations of Greek poetry by the obscure poet Theophylactus Simocattes.
- Kerr biography
- He won prizes in Mathematics, Greek and Natural Philosophy.
- Latin plays an unimportant part in the training and Greek, by regulations of the school, is expressly excluded.
- Al-Quhi biography
- However, it is in mathematics that he is more famous, being the leading figure in a revival and continuation of Greek higher geometry in the Islamic world.
- Of course, al-Quhi does not express the mathematics in these modern terms but rather in the usual classical geometry of ancient Greek mathematics.
- Adams biography
- John attended the nearby village school at Laneast, where he studied Greek and algebra, until he was twelve years old when he went to a private school at Devonport run by his cousin the Revd John Couch Grylls.
- He began his undergraduate mathematics course in October 1839 and graduated as Senior Wrangler (ranked top of the First Class) four years later having, rather remarkably, won the first prize in Greek testament every year.
- Truesdell biography
- He studied Latin and Ancient Greek and, using his time in various countries to acquire language skills, became proficient in German, French and Italian.
- (In my 62nd year, I started working on Attic and Homeric Greek, but was too old for such efforts.)
.
- Greaves John biography
- He also studied Greek and Arabic prompted by his broad interests.
- But it contained much more, for it was a deep archaeological study based on all the written sources that he could find, including Arabic, Persian and Greek manuscripts he had collected on his travels.
- Cleomedes biography
- Heath [A history of Greek mathematics I, II (Oxford, 1931).',2)">2] favours a date in the middle of the first century BC.
- As Heath comments [A history of Greek mathematics I, II (Oxford, 1931).',2)">2]:-
.
- Theon of Smyrna biography
- Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
.
- History Topics: Greek Astronomy
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- Talbot biography
- She was very interested in politics and fluent in French, Latin, and Greek.
- He entered Trinity College, Cambridge, in 1817 and there he won prizes for Greek verse, and graduated with the classical medal in 1821 being twelfth wrangler in mathematics (that is he was placed twelfth in the ranked list of First Class students).
- Menelaus biography
- It is also worth commenting that [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',3)">3]:-
.
- There are other works by Menelaus which are mentioned by Arab authors but which have been lost both in the Greek and in their Arabic translations.
- Macdonald William biography
- In his first year 1868-9 he studied English Literature, Greek 1, Latin 1, and Mathematics 1; in 1869-70 he studied Logic, Greek 2, Latin 2, and Mathematics 2; in 1870-71 he studied Moral Philosophy, Political Economy and Mathematics 3; in 1871-72 he studied Natural Philosophy and Chemistry; finally in 1872-73 he studied Natural Philosophy and Mathematics 3.
- Kirkman biography
- Thomas attended the grammar school in Bolton where he was taught Greek and Latin but no mathematics.
- He worked in his father's office, continuing his study of Greek and Latin in his own time and extending his knowledge of languages by also learning French and German.
- Pullar biography
- In his first year of study he took classes in Greek 1, Latin 1, and Mathematics 1.
- In 1879-80, his second year of study, Pullar took the courses Logic, Greek 2, and Latin 2.
- Dee biography
- There he studied Greek, Latin, philosophy, geometry, arithmetic and astronomy.
- We note that in his diaries Dee refers to himself as Δ, a clever pun on the fact that Δ is the Greek character for the letter "dee" and also a magical symbol.
- Young Laurence biography
- In addition to English he read French, Italian, German, Russian, Danish, Polish, Latin and Greek.
- In the last decade of his life he entertained himself by translating classic poetry from Ancient Greek into French, German, English and Italian.
- Porphyry biography
- In fact it was a clever pun since 'Porphyry' means 'purple' in Greek and he was given this name since he came from Tyre which was famous for the production of the royal purple dye and his name 'Malchus' meant 'king' = 'royal' = 'purple'.
- According to Heath [A history of Greek mathematics I, II (Oxford, 1931).',5)">5] Porphyry was:-
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- Leibniz biography
- Although he was taught Latin at school, Leibniz had taught himself far more advanced Latin and some Greek by the age of 12.
- Among the other topics which were included in this two year general degree course were rhetoric, Latin, Greek and Hebrew.
- Duhem biography
- Leaving the College Stanislas with outstanding achievements in Latin, Greek, science, mathematics and other subjects, he had to choose between studying at the Ecole Polytechnique which, in principle, prepared one to be an engineer, and the Ecole Normale, the more academic of the two.
- Duhem's mother, on the other hand, wanted him to study Latin and Greek at the Ecole Normale, principally because she feared that a study of science would turn him away from the Roman Catholic beliefs that she had instilled in her children.
- Neugebauer biography
- They established the great richness of Babylonian mathematics, far exceeding anything one could have guessed from Greek or Egyptian sources.
- History Topics: Greek Astronomy
.
- Murphy biography
- Murphy had amused himself by studying the classical Greek problems and believed that they were insoluble.
- He was ordained a deacon on 4 June 1831 and gave Greek lectures in 1832.
- Philip biography
- In his first year of study he took English Literature, Greek 1, Latin 1, and Mathematics 1.
- In session 1871-72 he studied Moral Philosophy, Political Economy, Greek 2, and Latin 3.
- Magiros biography
- Reidel Publishing Co., Dordrecht; published on behalf of the Greek Mathematical Society, Athens, 1985).
- Reidel Publishing Co., Dordrecht; published on behalf of the Greek Mathematical Society, Athens, 1985), ix.',3)">3], talks about Magiros as a teacher:-
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- Amringe biography
- Van Amringe taught at Columbia while an undergraduate but not, as one might expect, in the Department of Mathematics but rather in the Department of Greek.
- So brilliant and many-sided were his native powers and his attainments that even before graduation he had been tendered an instructorship in no fewer than five widely diverse departments: Greek, Latin, history, chemistry, mathematics.
- Serenus biography
- In the preface to the first of these Serenus gives his reasons for writing the work [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]:-
.
- As Heiberg comments in [Sereni Antinoensis opuscula (Leipzig, 1896).',3)">3], even though Greek geometry was in decline by this time mathematicians were sufficiently knowledgeable to find this definition funny.
- Callippus biography
- Heath writes [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',4)">4]:-
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- History Topics: Greek Astronomy
.
- Toeplitz biography
- An historical topic which interested him deeply was the relation between Greek mathematics and Greek philosophy.
- Uhlenbeck biography
- Although Uhlenbeck performed extremely well in his final school examinations in July 1918, he was not allowed to enter a university since his studies had not included Greek and Latin.
- He chose to study chemical engineering there but shortly after, when the rules were changed by the Dutch government so that Greek and Latin were no longer required for university entrance, he left the Institute of Technology.
- Reichardt biography
- Hans attended the Humanistic Gymnasium in Altenburg where he learnt the three ancient languages of Latin, Greek and Hebrew.
- This short book, written as a contribution to Gauss celebrations in 1977, covers rather more ground than its title suggests, for it starts with the Greek tradition and wends its way through some eighteenth-century figures and Gauss and then on to Riemann and Hilbert.
- Guo Shoujing biography
- 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.
- Grosseteste biography
- Grosseteste also made Latin translations of many Greek and Arabic scientific writings.
- Wilson Alexander biography
- The quality of the type produced by the foundry was outstanding and the finest of all was a Greek type.
- Jackson Frank biography
- 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.
- Shen Kua biography
- Unlike Greek astronomers who tried to explain all motions as circular, Shen proposed that planets moved in a willow leaf motion composed with circular motion round the earth.
- Sturm biography
- Charles-Francois's parents gave him a good education and at school he showed great promise, particularly in Greek and Latin poetry for which he had a remarkable talent.
- Rayleigh biography
- Since it refused to make chemical combinations it was called argon from the Greek word for inactive.
- Galois biography
- She taught him Greek, Latin and religion where she imparted her own scepticism to her son.
- Arf biography
- The Bulgarians defeated the main Ottoman forces, advancing towards Istanbul (then called Constantinople), and the Greeks occupied Thessaloniki.
- Takebe biography
- 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.
- Heisenberg biography
- He spent three years at that school but then in 1909 his father was appointed Professor of Middle and Modern Greek at the University of Munich.
- Lamy biography
- While teaching literature, grammar, Latin, Greek, history and geography at Juilly, Lamy was ordained to the priesthood in 1667.
- Schwartz biography
- At the lycee he attended in Paris, Schwartz excelled at both mathematics and the classical languages of Greek and Latin.
- Desargues biography
- He invented a new, non-Greek way of doing geometry, now called 'projective' or 'modern' geometry.
- Tinbergen biography
- These schools allowed entry to the university system after passing additional examinations in Latin and Greek and this Tinbergen did.
- Cavalieri biography
- The argument really centres around the fact that Guldin was a classical geometer following the methods of the ancient Greek mathematicians.
- Eisenhart biography
- He attended York High School but took the final year off school to prepare for entry to College undertaking independent study of Latin and Greek.
- Abraham biography
- Of course he knew geometry through the works of Euclid, but he also knew the contributions to geometry from other Greek texts such as Theodosius's Sphaerics in three books, On the Moving Sphere which is a work on the geometry of the sphere by Autolycus, Apollonius's Conics, and the later contributions by Heron of Alexandria and Menelaus of Alexandria.
- Jackson biography
- At this stage he was not aiming at a university education so, although he took a broad range of science subjects and modern languages, he did not take Latin or Greek one of which was compulsory at this time to enter university.
- Alcuin biography
- Most of the works of the ancient Greek mathematicians which have survived do so because of this copying process and it is the 'latest' version written in minuscule script which has survived.
- Rosellini biography
- He perfected his knowledge of French, Latin and Greek, with the help of his brother Ippolito, who by then was professor of Oriental Languages at the University of Pisa.
- Pearson biography
- His personal appearance was arresting: the typical Greek athlete, with finely cut features and a magnificent physique.
- Bortolotti biography
- We also note that recent work by Fowler has added much to our understanding of the concept of continued fractions as present in ancient Greek mathematics.
- McCowan biography
- John McCowan attended the University of Glasgow, taking classes in Latin and Greek in Session 1879-80.
- Le Tenneur biography
- He wished geometry to be Greek style, not in the style of Descartes and his followers.
- Mathieu Emile biography
- He excelled at school, first in classical studies showing remarkable abilities in Latin and Greek.
- Hurwitz biography
- However there was a difficulty -- Hurwitz did not have sufficient knowledge of Greek to satisfy the Faculty requirements! Luckily Gottingen had no such requirement and Hurwitz became a Privatdozent at the University of Gottingen after submitting his habilitation thesis there in 1882.
- Snedecor biography
- He was then appointed as an instructor at Selma Military Academy in 1905, moving after two years to Austin College, Sherman, Texas, where he taught mathematics and Greek.
- Ceva Giovanni biography
- He discovered one of the most important results on the synthetic geometry of the triangle between Greek times and the 19th Century.
- Walsh biography
- The falsehood of the Greek method of exhausted quantities, so celebrated throughout all ages, even in our own times, by the mathematicians, astronomers and philosophers of the world, as an admirable and refined invention.
- Schreier biography
- This school, which was excellent for mathematics and science, took a new approach emphasising English rather than Latin, Greek and French.
- Magnitsky biography
- He studied at the Slavo-Greco-Latin Academy in Moscow from 1685 until 1694 and there became fluent in Latin, Greek, German and Italian.
- McClintock biography
- As well as being a clergyman in the Methodist Episcopal Church, the Reverend McClintock taught mathematics, Greek, and Latin at Dickinson College, Carlisle, Pennsylvania.
- Zeckendorf biography
- There he studied the classical languages of Greek and Latin, and the modern languages of English and German.
- Humbert Pierre biography
- He had almost an ancient Greek attitude to scholarship and learning and indeed he did have a deep interest in history although it was in general more recent history than that of ancient Greece.
- Janovskaja biography
- One example of the need for mathematical rigour, Janovskaja claims, is the fact that the three classical Greek problems were not solved until they had been framed more rigorously.
- Savile biography
- On his return to Oxford in 1582 he became a Greek tutor to Queen Elizabeth.
- Maxwell biography
- Maxwell was not dux of the Edinburgh Academy, this honour going to Lewis Campbell who later became the professor of Greek at the University of St Andrews.
- Miller John biography
- He had taken a range of subjects including Latin, Greek, Mathematics and Natural Philosophy and his degree was awarded with First Class Honours in Mathematics and Physics.
- Gerbert biography
- The two had met in Rome in 972 where they were both attending the wedding of Otto I's son to a Greek princess.
- Clausen biography
- Holst was an amateur astronomer and mathematician and was able to teach Clausen these subjects as well as Latin and Greek.
- Hindenburg biography
- he took courses in medicine, philosophy, Latin, Greek, physics, mathematics, and aesthetics.
- Saccheri biography
- The topics taught there were Holy Scripture, Scholastic theology, mathematics, philosophy, Greek, Hebrew, the humanities, rhetoric and grammar.
- Cayley biography
- We are surrounded by some of the highest mountains in Scotland, among which the most beautiful is certainly Schiehallion famous in the history of gravitation; I have been trying in vain to induce Cayley to repeat Dr Maskelyne's experiments on its summit and to acquire to himself a never dying fame; but alas he has no desire for notoriety, and has a rooted aversion to experiments & calculations of all kinds: he is now sitting by my side carrying on those dreadful investigations commenced in the Mathematical Journal, in which having exhausted all the letters of the Greek & English alphabets, he is fain to turn his δελτα s tops turvy, and have recourse to the old English.
- Haughton biography
- The subjects of the examination were pure and applied mathematics, experimental physics, mental and moral philosophy, Greek language and literature, Latin language and literature, and Hebrew and cognate languages.
- Lloyd Bartholomew biography
- In addition to the positions we have mentioned above, in 1821, 1823 and 1825 he was elected regius professor of Greek.
- Bachet biography
- He is most famous for his Latin translation of Diophantus's Greek text Arithmetica (1621) in which Fermat wrote his famous 'Last Theorem' marginal note.
- Hankel biography
- improved his Greek by reading the ancient mathematicians in the original.
- Vega biography
- In addition to mathematics, he also studied Latin, Greek, religion, German, history, geography and science as this school.
- Hart biography
- The subjects of the examination were pure and applied mathematics, experimental physics, mental and moral philosophy, Greek language and literature, Latin language and literature, and Hebrew and cognate languages.
- Hippocrates summary
- Hippocrates was a Greek mathematician who worked on the classical problems of squaring the circle and duplicating the cube.
- Pappus summary
- Pappus is the last of the great Greek geometers and one of his theorems is cited as the basis of modern projective geometry.
- Euclid summary
- Euclid was a Greek mathematician best known for his treatise on geometry: The Elements .
- Diophantus summary
- Diophantus was a Greek mathematician sometimes known as 'the father of algebra' who is best known for his Arithmetica.
- Eudoxus summary
- Eudoxus was a Greek mathematician and astronomer who contributed to Euclid's Elements.
- Thales summary
- Thales was the first known Greek philosopher, scientist and mathematician.
- Kepler (print-only)
- At Tubingen, Kepler studied not only mathematics but also Greek and Hebrew (both necessary for reading the scriptures in their original languages).
- Abu Kamil (print-only)
- It is the combination of the geometric methods developed by the Greeks together with the practical methods developed by al-Khwarizmi mixed with Babylonian methods.
- Hippocrates (print-only)
- Proclus, the last major Greek philosopher, who lived around 450 AD wrote:-
.
- Picard Emile (print-only)
- Strangely he was a brilliant pupil at almost all his subjects, particularly in translating Greek and Latin poetry, but he disliked mathematics.
- Hamilton (print-only)
- By the age of five, William had already learned Latin, Greek, and Hebrew.
- Lagrange (print-only)
- At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.
- Aryabhata I (print-only)
- The conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found to be without foundation.
- Germain (print-only)
- Sophie pursued her studies, teaching herself Latin and Greek.
- Fontenelle (print-only)
- For example in De l'origine des fables (1724) he compared Greek and American Indian myths and suggested that there was a universal human predisposition toward mythology.
- Viete (print-only)
- However, Viete did not find Arabic mathematics to his liking and based his work on the Italian mathematicians such as Cardan, and the work of ancient Greek mathematicians.
- Galois (print-only)
- She taught him Greek, Latin and religion where she imparted her own scepticism to her son.
- Maxwell (print-only)
- Maxwell was not dux of the Edinburgh Academy, this honour going to Lewis Campbell who later became the professor of Greek at the University of St Andrews.
- Kepler biography
- At Tubingen, Kepler studied not only mathematics but also Greek and Hebrew (both necessary for reading the scriptures in their original languages).
- Abu Kamil biography
- It is the combination of the geometric methods developed by the Greeks together with the practical methods developed by al-Khwarizmi mixed with Babylonian methods.
- Picard Emile biography
- Strangely he was a brilliant pupil at almost all his subjects, particularly in translating Greek and Latin poetry, but he disliked mathematics.
- Hamilton biography
- By the age of five, William had already learned Latin, Greek, and Hebrew.
- Lagrange biography
- At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.
- Aryabhata I biography
- The conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found to be without foundation.
- Germain biography
- Sophie pursued her studies, teaching herself Latin and Greek.
- Fontenelle biography
- For example in De l'origine des fables (1724) he compared Greek and American Indian myths and suggested that there was a universal human predisposition toward mythology.
- Viete biography
- However, Viete did not find Arabic mathematics to his liking and based his work on the Italian mathematicians such as Cardan, and the work of ancient Greek mathematicians.
- Galois biography
- She taught him Greek, Latin and religion where she imparted her own scepticism to her son.
- Maxwell biography
- Maxwell was not dux of the Edinburgh Academy, this honour going to Lewis Campbell who later became the professor of Greek at the University of St Andrews.
- Suter biography
- In 1863 Suter entered the high school for Zurich Canton where he studied Latin and Greek.
- Newson biography
- She was taught at home by her mother, who taught herself Latin and Greek so that she could prepare her children for a university education.
- Woods biography
- used to boast that he was the only fisherman in Auckland with a knowledge of Latin and Greek, and he was certainly not going to let his son waste time learning such useless dead languages!
.
- Pascal Etienne biography
- He had a particular passion for science and mathematics but he was also knowledgeable in ancient languages, Greek literature, and the art of poetry.
- Scheiner biography
- Welser was a scholar of Greek and Latin with a passion for history and philology who corresponded with many Jesuit scholars, including Christopher Clavius.
- Lewis biography
- There he studied Latin, Greek, French, mathematics and physics.
- Angeli biography
- The approach followed by Angeli in all these works is that of his teacher Cavalieri and of Torricelli, so when Guldin and Tacquet attacked these methods and defended the approach of the ancient Greeks, Angeli disputed with them over indivisibles.
- Stern biography
- He had a particular talent for languages, learning Hebrew, Yiddish, Latin and Greek to perfection, as well as ancient oriental languages such as Chaldean and Syriac.
- Hamilton William biography
- William was educated at a variety of Scottish and English schools before entering the University of Glasgow when he was 12 years old to study Greek and Latin.
- Khayyam biography
- In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations.
- Neyman biography
- He contacted a Greek professor of international law who he knew to have been in Paris when he was there in 1926-27.
- Graves Charles biography
- The subjects of the examination were pure and applied mathematics, experimental physics, mental and moral philosophy, Greek language and literature, Latin language and literature, and Hebrew and cognate languages.
- Al-Kashi biography
- 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).
- Mitchell James biography
- He won medals or prizes in a wide range of subjects: mathematics, natural philosophy, Latin, Greek, logic, and psychology.
- Coulson biography
- Indeed, the very word electricity is derived from the Greek word for amber.
- Wu Wen-Tsun biography
- 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.
- Hutton James biography
- There he studied Latin, Greek and mathematics, and in November 1740, at the age of fourteen, he entered the University of Edinburgh.
- Gauss biography
- Gauss left Gottingen in 1798 without a diploma, but by this time he had made one of his most important discoveries - the construction of a regular 17-gon by ruler and compasses This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae.
- Kepler biography
- At Tubingen, Kepler studied not only mathematics but also Greek and Hebrew (both necessary for reading the scriptures in their original languages).
- Cataldi biography
- These were known to the ancient Greeks, and the next perfect number had been found in 1536 by Hudalrichus Regius who showed that 213 - 1 is prime giving 33350336 as the next perfect number (this had been discovered earlier by a number of mathematicians but their discoveries only became common knowledge comparatively recently).
- Abu Kamil biography
- It is the combination of the geometric methods developed by the Greeks together with the practical methods developed by al-Khwarizmi mixed with Babylonian methods.
- Study biography
- Eduard Study was the son of Carl Traugott Wilhelm Study, a teacher of Latin, Greek, German and history at the Coburg Gymnasium, and Caroline Therese Henriette von Langsdorff.
- Klingenberg biography
- Wilhelm attended schools in Berlin where he learnt Latin, Greek and French but he had to study mathematics on his own.
- Wylie biography
- He took part in activities organised by the University of the Third Age in Cambridge, including play-reading in the original Greek, impressing his fellow readers of plays until the end of his life.
- Maclaurin biography
- Maclaurin appealed to the geometrical methods of the ancient Greeks and to Archimedes' method of exhaustion in attempting to put Newton's calculus on a rigorous footing.
- Cantor Moritz biography
- However, Cantor saw that the Roman surveyors had played an important role in transmitting Egyptian and ancient Greek practical geometric methods to Europe in the Middle Ages.
- Kronecker biography
- students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.
- Roomen biography
- The first part contains a Latin translation by van Roomen of the Greek text of Archimedes' On the measurement of the circle.
- Al-Qalasadi biography
- Ignorance of the earlier contributions led historians to give too much credit to al-Qalasadi who in many ways displayed the same characteristics as the later ancient Greek mathematicians.
- Levy Paul biography
- Paul attended the Lycee Saint Louis in Paris and he achieved outstanding success winning prizes not only in mathematics but also in Greek, chemistry and physics.
- Antoine biography
- Good father, he watched over the upbringing of his children, speaking in French, in Latin, in Greek, teaching physics and mathematics well ..
- Lagny biography
- While at the College he composed Greek verse, and also studied mathematics texts such as Euclid's Elements and an algebra text by Jacques Pelletier which, Fontenelle writes [Histoire et Memoires de l\'Academie ..
- Al-Tusi Nasir biography
- Al-Tusi wrote many commentaries on Greek texts.
- Hilbert biography
- Although this was reputed to be the best school in Konigsberg, the emphasis was on Latin and Greek with mathematics considered as less important.
- Gassendi biography
- From 1611, Gassendi studied theology under Professor Raphaelis and, as part of the course, he learnt Greek and Hebrew.
- Brunelleschi biography
- Finally we should mention the fact, in the tradition of ancient Greek architects, that he made stage sets for shows and festivals.
- Murphy Gerard biography
- He took his teaching responsibility seriously and sought to inculcate the notions of precision and proof handed down by the ancient Greeks.
- Renyi biography
- However, he was still able to show his outstanding abilities by receiving an honorable mention in both a mathematics competition and in a Greek competition in the autumn of 1939.
- Dubreil-Jacotin biography
- Her mother was the daughter of M Rodon, who was a glass-blower from Briare in Loiret, although the family were originally of Greek origin.
- Hardy Claude biography
- He edited the Greek edition of Euclid and provided a Latin translation of the work and the commentary by Marin Mersenne.
- Nikodym biography
- He prepared himself for the exams in Latin and Greek and also obtained a classical high school diploma.
- Lindemann biography
- The problem of squaring the circle, namely constructing a square with the same area as a given circle using ruler and compasses alone, had been one of the classical problems of Greek mathematics.
- Kac biography
- Kac studied Latin and Greek at school as well as mathematics, physics and chemistry.
- Sporus biography
- Eutocius however supports Archimedes, writing (in Heath's translation, see for example [A History of Greek Mathematics (2 Vols.) (Oxford, 1921).',2)">2]):-
.
- Koopmans biography
- There he learnt mathematics, physics, chemistry, Latin, Greek, and three modern languages.
- Lawson biography
- In session 1887-88 he took the classes Greek 2, Latin 2, and Mathematics 1; in 1888-89 he took Logic, English Literature, and Mathematics 2; in session 1889-90 he took Natural Philosophy, Moral Philosophy and Political Economy, English Literature, Advanced Metaphysics, and Mathematics 3; in session 1890-91 he took Natural Philosophy, Chemistry, Natural History, and Physiology.
- Apery biography
- Roger Apery's father, Georges Apery (1887-1978), was born in Constantinople in 1887 but he was of Greek origin.
- Richard Louis biography
- While teaching at this lycee he became friendly with a student A J H Vincent, who later became a famous historian of Greek mathematics.
- Archibald biography
- He was brought up in the classical tradition with much emphasis on Latin and Greek.
- Sitter biography
- He is not a cold, dispassionate juggler of Greek letters, a balancer of equations, but rather an artist in whom wild flights of the imagination are restrained by the formalism of a symbolic language and the evidence of observation.
- Young Andrew biography
- He passed the Leaving Certificate examinations at Higher level in English, Latin, Greek, and Mathematics in 1907.
- Bassi biography
- She is said to have studied anatomy, natural history, logic, metaphysics, philosophy, chemistry, hydraulics, mechanics, algebra, geometry, ancient Greek, Latin, French, and Italian.
- Tibbon biography
- He translated into Hebrew many Arabic versions of Greek mathematical and astronomical works, including Euclid's Elements, Euclid's Data, Euclid's Optics, Menelaus's Spherics, Autolycus of Pitane's On the Moving Sphere, Ptolemy's Almagest as well as certain Arabic works such as al-Haytham's Configuration of the World (intended for the layman), and works by al-Ghazali, al-Zarqali, and others.
- Brown Alexander biography
- He sat the Scottish Leaving Certificate examinations and passed Higher English, Mathematics, Latin, and Greek in June 1893.
- Williams biography
- Her mother was a painter while her father was a mathematician who had graduated from Harvard and taught mathematics and Greek at a private boys school.
- Zenodorus biography
- History Topics: How do we know about Greek mathematicians?
.
- McQuistan biography
- After winning a Marshall Bursary, he entered the University of Glasgow in the same year and, after obtaining honours in Greek and Latin in April 1899, he graduated M.A.
- Picard Emile biography
- Strangely he was a brilliant pupil at almost all his subjects, particularly in translating Greek and Latin poetry, but he disliked mathematics.
- Schwarzschild biography
- There the motto runs that mathematics, physics, and astronomy constitute one knowledge, which, like the Greek culture, is only comprehended as a perfect whole.
- Fergola biography
- However, his interests turned towards the study of ancient Greek geometry and he wrote Nuovo metodo da risolvere alcuni problemi di sito e di posizione in 1786.
- Todhunter biography
- His sustained industry and methodical distribution of his time enabled him to acquire a wide acquaintance with general and foreign literature; and besides being a sound Latin and Greek scholar, he was familiar with French, German, Spanish, Italian, and also Russian, Hebrew, and Sanscrit.
- Aepinus biography
- He came from a famous family of theologians who were originally named Hoeck or Hoch but Franz's great-grandfather had changed the family name to its Greek form.
- Ferrand biography
- Her father, Auguste, was a school teacher of Latin and Greek.
- Anaximander biography
- ',17)">17] to be among the early Greek works that led to the modern study of trigonometry,
.
- Jagannatha biography
- So Jai Singh employed Jagannatha to make Sanskrit translations of the important Greek scientific works which at that time were only available in Arabic translations.
- Tarski biography
- He studied subjects such as Russian, German, French, Greek and Latin in addition to the standard school topics.
- Miller Kelly biography
- Miller was awarded a scholarship to study there but had to take a 3-year Preparatory Course covering Latin, Greek, and mathematics before attending the College of Arts and Sciences at Howard.
- Stein Philip biography
- But whether it was Greek Orthodox or Roman Catholic I do not know.
- Hamilton biography
- By the age of five, William had already learned Latin, Greek, and Hebrew.
- Burkill biography
- Three years later he entered St Paul's School, London, where he was an outstanding scholar of classical Greek and Latin as well as showing considerable talent for mathematics.
- Hooke biography
- At Westminster Hooke learnt Latin and Greek but, although he enjoyed speaking Latin, unlike his contemporaries he never wrote in Latin.
- Bacon biography
- Bacon strongly believed in this teaching by St Augustine and studied all the Greek and Arabic works he could lay his hands on.
- Lagrange biography
- At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.
- Mathews biography
- Mathews was an accomplished classical scholar; and besides Latin and Greek he was proficient in Hebrew, Sanskrit and Arabic.
- Wren biography
- This second plan was based on a Greek design which was rejected by the clergy as not in keeping with the proper form of a Christian church.
- Al-Samarqandi biography
- Such methods of enquiry were much used by the ancient Greeks.
- Bethe biography
- In an unusual twist, he wrote first from left to right on one line, and then from right to left on the following line, a manner of writing he much later learned was employed by the ancient Greeks on some of their tablets.
- Titchmarsh biography
- I learnt enough Latin to pass and enough Greek to fail.
- Cunitz biography
- It is a little difficult now to assess exactly how far her education took her, but Johann Kaspar Eberti, writing in Educated Silesian Women and Female Poets in 1727 long after her death, claimed that Cunitz mastered many languages, in particular Hebrew, Greek, Latin, German, Polish, Italian, and French.
- Luzin biography
- The aim of set theory is a question of great importance: can we regard a line atomistically as a set of points: incidentally this question is not new, but goes back to the Greeks.
- Aryabhata I biography
- The conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found to be without foundation.
- Hodge biography
- He opted to take optional courses on Latin and Greek instead of science.
- Dirac biography
- There was no Latin or Greek, something of which I was rather glad, because I did not appreciate the value of old cultures.
- Borda biography
- In the College he studied Greek, Latin until he reached the age of eleven but he learnt little of mathematics or science from the Barnabites.
- Mourey biography
- The text itself offers no information on its author; notably, it is without reference to Mourey's mathematical influences and in this respect it is similar to the work of the great Greek authors.
- De Beaune biography
- (1) Nothing in Greek is represented in the library.
- Werner biography
- His work on the duplication of the cube was not original but simply collected eleven methods which were known to the ancient Greeks.
- Rogosinski biography
- This was a humanistic gymnasium with strong emphasis on Latin and Greek at which Werner excelled.
- Stewart biography
- During this time, however, he continued to correspond with Simson, who was by now a friend rather than a teacher, on Greek geometry.
- Pell biography
- He was by this time an expert in Latin and Greek and, although we know little of his training in mathematics, we do know that he corresponded with Briggs about logarithms in the year in which he graduated with his B.A.
- Bernoulli Johann biography
- Hence Johann was not returning to Basel expecting the chair of mathematics, rather he was returning to fill the chair of Greek.
- Tacquet biography
- After this Tacquet, showing his versatility, spent a while teaching Greek and poetry at Bruges.
- Wilkins biography
- He was taught his Latin and Greek by Edward Sylvester, a noted Grecian, who kept a Private School in the Parish of All Saints in Oxford: His Proficiency was such, that at Thirteen Years of Age he entered a Student in New-Inn, in Easter-Term 1627.
- Sylvester biography
- His mastery of French, German, Italian, and Greek was often reflected in the mathematical neologisms - like "meicatecticizant" and "tamisage" - for which he gained a certain notoriety.
- Thomson James biography
- I was teaching eight hours a day at Dr Edgar's, and during the extra hours - often fagged and comparatively listless - I was reading Greek and Latin to prepare me for entering College, which I did not do till nearly two years after.
- Prandtl biography
- He excelled at Latin, Greek and particularly natural sciences.
- Plato biography
- History Topics: Greek Astronomy
.
- Jabir ibn Aflah biography
- In particular the author looks at his influence on the Persian astronomer Qutb al-Din al-Shirazi, who was a pupil of Nasir al-Din al-Tusi; on the Hispano-Arabian philosopher ibn Rushd, who is often known as Averroes, was born in Cordoba in 1126 and integrated Islamic traditions and Greek thought; and on Levi ben Gerson (sometimes known as Gersonides).
- De Moivre biography
- When he was eleven years old his parents sent him to the Protestant Academy at Sedan where he spent four years studying Greek under Du Rondel.
- Vitruvius biography
- Volume VI is on private houses of both Roman and Greek style.
- Granville biography
- I believe that math is in grave danger of joining Latin and Greek on the heap of subjects which were once deemed essential but are now, at least in America, regarded as relics of an obsolete, intellectual tradition ..
- Danti biography
- we know of no book or written document which has come down to us from ancient practitioners, although they were mot excellent, as is convincingly shown by the descriptions of the stage scenery they made, which was much prized both in Athens among the Greeks and in Rome among the Latins.
- Girard Albert biography
- In 1625 he prepared a revised edition of Stevin's Arithmetique but he also added to it translations from the Greek of Books 5 and 6 of Diophantus's Arithmetica as well as Stevin's Appendice.
- Euler biography
- He began his study of theology in the autumn of 1723, following his father's wishes, but, although he was to be a devout Christian all his life, he could not find the enthusiasm for the study of theology, Greek and Hebrew that he found in mathematics.
- Drysdale biography
- Drysdale passed English at Higer level in the Scottish Leaving Certificate examinations in June 1894, then, also at the Higher grade, Mathematics, Latin, and Greek in June of the following year.
- Born biography
- Max attended the Konig Wilhelm Gymnasium in Breslau, studying a wide range of subjects such as mathematics, physics, history, modern languages, Latin, Greek, and German.
- Gregory David biography
- These 'modern' theories were not taught in universities until much later and at this time even Cambridge was still teaching Greek natural philosophy.
- Germain biography
- Sophie pursued her studies, teaching herself Latin and Greek.
- Mansur biography
- Abu Nasr Mansur's reworking of the Spherics of Menelaus is particularly important since the Greek original of Menelaus work has been lost, although there are several Arabic versions.
- Ayyangar biography
- It establishes that Western writers, who had a bias towards Greek mathematicians, are wrong in saying that Hindus did not have their own system of numerals and that these numerals were spread by Arab merchants to their country.
- Von Neumann biography
- At the age of six, he was able to exchange jokes with his father in classical Greek.
- Hecke biography
- Caratheodory, who went from Gottingen to Berlin in 1918 left in the following year and, at the request of the Greek government, he took up a post at the University of Smyrna.
- Al-Jawhari biography
- He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated.
- Hellins biography
- While his nights were engaged at [the Royal Observatory at Greenwich] in stargazing for Dr Maskelyne, he was employed by day in studying Latin and Greek, which at length enabled him to get into holy orders.
- Arbuthnot biography
- He probably gave his son John a good grounding in Latin and Greek.
- Somerville biography
- In addition she studied botany and improved her knowledge of Greek.
- Nightingale biography
- The Nightingales gave their first born the Greek name for the city, which was Parthenope.
- Fontenelle biography
- For example in De l'origine des fables (1724) he compared Greek and American Indian myths and suggested that there was a universal human predisposition toward mythology.
- Al-Haytham biography
- The ancient Greeks used analysis to solve geometric problems but ibn al-Haytham sees it as a more general mathematical method which can be applied to other problems such as those in algebra.
- Abu'l-Wafa biography
- Early historians such as Moritz Cantor believed that there were opposing schools of authors, one committed to Indian methods, the other to Greek methods.
- Caramuel biography
- 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.
- Hadamard biography
- He excelled in particular in Greek and Latin.
- Al-Baghdadi biography
- Of course these properties of numbers had been studied by the ancient Greeks.
- Biancani biography
- Finally, we note that Biancani's writings on history, poetry and classical Greek and Latin have not survived.
- Graves John biography
- 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.
- Polya biography
- Following this he entered the Daniel Berzsenyi Gymnasium studying the classical languages of Greek and Latin as well as the modern language of German and of course Hungarian.
- Osipovsky biography
- Again in 1813, at the annual lecture sponsored by Kharkov University, he claimed that Bacon and Descartes had freed modern science from ancient Greek philosophy.
- Bouvelles biography
- In the following years he worked on mathematical topics, in particular trying to solve the classical Greek problem of squaring the circle.
- Hayes biography
- She took a range of subjects at Oberlin College; mathematics and science were her major topics but she also studied arts type subjects such as history, English literature, Greek and Latin.
- MacColl biography
- John MacColl was an educated man who taught his older children some Latin, Greek and mathematics.
- Jackson Dunham biography
- He excelled in high school, not only in science subjects but also in English literature, Latin, Greek and German.
- Al-Umawi biography
- One has to remember that these results are about decimal representations rather than about numbers themselves and show how an understanding of the decimal system was progressing at a time when Christian Europe (if I may call it that) had little interest in anything beyond the mathematics of the ancient Greeks.
- Bilimovic biography
- He completed his training in 1896 before moving to St Petersburg where he studied Latin and Greek at the Nikolayevsky engineering academy, named after Grand Duke Nicholas Nikolaevich of Russia.
- Wantzel biography
- Unhappy with the less academic nature of the school in 1828, Wantzel entered the College Charlemagne after being coached in Latin and Greek by a M Lievyns (whose daughter he was later to marry).
- Boussinesq biography
- From this uncle he learnt Latin and Greek as well as how to study on his own.
- Carslaw biography
- The breadth of his course in comparison to courses of today is shown by the fact that he also studied Latin, Greek, Moral Philosophy and Logic.
- Dixmier biography
- His father taught languages, French, Latin and Greek.
- Ezra biography
- The authors were called "paytanim" (from the Greek poietes, "poet"), their poems "piyyutim".
- Ursell biography
- At the Comenius Gymnasium, Fritz learnt Greek and Latin.
- Morley biography
- (It wasn't hard to gather that my father was working at geometry, and I knew pretty well what geometry was, because for a long time I had been drawing triangles and things; but when you examined the envelope he left behind, what was really mysterious was that there was hardly ever a drawing on it, but just a lot of calculations in Greek letters.
- Dezin biography
- He spoke English fluently and read easily in several languages which included German, French, Italian and Greek.
- Eudemus biography
- We are fortunate therefore that much of the knowledge that Eudemus had of the history of Greek mathematics before Euclid (it had to be before Euclid given the dates when Eudemus was writing) has reached us despite the fact that he book has not.
- Ricci biography
- Michelangelo quickly profited by these educational opportunities, quickly mastering Latin and Greek.
- Comessatti biography
- Probably no other country adheres so closely to the old Greek methods, nor guards so jealously the rich and varied traditions of its past.
- Agnesi biography
- She showed remarkable talents and mastered many languages such as Latin, Greek and Hebrew at an early age.
- FitzGerald biography
- telegraphy owes a great deal to Euclid and other pure geometers, to the Greek and Arabian mathematicians who invented our scale of numeration and algebra, to Galileo and Newton who founded dynamics, to Newton and Leibniz who invented the calculus, to Volta who discovered the galvanic coil, to Oersted who discovered the magnetic actions of currents, to Ampere who found out the laws of their action, to Ohm who discovered the law of resistance of wires, to Wheatstone, to Faraday, to Lord Kelvin, to Clerk Maxwell, to Hertz.
- Grandi biography
- Guido's family had a number notable people in it, perhaps the most significant being Lorenzo, a maternal uncle, who was a physician and professor of Greek at the University of Bologna, and the 17th century writer Domenico Legati.
- Brun biography
- A summary of pre-historic, Egyptian, Babylonian, Hindu and Arabic mathematics fills the first 80 pages, while on the next 100 pages the life and work of about two dozen Greek mathematicians is described.
- Al-Jayyani biography
- Neither Euclid nor any other Greek mathematician would have considered "number" as a geometrical magnitude, but al-Jayyani needs the notion for his definition of ratio which follows the Arabic idea of number.
- Atkinson biography
- He also was proficient in Latin and Ancient Greek, and had some knowledge of French, Italian, and Spanish.
- Schooten biography
- Van Schooten was also familiar with the classical Greek texts having read Frederico Commandino's editions of works by Archimedes, Apollonius and Pappus.
- Edgeworth biography
- Besides we owe him something, like a good German he knew that the Greek k is not a modern c, and, if any of you at any time wonder where the k in Biometrika comes from, I will frankly confess that I stole it from Edgeworth.
- Karpinski biography
- She graduated from Cornell in 1904 and, during 1904-05, she taught physics, chemistry, Greek and Latin at a school in New Jersey.
- Zeno of Sidon biography
- Heath writes in [A history of Greek mathematics I, II (Oxford, 1931).',2)">2] regarding comments by Proclus concerning Zeno:-
.
- Brouwer biography
- He had not studied Greek or Latin at high school but both were required for entry into university, so Brouwer spent the next two years studying these topics.
- Shafarevich biography
- He read of the Greek legends, the Russian fables and the German fables of the brothers Grimm.
- Viete biography
- However, Viete did not find Arabic mathematics to his liking and based his work on the Italian mathematicians such as Cardan, and the work of ancient Greek mathematicians.
- Tait biography
- Lewis Campbell, who later became the professor of Greek at the University of St Andrews, and James Clerk Maxwell were one year above Tait at the Academy.
- Comrie biography
- He first matriculated at the University of St Andrews in 1888 and in his first session he studied Greek 2, Latin 2, and Mathematics 2.
- Alison biography
- In 1879 he entered Edinburgh University, and during the sessions from 1879 to 1884 gained distinction in Latin, Greek, logic, and mathematics.
- Briggs biography
- Henry Briggs attended a grammar school near Warley Wood where he became highly proficient at Greek and Latin.
- Campanus biography
- His Euclid was almost the canonical version until the sixteenth century, when it was gradually superseded by translations made directly from the Greek.
History Topics
- Squaring the circle
- Ancient Greek index
.
- There are three classical problems in Greek mathematics which were extremely influential in the development of geometry.
- The problem of squaring the circle in the form which we think of it today originated in Greek mathematics and it is not always properly understood.
- The methods one was allowed to use to do this construction were not entirely clear, for really the range of methods used in geometry by the Greeks was enlarged through attempts to solve this and other classical problems.
- Pappus, writing in his work Mathematical collection at the end of the period of Greek development of geometry, distinguishes three types of methods used by the ancient Greeks (see for example [Greek mathematical works (London, 1939).',5)">5]):-
.
- The ancient Greeks, however, did not restrict themselves to attempting to find a plane solution (which we now know to be impossible), but rather developed a great variety of methods using various curves invented specially for the purpose, or devised constructions based on some mechanical method.
- Plutarch, in his work On Exile which was written in the first century AD, says [Selections illustrating the history of Greek mathematics : Vol 1 (From Thales to Euclid) (London, 1967).',4)">4]:-
.
- Two characters are speaking, Meton is the astronomer (see D Barrett (trs.), Aristophanes, Birds (London, 1978) or [Selections illustrating the history of Greek mathematics : Vol 1 (From Thales to Euclid) (London, 1967).',4)">4] for a shorter quote):-
.
- Indeed the Greeks invented a special word which meant 'to busy oneself with the quadrature'.
- For references to squaring the circle to enter a popular play and to enter the Greek vocabulary in this way, there must have been much activity between the work of Anaxagoras and the writing of the play.
- [Oenopides] may have been the first to lay down the restriction of the means permissible in constructions with ruler and compasses which became a canon of Greek geometry for all plane constructions..
- In fact it is a rather remarkable fact that the Greeks did not produce fallacious 'proofs' that the circle could be squared by plane methods.
- Sadly later mathematicians did not follow the good example shown by the ancient Greeks and indeed many claimed incorrectly to have discovered a 'ruler and compass' proof.
- Themistius states [A history of Greek mathematics I (Oxford, 1931).',1)">1]:-
.
- Archimedes gives the following definition of the spiral in his work On spirals (see [Greek mathematical works (London, 1939).',5)">5] for example):-
.
- Now we leave the ancient Greek period and look at later developments but the first comment we should make is that the Greeks were certainly not the only ones to be interested in squaring the circle at this time.
- Some time later the Arab mathematicians were, like the Greeks, fascinated by the problem.
- Although this treatise is of great historical interest, it does show how European mathematics at the time was far behind the ancient Greeks in depth of understanding.
- Again it is worth commenting that the ancient Greeks basically knew that the circle could not be squared by plane methods, although they stood no chance of proving it.
- The mechanical methods of the Greeks certainly appealed to Leonardo who thought about mathematics in a very mechanical way.
- Ancient Greek index
.
- Greek numbers
- Greek number systems
.
- Greek index
.
- There were no single Greek national standards in the first millennium BC.
- We should say immediately that the ancient Greeks had different systems for cardinal numbers and ordinal numbers so we must look carefully at what we mean by Greek number systems.
- Also we shall look briefly at some systems proposed by various Greek mathematicians but not widely adopted.
- The first Greek number system we examine is their acrophonic system which was use in the first millennium BC.
- However this is simply a consequence of changes to the Greek alphabet after the numerals coming from these letters had been fixed.
- Here is 1-10 in Greek acrophonic numbers.
- 1-10 in Greek acrophonic numbers.
- We have already seen that that Greek acrophonic numbers had a special symbol for 5.
- Different forms of 50 in different Greek States.
- We know that the ancient Greeks had a somewhat different idea because the numbers were used in slightly different forms depending to what the number referred.
- We now look at a second ancient Greek number system, the alphabetical numerals, or as it is sometimes called, the 'learned' system.
- It is worth noting that the Greeks were one of the first to adopt a system of writing based on an alphabet.
- The Greek alphabet used to write words was taken over from the Phoenician system and was quite close to it.
- We will not examine the forms of the Greek letters themselves, but it is certainly worth stressing how important this form of writing was to be in advancing knowledge.
- There are 24 letters in the classical Greek alphabet and these were used together with 3 older letters which have fallen out of use.
- How did the Greeks represent numbers greater than 9999? Well they based their numbers larger than this on the myriad which was 10000.
- Greek index
.
- http://www-history.mcs.st-andrews.ac.uk/HistTopics/Greek_numbers.html
.
- Greek astronomy
- Greek astronomy
.
- Ancient Greek index
.
- It is important to realise that Greek astronomy (we are interested in the topic during the 1000 years between 700 BC and 300 AD) did not involve physics.
- Indeed, as Pannekoek points out in [A history of astronomy (New York, 1989).',7)">7], a Greek astronomer aimed only to describe the heavens while a Greek physicist sought out physical truth.
- The Greeks began to think of philosophy from the time of Thales in about 600 BC.
- Thales himself, although famed for his prediction of an eclipse, probably had little knowledge of astronomy, yet he brought back from Egypt knowledge of mathematics into the Greek world and possibly also some knowledge of Babylonian astronomy.
- Hesiad, one of the earliest Greek poets, often called the "father of Greek didactic poetry" wrote around 700 BC.
- However, the Greeks relied mainly on the moon as their time-keeper and frequent adjustments to the calendar were necessary to keep it in phase with the moon and the seasons.
- Whether he learnt of the 12 signs of the zodiac from scholars in Mesopotamia or whether his discoveries were independent Greek discoveries is unknown.
- Again we do not know if the 19 year cycle was an independent discovery or whether Greek advances were still based on earlier advances in Mesopotamia.
- Meton's calendar never seems to have been adopted in practice but his observations proved extremely useful to later Greek astronomers such as Hipparchus and Ptolemy.
- The beginning of the 4th century BC was the time that Plato began his teachings and his writing was to have a major influence of Greek thought.
- I see no need for considering Greek philosophy as an early stage in the development of science ..
- The real "Greek miracle" is the fact that a scientific methodology was developed, and survived, in spite of a widely admired dogmatic philosophy.
- In fact Eudoxus marks the beginning of a new phase in Greek astronomy and must figure as one of a small number of remarkable innovators in astronomical thought.
- His sun-centred universe found little favour with the Greeks, however, who continued to develop more and more sophisticated models based on an earth centred universe.
- These are interesting observations since the work of Timocharis and Aristyllus strongly influenced the most important of all of the Greek astronomers, namely Hipparchus, who made his major contribution about 100 years later.
- Ancient Greek index
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- http://www-history.mcs.st-andrews.ac.uk/HistTopics/Greek_astronomy.html
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- Greek sources I
- How do we know about Greek mathematics?
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- Ancient Greek index
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- There are two separate articles in this archive: "How do we know about Greek mathematics?" and How do we know about Greek mathematicians?.
- There is a common belief that the question posed in this article, about Greek mathematics rather than Greek mathematicians, is easy to answer.
- Perhaps all we need to do to answer it is to read the mathematical treatises which the Greek mathematicians wrote.
- We might think, very naively, that although some of the origainal texts have been lost there should be plenty left for us to be able to gain an excellent picture of Greek mathematics.
- The truth, however, is not nearly so simple and we will illustrate the way that Greek mathematical texts have come down to us by looking first at perhaps the most famous example, namely Euclid's Elements.
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- When we read Heath's The Thirteen Books of Euclid's Elements are we reading an English translation of the words which Euclid wrote in 300 BC? In order to answer this question we need to examine the way the Elements has reached us, and, more generally, how the writings of the ancient Greek mathematicians have been preserved.
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- The Greeks, however, began to use papyrus rolls on which to write their works.
- It was not used by the Greeks, however, until around 450 BC for earlier they had only an oral tradition of passing knowledge on through their students.
- It is easy to see, therefore, why no complete Greek mathematics text older than Euclid's Elements has survived.
- vi) The first versions of the Elements to appear in Europe in the Middle Ages were not translations of any of any of these Greek texts into Latin.
- At this time no Greek texts of the Elements were known and the only versions of the Elements were those which had been translated into Arabic.
- The relations between the different versions of a large number of Greek mathematical manuscripts was brilliantly worked out by the Danish scholar J L Heiberg towards the end of the 19th century.
- Let us now turn to the works of perhaps the greatest of the Greek mathematicians, namely Archimedes.
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- William of Moerbeke (1215-1286) was archbishop of Corinth and a classical scholar whose Latin translations of Greek works played an important role in the transmission of Greek knowledge to medieval Europe.
- He had two Greek manuscripts of the works of Archimedes and he made his Latin translations from these manuscripts.
- The first of the two Greek manuscripts has not been seen since 1311 when presumably it was destroyed.
- Up until 1899 Heiberg had found no sources of Archimedes' works which were not based on the Latin translations by William of Moerbeke or on the copies of the second Greek manuscript which he used in his translation.
- The two main reasons to do this were either cost, it was cheaper to reuse an old parchment rather than purchase a new one, or often Greek texts were deliberately destroyed for it was considered by some Christians to be a holy act to destroy a pagan text and replace it by a Christian one.
- The Archimedes palimpsest had been copied in the 10th century by a monk in a Greek Orthodox monastery Constantinople.
- Ataturk faced local uprisings, official Ottoman forces opposed to him, and Greek armed forces.
- However, Turkey was declared a sovereign nation in January 1921 but, later that year, the Greek armies made major advances almost reaching Ankara.
- The survival of the library of the Metochion of the Holy Sepulchre in Istanbul could not be guaranteed amid the fighting, and head of the Greek Orthodox Church requested that the books from the library be sent to the National Library of Greece to ensure their safety.
- Ancient Greek index
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- http://www-history.mcs.st-andrews.ac.uk/HistTopics/Greek_sources_1.html
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- Rose literature
- Herbert Jennings Rose's Greek mathematical literature
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- Herbert Jennings Rose was Professor of Greek in the United College of St Salvator and St Leonard of the University of St Andrews.
- His famous text, A handbook of Greek literature (Methuen & Co, London, 1934) ran to several editions and included information on the literature of mathematics and related topics.
- We have also removed the Greek titles of the works; in a few cases no translation is offered and we have simply replaced the Greek with 'Greek Title':
.
- Of the 'Greek Title' we know that it dealt with conic sections, and consisted of two books.
- This and other questions which interested him involved finding a better system of numeration than that generally in use among Greeks; from this sprung a popular essay, addressed to Gelon, son of Hieron II of Syracuse, known as The Sand-reckoner, showing that it was perfectly possible to reckon the number of grains of sand required to fill the whole universe, by employing an ingenious system of his own for expressing high figures, and therefore that the common proverb 'numberless as the sands' was wrong.
- Hence it is not surprising that the very obscure Kleostratos, said to have been the first Greek to mention the signs of the zodiac, owes something - how much is a point as disputable as his date, which is variously computed at the sixth century or the fourth - to the Babylonians also.
- At all events, their observations were, at least in some measure, accessible to Greek astronomers after Alexander.
- Hence the constellations, especially those of the zodiac, become familiar subjects at this epoch, though scattered mentions of them are to be found from Homer onwards, and the differences between the Greek and the non-Greek picture of the heavens a matter for comment.
- It is a pardonable exaggeration to trace to him almost everything in Roman and later Greek writers which deals with any of his subjects and cannot be definitely assigned to anothe source; but it is an exaggeration nevertheless, and criticism is beginning to correct the too exuberant assumptions of enthusiasts for this highly important, but unfortunately lost writer.
- He probably was born about 135, or at most some eight or nine years sooner; he lived in Rhodes for some time from about 97 onwards, till Panaitios died; after that he set out on his travels, which included the western countries (Gaul and Spain) then almost unknown to Greeks.
- Poseidonios' own astronomical studies were probably the result of his interest in philosophy in general rather than this science in particular, but there seems little doubt that he wrote a considerable work, the 'Greek Title', besides some smaller essays; of the former we have perhaps a sort of synopsis in the compendium of astronomy by Kleomedes, a writer otherwise unknown.
- Another development of mathematics was Mechanics, a department of knowledge concerning which the Greeks seem to have written comparatively little.
- Not unconnected with mathematics is Music, and it may be mentioned here that Ptolemy wrote three books of Harmonies (i.e., musical theory, not what our musicians call harmony, for that did not exist in Greek music), which still survive, and another writer, perhaps of the third century A.D., Aristeides Quintilianus, also produced three books, On Music.
- They write in the Greek of their own day, that is in the 'common dialect', for the most part; Archimedes uses his native Sicilian Doric for several works (in the case of some which are in the common dialect, it may be suspected that they were originally in Doric also).
- apparently much interpolated by those who used it as a school-book, falsely attributed to Skylax of Karyanda, but really composed about the thirties of the fourth century B.C., the 'Greek Title', which takes the reader around the shores of the entire civilized world as known to the author, diversifying the list of names by notes on ethnology, local myths and the like.
- More important than any of these is the geography, or rather gazetteer, of Ptolemy, in eight books, under the general title of 'Greek Title'.
- Doubling the cube
- Ancient Greek index
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- There are three classical problems in Greek mathematics which were extremely important in the development of geometry.
- It is fair to say that although the problem of squaring the circle was to become the most famous in more modern times, certainly among amateur mathematicians, the problem of doubling the cube was certainly the more famous in the time of the ancient Greeks.
- Theon of Smyrna quotes a work by Eratosthenes (see Heath [A history of Greek mathematics I (Oxford, 1931).',2)">2]):-
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- Eratosthenes, in his work entitled Platonicus relates that, when the god proclaimed to the Delians through the oracle that, in order to get rid of a plague, they should construct an altar double that of the existing one, their craftsmen fell into great perplexity in their efforts to discover how a solid could be made the double of a similar solid; they therefore went to ask Plato about it, and he replied that the oracle meant, not that the god wanted an altar of double the size, but that he wished, in setting them the task, to shame the Greeks for their neglect of mathematics and their contempt of geometry.
- This story relates an episode in Greek mythology rather than to historical facts.
- The origins of the problem of doubling the cube may be somewhat obscure as we have just seen, but there is no doubt that the Greeks had known for a long time how to solve the problem of doubling the square.
- Heath also suggests in [A history of Greek mathematics I (Oxford, 1931).',2)">2] that Hippocrates may have come to the idea from number theory for he quotes Euclid's Elements Book VIII:-
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- Heath writes [A history of Greek mathematics I (Oxford, 1931).',2)">2]:-
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- However, Heath [A history of Greek mathematics I (Oxford, 1931).',2)">2] suggests that Eudoxus was:-
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- He erected a column at Alexandria dedicated to King Ptolemy with an epigram inscribed on it relating to his own mechanical solution to the problem of doubling the cube [A history of Greek mathematics I (Oxford, 1931).',2)">2]:-
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- Details of the construction is given in [A history of Greek mathematics I (Oxford, 1931).',2)">2].
- Although these many different methods were invented to double the cube and remarkable mathematical discoveries were made in the attempts, the ancient Greeks were never going to find the solution that they really sought, namely one which could be made with a ruler and compass construction.
- However, there was no way that the ancient Greeks could ever have proved such a result since it required mathematics well beyond anything they developed.
- It is fair to say, however, that although they could not prove that a ruler and compass construction was impossible the best of the ancient Greek mathematicians knew intuitively that indeed it was impossible.
- Ancient Greek index
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- Trisecting an angle
- Ancient Greek index
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- There are three classical problems in Greek mathematics which were extremely influential in the development of geometry.
- Certainly in ancient Greek times doubling of the cube was the most famous, then in more modern times the problem of squaring the circle became the more famous, especially among amateur mathematicians.
- There are a number of ways in which the problem of trisecting an angle differs from the other two classical Greek problems.
- Pappus in his Mathematical collection writes (see for example [Greek mathematical works (London, 1939).',3)">3]):-
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- The ancient Greeks would certainly have wanted to divide angles into any required ratio for once that is possible the construction of a regular polygon of any number of sides would become possible.
- The construction of regular polygons using ruler and compass was certainly one of the major aims of Greek mathematics and it was not until the discoveries of Gauss that further polygons were constructed with ruler and compass which the ancient Greeks had failed to find.
- Now one of the reasons why the problem of trisecting an angle seems to have attracted less in the way of reported solutions by the best ancient Greek mathematicians is that the construction above, although not possible with an unmarked straight edge and compass, is nevertheless easy to carry out in practice.
- So as a practical problem there was little left to do although the Greeks still were not satisfied in general with mechanical solutions from a purely mathematical point of view they did not find them.
- Heath writes in [A history of Greek mathematics I (Oxford, 1931).',1)">1]:-
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- Pappus tell us about the conchoid of Nicomedes in his Mathematical collection (see [Greek mathematical works (London, 1939).',3)">3] for example).
- Heath comments on how this passage in Pappus is interesting with regard to the Greek development of conics.
- He writes [A history of Greek mathematics I (Oxford, 1931).',1)">1]:-
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- The passage from Pappus from which this solution is taken is remarkable as being one of three passages in Greek mathematical works still extant ..
- These constructions described by Pappus show how the Greeks 'improved' their solutions to the problem of trisecting an angle.
- Ancient Greek index
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- Arabic mathematics
- In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks.
- There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century.
- The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.
- Arabic science only reproduced the teachings received from Greek science.
- There began a remarkable period of mathematical progress with al-Khwarizmi's work and the translations of Greek texts.
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- He encouraged scholarship and the first translations of Greek texts into Arabic, such as Euclid's Elements by al-Hajjaj, were made during al-Rashid's reign.
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- The most important Greek mathematical texts which were translated are listed in [Encyclopaedia Britannica.',17)">17]:-
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- The more minor Greek mathematical texts which were translated are also given in [Encyclopaedia Britannica.',17)">17]:-
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- It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry.
- Ibrahim ibn Sinan (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world.
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- Sundials
- Ancient Greek index
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- In the time of the ancient Greeks and Romans, the earth was considered the centre of the universe, which was itself a sphere containing all the stars.
- Before the Greeks developed the sundial into the forms Vitruvius lists, the more ancient civilizations of Egypt and Mesopotamia had shadow measuring devices as early as 1500 B.C.
- This is perhaps the crudest order of gnomon use and provides little of either theoretical or empirical interest for the Greeks."[Timing the sun in Egypt and Mesopotamia.
- In the Greek world, the earliest sundials "consisted of a gnomon in the form of a vertical post or peg set in a flat surface, upon which the shadow of the gnomon served to indicate the time."[Early Sundials and the Discovery of the Conic Sections.
- The equinoctial curve is a line at every latitude except the poles.[Greek and Roman sundials / Sharon L Gibbs.
- The paths of the tip of the gnomon's shadow as traced out on these horizontal sundials formed a pattern resembling an axe called a pelekinon (derived from the Greek word for axe).[Early Sundials and the Discovery of the Conic Sections.
- In addition to the centre noon line, additional oblique lines were added on either side to denote the hours of daylight before and after noon [Greek and Roman sundials / Sharon L Gibbs.
- Evidence does exist which suggests methods of projection were used to determine the hour points [Greek and Roman sundials / Sharon L Gibbs.
- These daily arcs were all parallel, and the arc of the equinox was half of a circle with the same centre as the hemisphere (a great circle).[Greek and Roman sundials / Sharon L Gibbs.
- These were great circles which ran perpendicular to the equinoctial circle [Greek and Roman sundials / Sharon L Gibbs.
- It was only when the dial was to serve as a calendar that these lines needed to correspond to the equinoxes and solstices.[Greek and Roman sundials / Sharon L Gibbs.
- Ancient Greek index
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- Greek astronomy references
- References for: Greek astronomy
.
- D R Dicks, Early Greek Astronomy to Aristotle (London, 1970).
- G Abraham, Mean sun and moon in ancient Greek and Indian astronomy, Indian J.
- J L Berggren, The relation of Greek spherics to early Greek astronomy, in Science and philosophy in classical Greece (New York, 1991), 227-248.
- B R Goldstein, The obliquity of the ecliptic in ancient Greek astronomy, in Theory and observation in ancient and medieval astronomy (London, 1985), 12-23.
- B R Goldstein, The obliquity of the ecliptic in ancient Greek astronomy, Arch.
- B R Goldstein and A C Bowen, A new view of early Greek astronomy, Isis 74 (273) (1983), 330-340.
- B R Goldstein and A C Bowen, The introduction of dated observations and precise measurement in Greek astronomy, Arch.
- G Hon, Is there a concept of experimental error in Greek astronomy?, British J.
- Jones, A Greek Saturn table, Centaurus 27 (3-4) (1984), 311-317.
- A Jones, Babylonian and Greek astronomy in a papyrus concerning Mars, Centaurus 33 (2-3) 1990), 97-114.
- A Jones, On Babylonian astronomy and its Greek metamorphoses, in Tradition, transmission, transformation (Leiden, 1996), 139-155.
- A Jones, The adaptation of Babylonian methods in Greek numerical astronomy, Isis 82 (313) (1991), 441-453.
- R Mercier, Newly discovered mathematical relations between Greek and Indian astronomy, in Proceedings of the Symposium on the 1500th Birth Anniversary of Aryabhata I, Indian J.
- D Pingree, The recovery of early Greek astronomy from India, J.
- C W Rufus, Greek astronomy - its birth, death, and immortality, J.
- B L van der Waerden, Greek astronomical calendars.
- B L van der Waerden, The Great Year in Greek, Persian and Hindu astronomy, Arch.
- B L van der Waerden, The heliocentric system in Greek, Persian and Hindu astronomy, in From deferent to equant (New York, 1987), 525-545.
- B L van der Waerden, The motion of Venus, Mercury and the Sun in early Greek astronomy, Arch.
- B L van der Waerden, Greek astronomical calendars.
- [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_astronomy.html]
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- Greek astronomy references
- References for: Greek astronomy
.
- D R Dicks, Early Greek Astronomy to Aristotle (London, 1970).
- G Abraham, Mean sun and moon in ancient Greek and Indian astronomy, Indian J.
- J L Berggren, The relation of Greek spherics to early Greek astronomy, in Science and philosophy in classical Greece (New York, 1991), 227-248.
- B R Goldstein, The obliquity of the ecliptic in ancient Greek astronomy, in Theory and observation in ancient and medieval astronomy (London, 1985), 12-23.
- B R Goldstein, The obliquity of the ecliptic in ancient Greek astronomy, Arch.
- B R Goldstein and A C Bowen, A new view of early Greek astronomy, Isis 74 (273) (1983), 330-340.
- B R Goldstein and A C Bowen, The introduction of dated observations and precise measurement in Greek astronomy, Arch.
- G Hon, Is there a concept of experimental error in Greek astronomy?, British J.
- Jones, A Greek Saturn table, Centaurus 27 (3-4) (1984), 311-317.
- A Jones, Babylonian and Greek astronomy in a papyrus concerning Mars, Centaurus 33 (2-3) 1990), 97-114.
- A Jones, On Babylonian astronomy and its Greek metamorphoses, in Tradition, transmission, transformation (Leiden, 1996), 139-155.
- A Jones, The adaptation of Babylonian methods in Greek numerical astronomy, Isis 82 (313) (1991), 441-453.
- R Mercier, Newly discovered mathematical relations between Greek and Indian astronomy, in Proceedings of the Symposium on the 1500th Birth Anniversary of Aryabhata I, Indian J.
- D Pingree, The recovery of early Greek astronomy from India, J.
- C W Rufus, Greek astronomy - its birth, death, and immortality, J.
- B L van der Waerden, Greek astronomical calendars.
- B L van der Waerden, The Great Year in Greek, Persian and Hindu astronomy, Arch.
- B L van der Waerden, The heliocentric system in Greek, Persian and Hindu astronomy, in From deferent to equant (New York, 1987), 525-545.
- B L van der Waerden, The motion of Venus, Mercury and the Sun in early Greek astronomy, Arch.
- B L van der Waerden, Greek astronomical calendars.
- http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_astronomy.html
.
- Zero
- 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.
- The Greeks however did not adopt a positional number system.
- 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".
- Neugebauer, however, dismisses this explanation since the Greeks already used omicron as a number - it represented 70 (the Greek number system was based on their alphabet).
- 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.
- Greek sources II
- How do we know about Greek mathematicians?
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- Ancient Greek index
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- There are two separate articles: How do we know about Greek mathematics? and "How do we know about Greek mathematicians?".
- Before reading this second article on how we can find out about the lives of the ancient Greek mathematicians, it will help if the reader first looks at the previous article on how the works of these mathematicians have reached us.
- Some mathematicians added a date to their work and this has been preserved during the copying process described in the article How do we know about Greek mathematics?.
- First let us see what facts Heath knew when he wrote his famous book [A history of Greek mathematics I, II (Oxford, 1921).',1)">1], A history of Greek mathematics, which he began in 1913.
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- In the 5th century AD, Proclus wrote his Commentary on Euclid which is our principal source of knowledge about the early history of Greek geometry.
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- Hence, Heath can deduce that [A history of Greek mathematics I, II (Oxford, 1921).',1)">1]:-
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- Eutocius's commentary on Archimedes On the Sphere and Cylinder II includes a quotation from Diocles solving the following problem of Archimedes (see for example [A history of Greek mathematics I, II (Oxford, 1921).',1)">1]):-
.
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- Eutocius states that the quote he gives is from Diocles' On burning mirrors but at the time Heath wrote his book no version of Diocles's text had been found, either in Greek or Arabic.
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- Heath deduces from the quotes in Eutocius that Diocles [A history of Greek mathematics I, II (Oxford, 1921).',1)">1]:-
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- Heath also writes [A history of Greek mathematics I, II (Oxford, 1921).',1)">1]:-
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- Before continuing with the description of what has been discovered since Heath wrote his A history of Greek mathematics it is reasonable to ask: Does it matter when Diocles lived? The answer is certainly "yes" in deciding how fine a mathematician Diocles was.
- Let us now turn to other ways to gain information about the ancient Greek mathematicians.
- It was a Greek city some 25 km from the Aegean Sea; today the town of Bergama, Izmir, Turkey stands on the site.
- One papyrus states [A history of Greek mathematics I, II (Oxford, 1921).',1)">1]:-
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- Ancient Greek index
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- http://www-history.mcs.st-andrews.ac.uk/HistTopics/Greek_sources_2.html
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- Infinity
- This is particularly true in ancient Greek times, as Knorr writes in [Infinity and continuity in ancient and medieval thought (Ithaca, N.Y., 1982), 112-145.',26)">26]:-
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- The interaction of philosophy and mathematics is seldom revealed so clearly as in the study of the infinite among the ancient Greeks.
- The early Greeks had come across the problem of infinity at an early stage in their development of mathematics and science.
- We will come to Cantor's ideas towards the end of this article but for the moment let us consider the effect Aristotle had on later Greek mathematicians by only allowing the potentially infinite, particularly on Euclid; see for example [Konzepte des mathematisch Unendlichen im 19.
- We should strongly emphasise, however, that this is not the way that the ancient Greeks looked at the method.
- Recently, however, evidence has come to light which suggests that not all ancient Greek mathematicians felt constrained to deal only with the potentially infinite.
- We suspect there may be no other known places in Greek mathematics - or, indeed, in ancient Greek writing - where objects infinite in number are said to be "equal in magnitude".
- History overview
- The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC.
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- The major Greek progress in mathematics was from 300 BC to 200 AD.
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- This work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics.
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- From about the 11th Century Adelard of Bath, then later Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into Europe.
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- Of course I am not suggesting that Barrow should receive the credit for discovering the calculus, I'm merely pointing out that the calculus comes out of a long period of progress starting with Greek mathematics.
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- Calculus history
- The first steps were taken by Greek mathematicians.
- To the Greeks numbers were ratios of integers so the number line had "holes" in it.
- They got round this difficulty by using lengths, areas and volumes in addition to numbers for, to the Greeks, not all lengths were numbers.
- Leucippus, Democritus and Antiphon all made contributions to the Greek method of exhaustion which was put on a scientific basis by Eudoxus about 370 BC.
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- However Archimedes, around 225 BC, made one of the most significant of the Greek contributions.
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- Luca Valerio (1552-1618) published De quadratura parabolae in Rome (1606) which continued the Greek methods of attacking these type of area problems.
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- His method consisted of thinking of areas as sums of lines, another crude form of integration, but Kepler had little time for Greek rigour and was rather lucky to obtain the correct answer after making two cancelling errors in this work.
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- Greek sources I references
- References for: How do we know about Greek mathematics?
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- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_sources_1.html
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- Greek sources II references
- References for: How do we know about Greek mathematicians?
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- T L Heath, A history of Greek mathematics I, II (Oxford, 1921).
- http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_sources_2.html
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- Greek sources I references
- References for: How do we know about Greek mathematics?
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- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_sources_1.html]
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- Greek sources II references
- References for: How do we know about Greek mathematicians?
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- T L Heath, A history of Greek mathematics I, II (Oxford, 1921).
- [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_sources_2.html]
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- Greek numbers references
- References for: Greek number systems
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- http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_numbers.html
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- Greek numbers references
- References for: Greek number systems
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- [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greek_numbers.html]
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- Mathematics and Architecture
- Now symmetry to a mathematician today suggests an underlying action of a group on a basic configuration, but it is important to realise that the word comes from the ancient Greek architectural term "symmetria" which indicated the repetition of shapes and ratios from the smallest parts of a building to the whole structure.
- It should now be clear what the belief that "all things are numbers" meant to the Pythagoreans and how this was to influence ancient Greek architecture.
- After the Greek victory over the Persian at Salamis and Plataea the Greeks did not begin the reconstruction of the city of Athens for several years.
- Only after the Greek states ended their fighting in the Five Years' Truce of 451 BC did the conditions exist to encourage reconstruction.
- The naos, which in Greek temples is the inner area containing the statue of the god, is 21.44 m wide and 48.3 m long which again is in the ratio 4 : 9.
- Of course many have argued that the golden number can be found in the proportions of the human body so it may be that the evidence found today for the golden number in ancient Greek temples is explained by its relation to human proportions.
- The wave theory of sound was Greek, its application to the acoustics of a hall typically Roman.
- Indian numerals
- However we must not forget that many countries use symbols today which are quite different from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and unless one learns these symbols they are totally unrecognisable as for example the Greek alphabet is to someone unfamiliar with it.
- 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.
- 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.
- Some historians believe that the Babylonian base 60 place-value system was transmitted to the Indians via the Greeks.
- We have commented in the article on zero about Greek astronomers using the Babylonian base 60 place-value system with a symbol o similar to our zero.
- However, the same story in Lalitavistara convinces Kaplan (see [The nothing that is : a natural history of zero (London, 1999).',3)">3]) that the Indians' ideas of numbers came from the Greeks, for to him the story is an Indian version of Archimedes' Sand-reckoner.
- Measurement
- The Greeks used as their basic measure of length the breadth of a finger (about 19.3 mm), with 16 fingers in a foot, and 24 fingers in a Greek cubit.
- These units of length, as were the Greek units of weight and volume, were derived from the Egyptian and Babylonian units.
- The Romans adapted the Greek system.
- Part of the problem was that Greek and Latin prefixes like kilo- and centi- had been proposed to help make the new system internationally acceptable but were strongly disliked in France.
- In November 1800 an attempt was made to make the system more acceptable by dropping the Greek and Latin prefixes and reinstating the older names for measures but with new metric values.
- In 1830 Belgium became independent of Holland and made the metric system, together with its former Greek and Latin prefixes, the only legal measurement system.
- Ptolemy mss.html
- Ancient Greek index
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- Unfortunately, [Studies in Greek manuscript tradition, Amsterdam :: Hakkert.',1)">1]
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- Latin texts first appeared in the Renaissance, no doubt translated from the Greek manuscripts of the day.)[Isis 22(2) (1935) 533-539.',3)">3]
.
- The nature of such errors point to a manuscript that utilized capital letters, which were used in manuscripts prior to the ninth century[Studies in Greek manuscript tradition, Amsterdam :: Hakkert.',1)">1].
- While a certain amount of errors in transmission are to be expected (noted as early as the seventh century[Studies in Greek manuscript tradition, Amsterdam :: Hakkert.',1)">1]), emendations have also been made which are not errors, as can be seen by comparing this family to others.
- In places, the Greek has been corrected or reworded, corruptions from previous traditions have been fixed, and errors in the data possibly dating all the way back to Ptolemy have been adjusted.
- Neither manuscript version A nor B could have contained the maps, which must always have occupied large sheets.[Studies in Greek manuscript tradition, Amsterdam :: Hakkert.',1)">1] For the world map to accommodate all of the cities listed in the catalogue, at best it would need to be one by two metres.
- Further evidence that the maps do not originate with Ptolemy has been put forward by Diller: [Studies in Greek manuscript tradition, Amsterdam :: Hakkert.',1)">1]
.
- Ancient Greek index
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- Greeks poetry references
- [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greeks_poetry.html]
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- Greeks poetry
- http://www-history.mcs.st-andrews.ac.uk/HistTopics/Greeks_poetry.html
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- Greeks poetry references
- http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Greeks_poetry.html
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- Real numbers 1
- We should begin a discussion of real numbers by looking at the concepts of magnitude and number in ancient Greek times.
- Before continuing to describe advances in ideas concerning numbers, it should be mentioned at this stage that the Egyptians and the Babylonians had a different notion of number to that of the ancient Greeks.
- The Babylonians looked at reciprocals and also at approximations to irrational numbers, such as √2, long before Greek mathematicians considered approximations.
- We should note that Euclid, as earlier Greek mathematicians, did not consider 1 as a number.
- His first thesis was to argue against the Greek idea that 1 is not a number but a unit and the numbers 2, 3, 4, ..
- Cartography
- The earliest ancient Greek who is said to have constructed a map of the world is Anaximander, who was born in 610 BC in Miletus (now in Turkey), and died in 546 BC.
- Of course, although only a very limited portion of the Earth was known to these ancient Greeks, the shape of the Earth was always going to be of fundamental importance in world maps.
- The final ancient Greek contribution we consider was the most important and, unlike that of Strabo, was written by a noted mathematician.
- He studied Greek, Latin and mathematics and, strongly influenced by Gerardus Mercator, went on to open a map making business.
- Christianity and Mathematics
- In order to understand the subtleties of the interaction between Christianity and the mathematical sciences through the 16th and 17th Centuries we need to go back to the Greek philosophers long before the rise of the Christian church.
- The two ancient Greek philosophers who exerted the most influence on Christianity were Plato and Aristotle.
- We have discussed Greek philosophers at some length in an article on Christianity and the mathematical sciences.
- Copernicus in the 16th century is often seen as providing the revolution which would overturn this ancient Greek world-view.
- Firstly Copernicus himself was very much in the mould of the ancient Greek mathematicians, and secondly his book On the Revolutions of the Heavenly Spheres had little impact when fist published.
- Copernicus used observational data from the Greek astronomers, and stuck rigidly to Aristotle's belief that perfect motion of the heavenly bodies had to be circular.
- His respect for the ancient Greek system is reflected in his own words (see for example [Rebuilding the matrix (Oxford, 2001).',1)">1]):-
.
- Burt Thompson
- He was born on 2nd May 1860, the son of D'Arcy Wentworth Thompson, previously classical master at the Edinburgh Academy and at that time Professor of Greek in Queen's College, Galway.
- D'Arcy Thompson pursued his investigations "according to the accepted discipline of the physical sciences," in the light of properties of matter and forms of energy, and elucidated, often in the simpler mathematics of the Greeks, the mathematical aspects of organic form.
- His contribution to pure classical scholarship is seen in his Glossary of Greek Birds (1895), Aristotle: Historia Animalium (a translation, in 1910), and his Glossary of Greek Fishes (1947).
- Doubling the cube references
- M R Cohen and I E Drabkin (trs.), A source book in Greek science (Harvard, 1948).
- T L Heath, A history of Greek mathematics I (Oxford, 1931).
- I Thomas, Selections illustrating the history of Greek mathematics : Vol 1 (From Thales to Euclid) (London, 1967).
- I Thomas, Greek mathematical works (London, 1939).
- K Saito, Doubling the cube : a new interpretation of its significance for early Greek geometry, Historia Math.
- Doubling the cube references
- M R Cohen and I E Drabkin (trs.), A source book in Greek science (Harvard, 1948).
- T L Heath, A history of Greek mathematics I (Oxford, 1931).
- I Thomas, Selections illustrating the history of Greek mathematics : Vol 1 (From Thales to Euclid) (London, 1967).
- I Thomas, Greek mathematical works (London, 1939).
- K Saito, Doubling the cube : a new interpretation of its significance for early Greek geometry, Historia Math.
- Indian mathematics
- Yavanesvara, in the second century AD, played an important role in popularising astrology when he translated a Greek astrology text dating from 120 BC.
- He introduced trigonometry in order to make his astronomical calculations, based on the Greek epicycle theory, and he solved with integer solutions indeterminate equations which arose in astronomical theories.
- The Hindu astronomers did not possess a general method for solving problems in spherical astronomy, unlike the Greeks who systematically followed the method of Ptolemy, based on the well-known theorem of Menelaus.
- Chinese overview
- 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.
- Classical light
- The study of light has been a major topic in the study of mathematics and physics from ancient Greek times up to the present day.
- The early Greek ideas on natural philosophy, and in particular on the nature of light, would influence the world for two thousand years.
- However, without making major advances on the Greeks, some Europeans did make some improvements.
- Weil family
- Toys are banned at home and instead of the chatter of children they communicate in ancient Greek or rhyming couplets [The apprenticeship of a mathematician (Basel, 1992).',3)">3]:-
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- By age twelve Andre had taught himself ancient Greek and Sanskrit.
- Simone learnt ancient Greek and a number of modern languages in her early teens.
- preferred topics of conversation at the Weils dinner table - music, literature, and Andre's favourite hobby, the collecting rare editions of Greek and Latin texts - were occasionally held in the family's second languages, German and English.
- Squaring the circle references
- T L Heath, A history of Greek mathematics I (Oxford, 1931).
- I Thomas, Selections illustrating the history of Greek mathematics : Vol 1 (From Thales to Euclid) (London, 1967).
- I Thomas, Greek mathematical works (London, 1939).
- A Wasserstein, Some early Greek attempts to square the circle, Phronesis 4 (1959), 92-100.
- Squaring the circle references
- T L Heath, A history of Greek mathematics I (Oxford, 1931).
- I Thomas, Selections illustrating the history of Greek mathematics : Vol 1 (From Thales to Euclid) (London, 1967).
- I Thomas, Greek mathematical works (London, 1939).
- A Wasserstein, Some early Greek attempts to square the circle, Phronesis 4 (1959), 92-100.
- Jaina mathematics
- As with much research into Indian mathematics there is interest in whether the Indians took their ideas from the Greeks.
- It was not until the works of Aryabhata that the Greek ideas of epicycles entered Indian astronomy.
- Set theory
- Nine chapters
- 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.
- 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.
- Tait's scrapbook
- 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.
- 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.
- EMS History
- Up to 1860 an undergraduate in a Scottish university (St Andrews (founded 1411), Glasgow (1450), Aberdeen (1494), and Edinburgh (1582)) studied for an M.A., essentially a set course consisting of English, Latin, Greek, Mental Philosophy, Mathematics and Natural Philosophy.
- The work was done, and well done; the needs of his fellow-students of Greek mathematics were sufficiently met; and what was lost was a matter for him alone.
- A few years later, in a less direct and poignant way, Mackay found himself forestalled by the publication of Allman's "Greek Geometry".
- Arabic numerals
- , of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians, and of their valuable methods of calculation which surpass description.
- If those who believe, because they speak Greek, that they have arrived at the limits of science, would read the Indian texts, they would be convinced, even if a little late in the day, that there are others who know something of value.
- Trisecting an angle references
- T L Heath, A history of Greek mathematics I (Oxford, 1931).
- I Thomas, Selections illustrating the history of Greek mathematics : Vol 1 (From Thales to Euclid) (London, 1967).
- I Thomas, Greek mathematical works (London, 1939).
- Debating topics
- One is tempted to say "Of course it is." But the ancient Greeks did not consider 1 to be a number.
- Is it a better number system than the Greek or Roman number system? Why?
.
- Trisecting an angle references
- T L Heath, A history of Greek mathematics I (Oxford, 1931).
- I Thomas, Selections illustrating the history of Greek mathematics : Vol 1 (From Thales to Euclid) (London, 1967).
- I Thomas, Greek mathematical works (London, 1939).
- Trigonometric functions references
- G J Toomer, The chord table of Hipparchus and the early history of Greek trigonometry, Centaurus 18 (1973/74), 6-28.
- B L van der Waerden, On Greek and Hindu trigonometry, Bull.
- Ledermann interview
- Latin started at age 9, then Greek came later on but mathematics at age 11.
- Then came Greek and French but I couldn't manage French very well.
- Weather forecasting
- The ancient Greeks were the first to develop a more scientific approach to explaining the weather.
- 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.
- Babylonian Pythagoras references
- E M Bruins, Square roots in Babylonian and Greek mathematics, Nederl.
- Greek Math.
- Kepler's Planetary Laws
- Since Greek times, the accepted description of the planetary system had been a geometrical one, known as the Ptolemaic theory (a geocentric configuration), which supposed that the Earth was fixed at the centre of the universe, with the Moon, the Sun, and the five known (naked-eye) planets revolving round it.
- Kepler always showed the greatest respect for his Greek predecessors, and read their works thoroughly, selecting material that he could incorporate into his new astronomical synthesis.
- Water-clocks
- Ancient Greek index
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- Ancient Greek index
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- Trigonometric functions references
- G J Toomer, The chord table of Hipparchus and the early history of Greek trigonometry, Centaurus 18 (1973/74), 6-28.
- B L van der Waerden, On Greek and Hindu trigonometry, Bull.
- Trigonometric functions
- Mathematics and Art
- There is little doubt that a study of the development of ideas relating to perspective would be expected to begin with classical times, and in particular with the ancient Greeks who used some notion of perspective in their architecture and design of stage sets.
- we know of no book or written document which has come down to us from ancient practitioners, although they were mot excellent, as is convincingly shown by the descriptions of the stage scenery they made, which was much prized both in Athens among the Greeks and in Rome among the Latins.
- Prime numbers
- Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.
- In about 200 BC the Greek Eratosthenes devised an algorithm for calculating primes called the Sieve of Eratosthenes.
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- Babylonian Pythagoras references
- E M Bruins, Square roots in Babylonian and Greek mathematics, Nederl.
- Greek Math.
- Maxwell's House
- Then it became obvious that this boy had a brain for mathematics, and his self-confidence grew so much that he also began to do well in Latin and Greek.
- Let Pedants seek for scraps of Greek,
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- Bourbaki 1
- It would appear that Nicolas was a classical reference to an ancient Greek hero from whom General Bourbaki was descended.
- Cosmology
- But it was the Ancient Greeks who were the first to build a cosmological model within which to interpret these motions.
- Egyptian mathematics
- Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical arithmetic.
- references
- Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.
- Perfect numbers
- Among the many Arab mathematicians to take up the Greek investigation of perfect numbers with great enthusiasm was Ismail ibn Ibrahim ibn Fallus (1194-1239) who wrote a treatise based on the Introduction to arithmetic by Nicomachus.
- Water-clocks references
- S L Gibbs, Greek and Roman sundials / Sharon L Gibbs.
- Classical time
- Certainly St Augustine was right to feel that his ideas are less than satisfactory, yet that said, he had thought more deeply about time than anyone seems to have done before him including the greatest of the Greek philosophers, and more deeply than anyone else seems to have done during the following one thousand years.
- Indian numerals references
- K Vogel, Uses of letters and Indian numerals in Byzantium (Greek), Neusis No.
- Indian numerals references
- K Vogel, Uses of letters and Indian numerals in Byzantium (Greek), Neusis No.
- Pell's equation references
- B L van der Waerden, Pell's equation in the mathematics of the Greeks and Indians (Russian), Uspehi Mat.
- Babylonian Pythagoras
- references
- Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.
- Modern light
- The study of light from ancient Greek times up to the revolutionary breakthrough by Maxwell is studied in the article Light through the ages: Ancient Greece to Maxwell.
- Pell's equation references
- B L van der Waerden, Pell's equation in the mathematics of the Greeks and Indians (Russian), Uspehi Mat.
- Mathematical games
- Sundials references
- S L Gibbs, Greek and Roman sundials / Sharon L Gibbs.
- Arabic mathematics references
- S Pines, Studies in Arabic versions of Greek texts and in Medieval science (Leiden, 1986).
- references
- Arabic mathematics references
- S Pines, Studies in Arabic versions of Greek texts and in Medieval science (Leiden, 1986).
- function concept
- If we move forward to Greek mathematics then we reach the work of Ptolemy.
- Orbits
- Although the motions of the planets were discussed by the Greeks they believed that the planets revolved round the Earth so are of little interest to us in this article although the method of epicycles is an early application of Fourier series.
Go directly to this paragraph
- Longitude1
- Sundials references
- S L Gibbs, Greek and Roman sundials / Sharon L Gibbs.
- Indian Sulbasutras
- This in itself gives us a problem, for we do not know if these people undertook mathematical investigations for their own sake, as for example the ancient Greeks did, or whether they only studied mathematics to solve problems necessary for their religious rites.
- Water-clocks references
- S L Gibbs, Greek and Roman sundials / Sharon L Gibbs.
- Longitude2
- This variation between clock time and sundial time is known as the Equation of Time or the Equation of Natural Days and had been known to the Greeks and Arabs many centuries earlier (although of course the reason for the variation was not then understood).
- references
- Diller, A., Studies in Greek manuscript tradition, Amsterdam :: Hakkert.
Famous Curves
- Circle
- The greeks considered the Egyptians as the inventors of geometry.
- One of the problems of Greek mathematics was the problem of finding a square with the same area as a given circle.
- Spiric
- After Menaechmus constructed conic sections by cutting a cone by a plane, around 150 BC which was 200 years later, the Greek mathematician Perseus investigated the curves obtained by cutting a torus by a plane which is parallel to the line through the centre of the hole of the torus.
- Conchoid
- The name means shell form and was studied by the Greek mathematician Nicomedes in about 200 BC in relation to the problem of duplication of the cube.
- Trisectrix
- Like so many curves it was studied to provide a solution to one of the ancient Greek problems, this one is in relation to the problem of trisecting an angle.
Societies etc
- Hellenic Mathematical Society
- The Society began publication of the Bulletin of the Greek Mathematical Society in 1919 and Remoundos was a member of the editorial board.
- Metaxas, now a dictator, tried to bring back the values of ancient Greece and imposed his wishes on all aspects of Greek life.
- Other Web sitesSociety Web-site (in Greek)
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- National Academy of Sciences of Italy
- Spallanzani served as professor of logic, metaphysics, and Greek, then as professor of physics at the University of Modena before gaining worldwide recognition as a physiologist while holding a chair at the University of Pavia.
- He worked at the universities of Reggio and Modena as professor of logic, metaphysics and Greek, then at the University of Pavia as professor of natural history.
- References for Plato
- J P Lynch, Aristotle's School : A study of a Greek educational institution (Berkeley, 1972).
- BMC 1984
- Fowler, DA new interpretation of early Greek mathematics
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- Clay Award
- for finding, jointly with two undergraduate students, an algorithm that solves a modern version of a problem going back to the ancient Chinese and Greeks about how one can determine whether a number is prime in a time that increases polynomially with the size of the number
.
- Academy of Plato
- What is reported above the door of the Academy is exactly the same Greek words except "unfair or unjust" has been replaced by "non-geometrical".
References
- References for Thales
- W K C Guthrie, The Greek Philosophers: From Thales to Aristotle (1975).
- T L Heath, A History of Greek Mathematics I (Oxford, 1921).
- C H Kahn, Anaximander and the origins od Greek cosmology (Indianapolis, 1994).
- S Sambursky, The physical world of the Greeks (London, 1956).
- E Stamatis, The pre-Socratic philosophers : Thales of Miletus, the great scholar and philosopher (Greek), Episteme kai Techne 116 (1959).
- References for Knorr
- M Caveing, Review: The Evolution of the Euclidean Elements : A Study of the Theory of incommensurable Magnitudes and its significance for early Greek Geometry, by Wilbur Richard Knorr, Revue d'histoire des sciences 29 (2) (1976), 180-183.
- D Fowler, Review: The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry, by Wilbur Richard Knorr, The Mathematical Gazette 60 (413) (1976), 229.
- S Unguru, Review: The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry, by Wilbur Richard Knorr, Isis 68 (2) (1977), 314-316.
- B L van der Waerden, Review: The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry, by Wilbur Richard Knorr, Historia Mathematica 3 (4) (1976), 497-499.
- References for Euclid
- http://www.britannica.com/biography/Euclid-Greek-mathematician
.
- H L L Busard (ed.), The Mediaeval Latin translation of Euclid's 'Elements' : Made directly from the Greek (Wiesbaden, 1987).
- T L Heath, A history of Greek mathematics 1 (Oxford, 1931).
- W Knorr, Problems in the interpretation of Greek number theory : Euclid and the 'fundamental theorem of arithmetic', Studies in Hist.
- References for Magiros
- Reidel Publishing Co., Dordrecht; published on behalf of the Greek Mathematical Society, Athens, 1985).
- Reidel Publishing Co., Dordrecht; published on behalf of the Greek Mathematical Society, Athens, 1985), 511-516.
- Reidel Publishing Co., Dordrecht; published on behalf of the Greek Mathematical Society, Athens, 1985), ix.
- Reidel Publishing Co., Dordrecht; published on behalf of the Greek Mathematical Society, Athens, 1985), 509-510.
- References for Plato
- W K C Guthrie, A History of Greek Philosophy 4 (1975), 5 (1978).
- E S Stamatis,The mathematicians of Plato's Academy (Greek) (Athens, 1982).
- D G Kontogiannes and E Ntziachrestos, The geometric and didactic concepts of Plato (Greek), in Mathematics - education and applications (Nicosia, 1997), 132-138.
- E Stamatis, The theory of sets by Plato (Greek), Prakt.
- References for Descartes
- I Sp Papadatos, Arithmetic polygons of the ancient Greeks and arithmetic polyhedra of Descartes (Greek) (Athens, 1982).
- References for Diophantus
- T L Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra (New York, 1964).
- T L Heath, A history of Greek mathematics 2 (Oxford, 1931).
- J Klein, Greek mathematical thought and the origin of algebra (London, 1968).
- References for Pythagoras
- T L Heath, A history of Greek mathematics 1 (Oxford, 1931).
- E S Stamatis, Pythagoras of Samos (Greek) (Athens, 1981).
- W K C Guthrie, A History of Greek Philosophy I (1962), 146-340.
- References for Aristotle
- W K C Guthrie, A history of Greek philosophy Volume 6, Aristotle : An encounter (Cambridge, 1981).
- T L Heath, A history of Greek mathematics 1 (Oxford, 1931).
- J P Lynch, Aristotle's school : A Study of a Greek Educational Institution (Berkeley, 1972).
- References for Democritus
- T Cole, Democritus and the Sources of Greek Anthropology, Amer.
- W K C Guthrie, A history of Greek philosophy (six vols.) (Cambridge, 1962-81).
- T L Heath, A History of Greek Mathematics I (Oxford, 1921).
- References for Leucippus
- C Bailey, The Greek Atomists and Epicurus (Oxford, 1928).
- J Barnes, Early Greek Philosophy (1987).
- W K C Guthrie, History of Greek Philosophy II (Cambridge, 1965).
- References for Archimedes
- T L Heath, A history of Greek mathematics II (Oxford, 1931).
- E S Stamatis, The burning mirror of Archimedes (Greek) (Athens, 1982).
- E S Stamatis, Reconstruction of the ancient text in the Sicilian Doric dialect of fifteen theorems of Archimedes which are preserved in the Arabic language (Greek), Bull.
- References for Eudoxus
- T L Heath, A History of Greek Mathematics I (Oxford, 1921).
- G Huxley, Eudoxian Topics, Greek, Roman and Byzantine Studies 4 (1963), 83-96.
- H Stein, Eudoxos and Dedekind : on the ancient Greek theory of ratios and its relation to modern mathematics, Synthese 84 (2) (1990), 163-211.
- References for Anaximander
- C H Kahn, Anaximander and the Origins of Greek Cosmology (New York, 1960).
- Ancient Greek Scientists : Anaximander,Technology Museum of Thessaloniki
.
- C H Kahn, On Early Greek Astronomy, The Journal of Hellenic Studies 90 (1970), 99-116.
- References for Callippus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- B L van der Waerden, Greek astronomical calendars.
- References for Ptolemy
- T L Heath, A Manual of Greek Mathematics (1963).
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Pappus
- I Bulmer-Thomas, Selections illustrating the history of Greek mathematics II (London, 1941).
- T L Heath, A History of Greek Mathematics II (Oxford, 1921).
- References for Zenodorus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- G J Toomer, The Mathematician Zenodorus, Greek Roman and Byzantine Studies 13 (1972), 177-192.
- References for Hypatia
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- L Cameron, Isidore of Miletus and Hypatia of Alexandria: On the Editing of Mathematical Texts, Greek, Roman and Byzantine Studies 31 (1990), 103-127.
- References for Dinostratus
- G J Allman, Greek geometry from Thales to Euclid (Dublin, 1889).
- T L Heath, A History of Greek Mathematics I (Oxford, 1921).
- References for Hipparchus
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- G J Toomer, The chord table of Hipparchus and the early history of Greek trigonometry, Centaurus 18 (1973/74), 6-28.
- References for Geminus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- B L van der Waerden, Greek astronomical calendars.
- References for Cleomedes
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- A Wasserstein, Some early Greek attempts to square the circle, Phronesis 4 (1959), 92-100.
- References for Heron
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- I Thomas, Selections illustrating the history of Greek mathematics II (London, 1941).
- References for Zeno of Elea
- W K C Guthrie, A History of Greek Philosophy (Vol.
- T L Heath, A history of Greek mathematics 1 (Oxford, 1931).
- References for Dionysodorus
- T L Heath, A History of Greek Mathematics II (Oxford, 1921).
- I Thomas, Selections illustrating the history of Greek mathematics II (London, 1941).
- References for Theaetetus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- G J Allman, Theaetetus, in Greek geometry from Thales to Euclid ((London-Dublin, 1889), 206-215.
- References for Menaechmus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- G J Allman, Greek Geometry from Thales to Euclid (Dublin-London, 1889), 153-179.
- References for Oenopides
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- D R Dicks, Early Greek Astronomy to Aristotle (London, 1970).
- References for Theon
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- D Pingree, An illustrated Greek astronomical manuscript : Commentary of Theon of Alexandria on the 'Handy tables' and scholia and other writings of Ptolemy concerning them, J.
- References for Bryson
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Archytas
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Eutocius
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Heraclides
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Autolycus
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Maurolico
- R Moscheo, Greek heritage and the scientific work of Francesco Maurolico, in Medieval and classical traditions and the renaissance of physico-mathematical sciences in the 16th century (Brepols, Turnhout, 2001), 15-22.
- References for Proclus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Marinus
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Antiphon
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Anthemius
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Hippocrates
- T L Heath, A History of Greek Mathematics I (Oxford, 1921), 182-202.
- References for Zu Chongzhi
- C-Y Chen, A comparative study of early Chinese and Greek work on the concept of limit, in Science and technology in Chinese civilization (Teaneck, NJ, 1987), 3-52.
- References for Abu Kamil
- M Levey, Some notes on the algebra of Abu Kamil Shuja, a fusion of Babylonian and Greek algebra, Enseignement Math.
- References for Theodorus
- T L Heath, A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212.
- References for Theon of Smyrna
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Hypsicles
- T L Heath, A History of Greek Mathematics I (Oxford, 1921).
- References for Aryabhata
- E G Forbes, Mesopotamian and Greek influences on ancient Indian astronomy and on the work of Aryabhata, Indian J.
- References for Nicomachus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Regiomontanus
- M Folkerts, Regiomontanus' role in the transmission and transformation of Greek mathematics, in Tradition, transmission, transformation, Norman, OK, 1992/1993 (Leiden, 1996), 89-113.
- References for Apollonius
- T L Heath, A History of Greek Mathematics (2 vols.) (Oxford, 1921).
- References for Dedekind
- H Stein, Eudoxos and Dedekind : on the ancient Greek theory of ratios and its relation to modern mathematics, Synthese 84 (2) (1990), 163-211.
- References for Eudemus
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Anaxagoras
- T L Heath, A history of Greek mathematics 1 (Oxford, 1931).
- References for Nicomedes
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Philon
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Hippias
- T L Heath, A History of Greek Mathematics I (Oxford, 1921).
- References for Zeno of Sidon
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Diocles
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Eratosthenes
- T L Heath, A History of Greek Mathematics (2 vols.) (Oxford, 1921).
- References for Al-Kindi
- M Moosa, Al-Kindi's role in the transmission of Greek knowledge to the Arabs, J.
- References for Chrysippus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Menelaus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Simplicius
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Aristaeus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Domninus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Perseus
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Sporus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Serenus
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Conon
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Theodosius
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Caratheodory
- N Sakellariou, Obituary: Constantin Caratheodory (Greek), Bull.
- References for Posidonius
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Fowler David
- C Zeeman, Obituary David Fowler : A leading expert on the history of early Greek mathematics and head of Warwick University's Mathematics Research Centre, The Guardian (Monday, 3 May 2004).
- References for Thymaridas
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Xenocrates
- T L Heath, A History of Greek Mathematics (2 Vols.) (Oxford, 1921).
- References for Empedocles
- W K C Guthrie, A History of Greek Philosophy II (Cambridge, 1965), 122-265.
- References for Aryabhata I
- E G Forbes, Mesopotamian and Greek influences on ancient Indian astronomy and on the work of Aryabhata, Indian J.
- References for Aristarchus
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
- References for Porphyry
- T L Heath, A history of Greek mathematics I, II (Oxford, 1931).
Additional material
- Rudio's talk
- This was at the time of the Greek colonies in Lower Italy and in Sicily.
- And the last Greek mathematician on Italian soil, Archimedes of Syracuse, was also the most brilliant of mathematicians in Antiquity.
- This put the final nail in the coffin for the science that he represented so brilliantly; along with him, mathematical research in Italy was wiped out for more than a millennium! For over the centuries, the great Roman nation did not produce a single noteworthy mathematician, at least by Greek standards.
- Indeed, we can hardly regard it as a coincidence that this nation, which showed little originality and was wholly dependent on the Greeks with respect to arts and literature, also showed such an incredibly poor predisposition for any mathematical speculation.
- There is a widespread belief that after the Arabs conquered Alexandria in 641 Greek culture found a new home in Constantinople, where many Alexandrian scholars had moved, and that here, in the capital of the Eastern Roman emperors and protected by them, Greek science was maintained and developed throughout the Mediaeval Ages.
- When scholarly Greek refugees brought classic manuscripts to safety to Italy, due to the Turks' advance and the conquest of Constantinople in 1453, the Occident became acquainted with the precious treasure of Greek culture, which initiated the renaissance of the sciences.
- A tale according to which the well-known Italian humanist Pico della Miranbola paid for a single Livius with an entire estate illustrates the enthusiasm with which those manuscripts were received, and how badly the intellectuals wanted to possess old Greek and also Roman manuscripts! But knowledge of Greek mathematics had already been brought to the civilized people of Europe via a different route.
- In addition, and just to mention it straight away: Greek mathematics only formed one arm of the gigantic stream of mathematical thoughts that flooded the Occident towards the end of the Mediaeval Ages.
- It was under this ruler, with whom we have been so familiar since our childhood through the tales of "Arabian Nights", but even more so under the Caliph Al-Ma'mun that a fruitful period of translating began, due to which we know many a Greek writing that might have been lost otherwise.
- The first Greek manuscripts that were translated to Arabic were Ptolemy's Σύnτaξiς, Euclid's Elements, Apollonius's conic sections and Archimedes's treatises on measuring the circle and on the sphere and cylinder.
- It derives from the Arabic article al and the Greek superlative μeγiστe, "greatest", which Μeγάλe ("great") Σύnτaξiς had gradually turned into.
- In the 12th and 13th centuries scholars from all over Europe flocked to the academies in Toledo, Seville, Cordoba and Granada to study the Greek classics and, most importantly, to translate them from Arabic to Latin.
- But I have already mentioned that Greek mathematics only formed one of the sources that were to amalgamate and develop into our modern mathematics.
- For, apart from knowledge of Antiquity, we also owe to them a first insight into the intellectual life of a people whose approach to mathematics was completely different to that of the Greeks, but not less sophisticated, and which complemented it very well: I am talking about the Indians.
- Due to their highly developed sense of aesthetics, the Greek almost exclusively investigated mathematical problems that could easily be visualized, i.e.
- We owe an invention to our predescessors at the banks of the Ganges, whose sophisticated mathematics was brought to us by the Arabs at the same time as that of the Greeks.
- Both the Greeks and the Romans, in fact, all Indo-Germanic peoples, used the decimal system.
- I do not even wish to remind you that this simple assumption, which forms the basis of our calculation today, has escaped a mathematically gifted people such as the Greeks.
- The Greek notation on the other hand is in a different league entirely.
- Thus, at the beginning of the 13th century we see the Christian Occident being in possession of the mathematics of two highly gifted people; one of which, the Greeks, represents the mathematics of Antiquity, and the other one, the Indians, represent the mathematics of the Mediaeval Ages.
- It was only now that mathematics had a more profound impact on Western culture, and that we can talk about a revival of the Greek and Indian spirit.
- On the other hand we notice that Greek geometry, culminating in Ptolemy's theories, is booming, particularly in Germany and Italy; and we marvel at the global scientific process that comes to a close in the mid-16th century, when the Ptolemaic world-view gives way to our modern one.
- Many have debated whether or not the ancient Greek and Roman painters knew the art of perspective.
- This was exemplified during a stay in Vienna in 1460, when the scholarly Cardinal Bessarion, one of the Greek refugees who had come to Italy from Constantinople, invited Peurbach to continue his astronomic studies in Rome.
- He plunged into the study of the Greek mathematicians with enthusiasm; as we know, these had only been known through translations from Arabic to Latin up until then, but now they were available to him in the original.
- When he left Italy in 1468, he was in possession of a whole collection of valuable Greek manuscripts; among them the manuscript of Ptolemy's σunτaξiζ in particular, a gift from Bessarion.
- For example, he wanted to publish printed, accurate original editions of all the Greek manuscripts that he had brought with him from Italy.
- Knorr's books
- A study of the theory of incommensurable magnitudes and its significance for early Greek geometry (1975).
- This book studies the history of the Greek theory of irrationals, which culminated, with Theaetetus and Eudoxus, in the geometric constructions and classification of ratios (medial, binomial, apotome, bimedial, etc.) in Euclid's Elements Book X.
- The explicit view adopted is that a study of incommensurable magnitudes was of interest in its own right to Greek geometers; Eudoxus' later foundational innovations in Book V do not concern the author, and so he deals with him only incidentally and then mainly in connection with his contribution to the older theory of irrationals.
- But anybody with a special interest in Greek mathematics, or who has been dissatisfied with the tentative accounts of pre-Euclidean mathematics given in more general histories, should make a special effort to study this book.
- It is a book to be taken seriously by students of ancient Greek mathematics.
- The issue of incommensurable magnitudes was of great importance in the science and Greek philosophy prior to Euclid.
- Ancient Sources of the Medieval Tradition of Mechanics: Greek, Arabic, and Latin studies of the balance (1982).
- In the second part of this study I will take up the problem of the character of the dependence of these works on the earlier Greek tradition of mechanics.
- Among the ancient authors who are discussed and studied, are fragments of Euclid, Archimedes, Menelaus, Philon of Byzantium, Heron of Alexandria, Pappus and Eutocius; among the medieval, especially the Arabic versions of Greek authors and the 'Liber de canonio', an anonymous treatise of the XII/XIII century, the Latin version of an original Greek text which is lost.
- The problems of cube duplication, angle trisection and circle quadrature have long attracted major interest among scholars of ancient Greek geometry.
- The basic idea behind this study of problem-solving in ancient Greek geometry by Dr Wilbur Knorr, Associate Professor of Stanford University, is that 'the corpus of solved problems forms an ordered sequence in which each problem can be reduced to those preceding'.
- There were three classic problems which fascinated the Greeks for centuries and provoked many solutions - the duplication of the cube, the squaring of the circle and the trisection of an angle.
- Knorr's study of these problems is the most detailed yet offered, and as he investigates every by-way it virtually amounts to a history of Greek geometry within his chronological limits.
- One of the standard topics in Greek mathematics is the unsolvability of three problems: the duplication of the cube, the trisection of an angle, and the squaring of the circle.
- Knorr proceeds instead by working his way exhaustively through the surviving texts on the subject from earliest Greek times through Plato and Euclid to the high points of Archimedes and Apollonius.
- The three traditional problems of Greek classical mathematics are of course the duplication of the cube (more generally the finding of two mean proportionals between two given quantities or magnitudes), the trisection of an arbitrary angle, and above all, the quadrature of the circle.
- To gain all insight possible into the generality as well as particular instances of [Greek geometrical] analysis is Knorr's brief in this book; and he vigorously and forthrightly pursues it.
- The material examined is not different from that described in Heath (A History of Greek Mathematics (1960)); he works from sources preserved in Arabic.
- This latest work of the author forms a comprehensive survey of the available literature on Greek problem-solving in geometry.
- In Greek geometry, a problem and a theorem are distinct objects.
- The author concentrates on the three classical problems of the Greeks, cube duplication, angle trisection and circle quadrature, mainly because of the sources which have survived.
- Nevertheless he leaves no ivy leaf unturned in his investigation into the other Greek geometrical problems and their solutions.
- Where did the impetus for these problems come from? Was it from inside or outside geometry? And what did the Greeks mean by "a solution"? Why did they continue to work on the problems once they had a solution? Was this something to do with the classification of problems into planar, solid and linear? And what was this ridiculous classification anyway? Or had this continual effort something to do with "mechanical" solutions, as opposed to "geometrical" or "demonstrative" ones? Is a circle not mechanical simply because Euclid postulated its existence? And talking of the circle, what was the special significance of ruler and compass constructions? The author examines the whole fascinating field in an attempt to produce a unified whole.
- The book is essential reading for anyone wishing to understand the complexities, technical, philosophical, textual and chronological, of the geometric problems of the Greeks.
- Knorr ranges chronologically from late antiquity to medieval Europe and deals with texts in Greek, Arabic, Hebrew, and Latin.
- In the main it is a minute examination of the writings concerned with the three classic problems of ancient Greek mathematicians - the doubling of the cube, the trisection of an angle, and the squaring of the circle, and not merely the ancient Greek writings but the Arabic, Latin and Hebrew translations.
- 9-245) special attention is paid to treatments within the Greek cultural ambit of the mathematical problems of duplicating the cube and trisecting the angle.
- It is entitled "The Textual Tradition of Archimedes' Dimension of the Circle" and traces the fortuna of this short work (or rather the Archimedean prototype from which, Knorr holds, the extant Greek text derives at some remove) in Greek, Arabic, and medieval Latin, although ranging rather more widely than the part's title would suggest.
- This book is devoted to the ancient and medieval tradition of three great problems of Greek geometry: the duplication of the cube, the trisection of the angle and the quadrature of the circle.
- In Part I, entitled "Ancient texts on geometric problems", Knorr presents translations with detailed commentaries of all extant Greek texts on the duplication of the cube, the construction of two mean proportionals between two given lines and the trisection of an angle.
- It was thus bound to suffer most modification in the light of the instructional aim of editors." Knorr discusses numerous Greek, Arabic, Latin and Hebrew versions of this work and of related texts.
- He argues that the extant Greek text of the Dimension of the circle is an abstract of a much larger Archimedean original, which he characterizes as "a body of properties on the circle comparable in format and scope to Archimedes' writings on the sphere".
- D'Arcy Thompson on Greek irrationals
- D'Arcy Thompson on Greek irrationals
.
- The original contains some Greek quotes.
- As part of the purpose of the article is to discuss the translation of these pieces of Greek text, this give some difficulty in presenting the article for readers who do not know any Greek.
- Where we have replaced a Greek word, a Greek phrase, or a Greek passage by a translation, we enclose the material in square brackets and [colour it red].
- In particular he compares the account of the irrational numbers given in the Epinomis with the descriptions (well-known to students of Greek mathematics) which Theon, lamblichus and Proclus give of the so-called 'side and diagonal numbers'; and he shows that, somehow or other, these side and diagonal numbers are connected with what Plato means by the 'One and the Great-and-small' as constituents of Number.
- It is just worth mentioning that what we here call the diagonal is called in Greek the diameter; it is the diagonal of the completed square (or parallelogram), and the diameter of the circle in which it can be described.
- Carry it on to ten or twenty terms, and it becomes a troublesome matter to evaluate; while the Greek side-and-diagonal numbers may be carried as far as you please, and still require only the easiest arithmetic.
- The Greek table has another advantage over our continued fraction, in that it obviously is just what it purports to be, namely an arithmetization of the corresponding geometrical figure.
- Secondly, the approximations are alternately on one side or the other, a little more or a little less than the number at which we aim; and herein lies the technical meaning in Greek arithmetic of 'excess and defect', (literally [a falling short and an exceeding]).
- This point, this precise nature of the agency of the 'One', and the simple explanation which it involves of the precise meaning of [to define] or [to equal], both seem to me to be made clear by our study of the Greek side-and-diagonal series; but the point is lost as soon as we replace that formula by the continued fractions of our modem arithmetic.
- Prof Taylor, so it seems to me, has treated the successive convergents of the continued fraction as identical with the successive fractions of the Greek series.
- Similar tables can be constructed, as the Greeks well knew, for other square roots; and the way to construct them is in each case easy to discover.
- (as we have seen that the Greeks did), was neither an easier nor a harder operation than to pass from that same table where the values of √2 are 'equalized' by 1, to the extended table where they are 'equalized' by 2, or powers of 2.
- Yet there is no account of it, nor the least allusion to it, in all the history of Greek mathematics; and it is commonly believed to have been first made known by the great arithmetician who introduced the Arabic numerals into the Christian world.
- It is inconceivable that the Greeks should have been familiarly acquainted with the one and yet unacquainted with the other of these two series, so simple, so interesting and so important, so similar in their properties and so closely connected with one another.
- Between them they arithmeticize what is admittedly the greatest theorem, and what is probably the most important construction, in all Greek geometry.
- Both of them hark back to themes which were the chief topics of discussion among Pythagorean mathematicians from the days of the Master himself; and both alike are based on the arithmetic of fractions, with which the early Egyptian mathematicians and doubtless the Greek also were especially familiar.
- ; let τ be this Golden Mean - [section (literally a cut)], the Section, as the Greeks called it.
- Rose's Greek mathematical literature
- Herbert Jennings Rose's Greek mathematical literature
.
- Herbert Jennings Rose was Professor of Greek in the United College of St Salvator and St Leonard of the University of St Andrews.
- His famous text, A handbook of Greek literature (Methuen & Co, London, 1934) ran to several editions and included information on the literature of mathematics and related topics.
- We have also removed the Greek titles of the works; in a few cases no translation is offered and we have simply replaced the Greek with 'Greek Title':
.
- Of the 'Greek Title' we know that it dealt with conic sections, and consisted of two books.
- This and other questions which interested him involved finding a better system of numeration than that generally in use among Greeks; from this sprung a popular essay, addressed to Gelon, son of Hieron II of Syracuse, known as The Sand-reckoner, showing that it was perfectly possible to reckon the number of grains of sand required to fill the whole universe, by employing an ingenious system of his own for expressing high figures, and therefore that the common proverb 'numberless as the sands' was wrong.
- Hence it is not surprising that the very obscure Kleostratos, said to have been the first Greek to mention the signs of the zodiac, owes something - how much is a point as disputable as his date, which is variously computed at the sixth century or the fourth - to the Babylonians also.
- At all events, their observations were, at least in some measure, accessible to Greek astronomers after Alexander.
- Hence the constellations, especially those of the zodiac, become familiar subjects at this epoch, though scattered mentions of them are to be found from Homer onwards, and the differences between the Greek and the non-Greek picture of the heavens a matter for comment.
- It is a pardonable exaggeration to trace to him almost everything in Roman and later Greek writers which deals with any of his subjects and cannot be definitely assigned to anothe source; but it is an exaggeration nevertheless, and criticism is beginning to correct the too exuberant assumptions of enthusiasts for this highly important, but unfortunately lost writer.
- He probably was born about 135, or at most some eight or nine years sooner; he lived in Rhodes for some time from about 97 onwards, till Panaitios died; after that he set out on his travels, which included the western countries (Gaul and Spain) then almost unknown to Greeks.
- Poseidonios' own astronomical studies were probably the result of his interest in philosophy in general rather than this science in particular, but there seems little doubt that he wrote a considerable work, the 'Greek Title', besides some smaller essays; of the former we have perhaps a sort of synopsis in the compendium of astronomy by Kleomedes, a writer otherwise unknown.
- Another development of mathematics was Mechanics, a department of knowledge concerning which the Greeks seem to have written comparatively little.
- Not unconnected with mathematics is Music, and it may be mentioned here that Ptolemy wrote three books of Harmonies (i.e., musical theory, not what our musicians call harmony, for that did not exist in Greek music), which still survive, and another writer, perhaps of the third century A.D., Aristeides Quintilianus, also produced three books, On Music.
- They write in the Greek of their own day, that is in the 'common dialect', for the most part; Archimedes uses his native Sicilian Doric for several works (in the case of some which are in the common dialect, it may be suspected that they were originally in Doric also).
- apparently much interpolated by those who used it as a school-book, falsely attributed to Skylax of Karyanda, but really composed about the thirties of the fourth century B.C., the 'Greek Title', which takes the reader around the shores of the entire civilized world as known to the author, diversifying the list of names by notes on ethnology, local myths and the like.
- More important than any of these is the geography, or rather gazetteer, of Ptolemy, in eight books, under the general title of 'Greek Title'.
- Stringham address
- The prehistoric period of mathematics belongs to the centuries immediately preceding the earliest development of Greek philosophy, and appears to have been first cultivated in connection with land surveying and astronomy in Egypt and Assyria.
- It had its beginnings as a science in the latter part of the seventh century, B.C., at Miletus, where the Greek philosopher Thales, who had travelled extensively in foreign countries and had resided for a time in Egypt, first taught geometry deductively; but nearly contemporaneously also at Crotona, where Pythagoras first established the principles of the doctrine of proportion, and laid the foundations of a goodly part of elementary geometry.
- Thenceforward, so long as Greek civilization existed anywhere, the study of geometry never ceased to be an important factor in the world's intellectual progress, and for more than twelve centuries the continuity of its study was unbroken.
- Near the close of the Greek period, during which geometry, from the merest beginnings, had grown into a comprehensive system, the first published accounts of an algebraic analysis appear, the Arithmetica of Diophantos, the last contribution of the great Alexandrian school to mathematical science.
- But even if the work of Diophantos had been constructed upon correct scientific principles, it would still have passed speedily into oblivion, for the light of Greek civilization was already feeble and flickering and was soon completely extinguished by the fanaticism and bigotry that for seven centuries enveloped Europe in almost total darkness.
- With the disappearance of the Greek schools there seemed to be no hope for the further cultivation of mathematics as a science on the face of the earth, for nowhere else in the ancient world, up to that time, had any results of a high order been achieved.
- The Arithmetica of Diophantos had been published (conjecturally) in the fourth century of our era, and the Alexandrian school continued in existence until the Mohammedan conquest in 641 A.D., during which time mathematics was still cultivated, though feebly, in the form of commentary or perfunctory study, and without originality or fruitful result; and it was more than a century and a half before the final catastrophe - the capture of Alexandria and the burning of the great library by the Mohammedans - that there appeared in India a work on algebra and trigonometry by the astronomer Aryabhatta, of which no Greek mathematician of the earlier centuries would have been capable.
- To what extent they were indebted to the Greeks for the raw materials out of which they constructed their algebra is not known, nor is it of consequence, for since the Greeks never succeeded in constructing an organic system of algebra, the indebtedness could, in any event, have been but small.
- Though the Indians had shown themselves to be consummate algebraists, they were in no sense geometers; and far from adding any new thought to the science which the Greeks had created, they apprehended with difficulty and with much blundering what of geometry they received from Greek sources.
- The Greeks had created geometry, but they could not invent algebra; the Indians invented algebra, but they could not add one proposition of importance to geometry.
- The Arabs, however, were not good conservers, or compilers, of the scientific knowledge accessible to them from Greek and Indian sources, and transmitted it in such imperfect form that many important principles had either to be rediscovered, or sought for in the Greek or Sanskrit, before Europe full possession of the mathematical knowledge which Greece on the one hand and India on the other had contained.
- Algebra followed the Indian model, which had been recovered with much difficulty by the help of the Arabs, and, barring a few geometrical constructions of algebraic equations, mere translations into modern forms of expression of well-known Greek demonstrations, did not get beyond the limited definition of algebraic quantity as rational or commensurable number.
- A straight line stands as the representative of any such quantity, an interpretation wholly repugnant to Greek geometry.
- So long as his practice conformed to the canon which, from time immemorial, had been his guide in the geometrical interpretation of quantity, x2 could mean for the Greek only a square, an area, never a straight line.
- But his orthodoxy was the barrier to his further progress, and from the abandonment of that orthodoxy the modern mathematician dates the possibility of achievement beyond the limits of investigation which the Greeks had set for themselves.
- By tradition, algebra became a mere mechanical device for turning out practical results, by careless reasoning errors crept into the explanation of its principles, and through incompetent compilers were perpetuated in the form of current literature, and thus, instead of becoming a classic like the geometry handed down to us from the Greeks in the form of Euclid's elements, algebra became a collection of processes practically exemplified and of principles inadequately explained.
- But standing, as we do now, near the close of the twenty-fifth century since mathematics became a science, and taking advantage of the discoveries of our predecessors, we may pass across the entire field that outlines the foundations of algebraic science, and seizing only upon those principles that mark its epochs of advance and are cardinal, construct an algebraic system which shall have all the logical rigour and completeness of Greek geometry; and for the accomplishment of this task no serious difficulty longer stands in our way.
- R L Wilder: 'Cultural Basis of Mathematics II
- He would include a great deal about computing with numbers and solving equations; but there wouldn't be any geometry as the Greek understood it in his history, simply because it had never been integrated with the mathematics of his culture.
- On the other hand, if A were a Greek of 200 A.D., his history of mathematics would be replete with geometry, but there would be little of algebra or even of computing with numbers as the Chinese practiced it.
- Here is a subject which, despite the dependence of the Greeks on logical deduction, and despite the fact that mathematicians, such as Leibniz and Pascal, have devoted considerable time to it on its own merits, has been given very little space in histories of mathematics.
- Through the centuries it developed along slender arithmetic and algebraic lines, with no hint of geometry as the Greeks developed it.
- One is tempted to speculate what might have happened if the Babylonian zero and method of position had been integrated with the Greek mathematics - would it have meant that Greek mathematics might have taken an algebraic turn? Its introduction into the Chinese mathematics certainly was not productive, other than in the slight impetus it gave an already computational tendency.
- That the Greek mathematics was a natural concomitant of the other elements in Greek culture, as well as a natural result of the evolution and diffusion processes that had produced this culture in the Asia Minor area, has been generally recognized.
- Not only was the Greek culture conducive to the type of mathematics that evolved in Greece, but it is probable that it resisted integration with the Babylonian method of enumeration.
- For if the latter became known to certain Greek scholars, as some seem to think, its value could not have been apparent to the Greeks.
- More Smith History books
- To the reviewer t he failure of the book under review consists in ascribing to Greek and Roman influence developments and details only remotely connected with classical ideas and further in depreciating the contributions of other people.
- Greek learning, as scholars know it, has few defenders nowadays, and fewer devotees; but as practical men we are curious to know the precise extent of our classical inheritance, the value in dollars of the wisdom of the Greeks.
- The volume gives a very appreciative popular account of the mathematical contributions by the Greeks and the Romans, and brings out a number of historical facts which are not usually found in a history of mathematics.
- Mathematics has been called a Greek science, not only by those who find it difficult but also by those who are in position to understand its nature and who are familiar with the fundamental contributions of the Greeks along this line.
- It should, however, not be assumed that the Greeks developed the greater part of the mathematics of our times.
- The courses then began to include algebra, Euclid, trigonometry, calculus, conic sections (generally by the Greek method), astronomy, and "natural philosophy" (physics).
- Knorr's papers
- A close examination of these computations will affirm that the classical Greek mathematics, dominated by an interest in theoretical geometry, yet included expertise in practical arithmetic.
- In the ancient Egyptian and Greek arithmetical traditions, computations with fractions typically resorted to the mode of unit-fractions.
- Although the Greek evidence is rarely brought to bear on this question, the present study seeks to show that this evidence strongly upholds van der Waerden's position.
- In itself the Greek tradition is notably ambivalent, on the one hand revealing continuity with the more ancient Egyptian technique of unit-fractions, but on the other hand making frequent use of the general manner of fractions, equivalent to that familiar in modern school arithmetic.
- Through comparison of the various stages of the ancient computational tradition, one can discern different strata corresponding to differences in the time and manner of contact between the Greek and the Egyptian traditions.
- On the transmission of geometry from Greek into Arabic.
- Indeed, one might say, if with a bit of hyperbole, that documents of this kind have revolutionary implications for the study of the transmission of Greek geometry into Arabic.
- We thus have a clear instance where an Arabic document produced as a translation from the Greek was subsequently perceived to be an original contribution due to an Arabic geometer.
- Whenever we come upon a technical treatment securely within the compass of Greek methods and format, we must seriously consider whether that treatment was modelled after a prototype in Greek, even when no such Greek document is now known.
- It was cited frequently by the ancient mathematical commentators, was known in the Latin Middle Ages in translations both from the Arabic and directly from the Greek, and remains prominent in all general discussions of Archimedes' geometry to this day.
- Since the Greek prototype is lost, the only surviving witness of the Archimedean work is based on the Arabic translation made by Thabit ibn Qurra in the 9th century, as reedited in the 18th century under the title 'Book of the construction of the circle divided into seven equal parts, by Archimedes'.
- The other, J L Heiberg, only five years beyond his doctoral dissertation at Copenhagen, had already completed a new critical edition of the Greek text of Archimedes and was in the process of issuing a new critical edition of the Greek text of Euclid.
- They reveal how the medieval tradition of the text advocated by Klamroth displays textual superiority to the Greek tradition promoted by Heiberg.
- Semple and Kneebone: 'Algebraic Projective Geometry
- Geometry is commonly regarded as having had its origins in ancient Egypt and Babylonia, where much empirical knowledge was acquired through the experience of surveyors, architects, and builders; but it was in the Greek world that this knowledge took on the characteristic form with which we are now familiar.
- The Greek geometers were not only interested in the facts as such, but were intensely interested in exploring the logical connexions between them.
- Now for the Greeks, we must remember, geometry meant study of the space of ordinary experience, and the truth of the axioms of geometry was guaranteed by appeal to self-evidence.
- If, in fact, we turn back once again to Greek geometry, we may recall that the geometrical knowledge with which the Greeks began was derived ultimately from measurements made upon rigid bodies, and was therefore essentially a knowledge of shapes.
- In the Elements, as in all Greek treatises, euclidean geometry is treated synthetically, and synthetic treatments of projective geometry are to be found in a number of modern books on the subject.
- Heath: Everyman's Library 'Euclid' Introduction
- He was an enthusiastic admirer of the Greek geometers and spent the best part of his life in studying and elucidating them.
- Unfortunately, the editio princeps of the Greek text by Grynaeus (Basel 1533) and the various translations from the Greek down to the end of the eighteenth century (including that of Commandinus) depended upon MSS.
- Those who wish to sample the original Greek text of Euclid, which is well worth while, may be referred to Euclid in Greek, Book I, with Introduction and Notes (Cambridge 1920).
- Of recent histories of Greek mathematics in general we may mention James Gow's A Short History of Greek Mathematics, 1884 (now out of print), and the present writer's History of Greek Mathematics, 2 vols., 1921, and Manual of Greek Mathematics, 1931 (Clarendon Press).
- The propositions embody, in fact, the general method known as the "application of areas," which was of vital consequence to the Greek geometers, being the geometrical equivalent of the solution of the general quadratic equations ax ± bx2/c = S so far as they have real roots.
- Nowadays we solve such equations by algebra; the Greek geometers, however, had no algebraical notation, and hence they had to invent a sort of geometrical algebra.
- Mathematicians and Music 2.1
- Our first consideration is therefore to be given to the homophonic music of the Greeks: for in music as in mathematics the period of real development began in the sixth century B.C.
- And last among the Greeks to whom we shall refer is the celebrated mathematician, astronomer and geographer, Claudius Ptolemy, who flourished in the second century of the Christian era.
- Some interesting suggestions have been made by Paul Tannery as to the possible role of Greek music in the development of pure mathematics.
- In my brief sketch of the work done by the Greeks, I have not intended to give you any idea of their music, but merely to select a few illustrations of the manner in which their music is connected with mathematics.
- Of course where delicacy in any artistic observations made with the senses come into consideration, moderns must look upon the Greeks in general as unsurpassed masters.
- But the varied gradations of expression, which moderns attain by harmony and modulation, had to be effected by the Greeks and other nations that used homophonic music by a more delicate and varied gradation of tonal modes.
- Heinrich Tietze on Numbers, Part 2
- The Greeks perceived that there are segments which are not in the ratio of two whole numbers, and therefore have no common measure or are incommensurable.
- One must thoroughly understand that the spirit of classical Greek mathematics was based on exactitude and went far beyond the practical considerations of surveying.
- Greek mathematics could compare two segments as equal (or larger or smaller); what it lacked was the concept of assigning numbers to lengths of segments incommensurable with a prescribed unit segment.
- While now we can simply say: If the side AB is the unit of length, the length of the diagonal AC is equal to √2, this would have been impossible for the Greeks because √2 does not exist in the domain of rational numbers.
- We can write the sum EB as 1 + √2; the Greeks could add the two segments geometrically, but could not add them arithmetically.
- D'Arcy Thompson on Plato and Planets
- The original contains many Greek quotes.
- As part of the purpose of the article is to discuss the translation of the Greek text, this gives some difficulty in presenting the article for readers who do not know any Greek.
- Where we have replaced a Greek word, a Greek phrase, or a Greek passage by a translation, we enclose the material in square brackets and [colour it red].
- Here D'Arcy Thompson is quoting a Greek passage from The Republic.
- We might expect to elucidate the passage in two ways, either by literal interpretation of the Greek or by seeking a clue in the facts and language of Astronomy.
- While writing on this subject let me add, in parenthesis, that the very ancient and very decorative 'Greek key-pattern' seems to me to be nothing more nor less than an archaic representation of a planet's apparent course, a series of simplified hippopedes.
- W H and G C Young
- Her father doubled its value for her, and in April 1889 she went up to Cambridge to pass the Little-go examination which included Greek: she obtained a First Class and was cross at the unnecessary effort! Little did she know how she would later love Greek..
- I persuaded the School authorities to let me skip a year, to avoid the 1st year of English, and a trimester later to allow me to switch to the Greek section.
- My interest in Greek-- my passion for it-- was infectious, not only among my classmates, but also in my own family.
- My two eldest sisters started Greek too, and they sent me postcards in Greek whenever we were apart.
- Marion --one of a few girls allowed, on a trial basis, to attend my School, where she joined the section "mathematiques speciales" in the class above mine and was 2nd -- had no time for things like Greek and was soon earning money from Tutoring and using it to pay for her own clothes, she was anxious to be financially independent.
- We never heard of this episode-- I learnt of it only 60 years later! All Janet posted was the usual brave little postcard to me in Greek.
- Airy on Thales' eclipse
- It refers to a point of time which connects in a remarkable way the history of Asia Minor and the Greek colonies settled there with the history of the great Eastern empires.
- On the north coast there is one, of which the difficulties were well known from the retreat of the ten thousand Greeks.
- Such perversions however occur in the Persian poetical history with regard to other names, which there is reason to believe are correctly given by the Greeks.
- The name Xerxes, for instance, has been found by Colonel Rawlinson in the Behistun inscriptions under the form Khshayarsha, of which the Greek Xerxes was probably a fair oral representation; whereas the name preserved in the poetical history is Isfundear.
- Bell books
- There is too much logic, too much stress on the logical basis of mathematics and too much on those parts of mathematics which depend on logical assumptions; too much also about the Greeks and their way of thinking.
- No one but a person of both wide and deep scholarship could have written a volume which is not only an intensely human document, but at the same time the story of the steady march of mathematical ideas from the time of the Greeks to the present.
- In this book two divergences of opinion which keep scientists divided in our days are traced back to their alleged origin in Greek thought.
- Bell is a wonderful storyteller, and his subject, the Greek philosopher and mathematician Pythagoras, supplies him with ample material to demonstrate his skill.
- D'Arcy Thompson by David Burt
- He was born on 2nd May 1860, the son of D'Arcy Wentworth Thompson, previously classical master at the Edinburgh Academy and at that time Professor of Greek in Queen's College, Galway.
- D'Arcy Thompson pursued his investigations "according to the accepted discipline of the physical sciences," in the light of properties of matter and forms of energy, and elucidated, often in the simpler mathematics of the Greeks, the mathematical aspects of organic form.
- His contribution to pure classical scholarship is seen in his Glossary of Greek Birds (1895), Aristotle: Historia Animalium (a translation, in 1910), and his Glossary of Greek Fishes (1947).
- Gibson History 7 - Robert Simson
- Simson is known almost solely as an exponent of the Greek geometry, but I think it is worth noticing that when he was still a young man he showed a full command of one branch of analysis.
- It is perhaps within the mark to say that every English edition of Euclid till near the close of last century (with the exception of Williamson's) was not merely influenced by Simson's work but was in all essentials based on Simson's text and not on the Greek text.
- He was simply steeped in the ancient geometry and one should be very sure of one's ground before questioning any deliberate judgment of Simson's on the facts of any Greek textbook.
- To the study of Greek geometry Simson may almost be said to have dedicated his life, and he found ample scope for his ingenuity in his effort to recover some of the more important of the treatises of Euclid and Apollonius that had been lost but whose contents had been to a certain extent described by Pappus.
- Trail was neither unprejudiced nor very critical, but he had a competent knowledge of Greek geometry and was thoroughly familiar with Simson's work, and I quote his estimate of the edition:- "Such is the elegance of method and the ingenious contrivance of demonstration in this work that he has truly exhibited a copy, or at least very nearly a copy, of the work of Apollonius, that little regret need be had for the loss of the original." We need not indorse this eulogium in its entirety but it is not altogether wide of the mark.
- Heiberg is less enthusiastic, though I do not attach quite the same weight to Heiberg's views in this connection as in other fields in which he has rendered such great service to Greek geometry.
- I cannot here enter at all into the matter; the literature of the subject is considerable, and I may refer to Heath's Greek Mathematics, Vol.
- EMS obituary
- The rector was a really competent mathematician; but "his culture was not that of science alone, for he was widely read in literature, and he could grapple with the philosophical and theological questions of the day." An Academy boy had excellent opportunities for a sound training in English, Latin, and Greek, as well as in Mathematics and Science, and when Mackay matriculated at St Andrews in 1859, he was admirably prepared for entering on University studies.
- He had, besides, a scholarly knowledge of Latin and Greek, so that when he had fairly entered on historical studies he had at command a linguistic equipment such as professed historians do not always possess.
- From an early period his attention was directed to the Greek geometers, and one of the disappointments of his life is associated with these earlier studies.
- The work was done, and well done; the needs of his fellow-students of Greek mathematics were sufficiently met; and what was lost was a matter for him alone.
- A few years later, in a less direct and poignant way, Mackay found himself forestalled by the publication of Allman's Greek Geometry.
- Heath: 'Mathematics in Aristotle' Preface
- Mathematics in Aristotle is the result of work done during the last years of my husband's life, which he devoted to reading all that had been written on the subject and to making his own translations from the Greek.
- For the reading of the proofs I am indebted to him and to Mr Ivor Thomas, M.P., whose knowledge of Greek mathematics has been of the greatest assistance.
- Having no Greek myself, I have to thank Mr.
- Kerry Downes and Mr P R W Holmes for their assistance in transcribing the Greek quotations, and also my son Geoffrey T Heath who, since his demobilization, has helped me in reading proofs and revising Greek quotations, mathematical figures, and formulae.
- Turnbull lectures on Colin Maclaurin
- Simson had been inspired by Halley, the astronomer, to study the geometry of the Greeks at first hand, a task which he readily undertook; and so fully did he enter into the spirit of their work that he reached an explanation of what Euclid and Pappus meant by certain obscure allusions to their method of porisms.
- Folkes (1690-1754), who eventually succeeded Newton as a President of the Royal Society, was an antiquary as well as a man of science - a choice youth of penetrating genius and master of the beauties of the best Roman and Greek writers.
- Though he did not live to see this, one of the first to contribute to the newly formed society was his son John (1734-1796), Lord Dreghorn, a Scottish judge and advocate, a man of considerable literary ability and satirical wit, who published in the first volume of the Transactions an ingenious proof that the Greeks never took Troy.
- Horace Lamb addresses the British Association in 1904, Part 2
- The notion in question is a convenient fiction, and is a striking testimony to the ascendancy which Greek Mathematics have gained over our minds, but I do not think that more can be said for it.
- Whoever he was, the man who first projected the world into two dimensions, and proceeded to fence off that part of it which was reindeer from that which was not, was certainly under the influence of a geometrical idea, and had his feet in the path which was to culminate in the refined idealisations of the Greeks.
- But the Greek mind loved definiteness, and discovered that if we agree to speak of lines as if they had no breadth, and so on, exact statements became possible.
- Association 1904 Part 2.html
- Science at St Andrews
- What was to be taught, first as a general foundation and then in particular subjects such as law and mathematics, was laid down by Act of Parliament, when in 1579 it was declared that after a prescribed course in Latin, Greek, and elocution "the fourth Regent sall teach in Greek samekle of the phisikis as is neidfull in the spheir," and that "the mathematician now in St Salvator's college sall reid within the same four lessons ouklie in the mathematick sciences in sic dayis and houres as sall be appointed." It would seem that the allusion is to Homer Blair, who entered the college as a fellow-bejant with John Napier and later became a lecturer in mathematics.
- In the 500th year of the College we also celebrate the 400th anniversary of the birth of John Napier, whose invention of logarithms was the first essentially new mathematical advance that took place since the time of the ancient Greeks.
- Miller graduation address
- But the serene philosopher, endowed as he was with a double portion of Greek moderation and calmness of spirit, replied that he had nothing new to offer them beyond the principles which he had striven to inculcate during the whole period of their intercourse as disciples and master.
- The Hebrews and Greeks have solved man's relation to the eternal mystery - the one in its religious, the other in its philosophical aspect; each has come as near the perfect solution as, perhaps, it is possible for the human mind to reach.
- In Greek mythology we learn that Antaeus, the giant, in wrestling with Hercules received new vigour whenever he touched his mother earth; but Hercules, discovering the secret of his strength, lifted him into the air and squeezed him to death in his herculean grasp.
- Mathematicians and Music 2.2
- As the Roman absorbed the Greek, so the Christians accepted the Roman organization of learning.
- He should be taught also a good hand-writing, astrology, and when he is older, Greek and Latin.
- The first of these is a Greek and Latin edition of Ptolemy's Harmony, and Porphyry's third century commentary on the same, with an extensive appendix by Wallis on ancient and modern music.
- Then comes the only published text, with Latin translation, of a musical work by Manuel Bryenne, a fourteenth century Greek, four manuscripts of whose work are to be found at the Bodleian.
- Students in 1711
- In addition to mathematics they studied Greek through all four years.
- In their final year the topics were Greek, ethics and physics.
- He wrote in Latin (with some Greek) clearly thinking his father would be more likely to send the sporting equipment if he showed good scholarship.
- This fear has made me desire to be excused from Greek or Mathematics till the end of this month.
- Weil on history
- In 1621, the Greek text of Diophantus was published by Bachet, along with a useful.
- In contrast with this, the mere fact that Fermat initially had to draw his main inspiration from a Greek author of the third century, only lately come to light, points to the entirely different character of much of the earlier mathematical work as well as to the frequent disappearance and re-appearance of essential sources of knowledge in former times.
- About ancient mathematics (whether Greek or Mesopotamian) and medieval mathematics (Western or Oriental), the would-be historian must of necessity confine himself to the description of a comparatively small number of islands accidentally emerging from an ocean of ignorance, and to tenuous conjectural reconstructions of the submerged continents which at one time must have bridged the gaps between them.
- Still, what hope is there of our ever getting, say, a full picture of early Greek geometry? In the third century B.C., Eudemos (not himself a mathematician) wrote in four " books" a history of geometry, some fragments of which have been preserved.
- A N Whitehead: 'Autobiographical Notes
- Latin began at the age of ten years, and Greek at twelve.
- Holidays excepted, my recollection is that daily, up to the age of nineteen and a half years, some pages of Latin and Greek authors were construed, and their grammar examined.
- We read the Bible in Greek, namely, with the Septuagint for the Old Testament.
- Such Scripture lessons, on each Sunday afternoon and Monday morning, were popular, because the authors did not seem to know much more Greek than we did, and so kept their grammar simple.
- Enriques' reviews
- Enriques's collection of essays on Problems of Elementary Geometry has for its object to explain as simply and intelligibly as possible precisely what modern Mathematics has to say in correction, in explanation, and in completion of the old Greek Geometry.
- It is encouraging, however, to observe that the greatest textbook that the world has ever seen-The Elements of Euclid- is by no means dead, and that in the region of sound scholarship it continues to stand as a monument to the achievements of the Greek mind.
- These three serial manuals deal in six chapters with Greek scientific thought over the period of the few hundred years which includes the Ionian "physiologers" or naturalists, the Pythagoreans, the Eleatics, Empedocles, Anaxagoras, and the atomists.
- They show themselves well at home in the cultural context of Greek thought.
- Weil reviews
- In 'A Mathematicians's Apology' (1940), the English number theorist G H Hardy quoted his friend J E Littlewood as saying that the ancient Greek geometers "are not clever schoolboys or scholarship candidates, but Fellows of another college." In Number Theory, Andre Weil looks penetratingly at the accomplishments of leading "Fellows" who investigated the properties of integers prior to Karl Gauss.
- The chapters are (i) a quick trip through the ancient world, with particular attention to the contributions of the Mesopotamians, Greeks and Indians; (ii) a visit with Fermat (1601-1665); (iii) an extended stay with Euler (1707-1783); and (iv) brief stops to see Lagrange (1736) and Legendre (1752-1833).
- P G Tait's obituary of Listing
- Thus the Roman V becomes, by inversion, the Greek L - the Roman R perverted becomes the Russian (insert pict); the Roman L, perverted and inverted, becomes the Greek G.
- He points out in great detail the confusion which has been introduced in botanical works by the want of a common nomenclature, and finally proposes to found such a nomenclature on the forms of the Greek d and l.
- Mathematics at Aberdeen 1
- The Arts curriculum was to be widened with greater emphasis on Arithmetic and Geometry and to include teaching of Greek, Latin, Hebrew, Physiology, Geography, Astronomy and History, whilst Philosophy played a more minor role.
- The first year was occupied by Latin, Elementary Greek and Logic; the second mainly Logic with the writing and declaiming of Latin and Greek.
- Andrew Forsyth addresses the British Association in 1905
- The Italian mathematicians, of whom Cavalieri is the least forgotten, were developing Greek methods of quadrature by a transformed principle of indivisibles; but the infinitesimal calculus was not really in sight, for Newton and Leibniz were yet unborn.
- Some mathematics could be had, cumbrous arithmetic and algebra, some geometry lumbering after Euclid, and a little trigonometry; but these were mainly the mathematics of the Renaissance, no very great advance upon the translated work of the Greeks and the transmitted work of the Arabs.
- Broadbent's papers
- Euclid wrote (among other things) a masterpiece, 'The Elements of Geometry', and he wrote it for the cream of the adults of a race marked by a peculiar, perhaps a unique characteristic: the Greeks liked thinking.
- 'The Elements', with its orderly system, its brilliant logical triumphs in the theory of parallels and the theory of proportion (for the more Euclid's treatment is examined, the more impressive it becomes, in spite of gaps which later work has revealed), was a peak of Greek culture.
- Madras College exams
- The Greek and Latin classes of Dr Woodford occupied the visitors during the greater part of Wednesday.
- The five classes for Latin and the three for Greek were put through a very searching examination, being questioned not only by the master, but repeatedly and severely by Professors Pillans and Dunbar.
- Professor Dunbar spoke with high approbation both of the extent of the course of study set forth in the synopsis, and of the success with which the course had been prosecuted, as evinced by the difficult and complicated questions answered, he bestowed especial commendation upon the Greek classes, and he bore his testimony to the merits of Dr Woodford for accurate and philosophical acquaintance with the principles of the language, as well as for extensive and critical study of the classical authors.
- Dehn on Aristotle
- In pre-Greek times, there seems to have been no concept of formulated proof of results, at least, as far as we know today, and it is only in our time that the axiomatic method and the concept of rigorous proof has gradually come into being, i.e.
- Pre-Greek mathematics had primitive knowledge about integers and simple geometrical forms like points, straight lines, planes, etc.
- Aristotle's ideas and approaches had little influence on Greek mathematics.
- James Jeans: 'Physics and Philosophy' II
- (1) The Greek explanation that all motion tends to be circular because the circle is the perfect figure geometrically, an explanation which remained in vogue at least until the fifteenth century, notwithstanding its being contrary to the facts.
- To the Greek mind the supposed fact that the stars or planets moved in perfect geometrical figures provided a completely satisfying explanation of their motion - the world was a perfection waiting only to be elucidated, and here was a bit of the elucidation.
- It is not surprising that such explanations also should have been attempted from Greek times on, for, after all, our hairy ancestors had to think more about muscular force than about perfect circles or geodesics.
- George Chrystal's Second Promoter's Address
- In the draft ordinance for Arts degrees, while higher standards had been imposed on Latin and Greek as graduation subjects, nothing of the kind had been done for the Mathematical Departments.
- The young scholar who can readily construct a piece of Crabbed Greek but who cannot write a sentence of decent English, who prides himself of his ignorance of the first principles of mathematics, and who knows nothing of the history of human thought, has not caught the " Spirit of the Place", as little has the solver of multitudinous problems whose interest never wanders beyond the boards of a mathematical text book.
- If you have fully taken the advantage of all your opportunities; if you have marshalled out of the past with Professor Masson the noble company of English authors; appreciated with Professors Goodhart and Butcher the graceful humour of Horace and the stately rhythm and lofty wisdom of the Greek tragedians; in my own department followed for an hour the steps of Archimedes, and pondered, however superficially, the problems that engaged the mind of Newton, and finally, with Professors Seth and Calderwood, examined the manifold of its own experience - if any of you have done all this, and made no progress towards Shakespeare's manliness, then I fear that you must be classed in the category so pithily described by Burns as those that "gang in strike and come out asses"? I am tempted before concluding to say a word or two of a personal character.
- Halmos: creative art
- The celebrated Greek problems (trisect the angle, square the circle, duplicate the cube) are of this kind, and the irrepressible mathematical amateur to the contrary notwithstanding, mathematicians are no longer trying to solve them.
- The catch is that the original Greek formulation of the problem is more stringent: it requires a construction that uses ruler and compasses only.
- It takes some careful legalism (or some moderately pedantic mathematics) to formulate really precisely just what was and what wasn't allowed by the Greek rules.
- G A Miller - A letter to the editor
- For instance, recent discoveries relating to the finding of at least one root by the ancient Babylonians of certain numerical quadratic and cubic equations throws new light on the history of algebra and on the contributions made by the Greeks and the Arabians towards the solution of algebraic equations.
- It is therefore far from the truth to say that "the general quadratic as we know it today was thus fully mastered by Greek mathematicians" - D E Smith, History of Mathematics, Volume 1 (1923), page 126.
- Bell papers
- My interest in mathematics began with two school prizes, one in Greek, the other for physical laboratory, both richly bound in full calf.
- The Greek prize was Clerk Maxwell's classic on electricity and magnetism, the other, Homer's Odyssey.
- My cousin got the prize for Greek, I got the other.
- EMS obituary
- There are no exceptions to the rule that (Greek quote meaning "God always geometrises").
- These Greek words "God always geometrises" are attributed by Plutarch to Plato, and it is through this particular fact, that D'Arcy Thompson passed on to me as I was about to give a lecture in my first month at the United College, that our friendship began.
- During a walk he threw out the suggestion that Greece owed more to Ancient Egypt both physically and intellectually than we think, in evidence of which he pointed out how frequently the name of a famous Greek began with the letter P.
- Ball papers
- Primarily Pythagoras was a religious teacher and philosopher, inculcating a stern system of morality, greatly superior to any system then known to the Greeks.
- He thought this experiment might be well worth making in cases where authorship was in question, and in particular in the case of the Greek text of some of the books of the New Testament, but as far as I know the test has never been applied.
- Heinrich Tietze on Numbers
- Number symbols (see Arabic_numerals, Babylonian_numerals, Egyptian_numerals, Greek_numbers, Indian_numerals, Mayan_mathematics) have also differed widely.
- [0ne would have to consider how the consequent change in manual skill would have influenced the whole history of tools and weapons and the division of peoples according to their linguistic roots; if not indeed a whole new world.] For convenience, we shall use the word 'year' for twelve and shall borrow the tenth and eleventh letters of the Greek alphabet, kappa (k) and lambda (l), for the patterns with six fingers of one hand and four of the second hand (6 + 4), and six fingers of one hand and five of the second hand (6 + 5), respectively.
- E C Titchmarsh: 'Aftermath
- The question is sometimes asked, what have mathematicians discovered in modern times which would have been completely new and strange to the Greeks? One of the best answers to this is, the theory of functions of a complex variable.
- Cauchy's theory of functions of a complex variable would have surprised the Greeks very much, and surely it would have delighted them too.
- EMS obituary
- He was educated at John Street School, the Free Church Training College for Teachers and at the University where, after winning prizes in the classes of Latin, Greek, Mathematics and Natural Philosophy, he graduated M.A.
- He retired before the usual age in order to be able to extend his study of the Classics, for which he had a deep and abiding love, and to visit some of the places of importance in Latin and Greek lore.
- Education in St Andrews in 1849
- The eight professorships are devoted to the inculcating of Latin, Greek, Mathematics, Logic and Rhetoric, Medicine, Moral Philosophy, Natural Philosophy, and Civil History.
- He appointed the provost and first and second ministers of the city, and Dr Alexander, Professor of Greek, and on the decease of the latter, the Sheriff-depute of Fife, trustees for the management of the College, and enjoined that the Madras or monitorial method of tuition should be adopted and followed in the institution.
- Kepler's Planetary Laws
- Since Greek times, the accepted description of the planetary system had been a geometrical one, known as the Ptolemaic theory (a geocentric configuration), which supposed that the Earth was fixed at the centre of the universe, with the Moon, the Sun, and the five known (naked-eye) planets revolving round it.
- Kepler always showed the greatest respect for his Greek predecessors, and read their works thoroughly, selecting material that he could incorporate into his new astronomical synthesis.
- Kepler's Planetary Laws
- Since Greek times, the accepted description of the planetary system had been a geometrical one, known as the Ptolemaic theory (a geocentric configuration), which supposed that the Earth was fixed at the centre of the universe, with the Moon, the Sun, and the five known (naked-eye) planets revolving round it.
- Kepler always showed the greatest respect for his Greek predecessors, and read their works thoroughly, selecting material that he could incorporate into his new astronomical synthesis.
- D'Arcy Thompson's family
- This week my father and I, between us, are a hundred years old! But how did they happen to make that little mariner into a Professor of Greek, whose forefathers had always been sailormen? And however has it come to pass that I, his son, fetch up in the Port of Cardiff after a hundred years, with a Master's ticket in my pocket, and find myself for one short cruise, for one proud moment, Captain of the Ship?
.
- In 1863, after spending twelve years as a classics teacher at the Edinburgh Academy, Thompson moved to Ireland when offered the chair of Greek in Queen's College, Galway.
- Heath: 'The thirteen books of Euclid's Elements' Preface
- A new translation from the Greek was necessary for two reasons.
- In the matter of notes, the edition of the first six Books in Greek and Latin with notes by Camerer and Hauber (Berlin, 1824-5) is a perfect mine of information.
- Smith's History Papers
- It was furthermore appropriate that the discovery should be made by Professor Heiberg, facilis princeps among all workers in the field of editing the classics of Greek mathematics, and an indefatigable searcher of the libraries of Europe for manuscripts to aid him in perfecting his labours.
- Note on a Greek Papyrus in Vienna.
- Finkel's Solution Book
- It yet remains for me to express my thanks to my colleague and friend, Prof F A Hall, of the Department of Greek, for making corrections in the Greek terms used in this edition,
.
- De Coste on Mersenne 1.html
- Boyer's books
- In this book the author traces the development of calculus concepts from the time of the early Greeks up to the end of the nineteenth century.
- In fact, its scope is wider, it is more nearly a history of optics from the Greeks until the beginning of the last century.
- Ball books
- The first chapter contains a brief statement of what is known concerning the mathematics of the Egyptians and Phoenicians; this is introductory to the history of mathematics under Greek influence.
- The subsequent history is divided into three periods: first, that under Greek influence, chapters ii to vii; second, that of the middle ages and renaissance, chapters viii to xiii; and lastly that of modern times, chapters xiv to xix.
- H Weyl: 'Theory of groups and quantum mechanics'Preface to First Edition
- Occidental mathematics has in past centuries broken away from the Greek view and followed a course which seems to have originated in India and which has been transmitted, with additions, to us by the Arabs; in it the concept of number appears as logically prior to the concepts of geometry.
- But the present trend in mathematics is clearly in the direction of a return to the Greek standpoint; we now look upon each branch of mathematics as determining its own characteristic domain of quantities.
- George Temple's Inaugural Lecture I
- In the later editions it seems as if some Gothic wings had been added to a Greek temple.
- Graves on Hamilton
- He was a successful competitor on two occasions for the Vice-Chancellor's Prizes, proposed for the authors of English poems; and his examiners placed on record a judgment indicating their sense of his extraordinary proficiency in the Greek language, as well as a similar attestation of his remarkable attainments, for an undergraduate, in astronomical science.
- Blumenthal on Annalen
- I am trying to move into new areas of study in the theory of functions of several variables and in ancient Greek mathematics.
- Jordan and Deutsche Mathematik
- First of all, the stylistic differences between Greek and Western mathematics emphasized by Oswald Spengler [in his book The Decline of the West] should not become overvalued in their importance: the correctness of mathematical theorems is completely independent of them.
- Gibson History 9 - Colin Maclaurin
- After an interesting introduction in which he reviews the methods of exhaustion of the Greek geometers and the method of indivisibles of Cavellerius [Cavalieri] - an exposition marked by accuracy and breadth of view - he proceeds in Book I to explain and develop the general theory, making use of the conception of a velocity and keeping algebraic symbolism and calculation as far as possible in the background.
- Leslie works
- Though so devoted an admirer of the ancient Greeks, his constructive bent makes his geometry practical.
- Gregory-Collins correspondence
- Fermat's Diophantus will not be extant ths year yet, and I am afraid Borelli will be disturbed in Sicily from publishing Maurolico's Archimedes with his own annotations, for Turks having got together a great naval force, it is said 80 sail of galleys and 40 sail of ships besides those of Barbary, and two Greeks were executed as spies for surveying the Plains of Palermo: Alteiri is chosen Pope a man 74 years of age, deaf and dark ..
- Mathematics at Aberdeen 3
- The professor of Greek took the bajan or first year, then for the remaining three years a regent or professor of philosophy moved on with his class, teaching the entire course.
- Centenary of John Leslie
- He quoted Latin, and even Greek, copiously in later life.
- EMS obituary
- As a student at St Andrews University he was placed in the Honours List of all his classes (which included Greek, Humanity, Rhetoric and English Literature, Moral Philosophy and Political Economy) obtaining first place in Mathematics, Natural Philosophy, Chemistry, Logic and Metaphysics, Anatomy and Physiology.
- Horace Lamb addresses the British Association in 1904
- His treatise on the history of Greek Geometry, full of learning and sound mathematical perception, is written with great simplicity and an entire absence of pedantry or dogmatism.
- Gibson History 2 - Mathematics in the schools
- In many of the parish schools the schoolmasters were competent to impart advanced instruction in Latin, Greek and Mathematics and were proud of such pupils as took advantage of the opportunities offered.
- Newton's Arian beliefs
- He discovered that the final phrase 'and these three are one' was not present in any Greek version that he studied.
- Poincaré on the future of mathematics
- For the Greeks a good solution was one that employed only rule and compass; later it became one obtained by the extraction of radicals, then one in which algebraical functions and radicals alone figured.
- G H Hardy addresses the British Association in 1922, Part 2
- The problem belongs to the theory of the so-called 'perfect' numbers, which has exercised mathematicians since the times of the Greeks.
- Whiston's comments on Berkeley's 'Treatise
- It is the pretended metaphysical science itself, derived from the sceptical disputes of the Greek philosophers, not those particular great men who have been unhappily imposed on by it, that I complain of.
- Born Inaugural
- The idea that a science can be logically reduced to a small number of postulates or axioms is due to the great Greek mathematicians, who first tried to formulate the axioms of geometry and to derive the complete system of theorems from them.
- Coulson: 'Electricity
- Indeed, the very word electricity is derived from the Greek word for amber.
- Gillespie: 'Integration
- The integral calculus may be said to have been begun by the Greek mathematicians who strove to evaluate the area of a circle.
- Colin Maclaurin
- At the age of 11 Colin, already proficient in Latin and Greek, entered Glasgow University, and graduated M.A.
- J Ruska on Heinrich Suter
- Therefore he already decided while on the "Zuricher Industrieschule" to privately learn Latin and Greek; this was necessary since the school offered modern languages only.
- Women mathematicians by Dubreil-Jacotin
- Yet, it is as early as Greek antiquity that the first woman who can be considered as a mathematician makes her appearance: Hypatia, born at Alexandria in the year 370 of our era.
- De Coste on Mersenne
- Thus he devoted the best part of his life to this holy exercise, having never let a day pass without reading the Bible and some Greek or Latin Father.
- DArcy Thompson knighted
- Born in 1860, he is a son of D'Arcy Wentworth Thompson, Professor of Greek, Queen's College, Galway.
- G H Hardy addresses the British Association in 1922
- The problem belongs to the theory of the so-called 'perfect' numbers, which has exercised mathematicians since the times of the Greeks.
- Mathematicians and Music 3
- Again, is it not within the realms of possibility that some day the inadequacies of the present vehicle of musical expression may lead us to revive some of the ideals of Greek music during the golden period of Aristoxenus?
.
- H M Macdonald addresses the British Association in 1934
- Among the Greeks Empedocles was an exponent of the first view, while Aristotle supported the second view.
- Ernest Hobson addresses the British Association in 1910
- This system which, when scrutinised, affords the simplest illustration of the importance of Mathematical form, has become so much an indispensable part of our mental furniture that some effort is required to realise that an apparently so obvious idea embodies a great invention; one to which the Greeks, with their unsurpassed capacity for abstract thinking, never attained.
- Cajori: 'A history of mathematics' Introduction
- The chemist smiles at the childish efforts of alchemists, but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindus as useful and admirable as any research of today.
- Smith Major History books
- There are also the names of Archimedes, who deemed it a worthy labour to improve the number system of the Greeks; of Euclid, whose contributions were by no means confined to geometry; of Nicomachus and lamblichus, who represent the declining Hellenic civilization, and of Psellus, who was a witness of its final decay.
- Smith Autograph Papers
- Professor Alexander Macfarlane tells us of his considerable advancement in arithmetic at the age of three, of his ability to read Latin, Greek, and Hebrew at the age of five, and of his intimate familiarity with Italian, Sanskrit, and Arabic at the age of ten.
- Graves's papers
- (date of paper 1885) On Two Fragments of a Greek Papyrus, Hermathena 5 (11) (1885), 237-257.
- Mark Kac on education, physics and mathematics
- you had to take six years of Latin and four years of Greek and no nonsense about taking soul courses or folk music, or all that.
- Mathematics at Aberdeen 4
- The semi year was 'to be employed (besides reading some Greek as usual) in a Course of Mathematicks both Speculative & Practicall & in ane Introduction to all the Branches of Natural History'.
- Smith's Obituaries and Biographies
- Mr Kaye aroused a great deal of opposition on the part of Hindu mathematicians because of his conclusions that the early writers were dependent solely upon Greeks and showed little originality except framing fancifully worded problems.
- Edmund Whittaker: 'Physics and Philosophy
- Most of the Greek philosophers believed the heavenly bodies to be alive, divine and incorruptible, knowing no change other than circular and uniform motions, and governing the affairs of men.
- Stringham books
- Thus, instead of becoming a classic, like the geometry handed down to us from the Greeks, in the form of Euclid's Elements, algebra has become a collection of processes practically exemplified and of principles inadequately explained.
- Sommerville: 'Geometry of n dimensions
- This stage had been reached when Greek geometry started.
- Plato Mathematics
- From their efforts stem many of the greatest achievements of Greek mathematical science down to Ptolemy.
- Recollections of Mary Somerville
- and also the great Greek dramatists, whose tragedies she read fluently in the original, being a good classical scholar.
- Gregory tercentenary
- The honorary graduands were presented by Professor H J Rose, of the Chair of Greek, the LL.D.
- Andrew Forsyth addresses the British Association in 1905, Part 2
- As you are aware, the elements of Euclid have long been the standard treatise of elementary geometry in Great Britain; and the Greek methods, in Robert Simson's edition, have been imposed upon candidates in examination after examination.
- Clifford's 'Lectures and Essays'
- He read modern Greek fluently, and at one time he was curious about Sanskrit.
- The Edinburgh Mathematical Society: the first hundred years
- He was chief mathematical master at The Edinburgh Academy and was an able mathematician with a scholarly interest in the early Greek geometers.
- Henry Baker addresses the British Association in 1913
- but the song, once so full of dread, how much it owes to the highest refinements of his craft, from at least the time of the Greek devotion to the theory of conic sections; how much, that is, to the harmony that is in the human soul.
- Turnbull lectures on Colin Maclaurin, Part 2
- It was the first logical and systematic account of fluxions, and in point of rigour could hold its own with the geometrical method of exhaustions of the Greeks and the subsequent work of Cauchy and of Weierstrass.
- H M Macdonald addresses the British Association in 1934, Part 1
- Among the Greeks Empedocles was an exponent of the first view, while Aristotle supported the second view.
- Fields Medal Letter
- Because of the international character the language to be employed it would seem should be Latin or Greek? The design has still to be definitely determined.
- James Jeans: 'Physics and Philosophy' I
- Thus to primitive man the sun was a life-giving god - to the Greeks the horse-drawn chariot of a god - while a later and less pagan age supposed that angels had been entrusted with the task of pushing along the sun, moon and planets, and of maintaining the motion of the celestial spheres to which the more distant stars were supposed to be affixed.
- Al-Biruni: 'Coordinates of Cities
- read in some Greek books that one degree of the meridian is equivalent to 500 stadia ..
- Mathematics at Aberdeen 2
- In 1700 a regulation fixing one of the regents to teach Greek in the first year, opened the way towards a modification of the regenting system and specialist teaching.
Quotations
- Quotations by Al-Biruni
- for which reason I began searching for a number of demonstrations proving a statement due to the ancient Greeks ..
- Quotations by Anaxagoras
- The Greeks are wrong to recognize coming into being and perishing; for nothing comes into being nor perishes, but is rather compounded or dissolved from things that are.
- A quotation by Neugebauer
- It seems to me that all the evidence points to Apollonius as the founder of Greek mathematical astronomy.
- Quotations by Bochner
- The word "mathematics" is a Greek word and, by origin, it means "something that has been learned or understood," or perhaps "acquired knowledge," or perhaps even, somewhat against grammar, "acquirable knowledge," that is, "learnable knowledge," that is, "knowledge acquirable by learning."
.
- Quotations by Whitehead
- Mathematics as a science, commenced when first someone, probably a Greek, proved propositions about "any" things or about "some" things, without specifications of definite particular things.
- Quotations by Galton
- I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of Frequency of Error." The law would have been personified by the Greeks and deified,if they had known of it.
- Quotations by Weyl
- Without the concepts, methods and results found and developed by previous generations right down to Greek antiquity one cannot understand either the aims or achievements of mathematics in the last 50 years.
- Quotations by Heath
- A History of Greek Mathematics (1921).
Chronology
- Mathematical Chronology
- Greeks begin to use written numerals.
- Pappus of Alexandria writes Synagoge (Collections) which is a guide to Greek geometry.
- Metrodorus assembles the Greek Anthology consisting of 46 mathematical problems.
- There Greek and Indian mathematical and astronomy works are translated into Arabic.
- Gherard of Cremona begins translating Arabic works (and Arabic translations of Greek works) into Latin.
- Recorde translates and abridges the ancient Greek mathematician Euclid's Elements as The Pathewaie to Knowledge.
- Bachet publishes his Latin translation of Diophantus's Greek text Arithmetica.
- Jones introduces the Greek letter π to represent the ratio of the circumference of a circle to its diameter in his Synopsis palmariorum matheseos (A New Introduction to Mathematics).
- Maclaurin publishes Treatise on Fluxions which aims to provide a rigorous foundation for the calculus by appealing to the methods of Greek geometry.
- Chronology for 1AD to 500
- Pappus of Alexandria writes Synagoge (Collections) which is a guide to Greek geometry.
- Metrodorus assembles the Greek Anthology consisting of 46 mathematical problems.
- Chronology for 500 to 900
- Metrodorus assembles the Greek Anthology consisting of 46 mathematical problems.
- There Greek and Indian mathematical and astronomy works are translated into Arabic.
- Chronology for 1740 to 1760
- Maclaurin publishes Treatise on Fluxions which aims to provide a rigorous foundation for the calculus by appealing to the methods of Greek geometry.
- Chronology for 1500 to 1600
- Recorde translates and abridges the ancient Greek mathematician Euclid's Elements as The Pathewaie to Knowledge.
- Chronology for 1100 to 1300
- Gherard of Cremona begins translating Arabic works (and Arabic translations of Greek works) into Latin.
- Chronology for 1700 to 1720
- Jones introduces the Greek letter π to represent the ratio of the circumference of a circle to its diameter in his Synopsis palmariorum matheseos (A New Introduction to Mathematics).
- Chronology for 500BC to 1AD
- Greeks begin to use written numerals.
- Chronology for 1600 to 1625
- Bachet publishes his Latin translation of Diophantus's Greek text Arithmetica.
This search was performed by Kevin Hughes' SWISH and Ben Soares' HistorySearch Perl script
JOC/BS August 2001