Search Results for geometry


Biographies

  1. Pieri biography
    • He entered the University of Bologna in 1880, having had the fees waived, taking the courses on 'Projective geometry' and 'Design for projective geometry' with Pietro Boschi.
    • He also took the 'Algebra and analytic geometry' course given by Cesare Arzela, and courses on physics, chemistry and mineralogy.
    • Luigi Bianchi taught him 'Algebraic functions' in 1882-83 and 'Differential geometry' in 1883-84.
    • He wrote the thesis, Studies in Differential Geometry, also supervised by Bianchi, which he submitted to the Scuola Normale Superiore in Pisa in September 1884.
    • After teaching for a year in Pisa, Pieri won the competition for a professorship at the Royal Military Academy in Turin where he became professor of projective and descriptive geometry in November 1886.
    • In 1888 he also was appointed an assistant to the chair of projective geometry at the University of Turin [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3]:- .
    • By [1890] he had published about ten works, including an edited translation of G K C von Staudt's 1847 'Geometrie der Lage' (Geometry of Position).
    • He taught projective geometry courses there for several years.
    • In 1891 he entered the competition for the chair of analytic and projective geometry at the University of Rome which was won by Guido Castelnuovo (Pieri came third equal).
    • Two years later he entered the competition for the chair of projective geometry at the University of Naples (Domenico Montesano won with Pieri fourth equal), and the competition for the chair of projective and descriptive geometry at the University of Turin (Luigi Berzolari won with Pieri second equal).
    • Pincherle invited Pieri to apply writing (see [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3]):- .
    • In Catania he taught projective geometry and descriptive geometry but also took on the teaching of projective geometry to give him a better salary.
    • During these years in Catania [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3]:- .
    • he produced two research papers on algebraic geometry, two book reviews, four papers on logic and foundations of arithmetic, and about seven on foundations of geometry, including two major works axiomatising Euclidean and complex projective geometry.
    • In fact, although Pieri's main area was projective geometry, and he is an important member of the Italian School of Geometers, however, after he moved to Turin, he became influenced by Giuseppe Peano at the University and Cesare Burali-Forti who was a colleague at the Military Academy.
    • Their influence had led Pieri to study the foundations of geometry.
    • In 1895 he set up an axiomatic system for projective geometry with three undefined terms, namely points, lines and segments.
    • He improved on results of Moritz Pasch and Giuseppe Peano and then, in 1905, he gave the first axiomatic definition of complex projective geometry which does not build on real projective geometry.
    • In 1898 Pieri had published the memoir The principles of the geometry of position through the Academy of Sciences of Turin.
    • He did not attend these conferences but submitted a paper to the first of these on Geometry considered as a purely logical system.
    • However, he recovered and after moving to Parma in 1908 he became both an ordinary professor and director of the School of Projective and Descriptive Geometry with Design.
    • After he had succeeded in getting his former student Beppo Levi appointed to Parma in 1909 the two cooperated in expanding the university [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3]:- .
    • Russell wrote (see for example [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3]):- .
    • Following Pieri's death from cancer in 1913, his colleague and former student Beppo Levi wrote (see for example [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3]):- .

  2. Klein biography
    • Felix Klein is best known for his work in non-euclidean geometry, for his work on the connections between geometry and group theory, and for results in function theory.
    • Plucker held a chair of mathematics and experimental physics at Bonn but, by the time Klein became his assistant, Plucker's interests had become very firmly rooted in geometry.
    • Klein received his doctorate, which was supervised by Plucker, from the University of Bonn in 1868, with a dissertation Uber die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form on line geometry and its applications to mechanics.
    • However in the year Klein received his doctorate Plucker died leaving his major work on the foundations of line geometry incomplete.
    • After five years at the Technische Hochschule at Munich, Klein was appointed to a chair of geometry at Leipzig.
    • The journal specialised in complex analysis, algebraic geometry and invariant theory.
    • It is a little hard to understand the significance of Klein's contributions to geometry.
    • During his time at Gottingen in 1871 Klein made major discoveries regarding geometry.
    • He published two papers On the So-called Non-Euclidean Geometry in which he showed that it was possible to consider euclidean geometry and non-euclidean geometry as special cases a projective surface with a specific conic section adjoined.
    • This had the remarkable corollary that non-euclidean geometry was consistent if and only if euclidean geometry was consistent.
    • The fact that non-euclidean geometry was at the time still a controversial topic now vanished.
    • Its status was put on an identical footing to euclidean geometry.
    • Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872), profoundly influenced mathematical development.
    • The Erlanger Programm gave a unified approach to geometry which is now the standard accepted view.
    • Transformations play a major role in modern mathematics and Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties.
    • In this way the Erlanger Programm defined geometry so that it included both Euclidean geometry and non-Euclidean geometry.
    • He owed some of his greatest successes to his development of Riemann's ideas and to the intimate alliance he forged between the later and the conception of invariant theory, of number theory and algebra, of group theory, and of multidimensional geometry and the theory of differential equations, especially in his own fields, elliptic modular functions and automorphic functions.
    • History Topics: Non-Euclidean geometry .

  3. Pogorelov biography
    • Pogorelov's interests were in geometry and his postgraduate studies at Moscow State University were jointly supervised by Nikolai Vladimirovich Efimov in Moscow and Aleksandr Danilovic Aleksandrov who was based in Leningrad.
    • Research on "geometry in the large" had begun in Russia when Stefan Cohn-Vossen, who had worked closely with Hilbert, emigrated there to escape from the Nazis.
    • But we have got ahead of ourselves in describing Pogorelov's career, for after the award of his Candidate's degree by Moscow State University in 1947, he was appointed to the Geometry Department of Institute of Mathematics at Kharkov State University.
    • or habilitation) by Kharkov State University in 1948 and, two years later, he was appointed to the Chair of Geometry at Kharkov University.
    • He was head of the Geometry Department of the Institute of Mathematics of Academy of Sciences of Ukraine in 1959-60 and, from 1960 until 2000, he headed the Geometry Department at the B Verkin Institute for Low Temperature Physics & Engineering of the Ukrainian Academy of Sciences.
    • His skills, as we have seen, were not confined to geometry since he had gained considerable engineering skills during his time in Moscow.
    • He has solved a number of key problems in geometry in the large, in the foundations of geometry, and in the theory of the Monge-Ampere equations, and he also has obtained remarkable results in the geometric theory of stability of thin elastic shells.
    • First we note that his textbooks are all in the general area of geometry and run to many editions.
    • They include: Differential geometry (1955); Lectures on analytic geometry (1957); and Lectures on the foundations of geometry (1966).
    • An historical sketch of the foundations of geometry.
    • The modern axiomatic construction of Euclidean geometry.
    • A study of the axioms of Euclidean geometry.
    • Lobachevskian geometry.
    • The extrinsic geometry of convex surfaces (1969).
    • Even the title of the monograph stresses that it can be regarded as a continuation and development of the results presented in the monograph by A D Aleksandrov "Intrinsic Geometry of Convex Surfaces" published in 1948.
    • For his secondary school geometry textbook he was given the title "Excellent Teacher of USSR" and received the A S Makarenko medal.
    • in May 2004, the Department of Geometry at the Institute for Low-Temperature Physics and Engineering of the Ukrainian National Academy of Sciences, together with the Chair of Geometry of the Kharkov National University, organized the International Seminar "Geometry in the Large" dedicated to the 85th anniversary of Academician Pogorelov.
    • Pogorelov was one of the brightest representatives of the branch of geometry created by A D Alexandrov.

  4. Reye biography
    • Under this influence, Reye's early interest in mathematical physics and meteorology turned to an interest in geometry while he continued to hold the lectureship in mathematical physics at Zurich.
    • Led towards geometry by his interest in mechanics he began to study graphical statics and avidly read von Staudt's work on geometry.
    • Culmann's fundamental monograph Die graphische Statik (Graphical Statics) was published in 1865 but Reye was by this time deeply involved in studying geometrical methods similar to the ideas of projective geometry contained in von Staudt's Geometrie der Lage (1847).
    • The first volume of Reye's own major work Geometrie der Lage (Geometry of position) was published in 1866 [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • A French translation with title Lecons sur la geometrie de position appeared in 1881-1882, an Italian edition with title La geometria di posizione was published in 1884 and an English translation Lectures on the Geometry of Position in 1898.
    • Only the first of these children was born in Zurich for, in 1870, the family left Zurich and moved to Aachen in Germany when Reye was appointed to the Chair of Geometry and Graphical Statics at the newly founded Polytechnikum in Aachen (now the Rheinisch-Westfalische Technische Hochschule).
    • Reye had been an assistant to Joseph Wolfgang von Deschwanden (1819-1866), the professor of descriptive geometry, and taught descriptive geometry at Zurich after Deschwanden became ill.
    • Reye also taught projective geometry believing that the two courses should be linked.
    • However, after Deschwanden's death a new appointment was made to the chair of descriptive geometry, namely Otto Wilhelm Fiedler (1832-1912), and Reye wrote (in the Preface to the 2nd edition of Geometrie der Lage):- .
    • Unfortunately, I am now denied contributing to the propagation of my favourite science as a teacher in the way I did before; because recently I was ruthlessly deprived of my course of lectures on projective geometry, so that it could be assigned to the newly appointed professor of descriptive geometry at his request.
    • However, he only worked in Aachen for two years before he was appointed to the Chair of Geometry and Mechanics at the Kaiser-Wilhelm University of Strasbourg.
    • Two mathematics chairs were founded at this time, the first in Mathematics which was filled by Elwin Christoffel and the second in Geometry and Mechanics which was filled by Reye.
    • This required him to give an inaugural address and he chose as his subject Die synthetische Geometrie im Altertum und in der Neuzeit (Synthetic geometry in antiquity and in modern times).
    • Reye's work in geometry included a study of conics, quadrics and projective geometry.
    • This book on spherical geometry did not, however, have the same impact as his classic text Geometrie der Lage.
    • Reye treated in detail the theory of conics and quadrics and of their linear systems, that of third degree surfaces and some of the fourth degree, as well as many quadratic congruences and aggregates taken from line geometry.
    • He was one of the leading geometers of his time, and he published a great deal on synthetic geometry.
    • Part of the reason must be that some of Reye's work was later interpreted in the geometry set up by Corrado Segre, in particular it was interpreted in the setting of Segre manifolds.
    • Finally we note that although David Hilbert had a high opinion of Reye's work, their approach to geometry was very different.
    • Hilbert proposed a purely axiomatic method in geometry while Reye rejects everything that cannot be justified with geometric intuition.

  5. Smith Karen biography
    • At high school she studied the usual mathematical topics to prepare her for university entrance, namely geometry, algebra, pre-calculus, and then Calculus (AB).
    • It was at high school that her interest in geometry began to blossom, an interest which would develop as her studies progressed [Cogito (09.02.2009).',5)">5]:- .
    • In high school, I essentially discovered for myself the projective plane, a two-dimensional geometry not unlike the geometry one studies in high school but in which parallel lines meet "at infinity".
    • My mathematical research is in the general area of algebra, more specifically, algebraic geometry and commutative algebra.
    • Algebraic geometry studies geometric objects called algebraic varieties that arise as zero sets of polynomials.
    • Algebraic geometry underlies many applications of mathematics to industry; these applications range from coding theory, which brings us the compact disc, to spline theory, which brings us computer graphics essential to certain medical applications and to the entertainment industry.
    • My own research is not motivated by any particular application of algebraic geometry, but rather by the inherent elegance and beauty of the subject.
    • The rings themselves are a rich topic of investigation, and are studied by commutative algebraists without particular regard for the geometry associated with them.
    • My own research is very much at the interface of commutative algebra and algebraic geometry.
    • It is also awarded for her more recent work which builds new bridges between commutative algebra and algebraic geometry via the concept of tight closure.
    • She received the award for her proposal "Interactions of commutative algebras with analysis, geometry and computer science." Here are some extracts from the abstract of her proposal:- .
    • This long term research and educational project includes plans for a new graduate course at the University of Michigan stressing the use of computers in algebraic geometry, and for a junior level seminar for majors in mathematics and computer science in which teams of undergraduate students led by a graduate student learn projective geometry through self-discovery.
    • She has received a number of other major awards from the National Science Foundation including Prime Characteristic Techniques in Commutative Algebra and Algebraic Geometry in 2000, and Enhancing the Research Workforce in Algebraic Geometry and its Boundaries in the Twenty-First Century (2005).
    • establishes a research training program in algebraic geometry and its boundary areas at the University of Michigan.
    • The explosive growth of algebraic geometry at the end of the twentieth century has made this a very exciting time to begin research in the field, but it has also made it difficult for young researchers to get started.
    • This project will increase the flow of broadly trained researchers in algebraic geometry and its boundary areas, therefore enhancing the training infrastructure and the research workforce in these vital areas of mathematics in the twenty-first century.
    • The course was aimed at introducing algebraic geometry to graduate students and mathematicians with little training in the topic.
    • In collaboration with Lauri Kahanpaa, Pekka Kekalainen, and William Traves, Smith wrote up the lectures and published them as the book An invitation to algebraic geometry (2000).
    • The book under review provides an entryway for those who would like to become more familiar with algebraic geometry, while it entertains the hope, justifiable in this case, that some of its readers will fall under the sway of this beautiful subject.
    • Karen Smith, a highly accomplished young commutative algebraist, was led by the momentum of her work and its results to apply her techniques to problems in algebraic geometry.
    • Students contemplating algebraic geometry as a field of specialization will also find this an attractive and instructive place to start.

  6. Yano biography
    • To understand the general theory of relativity one had to study differential geometry.
    • Kentaro immediately made the decision to study differential geometry at university.
    • He entered the university and, together with some fellow students who also were keen to learn about differential geometry and Riemannian geometry, he studied some of the major texts such as those by Schouten, Weyl, Eisenhart, Levi-Civita and Cartan.
    • After completing his undergraduate studies in 1934, Yano began to undertake research in differential geometry at the University of Tokyo.
    • After the war, he wrote a book Geometry of Connections in Japanese which was published in 1947.
    • It is not intended as a treatise for the expert, but the reviewer believes it to be one of the best for the beginner in the geometry of connections.
    • Yano wanted to learn of these methods so that he could study global differential geometry rather than the local version he had been studying up to that time.
    • While in the United States, Yano received an invitation to teach at the University of Rome and to attend the International Congress on Differential Geometry in Italy in 1953.
    • Yano participated in the American Mathematical Society meeting on Differential Geometry at the University of Washington in the summer of 1956.
    • In 1960 he spent six months in Hong Kong as a visiting professor and his lectures there were later written up and published as Differential geometry on complex and almost complex spaces (1965).
    • He was in Zurich in June 1960 for the International Symposium on Differential Geometry and Topology, spent April, May and June of 1961 at the University of Washington, then spent a month at the University of Southampton followed by two months at the University of Liverpool in 1962.
    • During a visit to the University of Illinois in 1968, Yano was able to complete work on Integral formulas in Riemannian geometry published in 1970.
    • In 1969, as well as a week at a meeting on Differential Geometry at Oberwolfach, Germany, he spent time at Queen's University in Canada.
    • The book Differential geometry in honour of Kentaro Yano was published for his retirement.
    • This book is a collection of papers in differential geometry contributed by friends, colleagues and former students of Professor Kentaro Yano to celebrate his sixtieth birthday and in observance of his retirement from his position as Dean of Science at Tokyo Institute of Technology.
    • To gauge the magnitude of his contribution to differential geometry, it suffices to recall his mathematical work, extending over the past four decades, which covers such diverse areas of geometry as affine, projective and conformal connections, geometry of Hermitian and Kahlerian manifolds, holonomy groups, automorphism groups of geometric structures, harmonic integrals, tangent and cotangent bundles, submanifolds, and integral formulas in Riemannian geometry.
    • He was in Korea for the Symposium on Differential Geometry at Seoul and Kyunpook Universities from 14 to 18 September, then he was in Rome to celebrate the 100th birthday of Levi-Civita at the Accademia Nazionale dei Lincei in December.
    • They are: (with Shigeru Ishihara) Tangent and cotangent bundles: differential geometry (1973); (with Masahiro Kon) Anti-invariant submanifolds (1976); (with Masahiro Kon) CR submanifolds of Kahlerian and Sasakian manifolds (1983); and (with Masahiro Kon) Structures on manifolds (1984).

  7. Lobachevsky biography
    • Since Euclid's axiomatic formulation of geometry mathematicians had been trying to prove his fifth postulate as a theorem deduced from the other four axioms.
    • Instead he studied geometry in which the fifth postulate does not necessarily hold.
    • Lobachevsky categorised euclidean as a special case of this more general geometry.
    • On 11 February 1826, in the session of the Department of Physico-Mathematical Sciences at Kazan University, Lobachevsky requested that his work about a new geometry was heard and his paper A concise outline of the foundations of geometry was sent to referees.
    • The text of this paper has not survived but the ideas were incorporated, perhaps in a modified form, in Lobachevsky's first publication on hyperbolic geometry.
    • He published this work on non-euclidean geometry, the first account of the subject to appear in print, in 1829.
    • In 1837 Lobachevsky published his article Geometrie imaginaire and a summary of his new geometry Geometrische Untersuchungen zur Theorie der Parellellinien was published in Berlin in 1840.
    • This last publication greatly impressed Gauss but much has been written about Gauss's role in the discovery of non-euclidean geometry which is just simply false.
    • There is a coincidence which arises from the fact that we know that Gauss himself discovered non-euclidean geometry but told very few people, only his closest friends.
    • Two of his friends were Farkas Bolyai, the father of Janos Bolyai (an independent discoverer of non-euclidean geometry), and Bartels who was Lobachevsky's teacher.
    • Also Laptev in [Dedicated to the memory of Lobachevskii 1 (Kazan, 1992), 35-40.',29)">29] has examined the correspondence between Bartels and Gauss and shown that Bartels did not know about Gauss's results in non-euclidean geometry.
    • There are other claims made about Lobachevsky and the discovery of non-euclidean geometry which have been recently refuted.
    • The story of how Lobachevsky's hyperbolic geometry came to be accepted is a complex one and this biography is not the place in which to go into details, but we shall note the main events.
    • In 1866, ten years after Lobachevsky's death, Houel published a French translation of Lobachevsky's Geometrische Untersuchungen together with some of Gauss's correspondence on non-euclidean geometry.
    • Beltrami, in 1868, gave a concrete realisation of Lobachevsky's geometry.
    • Weierstrass led a seminar on Lobachevsky's geometry in 1870 which was attended by Klein and, two years later, after Klein and Lie had discussed these new generalisations of geometry in Paris, Klein produced his general view of geometry as the properties invariant under the action of some group of transformations in the Erlanger Programm.
    • There were two further major contributions to Lobachevsky's geometry by Poincare in 1882 and 1887.
    • History Topics: Non-Euclidean geometry .

  8. Gromov biography
    • Around 1970, the world of differential geometry was astounded by the news that a young Russian by the name of Mikhael Gromov had proved that any noncompact differential manifold admits a Riemannian metric of positive sectional curvature, and also one of negative sectional curvature.
    • This is one of the papers referred to in the above quote by Wu, as are Gromov's papers (with V A Rokhlin) Imbeddings and immersions in Riemannian geometry (1970), and (with Ya M Eliashberg) Elimination of singularities of smooth mappings (1971).
    • The first edition of this book, published in French [Structures metriques pour les varietes riemanniennes (1981)], is considered one of the most influential books in geometry in the last twenty years.
    • In 1986 he was an invited plenary speaker at the International Congress of Mathematicians in Berkeley where he spoke on Soft and hard symplectic geometry.
    • This is a survey of the recent work on symplectic geometry, with emphasis on the author's own contributions ..
    • 'Soft' and 'hard' in this talk are limited to the framework of the global nonlinear analysis concerning the geometry of spaces of maps between smooth manifolds".
    • In 1985 Gromov was a plenary speaker at the British Mathematical Colloquium in Cambridge when he lectured on Differential geometry with and without infinitesimal calculus: anatomy of curvature.
    • These include: the Oswald Veblen Prize in Geometry from the American Mathematical Society (1981):- .
    • for his paper "Pseudo-holomorphic curves in symplectic manifolds", which revolutionized the subject of symplectic geometry and topology and is central to much current research activity, including quantum cohomology and mirror symmetry.
    • On the other hand, new techniques developed by Gromov for different purposes led to completely new kinds of problems: one can imagine the great variety of questions arising from the introduction of a natural geometric structure on the set of all (isomorphism classes) of Riemannian manifolds, or from the discovery of many new and remarkable invariants of manifolds (e.g., the K-area, the simplicial volume, the minimal volume, etc.), not to forget important new notions, such as that of hyperbolic groups, which is at the origin of major recent developments in differential geometry.
    • His contributions, including the introduction of a metric structure for families of various geometrical objects, have led to dramatic developments in geometry and many other fields of mathematics.
    • Gromov has pioneered entirely new disciplines in a variety of fields, including geometry and analysis, and has had a substantial impact on all the mathematical sciences.
    • Through the application of innovative ideas and radical nontraditional mathematical methods, he has also solved a great many complicated problems in modern geometry.
    • for his work in Riemannian geometry, which revolutionized the subject; his theory of pseudoholomorphic curves in symplectic manifolds; his solution of the problem of groups of polynomial growth; and his construction of the theory of hyperbolic groups.
    • in recognition of his profound and extraordinary insights whose influence extends far beyond the boundaries of his own field of geometry.
    • He is known for important contributions in many areas of mathematics, especially geometry.
    • Geometry is one of the oldest fields of mathematics; it has engaged the attention of great mathematicians through the centuries, but has undergone a revolutionary change in the last 50 years.
    • Mikhail Gromov has led some of the most important developments, producing profoundly original general ideas which have resulted in new perspectives on geometry and other areas of mathematics.
    • Gromov's name is forever attached to deep results and important concepts within Riemannian geometry, symplectic geometry, string theory and group theory.
    • [Gromov] has truly revolutionised geometry; laid the foundations of brand new fields, introduced spectacularly new viewpoints and a philosophy which makes his papers and thoughts unmistakable.

  9. Veblen biography
    • Under their direction he laid the basis for the important work he was later to achieve in the fields of foundations of geometry, projective geometry, topology, differential invariants and spinors.
    • His often quoted dissertation under Moore, on a system of axioms of Euclidean geometry, followed the trend of development of Pasch (1882) and Peano (1889, 1894) rather than that of Hilbert (1899) and Pieri (1899).
    • When Moore published On the projective axioms of geometry in the following year he acknowledged Veblen's contribution, writing:- .
    • Veblen's doctoral dissertation, supervised by Moore, was entitled A System of Axioms for Geometry and he was awarded his doctorate from the University of Chicago in 1903.
    • He presented twelve axioms for Euclidean geometry which he proved to be an complete system of axioms and he also proved the independence of the axioms.
    • During this time he effectively supervised the doctoral studies of Robert Moore, officially a student of Eliakim Moore's, who was awarded a doctorate in 1905 for a dissertation entitled Sets of Metrical Hypotheses for Geometry.
    • During this period Veblen worked on putting his own thesis into a form for publication and A system of axioms for geometry appeared as a 41 page paper in the Transactions of the American Mathematical Society in 1904.
    • Veblen's interest in the foundations of geometry led to his work on the axiom systems of projective geometry.
    • He published Finite projective geometries with W H Bussey in 1906, Collineations in a finite projective geometry (1907), and Non-Desarguesian and non-Pascalian geometries (1908).
    • Together with John Wesley Young he published Projective geometry (1910-18).
    • To this we can only reply that, in our opinion, an adequate knowledge of geometry cannot be obtained without attention to its foundations.
    • We believe, moreover, that the abstract treatment is particularly desirable in projective geometry, because it is through the latter that the other geometric disciplines are most readily coordinated.
    • from projective geometry than to derive projective geometry from one of them, it is natural to take the foundations of projective geometry as the foundations of all geometry.
    • Soon after Einstein's theory of general relativity appeared Veblen turned his attention to differential geometry.
    • His work The invariants of quadratic differential forms (1927) is a systematic treatment of Riemann geometry while his work, written jointly with his student Henry Whitehead, The foundations of differential geometry (1933) gives the first definition of a differentiable manifold.

  10. Von Staudt biography
    • This early research work was on determining the orbit of a comet but his interests at this time were also in geometry.
    • An important work on projective geometry, Geometrie der Lage was published in 1847.
    • It was the first work to completely free projective geometry from any metrical basis.
    • Another of his publications on projective geometry was Beitrage zur Geometrie der Lage (1856-60).
    • This deep thinker perceived two essential weaknesses in the synthetic geometry of his predecessors.
    • He developed geometry so as to meet these difficulties.
    • In this book Staudt tries to 'purify' the principles of projective geometry by removing all metrical notions.
    • Thereby he also raised synthetic geometry to a new level.
    • Together with Poncelet, Gergonne and Steiner, he belongs to the founders of projective and synthetic geometry.
    • von Staudt endeavoured to construct a geometry free from all metrical relations, and exclusively based upon relations of situation.
    • The author takes as his point of departure the harmonic properties of the complete quadrilateral and those of homologous triangles, proved solely by considerations of geometry of three dimensions, analogous to those of which the school of Monge has made so frequent a use.
    • Gian-Carlo Rota writes about Staudt's proof of the equivalence of synthetic and analytic projective geometry [Synthese 111 (2) (1997), 183-196.',19)">19]:- .
    • Every geometer is dimly aware of the equivalence of synthetic and analytic projective geometry; however, few geometers have ever bothered to look up the proof, let alone to remember it.
    • Theodor Reye, whose lectures on von Staudt's approach to projective geometry were first published in 1866, says in his preface that the austere language, the extreme abstractness of presentation, and the lack of diagrams have hindered the well-deserved recognition of von Staudt's work.
    • But it seems to me that it was probably Felix Klein, with his interest in the foundations of geometry and the so-called non-Euclidean geometries, who focused attention again on von Staudt.
    • Klein in 1873 claimed there was a gap in von Staudt's proof of a key result (what we now call the Fundamental Theorem of Projective Geometry), which could only be filled by an axiom of continuity.
    • It remained for later generations to appreciate the impact of von Staudt's work on the foundations of projective geometry.
    • Dirk Struik writes about a lecture Hans Freudenthal gave in Erlangen on 20 June 1967 on the occasion of the centenary of von Staudt's death (see [Geometry - von Staudt\'s point of view (Dordrecht-Boston, Mass., 1981), 401-425.',10)">10]).
    • looks, with modern eyes, at von Staudt's aim to found projective geometry independently of any metric assumptions, an approach closely approximating modern axiomatic form.
    • The conclusion is that von Staudt was the first to raise the question of foundations looking for purity of method in projective geometry, using his definition of projectivities by the invariance of harmonicity.

  11. Loria biography
    • He was awarded his doctorate in July 1883 for his dissertation on spherical geometry with the same title as his first publication.
    • He spent 1883-84 at the University of Pavia and during 1884-86 he was an assistant to D'Ovidio in algebra and analytic geometry at the University of Turin.
    • the geometry of straight lines and spheres, hyperspatial projective geometry, entities generated by algebraic correspondences between fundamental forms, and Cremona transformations in space.
    • Then, in November 1886 after a competition, he was appointed as an extraordinary professor of higher geometry in the University of Genoa, where he was to spend the rest of his academic life.
    • He was promoted to ordinary professor of higher geometry in November 1891, and continued to hold this chair until he retired on 1 August 1935 at the age of 73.
    • He was also "charge de cours" of higher analysis 1892-97, and "charge de cours" of descriptive geometry 1897-1935.
    • Although Loria continued to undertake research into geometry throughout his life, his main research activity was in the history of mathematics.
    • A full professor of higher geometry at the University of Genoa beginning in 1891, Loria wrote the history of mathematics as a mathematician writing for other mathematicians.
    • Loria's main works are devoted to the history of geometry, his area of research and teaching as a mathematician.
    • His first important historical work was an essay ['Nicola Fergola e la scuola di matematici che lo ebbe a duce' (1892)] on Nicolo Fergola (1753-1824), which marked the rediscovery of a virtually forgotten school of geometry that flourished in Naples during the early decades of the 19th century.
    • a target which the modern teaching of elementary geometry should reasonably aim at.
    • Geometry had to be taught in a rigorous but interesting way:- .
    • enemies as we are of any concession made at the sacrifice of geometric rigour, we are favourable on the other hand to any legitimate means of enlivening and keeping awake the interest of young people in geometry.
    • Geometry, he argued, was not a 'dead language', but a 'living language'.
    • Loria retired from the Chair of Higher Geometry at the University of Genoa in 1935.
    • He wrote no further works on geometry, but did continue to publish historical works up to his final contribution published in 1953 La matematica nel suo millenario sviluppo, ha seguito una direzione costante? .
    • In 1907 he received Binoux prize from the French Institute, then in 1922 he was awarded a second Binoux prize for Storia della geometria descrittiva dalle origini sino ai giorni nostri (History of Descriptive Geometry from Origins up to our Times).
    • With this concept in mind I forced myself to gain an exact knowledge of the vast scientific literature relating to descriptive geometry, without neglecting in the least the precursors of Monge.
    • And the French Institute in conferring the Binoux Prize [1922] for my 'History of Descriptive Geometry from Origins up to our Times' has accorded to my labours the most coveted and most significant reward to which I could aspire.

  12. Comessatti biography
    • He held the position of assistant in Descriptive Geometry, Analytic Geometry and Projective Geometry for twelve years.
    • they furnish a description of the geometry of this class of surfaces: number of real lines, topological form of the real part, minimal models over R.
    • In 1914 Comessatti was given the title of lecturer in Descriptive Geometry after writing twelve papers in the six years following the publication of his thesis.
    • He entered the competition for the chair of Algebraic Analysis and Analytic Geometry at the University of Cagliari and, having been successful, was appointed in 1920.
    • He continued to enter competitions for chairs and, in 1922, was declared the winner of the competition for the extraordinary chair of Projective Geometry and Descriptive Geometry at the University of Parma.
    • He was also appointed to the chair of Analytic Geometry and Projective Geometry at the University of Modena.
    • During the year 1922-23 he was called to the extraordinary professorship in Descriptive Geometry and its Applications at the University of Padua.
    • In the same year he was appointed to a full professorship in Descriptive Geometry and its Applications then, later in the same year, he was appointed to the Chair of Analytic and Projective Geometry.
    • He continued to hold this chair in Padua for the rest of his life, but during the years 1924-27 he taught analytic geometry, additional mathematics, and higher geometry at the University of Ferrara.
    • He also taught geometry courses at the University of Bologna during the years 1937-39.
    • The present volume, most of which had been used repeatedly by the author in mimeograph form previously, represents the work done in algebraic and projective geometry by the Italian university students during the first and second years.
    • This book would not be a suitable text for beginners in either analytic or projective geometry, according to the American methods of instruction, but is particularly valuable for reference and comparison after one has reached the maturity necessary to appreciate it.
    • In advanced courses intended for pure mathematicians, gave the treasures of his vast erudition, changing the topic every year, or giving it a different treatment, trying to impart knowledge to his students of the many vast fields of algebraic geometry and to inspire in them the desire for more knowledge and encouraging them to undertake their own research.

  13. Sasaki biography
    • Although in earlier years there were no mathematics texts in Japanese, by the time Sasaki attended High School there were Japanese texts on algebra, analytic geometry, trigonometry and calculus, all of which he studied.
    • These included several different geometry courses, including projective geometry, conformal geometry, non-Euclidean geometry, differential geometry, and synthetic geometry.
    • In addition Sasaki, who was by now becoming fascinated by differential geometry, read some classic differential geometry texts including ones by Blaschke, Eisenhart, Schouten, and Cartan.
    • He graduated in March 1935 and remained at Tohoku University to undertake research on differential geometry under Kubota's supervision.
    • During the early 1940s Sasaki wrote a major text Geometry of Conformal Connection in Japanese, completing the manuscript of the book in 1943.
    • Weyl opened the way to the conformal differential geometry of Riemannian spaces in which one studies the properties of the spaces invariant under the so-called conformal transformation of the Riemannian metric.
    • This book contains almost all the results mentioned above in the geometry of conformal connection.
    • Among the topics Sasaki contributed to over a long research career were Lie geometry of circles, conformal connections, projective connections, holonomy groups, Hermitian manifolds, geometry of tangent bundles and almost contact manifolds (now called Sasaki manifolds), global problems on curves and surfaces in various spaces.
    • He wrote a major text Differential geometry : Theory of surfaces which, S Funabashi, writes:- .
    • is a guide to differential geometry, illustrating the topics with the theory of surfaces.
    • The author's aim is to describe the method of study of global differential geometry, especially of the theory of two-dimensional surfaces immersed isometrically in a three-dimensional Euclidean space R3.
    • Most of the features for surfaces appearing in this book are closely related to topological geometry.

  14. Kerekjarto biography
    • This work was the first of its kind and inspired much later research in this new branch of geometry.
    • After Kerekjarto's visit to Gottingen in 1922, in the following year he gave courses at the University of Barcelona entitled Geometry and The theory of functions.
    • In 1925, Kerekjarto was appointed to full professorship of the Chair of Geometry and Descriptive Geometry at the University of Szeged.
    • The Department of Mathematics at that time consisted of the Mathematical Seminary and the Institute of Descriptive Geometry.
    • Also in his later work he preferred to work in topological problems which were closely connected with problems of classical geometry, theory of functions, etc.
    • It is enough to mention the topological characterisation of the homographic representations of the sphere and of the affine group of the plane, the foundations of complex projective geometry and theorems on the transitive groups of the line.
    • It was these methods which also led to fundamental results on topology and Euclidean and hyperbolic geometry in 3 dimensions.
    • His expertise in this new branch of geometry, topology, was recognised in, among other ways, his being asked to write the chapter on 'Topology' in the Encyclopedie Francaise.
    • From the very modest introduction one would never guess that this is one of the richest works on the foundations of geometry.
    • The second volume The Foundations of Geometry.
    • Projective Geometry was published in Hungarian in 1944.
    • This is the second volume of a treatise on the foundations of geometry; the preceding volume dealt with Euclidean geometry.
    • The classical projective geometry is developed in great detail so that this book can also be used as a text book.
    • The author's aim is to give a foundation of projective geometry on which it is possible to build either Euclidean, hyperbolic or elliptic geometry.
    • The greater part of the book is confined to the discussion of real projective geometry.
    • An analytical discussion of complex projective geometry is given separately.

  15. Semple biography
    • Introduction to algebraic geometry was published in 1949.
    • Roth and Semple also worked together setting up and running the London Geometry Seminar which operated for 40 years and provided one of the major focal points for geometry research throughout the world.
    • Semple also worked with Du Val who joined the London Geometry Seminar but they only wrote one joint paper.
    • Semple's work was on various aspects of geometry, in particular work on Cremona transformations and work extending results of Severi.
    • He wrote two famous texts Algebraic projective geometry (1952) and Algebraic curves (1959) jointly with G T Kneebone.
    • In the Preface of the first edition of Algebraic projective geometry the authors explain their approach to geometry:- .
    • Projective geometry is a subject that lends itself naturally to algebraic treatment, and we have had no hesitation in developing it in this way - both because to do so affords a simple means of giving mathematical precision to intuitive geometrical concepts and arguments, and also because the extent to which algebra is now used in almost all branches of mathematics makes it reasonable to assume that the reader already possesses a working knowledge of its methods.
    • In spite, however, of treating geometry algebraically, we have tried never to lose sight of the synthetic approach perfected by such geometers as von Staudt, Steiner, and Reye.
    • Around this time he seemed to become somewhat disillusioned with the direction that research in algebraic geometry was going and 1957 saw the publication of the last research paper that Semple would write for over ten years.
    • If Semple rather regretted the direction that research in algebraic geometry was taking, he certainly did not show it in his book Algebraic curves.
    • In the Preface to the book the authors argue convincingly for the importance of algebraic geometry:- .
    • the authors observe that the subject of algebraic geometry is one to which mathematicians have been powerfully drawn over a very long period of time.
    • One of the reasons of this enduring influence of algebraic geometry is the fact that the subject makes a strong appeal to the imagination in that it not only illuminates properties of geometrical figures that we are all able to draw or visualize but also extends the range of geometrical thinking far beyond the bounds of intuition, and one can think about it in a much deeper sense than that of granting formal assent to its conclusions.
    • It provides impressive evidence of the power of strictly classical projective geometry when applied to the right sort of problem.
    • In his own subject, he had in a high degree the gift of organising research, both for groups and for individuals; as a lecturer, he was much in demand and gave inspiration to many hundreds of students and other listeners; and, in his writing, he was able to pass on his love of geometry with infectious enthusiasm that few other authors have managed to achieve.
    • Semple and Kneebone's geometry .

  16. Taurinus biography
    • Of course, although he did not intend it to be so, he was then studying non-euclidean geometry.
    • Lambert noticed that, in this new geometry where the sum of the angles of a triangle was less than 180q, the angle sum of a triangle increased as the area of the triangle decreased.
    • Schweikart himself is famed for investigating this new geometry which he called astral geometry.
    • Taurinus not only corresponded on mathematical topics with his uncle but he also corresponded with Gauss about his ideas on geometry.
    • At first Taurinus tried to prove that Euclidean geometry was the only geometry but, in 1826, he accepted the lack of contradiction in other geometries.
    • In this last mentioned publication Taurinus accepts that a third system of geometry exists in which the sum of the angles of a triangle is less than 180q.
    • He called this geometry "logarithmic-spherical geometry" and he recognised the lack of a contradiction in this geometry as meaning that it was internally consistent.
    • Taurinus came up with the important idea that elliptic geometry could be realised on the surface of a sphere, an idea taken up by Riemann.
    • It showed that euclidean geometry held a unique dominating role.
    • This is an interesting sideways move since his original aim had been to prove that euclidean geometry was the unique geometry.
    • Finding that this was not so, he still wanted to demonstrate that euclidean geometry was "the" geometry.
    • A letter from Gauss to Taurinus discussing the possibility of non-Euclidean geometry.

  17. Klug biography
    • He taught there between 1874 and 1893, writing his first books on geometry, after which he taught at a secondary school in Budapest.
    • He also taught as a privatdocent in Synthetic Geometry at the University of Budapest from 1891.
    • After the award of his habilitation, Klug was appointed to the Joseph Franz University of Kolozsvar (now Cluj) as an extraordinary professor in Descriptive Geometry.
    • After two years he was appointed to the chair of Descriptive Geometry at the University of Kolozsvar, a position he held for nearly twenty years, until 1917, when he retired and moved back to Budapest.
    • Now I'm not allowed to tire them with geometry work.
    • Klug had a favourite subject, and that was projective geometry.
    • Projective plane geometry.
    • From the foundation of Leopold Klug - according to our information - the first aim will be to reward those students who have undertaken excellent work in the seminar in the field of Descriptive Geometry.
    • The second aim will be to reward those who show extraordinary interest and special talent in Geometry.
    • Dr Leopold Klug, the new patron of a university foundation, was born in Gyongyos in 1854, as a child of a Jewish family; in 1891 he became a privatdocent in synthetic geometry at the University of Budapest, and in 1897 he became an extraordinary professor at the University of Kolozsvar (Cluj).
    • He published his work entitled "The Elements of the Projective Geometry" with the support of the Hungarian Academy of Sciences.
    • While in the past two centuries geometers chose their themes from the field of Projective Geometry, particularly from the domain of second degree curves and surfaces, interest in this topic has significantly decreased nowadays.
    • Leopold Klug was an enthusiast for the flourishing of Projective Geometry, and within that, the synthetic method which inspired many great minds in the past.
    • As big as the love was with which he promoted projective geometry based on the synthetical method, equally big was the bitterness after he realized the decrease of interest in it.
    • The highlight of his work is represented by two textbooks: 'The Elements of Projective Geometry' and 'Projective Geometry' (1903).
    • In the third place there is his textbook entitled 'Descriptive Geometry', a work written with an excellent pedagogical sense and with a great choice of material, then the fourth one: 'A Synthetic Discussion of Third-Order Space Curves' (1881).
    • Besides these textbooks a large number of his articles enrich our geometry literature.

  18. Vagner biography
    • Go and study differential geometry under Professor Kagan.
    • The very spirit of modern geometry is close to that of relativity.
    • In 1922 Kagan had moved from Odessa to Moscow when the Department of Differential Geometry was founded at Moscow State University.
    • Kagan was the first Head of Department and he founded an important School of Differential Geometry there.
    • Vagner became his student in 1932 and wrote a thesis on the differential geometry of non-holonomic manifolds for his Candidate's Degree (equivalent to a Ph.D.).
    • Vagner was appointed to the Chair of Geometry at Saratov University after the award of the degree of Doctor of Science and he continued to work there until he retired in 1978.
    • Vagner started his research activity at the time when differential geometry was rapidly developing and providing a part of mathematical apparatus for general relativity.
    • All Vagner's research is connected with differential geometry and algebraization of its foundations.
    • Among Vagner's early papers we mention Differential geometry of non-linear non-holonomic manifolds in the three-dimensional Euclidean space (1940), The geometry of an (n-1)-dimensional non-holonomic manifold in an n-dimensional space (Russian) (1941), Geometric interpretation of the motion of non-holonomic dynamical systems (Russian) (1941), On the problem of determining the invariant characteristics of Liouville surfaces (Russian) (1941), and On the Cartan group of holonomicity for surfaces (1942).
    • He published a major 70 page paper General affine and central projective geometry of a hypersurface in a central affine space and its application to the geometrical theory of Caratheodory's transformations in the calculus of variations (Russian) in 1952.
    • Vagner published the book Geometria del calcolo delle variazioni in Italian in 1965 in which he gave a systematic treatment of his own approach to the geometry of the calculus of variations, which he developed during the years 1942-1952.
    • The quote by Schein above indicates how geometry led Vagner to study algebraic systems.
    • Let us quote Vagner's own words from the paper The foundations of differential geometry and modern algebra (Russian) (1963):- .
    • For contemporary differential geometry the concept of group is quite insufficient for the examination, from an algebraic point of view, of the basic concepts of the corresponding geometrical theories.
    • Moreover, algebraic problems arising in investigations concerning the foundations of contemporary differential geometry require the study of special algebraic systems which at present are not very seriously discussed.

  19. Enriques biography
    • Two years later, when he was thirteen, Federigo met geometry for the first time and became very enthusiastic about the subject.
    • Today, just imagine, he started a geometry course.
    • Of course, she was wrong about Enriques's new found love for geometry - it lasted his lifetime.
    • His thesis advisor had been Riccardo De Paolis (1854-1892), who had been appointed professor of higher geometry at the University of Pisa in 1881.
    • Giuseppe Bruno died early in 1893 and later that year a competition was announced to fill his chair of projective and descriptive geometry at the University of Turin.
    • Berzolari was appointed to the chair with Enriques ranked in fourth equal position behind Berzolari, Pieri and Del Re [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3]:- .
    • An extraordinary professorship (called professor straordinario) in descriptive and projective geometry became vacant at the University of Bologna when Domenico Montesano left Bologna to take up a chair at Naples in 1893.
    • Given the ranking in the previous competition, it was expected that Pieri or Del Re would be appointed and there was talk of appointing Pieri without a competition [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3]:- .
    • The competition for the permanent Bologna post did not take place until October 1896; Enriques was appointed to the chair of descriptive and projective geometry with Pieri coming a close second.
    • The long saga surrounding this appointment is described in detail in [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',3)">3].
    • Enriques made important contributions to geometry and to the history and philosophy of mathematics.
    • Another topic which Enriques worked on was differential geometry.
    • The foundations of mathematics had always interested Enriques, and, at Klein's request, he wrote an article on the foundations of geometry.
    • Enriques himself recalled how his interest in mathematics was due to a "philosophical infection caught at school." His interest in science problems (and geometry in particular) was never simply of a technical nature but was always motivated by questions of "general culture" and lively reflection on the role of scientific thought in human activity.
    • The mathematical community has evolved sophisticated ways of reading Enriques's work in algebraic geometry, and we see most of it as either correct or easy to put right.
    • In the following year he accepted the chair of higher geometry at the University of Rome where he founded the National Institute for the History of Science and the School of the History of Science.

  20. Bukreev biography
    • By the end of the 1890s Bukreev's research interests had moved somewhat and he began to undertake research into differential geometry; in 1900 he published A Course on Applications of Differential and Integral Calculus to Geometry.
    • Bukreev's work was broad and in addition to the areas of complex functions, differential equations, the theory and application of Fuchsian functions of rank zero, and geometry, he published papers on algebra such as On the composition of groups (1900).
    • He taught courses on analysis, differential and integral calculus and their applications to geometry, the theory of integration of differential equations, the theory of series, algebra, and other topics.
    • In 1933 the University of Kiev was restored and at this time a Department of Geometry was created in the Mechanics and Mathematics Faculty; the first head of this Department was Bukreev.
    • All the rays coming from a number of subjects: geometry, mechanics, physics, engineering ..
    • He was interested in both projective and non-Euclidean geometry, publishing about fifteen articles on this latter topic including Equidistant lines of constant geodesic curvature in the planimetry of Lobachevsky (1955) and Lobachevskian geometry (1957).
    • His most important book on non-Euclidean geometry was Non-Euclidean Planimetry in Analytic Terms which he published in 1951.
    • Realisation as geometry on the pseudosphere.
    • The book develops hyperbolic geometry from the point of view of differential geometry.
    • For 60 years, Boris Yakovlevic was the Head of Department at the University of Kiev, first in the chair of general mathematics, and then in the chair geometry.
    • Boris Yakovlevic worked on the theory of functions of a complex variable, on mathematical analysis, on algebra, on the calculus of variations, and on differential geometry.
    • Recently, he has focused on the geometry of Lobachevsky, promoting and developing the eternal ideas of the great Russian geometer.

  21. Forder biography
    • I bought his book [Foundations of Euclidean Geometry (1927)] ..
    • These are: The Foundations of Euclidean Geometry (1927), A School Geometry (1930), Higher Course Geometry (1931), The Calculus of Extension (1941), Geometry (1950), and Coordinates in Geometry (1953).
    • Although the Euclidean geometry is the oldest of the sciences and has been studied critically for over two thousand years, it seems there is no textbook which gives a connected and rigorous account of that doctrine in the light of modern investigations.
    • Grassmann's methods are of equal use in geometry, but this application is less widely appreciated.
    • Forder's 'The calculus of extension' not only presents a wealth of specific applications of the subject to geometry, that are either new or not readily obtainable elsewhere; but in addition furnishes an admirable and fresh exposition of the 'Ausdehnungslehre'.
    • It is a book that should be in the library of anyone who is interested in either algebra, the algebraic treatment of geometry, or vector and tensor analysis.
    • Geometry (1950) was reviewed by Donald Coxeter who was clearly fascinated by Forder's use of language:- .
    • The chapter on logical structure stresses the abstract nature of the order relation (ABC) by comparing it with the human relation "A prefers C to B." The possibility of coordinatising any descriptive geometry of three or more dimensions is epitomized in the statement that "we can create magnitudes from a mere muchness," and Archimedes' axiom in the statement that "you will always reach home, if you walk long enough." .
    • Coordinates in Geometry (1953) was reviewed by Marshall Hall, Jr.
    • The second chapter, without axioms of order deals with congruence of angles (crosses) and line segments (point pairs) in a Pappian geometry.
    • He loved mathematics with a burning intensity and most of all he loved geometry.

  22. Sperner biography
    • In winter 1929-30 Sperner gave his first lecture course 'Analytic geometry and algebra II' at the University of Hamburg.
    • Schreier intended eventually to publish his Hamburg lectures on analytical geometry and algebra in book form; and after his premature death in 1929 these were completed and edited by his pupil Sperner.
    • Three-quarters of the present first volume discuss the analytical geometry of n dimensions and the relevant algebra in a unified way; no special knowledge is assumed, but considerable demands are often made on the attention of the reader.
    • This final volume contains an introduction to the theory of (finite) matrices, which has already been published separately and is now out of print, together with a treatment of n-dimensional analytic geometry from the projective, affine and metric points of view, as far as the classification of quadrics.
    • One of its unique features is the simultaneous treatment of fundamental concepts in algebra and in affine and projective geometry.
    • The present skilful English translation has incorporated the "Vorlesungen uber Matrizen" (as in the German second edition), but has omitted the long final chapter on projective geometry in n dimensions.
    • The result is that the accent is much more on the algebraic developments than in the original and that the geometry serves more as illustration than as motivation.
    • This is also expressed in the changed title from which the reference to geometry has disappeared.
    • In 1961 the omitted material on projective geometry was translated into English and published as Projective Geometry of n Dimensions.
    • The reviewer feels that it is a great pity, as regards the teaching of projective geometry, that the present book has taken so long to appear.
    • Indeed it arrives at a period when the projective geometry in undergraduate courses is being whittled down in favour of vector space theory.
    • If it had appeared sooner this book would have done much to make projective geometry more attractive.
    • While at Bonn, he developed a type of generalised affine space, influenced by Hilbert's 'Foundations of geometry', which was taken up by other mathematicians, particularly those in Italy.
    • His group-theoretical proof of Desargues' Theorem remains a lasting, and much admired, result in absolute axiomatics; his theories of ordering functions and of weakly affine spaces are secure pieces of the discipline of the foundations of geometry.

  23. Aleksandrov Aleksandr biography
    • Delone's interests in the geometry of numbers and the structure of crystals soon began to attract Aleksandrov at least as much as his work in physics which was supervised by V A Fok.
    • His fourth work in 1934 was on geometry in the area of his first three papers.
    • Two years later he presented a thesis on geometry for a Master's degree which was on the topic of mixed volumes of convex bodies and he published six papers over the next couple of years on the results of this thesis.
    • This work generalises classical problems in differential geometry.
    • Aleksandrov was appointed as Professor of Geometry at Leningrad University in 1937.
    • In 1944 Aleksandrov returned to the University of Leningrad where he was Professor of Geometry.
    • In 1964 Aleksandrov left Leningrad and moved to Novosibirsk where he was appointed as Head of the Department of Geometry of the University of Novosibirsk.
    • He also became Head of the Department of Geometry of the Mathematical Institute of the Siberian Branch of the USSR Academy of Sciences.
    • Surveys 17 (6) (1962), 127-141.',10)">10] Aleksandrov's work in geometry is put into perspective:- .
    • [Aleksandrov] approached the differential geometry of surfaces [by extending the notion of the objects studied], extending the class of regular convex surfaces to the class of all convex surfaces ..
    • In order to solve concrete problems Aleksandrov had to replace the Gaussian geometry of regular surfaces by a much more general theory.
    • Aleksandrov constructed a theory of intrinsic geometry of convex surfaces on that basis.
    • Aleksandrov received many awards for his major contributions to geometry.
    • In 1942 he received the State Prize for his work in geometry, then in 1946 he was elected a Corresponding Member of the USSR Academy of Sciences.

  24. Poncelet biography
    • During his imprisonment he recalled the fundamental principles of geometry but, forgetting the details of what he had learnt from Monge, Carnot and Brianchon, he went on to develop projective properties of conics.
    • He called the notes that he made the 'Saratov notebook,' but it was only fifty years later that he incorporated much of what he had written in his treatise on analytic geometry Applications d'analyse et de geometrie (1862).
    • In his second chapter Poncelet attacks the problem of imaginary points in pure geometry with a courage and thoroughness ahead of anything shown by his predecessors.
    • Here we have the first announcement of one of the basic principles of metrical geometry.
    • Note that the work was subtitled "A work of utility for those studying the applications of descriptive geometry and geometric operations on land" and here one can see the influence of Monge's teaching.
    • This work contains fundamental ideas of projective geometry such as the cross-ratio, perspective, involution and the circular points at infinity.
    • He illustrated this technique by first noting the theorem from Euclidean geometry which states that the product of segments of intersecting chords in a circle is constant.
    • It is worth remarking that our term "projective geometry" comes from the title of this book, which is quite appropriate since Poncelet was one of the founders of modern projective geometry simultaneously discovered by Joseph Gergonne.
    • Let us look briefly at Andrei Nikolaevich Kolmogorov's description [Mathematics of the 19th Century: Geometry, Analytic Function Theory (Birkhauser, 1996).',4)">4]:- .
    • It pushed Poncelet away from his work on projective geometry and towards mechanics.
    • His work on projective geometry was too controversial, particularly following the attacks made on it earlier by Cauchy, for him to enter the Academy on the strength of these contributions.
    • Poncelet published many articles on geometry and mechanics in addition to those we have mentioned, particularly in Gergonne's Annales des Mathematique and Crelle's Journal.
    • (2) Fundamental properties of straight lines, circles, and conic sections (consisting of Geometry of ruler and transversals; Figures inscribed in and circumscribed around conic sections.

  25. Battaglini biography
    • The Scuola di Ponti e Strade (School of Bridges and Roads) was the only other public institution close by in which young people were able to study descriptive geometry, rational mechanics and then applied mathematics.
    • A school of classical geometry had been set up in Naples by Fergola and his pupil Flauti and it was so influential that it was able to prevent modern young geometers from obtaining posts.
    • Battaglini was named professor of higher geometry at the University of Naples in 1860.
    • Battaglini edited the journal, aimed at university students, which became the main outlet for papers in non-Euclidean geometry in Italy.
    • Many articles by Battaglini appear in the journal from 1863 onwards, but his first memoir on non-Euclidean geometry Sulla geometria immaginaria di Lobachevsky was published in 1867.
    • the part G Battaglini, often mentioned only for the foundation of his 'Giornale di Matematiche', has had in the elaboration and in the divulgation of non-Euclidean geometry.
    • By means of the study of his articles on hyperbolic geometry and of his unpublished specific correspondence with A Genocchi, we conclude by saying that the Neapolitan mathematician was interested not only in the technical development of non-Euclidean geometry but even in its foundational aspects and its philosophical implications.
    • The Neapolitan Hegelism, which by its emphasis on the notion of 'a priori' was inevitably opposite to the anti-metaphysical foundation of Lobachevsky's geometry, was a resistance to the affirmation of the new geometry in the academic Parthenopean culture.
    • On the contrary, the positivistic theory of knowledge was a theoretical reference nearer the principles of Lobachevskian geometry.
    • However, his main importance is his modern approach to mathematics which played a major role in invigorating the Italian university system, particularly in his efforts to bring the non-Euclidean geometry of Lobachevsky and Bolyai to the Italian speaking world.
    • Jules Houel played a similar role for non-Euclidean geometry in the French speaking world and the correspondence between the two (see [Riv.
    • In particular they debated the use of Euclid's Elements as a textbook for teaching elementary geometry in schools.

  26. Severi biography
    • He became fascinated by geometry and, under Corrado Segre's supervision, he went on to obtain his doctorate in 1900.
    • His doctoral thesis, Sopra alcune singolarita delle curve di un iperspazio, together with a series of other papers which he published had published while an undergraduate, deal with enumerative geometry, a subject which had been started by Hermann Schubert.
    • In 1904 Severi was appointed to the Chair of Projective and Descriptive Geometry at Parma.
    • The interview which led to this appointment must be unique in Italian university annals, for the selection committee insisted on Severi's giving a trial lesson in descriptive geometry at the blackboard (and this at a time when the candidate had publications which placed him in the front rank).
    • It was the good fortune of the Italian school of algebraic geometry to have this disinterested collaboration between 1890 and 1910.
    • There he began teaching a variety of courses from calculus to higher geometry.
    • His most important contributions are to algebraic geometry and we have seen in the quote above Castelnuovo's description of Severi's contributions in 1904-08.
    • Severi maintains a balance between geometry and analysis - he has actually made outstanding contributions to function theory.
    • After work on enumerative geometry, Severi turned to birational geometry of surfaces, a topic which Castelnuovo and Enriques has spent ten years developing before Severi began to work on it.
    • He introduced many concepts into geometry, for example the notion of algebraic equivalence.
    • You have shed a great light on geometry.
    • the first four of a planned six, contain 142 mathematical papers of Severi (1879-1961), principally concerning algebraic geometry (surfaces) and function theory of two (or more) complex variables.
    • it was as a teacher of geometry that Severi excelled.

  27. Eisenbud biography
    • V I Arnold once referred to [the] celebrated formula of Eisenbud-Levine, which links calculus, algebra and geometry, as a "paradigm" more than a theorem that provides a local manifestation of interesting global invariants and that "would please Poincare and Hilbert (also Euler, Cauchy and Kronecker, to name just those classical mathematicians, whose works went in the same direction)." Given this early work, it was natural for David's attention to turn to the study of singularities and their topology.
    • David next became interested in algebraic geometry, beginning a long and important collaboration with Joe Harris.
    • The language of modern algebraic geometry.
    • This book is intended to introduce basic notions of modern algebraic geometry.
    • These topics are important in the developing field of noncommutative geometry, where scheme-theoretic thinking is involved, and this book provides a good exposition for students.
    • His most significant book, however, was Commutative algebra: With a view toward algebraic geometry which was published in 1995.
    • Another was to provide algebraic geometers, commutative algebraists, computational geometers, and other users of commutative algebra with a book where they could find results needed in their fields, especially those pertaining to algebraic geometry.
    • But even more, Eisenbud felt that there was a great need for a book which did not present pure commutative algebra leaving the underlying geometry behind.
    • In his introduction he writes, "It has seemed to me for a long time that commutative algebra is best practiced with knowledge of the geometric ideas that played a great role in its formation: in short, with a view toward algebraic geometry." .
    • The geometry of schemes (2000), again written with Joe Harris:- .
    • is a wonderful introduction to the way of looking at algebraic geometry introduced by Alexandre Grothendieck and his school.
    • Finally we mention Eisenbud's 2005 book The geometry of syzygies.
    • A second course in commutative algebra and algebraic geometry and look forward to further fascinating books.

  28. Efimov biography
    • Benjamin Fedorovich Kagan held the chair of geometry at Moscow State University at this time and he was running a research seminar on vector and tensor analysis [Russian Math.
    • From then on, for almost half a century, Efimov devoted all his energies to geometry, mainly to the qualitative theory of surfaces, and he became one of the founders of the modern aspect of that branch of geometry.
    • However, he continued to work on his geometry research with the aim of submitting it for a doctorate (equivalent to a D.Sc.
    • The year 1934 when Efimov graduated, was highly significant for geometry research in Russia, for late in that year Stefan Cohn-Vossen emigrated there to escape from the Nazis in Germany.
    • Under his influence a school of "geometry in the large" was set up in Moscow and in Leningrad and Efimov's meeting with Cohn-Vossen, who had worked closely with Hilbert, was a major influence on setting the direction of his future research.
    • Many of the references listed below give detailed descriptions of Efimov's important contributions to geometry, but perhaps his achievements are best summed up by quoting from P S Aleksandrov [Russian Math.
    • Efimov's book with the largest number of editions is Higher Geometry (Russian) (1945).
    • This university textbook treats mainly the foundations of geometry.
    • The German edition Flachenverbiegung im Grossen (1957) collects together Efimov's early work on "geometry in the large" as G Y Rainich explains in a review:- .
    • Another teaching book, but one containing much not standard material for a work at undergraduate level, was Linear algebra and multidimensional geometry (Russian) (1970).
    • It was translated into English and published with the title Linear algebra and multidimensional geometry (1975).
    • Although this may be thought of as a book on linear algebra, the treatment is so geometrical that the subject matter merges naturally into higher-dimensional analytic geometry.

  29. Bachmann Friedrich biography
    • After further work on logic, including starting to prepare the correspondence between Gottlob Frege and Bertrand Russell for publication, Bachmann turned to questions on geometry.
    • He published Ein lineares Vollstandigkeitsaxiom in 1943 in which he investigated Hilbert's system of axioms for Euclidean geometry.
    • His scientific work at Kiel was mainly devoted to the axiomatic foundation of geometry.
    • In the short note Zur Begrundung der Geometrie aus dem Spiegelungsbegriff in Mathematische Annalen in 1951 he presented his well-known reduced version of A Schmidt's 'Axiomensystem der metrischen (absoluten) Geometrie' which is a system of axioms for absolute geometry based on line reflection only.
    • In this book Bachmann develops plane metric geometry by systematic use of reflections and the group of motions generated by them.
    • This remarkable book is essentially an elaboration of an idea of G Thomsen (The treatment of elementary geometry by a group-calculus, Math.
    • One soon begins to realize that such a geometry is not necessarily Euclidean.
    • It is more like the "absolute" geometry of Bolyai, in which a pencil of lines having a common perpendicular is not necessarily the same as a pencil of parallels.
    • In fact, the geometry may be regarded as a special kind of abstract group whose generators, called "points" and "lines," are involutory, with the distinction that , although the product of two lines may be a point, the product of two points is never a line.
    • Even this restriction is later waived so as to cover the case of a generalized elliptic geometry which admits an absolute polarity.
    • It is proved that the groups of plane absolute geometry are Hjelmslev groups.
    • This book is a must for all research workers in metric geometry and for all mathematicians interested in geometry and groups.

  30. Conforto biography
    • Chisini was able to give Conforto advice based on his own experiences for Chisini had studied engineering before finding that algebraic geometry was the right area for him.
    • He attended the Congress of the Italian Mathematical Union held in Bologna in April 1937 and gave the talk Sulle rigate razionali del quinto ordine in Section II (Geometry) which was published in the proceedings of the Congress.
    • As well as a series of interesting papers on algebraic geometry, he also became interested in the history of the topic and published Il contributo italiano al progresso della geometria algebrica negli ultimi cento anni in 1939.
    • In addition to his duties as an assistant in algebraic geometry, Conforto also worked at Mauro Picone's National Institute for the Applications of Calculus.
    • While he was in Foligno he received the news in November 1939 that he had been ranked first by the appointing commission set up to fill the chair of analytical geometry and descriptive geometry in Rome.
    • The range of Conforto's mathematical publications is great with contributions to algebraic geometry, projective geometry, and analytic geometry.
    • recalls briefly the well-known history of projective geometry, from Poncelet to von Staudt.
    • He stresses the role of perspective as developed in art (especially by the Italians) in the birth of Desargues' ideas in the 17th century, and the analogous influence of the drawing techniques promoted by Monge ("descriptive geometry") on his students and especially on Poncelet.
    • A special section is devoted to the Italian treatises on projective geometry, particularly those of Enriques and Severi.
    • In a closing section the author rightly insists on the influence of projective geometry on the concepts of modern mathematics, in introducing such general notions as transformation, correspondence, invariant, and duality, and in giving one of the first examples of a "hypothetico-deductive system", where fundamental notions are created, as it were, by the axioms of the theory.

  31. Bolyai biography
    • In fact he gave up this approach within a year for still in 1820, as his notebooks now show, he began to develop the basic ideas of hyperbolic geometry.
    • By 1824, however, there is evidence to suggest that he had developed most of what would appear in his treatise as a complete system of non-Euclidean geometry.
    • Bolyai gave him a draft of the materials which he was writing on the theory of geometry, probably because he hoped for some constructive comments from him.
    • denote by Σ the system of geometry based on the hypothesis that Euclid's Fifth Postulate is true, and by S the system based on the opposite hypothesis.
    • Most of the Appendix deals with absolute geometry.
    • The clearest reference in Gauss's letters to his work on non-euclidean geometry, which shows the depth of his understanding, occurs in a letter he wrote to Taurinus on 8 November 1824 when he wrote:- .
    • The assumption that the sum of the three angles of a triangle is less than 180° leads to a curious geometry, quite different from ours [i.e.
    • Euclidean geometry] but thoroughly consistent, which I have developed to my entire satisfaction, so that I can solve every problem in it excepting the determination of a constant, which cannot be fixed a priori.
    • What he did write concerned geometry and there are several ideas in this unpublished work which were ahead of their time such as notions of topological invariance.
    • In addition to his work on geometry, Bolyai developed a rigorous geometric concept of complex numbers as ordered pairs of real numbers.
    • Janos's paper was called Responsio and it was written to answer the question of whether the imaginary quantities used in geometry could be constructed.
    • He argued that it was not their construction that was important, rather it was their definition and role in geometry which were significant.
    • History Topics: Non-Euclidean geometry .

  32. Wu Wen-Tsun biography
    • It was under such influence that I investigated the possibility of proving geometry theorems in a mechanical way.
    • In 1977 Wu introduced a new way of studying geometry on a computer.
    • (It is worth noting that Ritt's approach was based on earlier ideas of van der Waerden.) With his new ideas Wu could take a problem in elementary geometry and transform it into an algebraic question about polynomials.
    • Desarguesian geometry and the Desarguesian number system.
    • Orthogonal geometry, metric geometry and ordinary geometry.
    • Mechanization of theorem proving in geometry and Hilbert's mechanization theorem.
    • The mechanization theorem of (ordinary) unordered geometry.
    • In 2000 Wu published Mathematics mechanization : Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equations-solving.
    • The researches in the first stage, started in 1947, are in pure mathematics, mainly in algebraic topology, occasionally also in algebraic geometry.
    • This resulted in a method of proving geometry theorems by means of computers.

  33. Blaschke biography
    • Josef was professor of descriptive geometry at the Landes Oberrealschule in Graz.
    • He favoured Steiner's approach to mathematics which was very much based on geometry and the belief that geometry alone stimulates thinking.
    • In Leipzig he became a close friend of Gustav Herglotz who was interested in partial differential equations, function theory and differential geometry, and succeeded Runge in Gottingen 10 years later.
    • Whereas earlier volumes of mine on differential geometry appeared in murky times, this book was completed as a dream of my youth was fulfilled, the union of my more narrowly seen homeland, Austria, with my larger homeland, Germany.
    • Blaschke's research was on various aspects of geometry.
    • One of the leading geometers of his time, Blaschke centered most of his research on differential and integral geometry and kinematics.
    • He wrote an important book on differential geometry Vorlesungen uber Differentialgeometrie (1921-1929) which was a major 3 volume work.
    • The first volume considered classical geometry, while the second volume was on affine differential geometry.
    • The third volume of Vorlesungen uber Differentialgeometrie considered geometry which originated from the action of various transformation groups, such as those of Mobius, Laguerre and Lie.
    • He also initiated the study of topological differential geometry, the study of invariant differentiable mappings.

  34. Aubin biography
    • As one of the major contributors to the resolution of nonlinear problems in geometry, Thierry Aubin has focused this monograph on a few significant questions on which he has been involved personally, rather than writing a broad treatise.
    • This book deals with certain nonlinear partial differential equations which arise from problems in global differential geometry.
    • Aubin updated and expanded this book, publishing the revised text as Some nonlinear problems in Riemannian geometry (1998).
    • In 2001, Aubin published A course in differential geometry.
    • This book provides an introduction to differential geometry, with principal emphasis on Riemannian geometry.
    • The aim of this book is to facilitate the teaching of differential geometry.
    • It covers topics every working mathematician (or theoretical physicist) ought to know: tensorial, differential and integral calculus on smooth manifolds, and basic Riemannian geometry.
    • To conclude, I believe this is an excellent textbook for a first course on basic differential geometry, very helpful to both the instructors and their students.
    • Thierry Aubin was a very important mathematician whose work had great influence on the fields of Differential Geometry and Partial Differential Equations.
    • In the past fifty years there has been a surge of work on problems that involve the interplay of differential geometry and analysis, particularly partial differential equations.
    • He had important insight on how to reduce geometry problems to proving fundamental inequalities.

  35. Barsotti biography
    • I first met Barsotti in Rome, when he was an assistant in geometry.
    • In those years the methods of modern algebra in algebraic geometry were almost unknown in Italy: in Rome it was only Barsotti who had already securely mastered them.
    • For a number of years intersection theory represented one of the most debated subjects in the field of algebraic geometry; also one of the main reasons for seeing in the whole structure of algebraic geometry an inherent flimsiness which even discouraged the study of this branch of mathematics.
    • This situation came to an end when the methods of algebra began to be successfully applied to geometry, mainly by van der Waerden and Zariski; in the specific case of intersection theory, a completely general and rigorous treatment of the subject was given by Chevalley in 1945.
    • This rebuilding of algebraic geometry on firm foundations has often taken a form quite different from what the classical works would have led one to expect.
    • The method by Andre Weil [in 'Foundations of algebraic geometry' (1946)] is another example of local theory.
    • In 1960 Barsotti moved to Brown University in Providence, Rhode Island but, while there, entered the competition for the chair of geometry and algebra at the University of Pisa.
    • Barsotti was not, however, to spend the rest of his career at Pisa for, in 1968, he moved to Padua to take up the chair of geometry at the university there.
    • He published a number of articles on the theta function after moving to Padua, for example Considerazioni sulle funzioni theta (1970) which aims to bring the classical theory of theta functions of several variables within the scope of abstract algebraic geometry.
    • Following his death in 1987, a Symposium in Algebraic Geometry was held in his honour from 24 to 27 June 1991 in Abano Terme.
    • The proceedings of the conference was published by Academic Press in 1994, see [Barsotti Symposium in Algebraic Geometry, Abano Terme, 1991, Perspect.

  36. Hsiung biography
    • Guy Grove not only reviewed this paper but also Hsiung's papers Sopra il contatto di due curve piane (1940), The canonical lines (1941), A graphical construction of the sphere osculating a space curve (1941), On the curvature form and the projective curvatures of a space curve (1942), Asymptotic ruled surfaces (1943), Projective differential geometry of a pair of plane curves (1943), Theory of intersection of two plane curves (1943), An invariant of intersection of two surfaces (1943), and Projective invariants of a pair of surfaces (1943).
    • We have already indicated the range of Hsiung's early work on projective geometry.
    • After he had spent time in Harvard as Whitney's research assistant, he began studying more global geometry problems.
    • A first course in differential geometry published in 1981 was reviewed by O Kowalski who writes that it:- .
    • is designed as a course of classical differential geometry for beginning graduate students or advanced undergraduate students.
    • In fact, the title of the book might well be A first course in differential and integral geometry.
    • Two further texts, written after he retired, are Almost complex and complex structures (1995) and A first course in differential geometry (1997).
    • The book is designed to introduce differential geometry via the study of curves and surfaces in E3.
    • The book provides a solid introduction to differential geometry of curves and surfaces.
    • This is his founding of the Journal of Differential Geometry in 1967.
    • Under the influence of this journal, differential geometry has become a very active branch of mathematics, with scope far exceeding its former classical one.
    • It is not necessary to heap accolades upon this contribution - the journal has long been recognized as a forum of publication of the finest papers in differential geometry.

  37. Sturm Rudolf biography
    • There he was taught by Schroter who encouraged him to study synthetic geometry.
    • In 1872 Sturm was appointed assistant professor at the Technical College in Darmstadt where he taught descriptive geometry and graphic statics.
    • In order to provide a good teaching book for his students, Sturm published a textbook Elemente der darstellenden Geometrie on descriptive geometry and graphical statics for his students in 1874.
    • Sturm wrote extensively on geometry and, other than the teaching textbook on descriptive geometry and graphical statics which we mentioned above and one other teaching text Maxima und Minima in der elementaren Geometrie which he published in 1910, all his work was on synthetic geometry.
    • He wrote a three volume work on line geometry published between 1892 and 1896, and a four volume work on projective geometry, algebraic geometry and Schubert's enumerative geometry the first two volumes of which he published in 1908 and the second two volumes in 1909.
    • Let us first comment on the three volume work, which was the biggest treatise ever to be written on line geometry.
    • The work in some respects represents the crowning achievement of synthetic geometry developed in Sturm's style.

  38. Eisenhart biography
    • Thomas Craig aroused my interest in differential geometry by his lectures and my readings of Darboux's treatises.
    • There are two stages in his work although it is all in differential geometry.
    • His first book A Treatise in the Differential Geometry of Curves and Surfaces , published in 1909, was on this topic and was a development of courses he had given at Princeton for several years.
    • The second stage started after 1921 when Eisenhart, prompted by Einstein's general theory of relativity and the related geometries, studied generalisations of Riemannian geometry.
    • He published Riemannian Geometry in 1926 and Non-Riemannian Geometry in 1927.
    • The book gave a presentation of the existing theory of Riemannian geometry after a period of considerable study and development of the subject by Levi-Civita, Eisenhart, and many others.
    • In 1933 Eisenhart published Continuous Groups of Transformations which continues the work of his earlier books looking at Lie's theory using the methods of the tensor calculus and differential geometry.
    • The new chapter began about 1920 with the extended studies of tensor analysis, Riemannian geometry and its generalizations, and the application of the theory of continuous groups to the new physical theories.
    • In fact he published 21 papers between 1951 and 1963, for example: Generalized Riemann spaces and general relativity (1953); A unified theory of general relativity of gravitation and electromagnetism (1956); The cosmology problem in general relativity (1960); and The Einstein generalized Riemannian geometry (1963).
    • Eisenhart had a long association with the American Mathematical Society being vice president in 1914, and Colloquium lecturer in 1925 when he lectured on non-Riemannian geometry.
    • For outside his family he had two "loves": differential geometry (as research and study) and education.

  39. Segre Beniamino biography
    • Beniamino graduated from Turin in 1923 having written a geometry dissertation on the double curves of symmetroids in S4 entitled Genera della curva doppia per la varieta di S4 che annulla un determinante simmetrico.
    • By 1931 when he was appointed to the Chair of Geometry at the University of Bologna he already had 40 publications in algebraic geometry, differential geometry, topology and differential equations.
    • This was a period during which he made exceptional research contributions on algebraic geometry but his interests also broadened, stimulated by discussions with Mordell and Kurt Mahler, to diophantine equations and the arithmetic of algebraic varieties.
    • Segre's output of research papers on geometry and related topics reached nearly 300 not counting a long list of other publications.
    • The second part deals no less attractively with the foundations of linear projective geometry in an arbitrary field.
    • An English edition, double the length of the original text, was published as Lectures on Modern Geometry in 1961.
    • Segre's contributions to geometry are many but, particularly in the latter part of his life, he is remembered for his study of geometries over fields other than the complex numbers.
    • By 1955 Segre was concentrating on geometries over a finite field and was producing results which we would now class as combinatorics rather than geometry.
    • Finally, we note that he was one of two invited speakers in the Geometry and Topology section of the International Congress of Mathematicians held in Cambridge, Massachusetts, USA in 1950 and a plenary speaker at the International Congress of Mathematicians held in Amsterdam in September 1954 when he gave the address Geometry upon an algebraic variety.

  40. Euclid biography
    • This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry.
    • Euclid must have studied in Plato's Academy in Athens to have learnt of the geometry of Eudoxus and Theaetetus of which he was so familiar.
    • someone who had begun to learn geometry with Euclid, when he had learnt the first theorem, asked Euclid "What shall I get by learning these things?" Euclid called his slave and said "Give him threepence since he must make gain out of what he learns".
    • Euclid's decision to make this a postulate led to Euclidean geometry.
    • Books one to six deal with plane geometry.
    • 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.
    • Book six looks at applications of the results of book five to plane geometry.
    • Books eleven to thirteen deal with three-dimensional geometry.
    • It was the primary source of geometric reasoning, theorems, and methods at least until the advent of non-Euclidean geometry in the 19th century.
    • History Topics: Non-Euclidean geometry .

  41. Yau biography
    • In fact, I felt I could understand my father's conversations better after I learned geometry.
    • Yau was awarded a Fields Medal in 1982 for his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge-Ampere equations.
    • S-T Yau has done extremely deep and powerful work in differential geometry and partial differential equations.
    • comes from algebraic geometry and involves proving the existence of a Kahler metric, on a compact Kahler manifold, having a prescribed volume form.
    • His derivation of the estimates is a tour de force and the applications in algebraic geometry are beautiful.
    • Another conjecture solved by Yau was the positive mass conjecture, which comes from Riemannian geometry.
    • In 1981 Yau was awarded The Oswald Veblen Prize in Geometry:- .
    • for his development of non-linear techniques in differential geometry leading to the solution of several outstanding problems.
    • As a result of Yau's work over the past twenty years, the role and understanding of basic partial differential equations in geometry has changed and expanded enormously within the field of mathematics.
    • His work has had, and will continue to have, a great impact on areas of mathematics and physics as diverse as topology, algebraic geometry, representation theory, and general relativity as well as differential geometry and partial differential equations.

  42. Bobillier biography
    • He taught the full range of mathematical topics, analytic geometry, descriptive geometry, trigonometry and statics.
    • I have tried to explain everything that it is necessary to understand in order to successfully follow courses in analytic geometry and in rational mechanics which were entrusted to me, requiring, however, that I did not exceed the level taught at the Ecole d'Arts et Metiers.
    • Loyal to Gaspard Monge's ideas, Bobillier treated geometric problems in a way akin to both analytic geometry and projective geometry.
    • He first set up a problem in the form of an equation in a particular case, simple enough so that the analytic geometry of his time could deal with it.
    • Also like his algebra text, his geometry book continued to appear in new editions long after his death (for example; a fourteenth edition was published in 1870) and his geometry book was adopted by the Ministry of Agriculture, Commerce and Public Works for use in Ecoles d'Arts et Metiers.
    • As we mentioned above, he had published a second edition of his geometry text in 1832 and a third edition two years later.
    • Some of the passages in his course in geometry are probably an early outline for this.
    • In kinematics there seem to be no known traces of the work Bobillier was doing toward the end of his life, although the passages in his book on geometry that treat this subject are still extant.

  43. Castelnuovo biography
    • There Castelnuovo was taught by Veronese who gave him an interest in geometry.
    • But I think he is still giving the Higher Geometry course.
    • In 1891 Castelnuovo was appointed to the Chair of Analytic and Projective Geometry at the University of Rome.
    • In Rome Castelnuovo was a colleague of Cremona but although he had given up active research he was still teaching the Higher Geometry course despite the fact that he had "not been interested in science for a long time", as Veronese had commented five years earlier.
    • After Cremona's death in 1903, Castelnuovo began to teach the advanced geometry courses.
    • Later in his career at Rome he taught a course on algebraic functions and abelian integrals in which he treated the theory of Riemann surfaces,and courses on non-euclidean geometry, differential geometry, interpolation and approximation, and probability theory.
    • Castelnuovo's most important work, however, was done in algebraic geometry, publishing Geometria analitica e proiettiva in 1903.
    • His areas of interest in geometry included the geometry of algebraic curves, linear systems of plane curves from the point of view of birational invariants, and the theory of surfaces.
    • A (7) 11 (2) (1997), 227-235.',4)">4], preserved the archive of Castelnuovo's papers and various historians of mathematics such as Gario and Conte have begun to study this material, see for example [Historia Mathematica 28 (2001), 48-53.',8)">8], and [Algebra and geometry (1860 - 1940) : the Italian contribution, Cortona, 1992, Rend.

  44. Cayley biography
    • The most important of his work is in developing the algebra of matrices, work in non-euclidean geometry and n-dimensional geometry.
    • Cayley developed the theory of algebraic invariance, and his development of n-dimensional geometry has been applied in physics to the study of the space-time continuum.
    • Cayley also suggested that euclidean and non-euclidean geometry are special types of geometry.
    • He united projective geometry and metrical geometry which is dependent on sizes of angles and lengths of lines.
    • His views of geometry were .
    • It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing demonstration: and that Lobachevsky constructed a perfectly consistent theory, wherein this axiom was assumed not to hold good, or say a system of non-Euclidean plane geometry.
    • But suppose the physical space of our experience to be thus only approximately Euclidean space, what is the consequence which follows? Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.
    • History Topics: Non-Euclidean geometry .

  45. Beltrami biography
    • He was appointed to the University of Bologna in 1862 as a visiting professor of algebra and analytic geometry.
    • Influenced by Cremona, Lobachevsky, Gauss and Riemann, Beltrami contributed to work in differential geometry on curves and surfaces.
    • His 1868 paper Essay on an interpretation of non-euclidean geometry which gives a concrete realisation of the non-euclidean geometry of Lobachevsky and Bolyai and connects it with Riemann's geometry.
    • Beltrami in this 1868 paper did not set out to prove the consistency of non-Euclidean geometry or the independence of the Euclidean parallel postulate.
    • Cremona worried that euclidean geometry was being used to describe non-euclidean geometry and he saw a possible logical difficulty in this.
    • Some of his work on physical topics relates to his non-euclidean geometry for he examined how the gravitational potential as given by Newton would have to be modified in a space of negative curvature.
    • He compared Saccheri's results with those of Borelli, Wallis, Clavius and the non-euclidean geometry of Lobachevsky and Bolyai.
    • History Topics: Non-Euclidean geometry .

  46. Titeica biography
    • He was promoted to professor of Analytical Geometry at Bucharest University on 4 May 1900.
    • Titeica's research contributions were mainly in geometry, in particular affine differential geometry.
    • Further investigations of such structures led Titeica to develop further beautiful theory which he set out in his book The projective differential geometry of lattices (1927).
    • He published Introduction to Differential Projective Differential Geometry of Curves in 1931.
    • As well as being famed for this geometrical research, Titeica also gave a famous geometry course at Bucharest University over many years.
    • Mihaileanu [\'Gheorghe Titeica and Dimitrie Pompeiu\' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Romanian) (Editura Acad.
    • Among these many topics were surfaces of constant curvature, ruled surfaces, metrical properties of space, minimal surfaces, Weingarten congruencies, conformal representation, and conformal geometry.
    • One would have left the courses of this apostle of geometry abiding by both his example of dignity and straightness that he was for his entire life and by the optimistic belief the mathematics, in general, and, especially, geometry have a high and admirable educational value for young people.
    • This man's life is split between the faculty, where his Analytical Geometry course flows like a river of clarity whose waters cannot be seen twice, the two magazines, and his scientific work.

  47. Arnold biography
    • Arnold has also made innumerable and fundamental contributions to the theory of differential equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magneto- hydrodynamics.
    • The areas are Dynamical Systems, Differential Equations, Hydrodynamics, Magnetohydrodynamics, Classical and Celestial Mechanics, Geometry, Topology, Algebraic Geometry, Symplectic Geometry, and Singularity Theory.
    • He published Problemes ergodiques de la mecanique classique (with A Avez) (1967), Ordinary differential equations (Russian) (1971), Mathematical methods of classical mechanics (Russian) (1974), Supplementary chapters to the theory of ordinary differential equations (Russian) (1978), Singularity theory (1981), Singularities of differentiable mappings (Russian) (with A N Varchenko and S M Gusein-Zade) (1982), Catastrophe theory (1984), Huygens and Barrow, Newton and Hooke (Russian) (1989), Contact geometry and wave propagation (1989), Singularities of caustics and wave fronts (1990), The theory of singularities and its applications (1991), Topological invariants of plane curves and caustics (1994), Lectures on partial differential equations (Russian) (1997), Topological methods in hydrodynamics (with B A Khesin) (1998), and Arnold problems (Russian) (2000).
    • In recognition of his basic contribution to the stability theory of Dynamical Systems, his pioneering work on singularity theory and seminal contributions to analysis and geometry.
    • Mentally challenged zealots of "abstract mathematics" threw all the geometry (through which connection with physics and reality most often takes place in mathematics) out of teaching.
    • The face of modern mathematics would be unrecognisable without his work in dynamical systems, classical and celestial mechanics, singularity theory, topology, real and complex algebraic geometry, symplectic and contact geometry, hydrodynamics, variation calculus, differential geometry, potential theory, mathematical physics, superposition theory, etc.

  48. Coolidge biography
    • His doctoral dissertation was supervised by Study and, in 1904, he was awarded his doctorate by Bonn University for a thesis entitled Die dual-projektive Geometrie im elliptischen und spharischen Raume (Dual-projective geometry in elliptical and spherical spaces).
    • Coolidge wrote many good texts on geometry including The Elements of Non-Euclidean Geometry (1909), A Treatise on the Circle and the Sphere (1916), The Geometry of the Complex Domain (1924) and A Treatise on Algebraic Plane Curves (1931).
    • The first four books listed above on geometry follow the style of Eduard Study and Corrado Segre but contain many original ideas due to Coolidge himself.
    • The book is in three parts: synthetic geometry, algebraic geometry, and differential geometry.
    • This is the book to be consulted by everyone who wants to know what we might call modern classical geometry and its history.
    • A great number of special topics are briefly or amply discussed, from the geometry of the spider's web to modern criticism of enumerative geometry, Douglas' work on the Plateau problem, quaternions and some tensor analysis.

  49. Sommerville biography
    • He had an original mind, and beneath his outward shyness considerable talents lay concealed: his intellectual grasp of geometry was balanced by a deftness in making models, and on the aesthetic side by an undoubted talent with the brush.
    • Sommerville worked on non-euclidean geometry and the history of mathematics.
    • the classification of all types on non-euclidean geometry (including those usually excluded as bizarre), the extension, involving the measurement of generalised angles in higher space, of Euler's Theorem on polyhedra, space filling figures, the classification of polytopes (i.e.
    • In 1911 he published Bibliography of non-Euclidean Geometry, including the Theory of Parallels, the Foundations of Geometry and Space of n Dimensions.
    • There are 1832 references to n-dimensional geometry.
    • Books which Sommerville published were Elements of Non-Euclidean Geometry (1914), Analytic Conics (1924), Introduction to Geometry of n dimensions (1929) and Three Dimensional Geometry (1934).
    • He also wrote 30 papers on combinatorial geometry.
    • Sommerville's Geometry of n dimensions .

  50. La Hire biography
    • La Hire set off for Venice in 1660 and there spent four years developing his artistic skills and learning geometry.
    • The interest in geometry arose from his study of perspective in art, but soon he was finding his mathematics classes more enjoyable than painting.
    • He continued to paint but his serious studies were devoted to geometry.
    • Bosse was an artist who was much older than La Hire, but had attended classes on geometry by Girard Desargues from 1641.
    • Thus La Hire deserves to be considered, after Pascal, a direct disciple of Desargues in projective geometry.
    • He began with their focal definitions and applied Cartesian analytic geometry t the study of equations and the solution of indeterminate problems; he also displayed the Cartesian method for solving certain types of equations by intersections of curves.
    • Although not a work of great originality, it summarises the progress achieved in analytical geometry during half a century and contained some interesting ideas, among them the possible extension of space to more than three dimensions.
    • He had, at that time, made no contributions to astronomy but Fontenelle [Oeuvres Completes de Fontenelle 1 (Paris, 1818), 257-266.',3)">3] suggests that his election was on the strength of his excellent publications in geometry.
    • Despite his interests across a whole range of scientific disciplines, La Hire remained fascinated by geometry.
    • In 1685 he published a comprehensive work on conic sections Sectiones conicae which contained a description of Desargues' projective geometry.
    • Although his rejection of the infinitesimal calculus may have rendered a part of his mathematical work sterile, his early works in projective, analytic, and applied geometry place him among the best of the followers of Desargues and Descartes.

  51. Le Paige biography
    • Le Paige undertook research for his doctorate in mathematics advised by Francois Folie, whose interests were mainly in descriptive geometry but also in astronomy, and he was also influenced by Catalan.
    • Some of these papers were on topics he had worked on before he settled on geometry as his main interest, for example there are papers on continued fractions, differential equations, the difference calculus, and Bernoulli numbers.
    • He began teaching geometry courses in 1879, taking over courses that had previously been taught by his research advisor Francois Folie.
    • There was a tradition of geometry at the University of Liege which created the background to Le Paige's early research.
    • Germinal Dandelin was professor of mining engineering at Liege from 1825 to 1830 and during that time he taught analytic geometry.
    • It was Jean-Baptiste Brasseur who learnt geometry from Dandelin at this time and he, in turn, taught course on Higher Geometry in which he outlined the theory of algebraic curves and surfaces.
    • Francois Folie was Brasseur's student and he took over teaching the Higher Geometry course in 1876.
    • Folie's approach to geometry was a purely geometrical one, but Le Paige approached the topic in a much more algebraic way, following the fashion that was beginning to take over the subject.
    • In particular Le Paige studied the geometry of algebraic curves and surfaces, building on this earlier work.
    • Le Paige's investigations touched mainly upon the geometry of algebraic curves and surfaces, and the theory of invariants and involutions.

  52. Baer biography
    • His mathematical work, some of which has been mentioned above, was wide ranging; topology, abelian groups and projective geometry.
    • He then generalised this to consider a new type of geometry, namely the lattice of subgroups of an abelian group.
    • In 1940 he introduced the concept of an injective module, then began studying group actions in geometry.
    • His algebraic formulation of geometry appeared in his paper A unified theory of projective spaces and finite abelian groups (1942).
    • His 1952 book Linear algebra and projective geometry presented a completely new approach to projective geometry.
    • to establish the essential structural identity of projective geometry and linear algebra.
    • Geometers of the older type may wonder about a book which neither mentions the favourite subjects of classical geometry nor uses any method of analysis, but nevertheless gives a deep insight into the background of geometry.
    • The results are obtained by a skilful combination of general algebra, lattice theory, and abstract set-theory with methods of classical synthetic geometry.

  53. De Franchis biography
    • He was awarded his laurea in mathematics in 1896, having been advised by Francesco Gerbaldi who had been appointed to the chair of analytic and projective geometry at the University of Palermo in 1890.
    • After entering the competition for a chair, he was appointed as professor of Algebra and Analytic Geometry at the University of Cagliari in 1905.
    • He moved to the University of Parma in following year on being appointed professor of Projective and Descriptive Geometry.
    • In 1909 he was appointed to the University of Catania where he remained until 1914 when he returned to Palermo on being appointed to succeed Guccia in the chair of Analytic and Projective Geometry.
    • All de Franchis's research contributions are in the area of algebraic geometry, but he was one of the first to use analytic methods in this area.
    • We have seen already that de Franchis's early work studied plane algebraic curves but after 1900 his interests turned more towards global algebraic geometry, working in the main areas of the Italian school.
    • However, he represented in a completely appropriately way the highest level of achievement reached by the Italian school of algebraic geometry.
    • This book contains a complete collection of the mathematical papers written by M de Franchis (Palermo, 1875-1946), one of the most interesting exponents of the Italian school of algebraic geometry at the beginning of the century.
    • De Franchis introduced and used implicitly some of the most important tools of modern algebraic geometry, such as characteristic classes and the Albanese map.
    • Some of de Franchis's results seem to suggest still future extensions which can reveal themselves to be useful for modern algebraic geometry.

  54. Durell biography
    • Before the outbreak of World War I, Durell published The arithmetic syllabus in secondary schools (1911) and Analysis and projective geometry (1911) in the Mathematical Gazette.
    • As well as writing articles for the Mathematical Gazette such as The use of limits in elementary geometry (1925) and The teaching of loci in the elementary geometry course to school certificate stage (1936), he was also actively involved with the committee work of the Mathematical Association and its report production.
    • He wrote reports The teaching of geometry in schools (1925), Memo from the Girls' Schools' Committee: Mathematics for girls (1926), and Questionnaire on the teaching of mathematics in evening continuation schools (1926).
    • Among the books he wrote around this time were: Readable relativity (1926), A Concise Geometry (1928), Matriculation Algebra (1929), Arithmetic (1929), Advanced Trigonometry (1930), A shorter geometry (1931), The Teaching of Elementary Algebra (1931), Elementary Calculus (1934), A School Mechanics (1935), and General Arithmetic (1936).
    • He contributed The teaching of loci in the elementary geometry course to school certificate stage (1936), On differentials (1936), Differentials (1937), A theorem in solid geometry (1941), The transition from school to university mathematics (1948), and The nature of main-school geometry (1949).
    • As secretary to the committee reporting in 1953 on the teaching of geometry in schools, he is described in [Mathematical Gazette 53 (1969), 312-313.',3)">3] as:- .

  55. Manin biography
    • He has written papers on: algebraic geometry including ones on the Mordell conjecture for function fields and a joint paper with V Iskovskikh on the counter-example to the Luroth problem; number theory including ones about torsion points on elliptic curves, p-adic modular forms, and on rational points on Fano varieties; and differential equations and mathematical physics including ones on string theory and quantum groups.
    • He has also written famous papers on formal groups, the arithmetic of rational surfaces, cubic hypersurfaces, noncommutative algebraic geometry, instanton vector bundles and mathematical logic.
    • In 1980 Manin and A I Kostrikin published the textbook Linear algebra and geometry in Russian.
    • In 1984 Manin published Linear algebra and geometry in Russian, it was translated into English in 1997.
    • Also in 1984 he published Gauge fields and complex geometry in Russian.
    • The first of these is the notion of twistor geometry and the other is that of supersymmetry.
    • In this book we find a beautiful blend of developments stemming from these two ideas written by a master expositor who uses the language of algebraic geometry to synthesize and unify the fundamental ideas involved.
    • Other books by Manin include Cubic forms: algebra, geometry, arithmetic (Russian) (1972), A course in mathematical logic (1977), Computable and noncomputable (Russian) (1980), Quantum groups and noncommutative geometry (1988), Topics in noncommutative geometry (1991), Frobenius manifolds, quantum cohomology, and moduli spaces (1999).

  56. Poincare biography
    • At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformation that I had used to define the Fuchsian functions were identical with those of non-euclidean geometry.
    • He also worked in algebraic geometry making fundamental contributions in papers written in 1910-11.
    • to make geometry ..
    • The same point is made again by Poincare when he wrote a review of Hilbert's Foundations of geometry (1902):- .
    • Poincare believed that one could choose either euclidean or non-euclidean geometry as the geometry of physical space.
    • for this reason he argued that euclidean geometry would always be preferred by physicists.
    • The breadth of his research led to him being the only member elected to every one of the five sections of the Academy, namely the geometry, mechanics, physics, geography and navigation sections.
    • Poincare on non-Euclidean geometry .
    • History Topics: A History of Fractal Geometry .

  57. Ferrand biography
    • Returning to Ferrand's career, she was promoted to full professor at Caen in 1948 and, later in the same year, she was appointed to the chair of calculus and higher geometry at the University of Lille, filling the chair left vacant when Bertrand Gambier (1879-1954) retired.
    • This is a textbook for the geometry part of the course "Mathematiques II" according to the new French undergraduate program.
    • This explains the fact that a textbook of differential geometry contains a chapter on non-euclidean geometry and one on the axiomatics of euclidean geometry, the first with a minimum, the second without any use of differential methods.
    • Volume I covered multivariable differential calculus, with a little differential geometry.
    • The second volume, published in 1974, covered analysis (multivariable differential calculus and one-variable integral calculus) while the third volume, published in the following year covered geometry with applications to mechanics.
    • Having produced these books intended for students, Ferrand then published Les fondements de la geometrie (1985) which was a text on the foundations of geometry intended for teachers of mathematics.
    • In 1964 Andre Lichnerowicz posed a famous conjecture on differential geometry in his paper Sur les transformations conformes d'une variete riemannienne compacte.
    • This led to her being an invited speaker in the Section on 'Differential Geometry and Analysis on Manifolds' at the International Congress of Mathematicians held in Vancouver in August 1974.

  58. Bianchi biography
    • He taught differential geometry at the University of Pisa where he was promoted a number of times, first to extraordinary professor in differential geometry, then after a competitive examination to extraordinary professor in projective geometry in 1886.
    • In the same year he was also appointed as extraordinary professor of analytic geometry, becoming a full professor of analytic geometry in 1890.
    • Luigi Bianchi made important contributions to differential geometry.
    • Bianchi also turns to non-Euclidean geometry, taking up the ideas of Beltrami; the surfaces of curvature zero in such geometry draw his special attention.
    • Bianchi's 'Lezioni di geometria differenziale' (1893), a new edition of autographed lectures published in 1886, is a systematic treatise of the theory of curves and surfaces, with special attention paid to more-dimensional geometry.
    • But even while he was writing on these topics, papers on surfaces were still appearing, and differential geometry absorbed nearly all his attention for the last twelve years or so of his life.

  59. Bartel biography
    • Following his graduation, he was appointed as an assistant in descriptive geometry to Placyd Zdzislaw Dziwinski (1851-1936), the professor of geometry, at Lwow Polytechnic, but he also enrolled as a student at the University of Lwow.
    • His interests were, however, broader than mathematics, and he took a course on the history of art given by Karl Dochlemann, the author of Projective geometry (1898) [Psychologie und Geschichte 5 (3-4) (April 1994).',2)">2]:- .
    • Bartel returned to his professorship at Lwow Polytechnic and in fact his book Descriptive geometry (Polish) was published around this time.
    • During these years when Bartel was again an academic, he applied his knowledge and skills in geometry to a historical study of perspective in European painting.
    • Let us return to make some comments on his book Descriptive geometry (Polish).
    • Here is an excellent introduction to a most attractive branch of descriptive geometry.
    • to convert them into problems in biorthoconal projection, and thus to surrender them to the other main branch of descriptive geometry.
    • The book is a comprehensive course in descriptive geometry, including what is called engineering drawing and concluding with the theory of shadows (pp.382-422).
    • The introductory chapter reminds the reader of the basic theorems (and their proofs) of solid geometry that will be made use of in the text.

  60. Boggio biography
    • He graduated on 8 July 1899 from Turin with 'high honours' in pure mathematics and was appointed in November as an assistant in projective and descriptive geometry to Mario Pieri at the University of Turin.
    • Pieri left Turin in 1900 and Boggio continued to teach projective and descriptive geometry.
    • While he tutored geometry at the university in his assistant position, Boggio was undertaking research in applied mathematics.
    • In 1918 Enrico D'Ovidio retired from his chair in Turin and Boggio took over teaching algebraic analysis and analytic geometry.
    • Boggio was director of the School of Algebra and Analytic Geometry in 1921-22.
    • On this point (which is the central point in their criticism of the application of geometry of curved space to physics) Burali-Forti and Boggio are behind those geometers who while using coordinates succeed in discriminating as to which expressions have a meaning independent of them.
    • Boggio taught Higher Geometry from 1938 to 1940, then both Higher geometry, and Analytic and Projective geometry in 1940-41.
    • After the war ended he taught Higher Geometry from 1945 to 1947, then Numerical Mathematics and Graph Theory in 1947-48.

  61. Postnikov biography
    • Most of this book consists of a well-written, self-contained text on homotopy theory and differential geometry, in preparation for chapters on Morse theory on finite dimensional manifolds, the variational theory of geodesics, and the study of path spaces by finite-dimensional approximation.
    • We have changed the chronological sequence of Postnikov's publications slightly to list finally the six textbooks which he wrote corresponding to six series of lectures on geometry given to undergraduate and graduate students at Moscow State University.
    • Linear algebra and differential geometry (1986).
    • Written by a famous author and mathematician with a worldwide reputation, a winner of the Lenin prize, the highest honour that a Soviet mathematician can be awarded, the book gives a very clear and at the same time rigorous presentation of different topics in linear algebra and their numerous applications to modern geometry and analysis.
    • Analytic geometry (1986).
    • Differential geometry (1988).
    • Riemannian geometry (1998).
    • This volume is the sixth in Postnikov's series of lecture notes in differential geometry, and provides an advanced overview of various topics in Riemannian geometry.
    • From 1954 until 1960 he lectured in the higher geometry and topology department at Moscow State University.

  62. Scorza biography
    • After obtaining his laurea, Scorza was appointed as an assistant to Eugenio Bertini, who held the chair of analytical and projective geometry at the University of Pisa.
    • Scorza spent the year in Turin assisting Corrado Segre, who held the chair of projective and descriptive geometry.
    • By the end of 1906, Scorza had begun to have contacts again with Bertini, Severi and other researchers in geometry and, given his outstanding early contributions, they were all keen to persuade him to return to his study of algebraic geometry.
    • Scorza's health improved markedly from 1906 and he was able to resume his research in algebraic geometry with a renewed vigour.
    • He was successful in the competition for the chair of projective and descriptive geometry at the University of Cagliari and took up his appointment in 1912.
    • This work is discussed by Guido Zappa in [Geometry Seminars 2005-2009 (Italian) (Univ.
    • Zappa also looks at this work by Scorza in [Geometry Seminars 2005-2009 (Italian) (Univ.
    • Despite the years of difficulty when he was a young man, Scorza published more than 160 works over his career, most importantly in algebraic geometry and related algebraic fields, but also very many general works and textbooks.
    • He was the author of two major reports for the Commission, namely L'insegnamento della matematica nelle Scuole e negli Istituti tecnici (1911) which examines mathematics teaching in technical schools and institutes, and Sui libri di testo di geometria per le scuole secondarie superiori (1912) which looked at geometry textbooks written at upper secondary school level.

  63. Artin Michael biography
    • Those were exciting times for algebraic geometry.
    • At IHES, Artin attended Alexander Grothendieck's seminars which had a major influence on the direction of his research in algebraic geometry at this time.
    • His contributions to algebraic geometry are beautifully summarised in the citation for the Lifetime Achievement Steele prize he received from the American Mathematical Society in 2002:- .
    • Michael Artin has helped to weave the fabric of modern algebraic geometry.
    • The point of the extension is that Artin's theorem on approximating formal power series solutions allows one to show that many moduli spaces are actually algebraic spaces and so can be studied by the methods of algebraic geometry.
    • Algebraic stacks and algebraic spaces appear everywhere in modern algebraic geometry, and Artin's methods are used constantly in studying them.
    • He has contributed spectacular results in classical algebraic geometry, such as his resolution (with Swinnerton-Dyer in 1973) of the Shafarevich-Tate conjecture for elliptic K3 surfaces.
    • With Mazur, he applied ideas from algebraic geometry (and the Nash approximation theorem) to the study of diffeomorphisms of compact manifolds having periodic points of a specified behaviour.
    • His main research area changed from algebraic geometry to noncommutative ring theory as he explained in his response to receiving the Steele Prize [Notices Amer.
    • an architect of the modern approach to algebraic geometry.

  64. Thales biography
    • [Thales] first went to Egypt and thence introduced this study [geometry] into Greece.
    • 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.
    • On the other hand B L van der Waerden [cience Awakening (New York, 1954).',16)">16] claims that Thales put geometry on a logical footing and was well aware of the notion of proving a geometrical theorem.
    • However, although there is much evidence to suggest that Thales made some fundamental contributions to geometry, it is easy to interpret his contributions in the light of our own knowledge, thereby believing that Thales had a fuller appreciation of geometry than he could possibly have achieved.
    • In many textbooks on the history of mathematics Thales is credited with five theorems of elementary geometry:- .
    • What is the basis for these claims? Proclus, writing around 450 AD, is the basis for the first four of these claims, in the third and fourth cases quoting the work History of Geometry by Eudemus of Rhodes, who was a pupil of Aristotle, as his source.
    • The History of Geometry by Eudemus is now lost but there is no reason to doubt Proclus.
    • Pamphile says that Thales, who learnt geometry from the Egyptians, was the first to describe on a circle a triangle which shall be right-angled, and that he sacrificed an ox (on the strength of the discovery).

  65. Moser William biography
    • During his 32 years in the Department of Mathematics and Statistics, Moser fashioned and taught his own geometry courses in an original manner, making collections of research problems along the way.
    • In order to promote his specialty, Moser became involved in the teaching of high school mathematics teachers, making films about geometry, organizing mathematical competitions and collecting and disseminating competition problems.
    • He has also taught NSF Summer Institutes for High School Teachers (1959-62) and participated in the College Geometry Project (1964-68) at the University of Minnesota, making beautiful films, one about Coxeter.
    • Finally we must say something about Moser's remarkable contributions in publishing surveys of problems in discrete geometry in both books and articles.
    • He edited Problems in discrete geometry (1980) which collected together 34 problems, each with references to preceding work.
    • In the following year he edited Research problems in discrete geometry which was a collection of 68 problems of combinatorial geometry including distance problems, covering and packing problems, lattice point problems, and visibility problems.
    • In 1989 the First Canadian Conference on Computational Geometry was held in Montreal and Moser presided over the two problem sessions publishing Problems, problems, problems in the conference proceedings.
    • This interest in problems in discrete geometry culminated in 2005 with the publication of a 500 page book Research problems in discrete geometry published jointly by Moser, Peter Brass and Janos Pach.

  66. Levi Beppo biography
    • Corrado Segre, who had himself studied at Turin with D'Ovidio, had been appointed to the Chair of Higher Geometry there in 1888.
    • He became Levi's thesis advisor, and Levi's thesis Sulla varieta delle cordi di una curva algebraica was a brilliant piece of work adding to the outstanding work of the Italian school of geometry.
    • He came third in this competition, the post going to Gino Fano [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',2)">2]:- .
    • In 1906 he was successful in a competition when he was appointed professor of descriptive and projective geometry at the University of Cagliari in Sardinia.
    • He remained at Cagliari, teaching analytic geometry, for four years until he was called to the chair of algebraic analysis at the University of Parma in 1910.
    • Although, for Albina [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',2)">2]:- .
    • As well as the chair of algebraic analysis, Levi also held the chair of analytic geometry and, for a year, the chair of mathematical physics too.
    • Before going to Parma he had already published over forty articles on topics ranging from algebraic geometry to logic, particularly working on the axiom of choice.
    • He takes no acquaintance with analytic geometry for granted.
    • As well as teaching courses on analysis, geometry and rational mechanics, he was very active in research, publishing around one third of all his work in Spanish.

  67. Robertson biography
    • However he did make outstanding contributions to differential geometry, quantum theory, the theory of general relativity, and cosmology [H P Robertson : January 27, 1903-August 26, 1961.
    • he was interested in the foundations of physical theories, differential geometry, the theory of continuous groups, and group representations.
    • His contributions to differential geometry came in papers such as: The absolute differential calculus of a non-Pythagorean non-Riemannian space (1924); Transformation of Einstein space (1925); Dynamical space-times which contain a conformal Euclidean 3-space (1927); Note on projective coordinates (1928); (with H Weyl) On a problem in the theory of groups arising in the foundations of differential geometry (1929); Hypertensors (1930); and Groups of motion in space admitting absolute parallelism (1932).
    • As a final illustration of Robertson's interest in the connection between geometry and physics we quote A H Taub's review of Robertson's paper The geometries of the thermal and gravitational fields which was published in the American Mathematical Monthly in 1950:- .
    • The author's purpose is to give "a study of the physical geometry of the gravitational and of the thermal fields, and of the reasons for the success of the former as opposed to the latter, as a basis for a tenable physical theory." The discussion of gravitation is given in terms of a four-dimensional scalar theory but the main ideas of Einstein's general theory are stressed.
    • However the author's purpose is not to give a physical theory of gravitation, but to discuss the geometry of a theory of the gravitational field.
    • He also discusses the geometry of the thermal field in terms of measurements made by rods after they have been allowed to come into thermal equilibrium with a heated medium.
    • It is shown that in the latter case the geometry depends on the material composing the measuring rods whereas in the gravitational case the geometry is universal because of the equivalence between inertial and gravitational mass.

  68. Folie biography
    • In 1857 he was appointed to teach courses in Algebra, Calculus and Analytic Geometry at the University of Liege.
    • It was a course he continued to give for a number of years but, in 1868, he gave up teaching the courses in Algebra, Calculus and Analytic Geometry.
    • Although he continued to be interested in applied mathematical topics, Folie had learnt about the latest developments in geometry from Jean-Baptiste Brasseur and soon he began to publish in this area.
    • In 1868, following Brasseur's death, he took over teaching the course on Higher Geometry.
    • His first research publication in geometry was the paper Sur quelques theoremes generaux de geometrie superieure (1869) but, more significantly, his first major publication in the Memoirs of the Royal Belgium Academy of Science was on geometry: Fondements d'une geometrie superieure cartesienne (1872).
    • Two years later he published three papers on geometry: Extension des theoremes analogues a celui de Pascal a des courbes tracees sur une surface quelconque; Quelques nouveaux theoremes sur les cubiques gauches; and Quelques nouveaux theoremes sur les courbes gauches du 4e ordre.
    • During these years when his research interests were firmly in the area of geometry he was thesis advisor to Constantin Le Paige who was awarded his doctorate on 28 July 1875.
    • Le Paige took over teaching the Geometry Course that Folie had been giving in 1879.
    • In fact, once he was appointed to the position of director of the Observatory he did not publish any further papers on geometry.

  69. Albanese biography
    • Another of his teachers at the School was Eugenio Bertini who was a leading researcher in algebraic geometry and Albanese undertook research for his doctorate under Bertini's supervision.
    • Five years later he moved to the University of Catania where he was appointed to the chair of Projective and Descriptive Geometry.
    • From 1929 until 1936 he held the chair of geometry at Pisa, a position which he could have continued to occupy for the rest of his life but he was sent to Brazil in 1936.
    • He considered the problem of resolution of singularities, a major problem in algebraic geometry, and produced some elegant results.
    • His name is remembered today for Albanese varieties used as a standard tool in algebraic geometry.
    • In other notes from 1924-1927 Albanese proved sufficient conditions for a surface to be rational; he resolved the problem of the base for the curves on a surface and undertook a general study of the geometry of manifolds.
    • This was in the years in which Albanese became professor of geometry at the University of Catania, going later to Palermo and then to Pisa.
    • Albanese spent the rest of his life in São Paulo, holding the chair of Analytical, Projective and Descriptive Geometry, except for the year 1942 when he returned to Pisa because of World War II.
    • created a very good mathematical library, especially rich in books of algebraic geometry.
    • geometry with Giacomo Albanese.

  70. Segre Corrado biography
    • In 1883 Segre was awarded his doctorate for a thesis on quadrics in higher dimensional spaces and was appointed as an assistant to the professor of algebra and to the professor of geometry at Turin.
    • In 1885 he was appointed as assistant in descriptive geometry.
    • In 1888 Segre succeeded D'Ovidio to the chair of higher geometry in Turin and he continued to hold this post for the next 36 years until his death.
    • Plucker's ideas on the geometry of ruled surfaces had been extended by Klein, and D'Ovidio lectured on this topic in session 1881-82.
    • D'Ovidio also included in these lectures results of Veronese on projective geometry and of Weierstrass on bilinear and quadratic forms.
    • In 1890 Segre looked at properties of the Riemann sphere and was led to a new area of representing complex points in geometry.
    • He introduced bicomplex points into geometry.
    • Motivated by the works of von Staudt, Segre considered a different type of complex geometry in 1912.
    • Through his teaching and publications, Segre played an important role in reviving an interest in geometry in Italy.
    • Segre's contribution to the knowledge of space assures him a place after Cremona in the ranks of the most illustrious members of the new Italian school of geometry.

  71. Federer biography
    • In 1948 Federer produced mimeographed notes for the course An Introduction to Differential Geometry that he was giving at Brown University.
    • It is not concerned with metric differential geometry but rather with the more primitive notions of differential forms, their integrals, de Rham's theorem and related matters.
    • In 1961, in collaboration with Bjarni Jonsson, Federer published the undergraduate text Analytic Geometry and Calculus.
    • Included in 'Analytic Geometry and Calculus' are the ordered pairs definition of relation and function; in equalities; absolute values; and the epsilon delta approach to limits and the wrapping function in trigonometry.
    • Although the authors claim that the reader needs only "the usual background of high school algebra and geometry, not necessarily including trigonometry," the generally sophisticated treatment of the classical topics in analytic geometry, logarithms, and trigonometry would not be wasting a student's time who happened to have, say, four years of mathematics in high school.
    • These advances have given us deeper perception of the analytic and topological foundations of geometry, and have provided new direction to the calculus of variations.
    • Some knowledge of elementary set theory, topology, linear algebra and commutative ring theory is prerequisite for reading this book, but the treatment is self-contained with regard to all those topics in multilinear algebra, analysis, differential geometry and algebraic topology which occur.
    • It has depth and beauty of its own, but its greatest worth should be in its effect on other areas of mathematics, e.g., differential geometry, differential topology, partial differential equations, algebraic geometry, potential theory.

  72. Weil biography
    • Weil's research was in number theory, algebraic geometry and group theory.
    • Beginning in the 1940s, Weil started the rapid advance of algebraic geometry and number theory by laying the foundations for abstract algebraic geometry and the modern theory of abelian varieties.
    • In fact Weil's work in this area was basic to work by mathematicians such as Yau who was awarded a Fields Medal in 1982 for work in three dimensional algebraic geometry which has major applications to quantum field theory.
    • Weil's work on bringing together number theory and algebraic geometry was highly fruitful.
    • He contributed substantially to topology, differential geometry and complex analytic geometry.
    • Also bringing these areas together was his work on the geometric theory of the theta function and Kahler geometry.
    • Weil's most famous books include Foundations of Algebraic Geometry (1946) and Elliptic Functions According to Eisenstein and Kronecker (1976).
    • Andre Weil: Algebraic Geometry .

  73. Van der Waerden biography
    • His doctoral thesis De algebraiese grondslagen der meetkunde van het aantal (The algebraic foundations of the geometry of numbers) was submitted to the University of Amsterdam and he defended it in the grand hall of the University on 24 March 1926.
    • He then began to publish a series of articles in Mathematische Annalen on algebraic geometry.
    • His work in algebraic geometry uses the ideal theory in polynomial rings created by Artin, Hilbert and Emmy Noether.
    • About ten years ago, van der Waerden, already eminent as an algebraist, began, in a series of papers in the Mathematische Annalen, to create rigorous foundations for algebraic geometry.
    • The implication-that there was some-thing unsound in the magnificent structure of Italian geometry-was vigorously contested by Severi.
    • Fortunately, van der Waerden continued his researches, but with the implicit sub-title, "An algebraist looks at algebraic geometry".
    • Van der Waerden worked on algebraic geometry, abstract algebra, groups, topology, number theory, geometry, combinatorics, analysis, probability theory, mathematical statistics, quantum mechanics, the history of mathematics, the history of modern physics, the history of astronomy and the history of ancient science.
    • Among his many historical books are Ontwakende wetenschap (1950) translated into English as Science Awakening (1954), Science Awakening II: The Birth of Astronomy (1974), Geometry and Algebra in Ancient Civilizations (1983), and A History of Algebra (1985).
    • 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).

  74. Zariski biography
    • However, even during my Rome period, my algebraic tendencies were showing and were clearly perceived by Castelnuovo who once told me: "You are here with us but are not one of us." This was said not in reproach but good naturedly, for Castelnuovo himself told me time and time again that the methods of the Italian geometric school had done all they could do, had reached a dead end, and were inadequate for further progress in the field of algebraic geometry.
    • Castelnuovo and Severi had encouraged Zariski to view Lefschetz's topological methods as being the road ahead for algebraic geometry, so between 1927 and 1937 Zariski frequently visited Lefschetz at Princeton.
    • At that time (1935) modern algebra had already come to life (through the work of Emmy Noether and the important treatise of B L van der Waerden), but while it was being applied to some aspects of the foundations of algebraic geometry by van der Waerden ..
    • the deeper aspects of birational algebraic geometry ..
    • At Johns Hopkins University between 1939 and 1940 Zariski carried out his project of applying modern algebra to the foundations of algebraic geometry.
    • After beginning his work in Italy in 1924 very much in the style of "Italian algebraic geometry," Zariski realised that the whole subject needed proper foundations.
    • His topological work concentrated mainly on the fundamental group; many of the ideas he pioneered were innovations in topology as well as algebraic geometry and have developed independently in the two fields since then.
    • In 1937 Zariski completely reoriented his research and began to introduce ideas from abstract algebra into algebraic geometry.
    • His use of the notions of integral independence, valuation rings, and regular local rings, in algebraic geometry proved particularly fruitful and led him to such high points as the resolution of singularities for threefolds in characteristic 0 in 1944, the clarification of the notion of simple point in 1947, and the theory of holomorphic functions on algebraic varieties over arbitrary ground fields.
    • All of Zariski's work has served as a basis for the present flowering of algebraic geometry and the current school uses his work and ideas in the modern development of the subject.

  75. Lacroix biography
    • Monge had been appointed by the National Convention on 11 March 1794 to the body that was put in place to establish the Ecole Centrale des Travaux Publics (soon to be called the Ecole Polytechnique), and he was appointed to teach a course in descriptive geometry on 9 November 1794.
    • Lacroix and Hachette assisted Monge in the work for his descriptive geometry course.
    • Lacroix was particularly well suited to assist with this course since he had been assembling material on descriptive geometry for several years, and he went on to publish Elements de geometrie descriptive.
    • In the first of these volumes Lacroix introduces for the first time the expression "analyic geometry" writing:- .
    • There exists a manner of viewing geometry that could be called geometrie analytique, and which would consist in deducing the properties of extension from the least possible number of principles, and by truly analytic methods.
    • Of the geometry texts of Lacroix and Lagrange, Lamande writes in [Physis Riv.
    • The authors, breaking with the intuitionism that had dominated eighteenth-century French treatises, updated the logic of geometry manuals.
    • These works illustrate the different paths that had opened for the teaching of geometry: the return to the Ancients or the rewriting and expansion of that material from an analytic or synthetic perspective.
    • It is interesting that Lacroix held the view that algebra and geometry:- .

  76. Brasseur biography
    • It was Dandelin who had a profound influence on Brasseur - he taught him geometry and encouraged him to undertake research on that topic.
    • Brasseur began giving courses on analytic geometry and descriptive geometry which were designed to complete the instruction of lieutenants of artillery and so allow them take the examinations of the Ecole Militaire necessary to rise to the rank of captain.
    • He had for a long time wanted an academic position and in 1832 he was appointed as a lecturer at the University of Liege to teach the course on Descriptive Geometry and also the course on Higher Analysis Applied to Geometry.
    • He based his geometry courses on the work of Gaspard Monge.
    • At this time Brasseur took over teaching the Analysis course, the Elementary Mathematics course, and was appointed to the Chair of Applied Mechanics and the Chair of Descriptive Geometry.
    • The memoir contains an original presentation of projective geometry.
    • He has had a profound influence on mathematics education at the University of Liege and was the founder of the School of Geometry that continued to shine brightly with his successors F Folie, C Le Paige and Fr Deruyter.

  77. D'Ovidio biography
    • In 1869 D'Ovidio, in collaboration with Achille Sannia, published a geometry text for schools.
    • In 1872 Eugenio Beltrami persuaded him to enter the competition for the Chair of Algebra and Analytic Geometry at the University of Turin.
    • He was a referee, with Eugenio Bertini and Giuseppe Veronese, for appointing the chair of analytic and projective geometry at the University of Rome in 1891.
    • In 1893 he was a referee for appointing the chair of projective and descriptive geometry at his own University of Turin; Luigi Berzolari was appointed.
    • Euclidean and non-Euclidean geometry were the areas of special interest to D'Ovidio.
    • It was a highly significant time for research in geometry with the work by Nikolai Ivanovich Lobachevsky, Janos Bolyai and Bernhard Riemann on non-Euclidean geometry becoming widely known and Felix Klein had put forward the general view of Euclidean and non-Euclidean geometries as invariants for transformation groups in the 'Erlanger Programm' in 1872.
    • In this work the he used concepts and methods from projective geometry to derive the metric functions in non-Euclidean n-dimensional spaces, paving the way for subsequent work of Giuseppe Veronese, Corrado Segre and his other students.
    • D'Ovidio and Corrado Segre built an important school of geometry at Turin.

  78. Uhlenbeck Karen biography
    • I decided Einstein's general relativity was too hard, but managed to learn a lot about geometry of space time.
    • Witten, who gave the next talk on Geometry and quantum field theory at the symposium said:- .
    • She described advances in geometry that have been achieved through the study of systems of nonlinear partial differential equations.
    • Among other things, she sketched some aspects of Simon Donaldson's work on the geometry of four-dimensional manifolds, instantons - solutions, that is, of a certain nonlinear system of partial differential equations, the self-dual Yang-Mills equations, which were originally introduced by physicists in the context of quantum field theory.
    • She has also served on the editorial boards of many journals; a complete list to date is Journal of Differential Geometry (1979-81), Illinois Journal of Mathematics (1980-86), Communications in Partial Differential Equations (1983- ), Journal of the American Mathematical Society (1986-91), Ergebnisse der Mathematik (1987-90), Journal of Differential Geometry (1988-91), Journal of Mathematical Physics (1989- ), Houston Journal of Mathematics (1991- ), Journal of Knot Theory (1991- ), Calculus of Variations and Partial Differential Equations (1991- ), Communications in Analysis and Geometry (1992- ).
    • For her many pioneering contributions to global geometry that resulted in advances in mathematical physics and the theory of partial differential equations.
    • Karen Uhlenbeck is a distinguished mathematician of the highest international stature, specialising in differential geometry, non-linear partial differential equations and mathematical physics.

  79. Veronese biography
    • In 1876 Veronese was appointed as assistant in analytical geometry on the strength of his paper on Pascal's hexagram which he had completed by this time.
    • Veronese was in contact with Klein who was about to take up a chair of geometry at the University of Leipzig.
    • Bellavitis died in November 1880 and his chair of algebraic geometry in Padua became vacant.
    • Freguglia, in [Geometry Seminars, 1996-1997 (Bologna, 1998), 253-277.',5)">5], describes Veronese's study of geometry in higher dimensions.
    • In 1880 Veronese described an n-dimensional projective geometry, showing that simplifications could be obtained in passing to higher dimensions.
    • This was a very original approach to higher-dimensional projective geometry that Veronese developed.
    • He is certainly considered to be one of the founders of that topic for with him what others had considered as linear algebra viewed geometrically became geometry.
    • Veronese provided both logical and psychological motivations for his approach which greatly influenced the Italian school of geometry for many years.

  80. Proclus biography
    • Proclus wrote Commentary on Euclid which is our principal source about the early history of Greek geometry.
    • In particular he certainly used the History of Geometry by Eudemus, which is now lost, as is the works of Geminus which he also used.
    • The notes on the postulates and axioms are preceded by a general discussion of the principles of geometry, hypotheses, postulates and axioms, and their relation to one another; here as usual Proclus quotes the opinions of all the important authorities.
    • Another interesting part of Proclus's commentary is his discussion of the critics of geometry.
    • it is against [the principles of geometry] that most critics of geometry have raised objections, endeavouring to show that these parts are not firmly established.
    • whereas others, like the Epicureans, propose only to discredit the principles of geometry.
    • Proclus and the history of geometry as far as Euclid .
    • History Topics: Non Euclidean geometry .

  81. Monge biography
    • At first Monge's post did not require him to use his mathematical talents, but Monge worked in his own time developing his own ideas of geometry.
    • The four memoirs that Monge submitted to the Academie were on a generalisation of the calculus of variations, infinitesimal geometry, the theory of partial differential equations, and combinatorics.
    • Not only was he a major influence in setting up the Ecole using his experience at Mezieres to good effect, but he was appointed as an instructor in descriptive geometry on 9 November 1794.
    • Monge's lectures on infinitesimal geometry were to form the basis of his book Application de l'analyse a la geometrie.
    • Another educational establishment, the Ecole Normale, was set up to train secondary school teachers and Monge gave a course on descriptive geometry.
    • He is considered the father of differential geometry because of his work Application de l'analyse a la geometrie where he introduced the concept of lines of curvature of a surface in 3-dimensional space.
    • He developed a general method of applying geometry to problems of construction.
    • [His] new approach addressed itself to the most profound, intimate and universal relations in space and their transformations, putting him in a position to interconnect geometry and analysis in a fertile, previously unheard-of fashion.
    • Minnesota (One of Monge's geometry theorems and its relationship to Desargues theorem) .

  82. Juel biography
    • Continuing with his doctoral studies at the University of Copenhagen he received a gold medal for a geometrical treatise in 1881 and was awarded his doctorate in 1885 for a dissertation on geometry entitled Contributions to the geometry of imaginary lines and imaginary planes (Danish).
    • He made substantial contributions to projective geometry and wrote an important book on the topic.
    • Juel's programme of research extended certain theorems on real manifolds of algebraic geometry to non-algebraic manifolds, even to non-analytic manifolds provided the concepts are generalised in a suitable way.
    • Many readers must have felt that if all that projective geometry could tell us of a problem involving a cubic equation was that it has at least one solution, and not more than three, then projective geometry had not by any means justified its claims to replace the ordinary algebraic kind.
    • About the strict axiomatic foundations of the real projective geometry with which he begins, he is content to be brief, and to refer the reader to the German version of Enriques' 'Lezioni sulla geometria projettiva'.
    • In essence, however, he allows himself purely projective axioms and an axiom of continuity, which, as is familiar, will suffice to build up what is from the algebraic point of view the coordinate geometry in the field of the real numbers and infinity.
    • The rest of the book follows the general lines of some portions of Reye - by no means the whole, since hardly any solid geometry is attempted - with due regard to the refinements introduced by the use of imaginary elements.

  83. Sintsov biography
    • Clebsch constructed the geometry of a ternary connex and applied it to the theory of ordinary differential equations.
    • Of course through his many years of research his interests varied but the main areas on which he worked were the theory of conics and applications of this geometrical theory to the solution of differential equations and, perhaps most important of all, the theory of nonholonomic differential geometry.
    • His classical work on the theory of connexes, of which he was one of the founders, and on nonholonomic differential geometry are well known far beyond the frontiers of our country.
    • The book in which the articles [Ja P Blank, D Z Gordevskii, A S Leibin and M A Nikolaenko (eds.), D M Sintsov, Papers on nonholonomic geometry (Kiev, 1972), 4-8.',2)">2] (written by Ja P Blank who was a student of Sintsov) and [Ja P Blank, D Z Gordevskii, A S Leibin and M A Nikolaenko (eds.), D M Sintsov, Papers on nonholonomic geometry (Kiev, 1972), 286-293.
    • ',5)">5] appear, contains a selection of the Sintsov's major works on nonholonomic geometry.
    • These were first published during the years 1927-1940 and include: A generalization of the Enneper-Beltrami formula to systems of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = 0 (1927); Properties of a system of integral curves of Pfaff's equation, Extension of Gauss's theorem to the system of integral curves of the Pfaffian equation Pdx + Qdy + Rdz = 0 (1927); Gaussian curvature, and lines of curvature of the second kind (1928); The geometry of Mongian equations (1929); Curvature of the asymptotic lines (curves with principal tangents) for surfaces that are systems of integral curves of Pfaffian and Mongian equations and complexes (1929); On a property of the geodesic lines of the system of integral curves of Pfaff's equation (1936); Studies in the theory of Pfaffian manifolds (special manifolds of the first and second kind) (1940) and Studies in the theory of Pfaffian manifolds (1940).
    • At Kharkov University, Sintsov created a school of geometry which became the leading school in this field in the Ukraine and has continued to flourish through the years still today being a leading centre.
    • There he studied the geometry of Monge equations and he introduced the important ideas of asymptotic line curvature of the first and second kind.

  84. Neuberg biography
    • At the University of Liege he taught analysis, higher algebra, descriptive geometry, projective geometry, analytic geometry, and the foundations of mathematics.
    • At the Ecole des Mine he taught higher algebra, analytic geometry and infinitesimal analysis.
    • Neuberg worked on the geometry of the triangle, showing great ingenuity, discovering many interesting new details but developing no large new theory.
    • Together with Henri Brocard and Emile Lemoine, he is considered as the creators of the geometry of the triangle and of the tetrahedron.
    • He was also involved in the study of various algebraic surfaces, including the cyclide of Dupin, geometry of ruled surfaces and curves arising from linkages.
    • In the early part of his career he published papers in the Nouvelles Annales de Mathematiques on inscribed and circumscribed conics, as well as on the geometry of the triangle.
    • He published his lessons on Infinitesimal Analysis, on Higher Algebra and on Analytic Geometry; these works were a great success.

  85. Hobbes biography
    • He was forty years old before he looked on geometry; which happened accidentally.
    • This made him in love with geometry.
    • To Hobbes mathematics was geometry and only geometry, and Wallis's Algebra he described as:- .
    • Hobbes's attempts to resolve three important mathematical controversies of the seventeenth century: the debates over the status of analytic geometry, disputes over the nature of ratios, and the problem of the 'angle of contact' between a curve and tangent.
    • was not the ignoramus in geometry that he is sometimes supposed.
    • In mathematics, he corrected some principles of geometry.
    • he solved some most difficult problems, which had been sought in vain by the diligent scrutiny of the greatest geometers since the very beginnings of geometry; namely these: .
    • And so, after I had given sufficient attention to the problem by different methods, which were not understood by the professors of geometry, I added this newest one.

  86. Joachimsthal biography
    • Steiner had been appointed as an extraordinary professor of geometry two years before Joachimsthal began his studies and he had already made a name for himself as a leading expert on projective geometry.
    • Jacobi had made important contributions to geometry and at the time that Joachimsthal was studying in Konigsberg, Jacobi was working on his theory of determinants.
    • At the University of Berlin Joachimsthal taught courses on analytic geometry and calculus, giving more advanced courses on the theory of surfaces, the calculus of variations, statics and analytic mechanics.
    • [Joachimsthal] taught, among other things, analytic geometry, differential geometry, and the theory of surfaces, in which - exceptional for the time - he operated with determinants and parameters.
    • He gave special lectures on geometry and mechanics for students of mining engineering and metallurgy.
    • However, he laid it aside probably because other areas of mathematics, namely the elements of analytic geometry, occupied him.
    • Joachimsthal applied the theory of determinants to geometry.

  87. Vashchenko biography
    • In particular he worked on the theory of linear differential equations, the theory of probability (see [A N Bogolyubov (ed.), On the history of the mathematical sciences 167 \'Naukova Dumka\' (Kiev, 1984), 36-39.',3)">3]) and non-euclidean geometry.
    • We also mention Vashchenko-Zakharchenko's Analytic geometry which he published in 1887.
    • Besides numerous and extensive notes, and additions to the text, designed to render Euclid's treatment of geometry more palatable to modern taste, and to fill up some lacunae in the old work, the author has prefixed to his translation a valuable dissertation on the axioms and postulates and on the so-called non-Euclidean geometry of Bolyai and Lobachevsky, of which a sufficiently full sketch is presented.
    • In fact I [EFR] can confirm that 100 years later, in the 1950s, Euclid was still being used as a textbook in Britain for I was taught geometry from Euclid at secondary school.
    • In 1880 Vashchenko-Zakharchenko professor of mathematics at the University of Kiev, and an active advocate of teaching geometry in Gymnasium according to Euclid, translated Euclid's 'Elements' into Russian with historical commentary.
    • Three years later he published the first volume of 'History of Mathematics' which was devoted mainly to geometry from antiquity to the Renaissance.
    • Despite the fact that Vashchenko-Zakharchenko's translation of Euclid was free and sometimes inaccurate, and that his 'History of Mathematics: Historical treatise on the development of geometry, Volume 1' was little more than a compilation of the works of Western European authors, especially Moritz Cantor, both work were of considerable importance.
    • Vashchenko-Zakharchenko wrote on other historical topics too; for example he wrote a history of the development of analytic geometry.

  88. Geiser biography
    • He was awarded his doctorate in July 1866 for his dissertation Beitrage zur synthetischen Geometrie (Contributions to synthetic geometry).
    • In Zurich he took on the duties of a professor after the death of Joseph Wolfgang von Deschwanden (1819-1866), the professor of descriptive geometry, until the chair could be filled.
    • At Zurich Polytechnikum, Geiser was appointed as an extraordinary professor in 1869, and then in 1873 he was appointed to a full professorship of higher mathematics and synthetic geometry, with special responsibility for teaching mathematics to engineering students and to mathematics students.
    • Geiser taught algebraic geometry (his own research topic), differential geometry and invariant theory at Zurich.
    • Geiser published on algebraic geometry and minimal surfaces.
    • He is also remembered by those working in algebraic geometry for his discovery of an involution, now named after him, which appears in his paper Zwei Geometrische Probleme (1867).
    • There devolved upon him, above all, the instruction of candidates for the teaching of algebraic geometry, differential geometry, and invariant theory.

  89. Snyder biography
    • This situation also occurs with the Plucker line-coordinates so that the parallel between line geometry in three-space and Lie's "Kugelgeometrie" was apparent.
    • During more than forty years at Cornell University, Professor Snyder has devoted himself whole-heartedly, and with high idealism, to improving the teaching of mathematics, to promoting the welfare of his students and guiding them into research, and to carrying on his own original work in the fields of geometry of the line and sphere, configurations of ruled surfaces, and birational transformations.
    • suggested that the department of mathematics introduce a regular course in descriptive geometry as an alternative for those Arts students who ought to have some mathematics, but who found the calculus too difficult or too unattractive to be studied with profit.
    • This work is not to overlap with the instruction given in descriptive geometry by the technical colleges of the university, but is planned to furnish an insight into the processes and methods of graphical representation of various kinds.
    • He published (with James McMahon) Treatise on Differential Calculus (1898), (with John I Hutchinson) Differential and Integral Calculus (1902), (with John H Tanner) Plane and Solid Geometry (1911), (with John I Hutchinson) Elementary Textbook on the Calculus (1912), and (with Charles H Sisam) Analytic Geometry of Space (1914).
    • It is probable that in no branch of elementary mathematics has there been such need of a good, teachable book as in the analytic geometry of space.
    • From 1924 to 1934 he was chairman of the National Research Council Committee on Rational Transformations and during this time five of his papers appeared under the title Selected Topics in Algebraic Geometry in the Bulletin No.
    • 63 (1928) and four papers under the title Selected Topics in Algebraic Geometry II.

  90. Schoen biography
    • for his work on the application of partial differential equations to differential geometry, in particular his completion of the solution to the Yamabe Problem in "Conformal deformation of a Riemannian metric to constant scalar curvature".
    • Schoen, 40, continues his research in differential geometry, nonlinear partial differential equations and the calculus of variations.
    • In 1994 they published Lectures on differential geometry.
    • The book contains significant results in differential geometry and global analysis; many of them are the works of the authors.
    • There are nine chapters in the book, with the last three chapters more like appendices, which focus on problems concerning different areas of differential geometry.
    • With details of proofs and background materials presented in a concise and delightful way, the book provides access to some of the most exciting areas in differential geometry.
    • He serves on the editorial boards of: the Journal of Differential Geometry, Communications in Analysis and Geometry, Communications in Partial Differential Equations, Calculus of Variations and Partial Differential Equations, and Communications in Contemporary Mathematics.
    • His research has fundamentally shaped geometric analysis, and his results form many cornerstones within geometry, partial differential equations and general relativity.

  91. Menaechmus biography
    • Amyclas of Heraclea, one of the associates of Plato, and Menaechmus, a pupil of Eudoxus who had studied with Plato, and his brother Dinostratus made the whole of geometry still more perfect.
    • Stobaeus tells the rather familiar story which has also been told of other mathematicians such as Euclid, saying that Alexander the Great asked Menaechmus to show him an easy way to learn geometry to which Menaechmus replied (see for example [Dictionary of Scientific Biography (New York 1970-1990).
    • O king, for travelling through the country there are private roads and royal roads, but in geometry there is one road for all.
    • 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.
    • Plutarch says that Plato disapproved of Menaechmus's solution using mechanical devices which, he believed, debased the study of geometry which he regarded as the highest achievement of the human mind.
    • 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.
    • 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]:- .

  92. Cesaro biography
    • Cesaro was particularly interested in lectures he attended given by Darboux on geometry and this led him to make his own studies of intrinsic geometry along similar lines.
    • infinite arithmetics, isobaric problems, holomorphic functions, theory of probability, and, particularly, intrinsic geometry.
    • Cesaro's main contribution was to differential geometry.
    • Influenced by Darboux while in Paris he formulated 'intrinsic geometry'.
    • Cesaro later pointed out that in fact his geometry did not use the parallel axiom so constituted a study of non-euclidean geometry.
    • In addition to differential geometry Cesaro worked on many topics such as number theory where, in addition to the topics we mentioned above, he studied the distribution of primes trying to improve on results obtained in this area by Chebyshev.
    • History Topics: A History of Fractal Geometry .

  93. Fano biography
    • In fact Castelnuovo had been appointed as D'Ovidio's assistant in Turin the year before Fano began his studies and Corrado Segre had been appointed to the chair of higher geometry in Turin the year that Fano entered the University of Turin.
    • This was an exciting place for research in geometry and it is not surprising that Fano was led to specialise in this area.
    • Twenty years earlier, in 1872, Klein had produced his synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872).
    • The Erlanger Programm gave the unified approach to geometry that is now the standard accepted view.
    • Fano's work was mainly on projective and algebraic geometry.
    • Fano was a pioneer in finite geometry and one of the first people to try to set geometry on an abstract footing.
    • Early studies deal with line geometry and linear differential equations with algebraic coefficients ..
    • Fano wrote many textbooks, examples of which are his famous geometry texts Lezioni di geometria descrittiva (1914) and Lezioni di geometria analitica e proiettiva (1930).

  94. Deligne biography
    • Andre Weil gave for the first time a theory of varieties defined by equations with coefficients in an arbitrary field, in his Foundations of Algebraic Geometry (1946).
    • Weil's work related questions about integer solutions to polynomial equations to questions in algebraic geometry.
    • This work brought together algebraic geometry and algebraic number theory and it led to Deligne being awarded a Fields Medal at the International Congress of Mathematicians in Helsinki in 1978.
    • The areas on which he has worked, in addition to algebraic geometry, are Hilbert's 21st problem, Hodge theory, theory of moduli, modular forms, Galois representations, L-series and the Langlands conjectures, and representations of algebraic groups.
    • for his fundamental research in algebraic geometry.
    • in recognition of his monumental contributions to algebraic geometry.
    • It has turned Grothendieck's philosophy of motives from a conjectural program into what is the driving force behind many of the most subtle areas of current algebraic geometry and arithmetic.
    • Through an unparalleled blend of penetrating insights, fearless technical mastery and dazzling ingenuity, Deligne has singlehandedly brought about a new understanding of the cohomology of varieties, both classical and in finite characteristic, with numerous applications to deep problems in geometry and number theory.
    • for major contributions to several important domains of mathematics (like algebraic geometry, algebraic and analytic number theory, group theory, topology, Grothendieck theory of motives), enriching them with new and powerful tools and with magnificent results such as his spectacular proof of the "Riemann hypothesis over finite fields" (Weil conjectures).

  95. Richmond biography
    • However his love for mathematics soon returned and, after writing a dissertation on algebraic geometry, he was awarded a Fellowship to King's College, Cambridge, in 1888.
    • His main work was in algebraic geometry.
    • Further, the Italians, Corrado Segre and Castelnuovo were opening the way into a vast unexplored field, geometry of more than three dimensions.
    • Nevertheless, methods of elementary algebra may still be employed with success both in geometry proper and in applications such as arithmetical properties of rational functions.
    • It is true that the scope of these methods is restricted, but there is compensation in the fact that when geometry is successful in solving a problem the solution is almost invariably both simple and beautiful.
    • Richmond's mathematical researches lay in the field of pure and algebraic geometry, though he also lectured to generations of undergraduates on differential geometry.
    • His forte lay in seeing the relations between apparently diverse theorems, and he was especially at home in the projective properties of figures in spaces of more than three dimensions, but he did not disdain the consideration of elementary theorems in plane geometry; he would remark a little sadly that his results were remote from the trends of modern geometry.

  96. Cartan biography
    • He was appointed as Professor of Rational Mechanics in 1920, and then Professor of Higher Geometry from 1924 to 1940.
    • Cartan worked on continuous groups, Lie algebras, differential equations and geometry.
    • His work is a striking synthesis of Lie theory, classical geometry, differential geometry and topology which was to be found in all Cartan's work.
    • He developed this theory between 1894 and 1904 and applied his theory of exterior differential forms to a wide variety of problems in differential geometry, dynamics and relativity.
    • From 1916 on he published mainly on differential geometry.
    • Klein's Erlanger Programm was seen to be inadequate as a general description of geometry by Weyl and Veblen and Cartan was to play a major role.
    • Cartan further contributed to geometry with his theory of symmetric spaces which have their origins in papers he wrote in 1926.

  97. Montesano biography
    • Given that Battaglini and Cremona were distinguished geometers, it is not surprising that Montesano's research was in geometry.
    • He lectured on projective geometry and his lecture notes were published as Lezioni di geometria proiettiva (1887).
    • By 1888 Montesano had 13 publications and was in a strong position when he entered the competition for the chair of descriptive and projective geometry at the University of Bologna.
    • In 1893 he entered the competition for the chair of projective geometry at the University of Federico II at Naples.
    • Two years later, Montesano was promoted to ordinary professor of higher geometry at Naples.
    • Let us note that filling the chair of descriptive and projective geometry at the University of Bologna, which Montesano vacated in 1893 when he went to Naples, remained unfilled until 1896 when a competition was held.
    • Montesano's research was almost all in geometry, and he followed the lines of research that he had begun under the influence of his two teachers Battaglini and Cremona.
    • One of the unsolved problems of modern algebraic geometry is the determination of involutions of minimum order, in particular, to determine whether involutions of order two exist that are certainly irrational.

  98. Griffiths biography
    • Part 1 Analytic geometry (American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2003), xiii-xiv.',6)">6] Griffiths describes his time as a research student in Princeton:- .
    • In those sessions I learned, among other things, Lie groups, sheaf cohomology, complex manifolds and differential geometry (the equivalence of the then-existing seven definitions of a connection).
    • The four volumes [1] on analytic geometry, [2] on algebraic geometry, [3] on variations of Hodge structures, and [4] on differential systems total 2600 pages yet present only a selection of Griffiths' work (many of his papers and all eleven of his books are omitted).
    • Though the papers selected cover a broad range of topics in complex analysis, algebraic geometry and differential equations ..
    • in 1978) Topics in algebraic and analytic geometry (1974); Entire holomorphic mappings in one and several complex variables (1976); Principles of algebraic geometry (1978); An introduction to the theory of special divisors on algebraic curves (1980); (with John W Morgan) Rational homotopy theory and differential forms (1981); Exterior differential systems and the calculus of variations (1983); (with Gary R Jensen) Differential systems and isometric embeddings (1987); Introduction to algebraic curves (1989); and (with Mark Green, a doctoral student of Griffiths' who was awarded his Ph.D.
    • Studies; Annals of Mathematics; Selecta Mathematica; Duke Mathematical Journal; Compositio Mathematica; and the Journal of Differential Geometry.

  99. Vranceanu biography
    • In 1929 Vranceanu moved to Cernauti University where he was appointed professor of analytical geometry, then still at Cernauti he was appointed professor of Differential and Integral Geometry in the following year.
    • In 1948 Vranceanu was appointed Head of Geometry and Topology at Bucharest University.
    • Meanwhile Vranceanu made new discoveries in global geometry.
    • He formed his own group of young geometers and together they wrote teaching texts, as well as the 4 volumes of a differential geometry text, later translated in German and French.
    • They cover all the branches of modern geometry, from the classical theory of surfaces to the notion of non-holonomic spaces which he discovered, creating efficient methods and solving fundamental problems.
    • Other topics he studied include the absolute differential calculus of congruences, analytical mechanics, partial differential equations of the second order, non-holonomic unitary theory, conformal connection spaces, metrics in spherical and projective spaces, Lie groups, global differential geometry, discrete groups of affine connection spaces, locally Euclidean connection spaces, Riemannian spaces of constant connection, differentiable varieties, embedding of lens spaces into Euclidean space, tangent vectors of spheres and exotic spheres, the equivalence method, non-linear connection spaces, and the geometry of mechanical systems.

  100. 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.
    • Pappus's major work in geometry is Synagoge or the Mathematical Collection which is a collection of mathematical writings in eight books thought to have been written in around 340 (although some historians believe that Pappus had completed the work by 325 AD).
    • 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.
    • There are, we say, three types of problem in geometry, the so-called 'plane', 'solid', and 'linear' problems.
    • This problem had a major impact on the development of geometry.
    • 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.
    • Pappus on analysis and synthesis in geometry .

  101. Briggs biography
    • In 1596 Briggs became the first professor of geometry at Gresham College, London which had just been founded.
    • In 1619 Savile founded a chair of geometry at Oxford because:- .
    • geometry is almost totally unknown and abandoned in England.
    • These include works on the geometry of Ramus and on Longomontanus's treatise on squaring the circle.
    • A treatise on geometry and another on arithmetic were similarly never published.
    • the mirror of the age for excellent skill in geometry.
    • Barrow, born the year the Briggs died, was appointed professor of geometry at Gresham College in 1662.
    • Savilian Geometry Professor1619 .

  102. Zeuthen biography
    • He decided to visit Paris and there he studied geometry with Chasles.
    • The first topic on which Zeuthen undertook research was enumerative geometry.
    • Up until 1875 Zeuthen worked almost exclusively on enumerative geometry.
    • He began to write on mechanics and he also made significant contributions to algebraic geometry, particularly the theory of algebraic surfaces.
    • The move towards rigour in geometry led to this theory being neglected for many years but recently some of the remarkable numerical results produced by it have been confirmed.
    • It looked in detail at the work of Descartes, Viete, Barrow, Newton and Leibniz as he traced the development of algebra, analytic geometry and analysis.
    • Kleiman gives an interesting biography of Zeuthen in [Enumerative algebraic geometry, Contemp.
    • This approach characterised his historical research equally with his work on enumerative methods in geometry.

  103. Connes biography
    • (4) application of the theory of C*-algebras to foliations and differential geometry in general.
    • After describing these contributions, Araki also notes other work of Connes such as his applications of operator algebras to differential geometry and his work on non-commutative integration theory which he published in 1979.
    • Connes' recent work has been on noncommutative geometry and he published a major text on the topic in 1994.
    • Connes continues to hold the Leon Motchane Chair at the Institut des Hautes Etudes Scientifiques and the Chair of Analysis and Geometry in the College de France.
    • for revolutionizing the field of operator algebras, for inventing modern non-commutative geometry, and for discovering that these ideas appear everywhere, including the foundations of theoretical physics.
    • for his penetrating work on the theory of operator algebras and for having been a founder of the non-commutative geometry.
    • He revolutionized the theory of operator algebras and was a primary founder of a new branch of mathematics - noncommutative geometry.
    • AMS (Work on non-commutative geometry) [registration required] .

  104. Cremona biography
    • There are eight book reviews and four historical articles intended to encourage research into geometry.
    • The apparatus of algebraic geometry is built upon polars, and these upon distances.
    • And the beginnings of enumerative geometry are here..
    • He wrote articles on such diverse topics as twisted cubics, developable surfaces, the theory of conics, the theory of plane curves, third- and fourth-degree surfaces, statics and projective geometry.
    • Cremona had a large influence on geometry in Italy.
    • Many of Steiner's proofs on synthetic geometry were revised and improved by Cremona.
    • One of the themes which were present in almost all his work throughout his career was projective geometry.
    • Cremona had many pupils who were to make major contributions to geometry, for example Bertini, Veronese and Guccia.

  105. Nagata biography
    • In Kyoto he joined the Kyoto School of Algebraic Geometry which was being developed by Yasuo Akizuki and attracting many talented young mathematicians.
    • For example Heisuke Hironaka, who won a Fields Medal in 1970, was in his final undergraduate year at Kyoto when Nagata arrived and he continued to work in the Kyoto School of Algebraic Geometry for the following four years.
    • Nagata played outstanding roles, especially in the 1950s and 1960s, in the development of commutative algebra and algebraic geometry.
    • A series of papers in the late 1950s on algebraic geometry over Dedekind domains laid the foundation for later developments of algebraic geometry in terms of schemes.
    • The completion of algebraic varieties - that is, embedding of algebraic varieties as open subvarieties of complete varieties - published in his paper in 1962, remains one of the basic techniques in algebraic geometry.
    • With Yosikazu Nakai, he published Algebraic geometry (Japanese) in 1957.
    • If you can read Japanese, know some algebraic geometry already and have enough nerves not to mind the five-page errata, then you will greatly enjoy reading this book.

  106. Baker biography
    • From 1903 to 1914 he also held the Cayley Lectureship in Mathematics, then from 1914 until he retired in 1936 he was Lowndean Professor of Astronomy and Geometry.
    • He also came in contact with the Italian School of geometry and made their work the subject of his 1911 London Mathematical Society presidential address.
    • From 1911 he studied birational geometry publishing his most important contribution, a six volume masterpiece Principles of Geometry from 1922 to 1925.
    • The first two volumes cover the foundations of Euclidean geometry and the introduction of a coordinate system, volume 3 studies solid geometry considering quadrics, cubic curves in space, and cubic surfaces.
    • In 1943 Baker published An Introduction to Plane Geometry which was reprinted in 1971.
    • He founded the Saturday afternoon seminar or 'tea party' which became the focus of activity in geometry.

  107. Saccheri biography
    • Perhaps Giovanni Ceva had the greatest mathematical influence for his passion for geometry, seen in his book De lineis rectis (1678), encouraged Saccheri to work in this area.
    • In this book, which Saccheri dedicated to Guzman who was the governor of Milan, he solved many problems in elementary geometry.
    • It was not a particularly significant work but showed that Saccheri was becoming deeply involved in thinking about Euclidean geometry.
    • In Euclides ab Omni Naevo Vindicatus (Euclid cleared of every defect), published in 1733, he did important early work on non-euclidean geometry, although he did not see it as such, rather an attempt to prove the parallel postulate of Euclid.
    • Readers who are familiar with the basics of non-Euclidean geometry may be rather puzzled by this statement for they will know of geometries in which the angles in a triangle add to more than two right angles, making it possible for the two angles in Saccheri's quadrilateral each to be greater than a right angle.
    • In this case, after 20 more propositions he was unable to obtain any contradiction and he developed many theorems of non-Euclidean geometry.
    • It is fair to say that the discovery of non-Euclidean geometry by Nikolai Lobachevsky and Janos Bolyai was not due to this masterpiece by Saccheri.
    • History Topics: Non-Euclidean geometry .

  108. Kuiper biography
    • Kuiper continued his studies at Leiden working for a doctorate with Willem van der Woude as his advisor and was awarded the degree in 1946 for his thesis Onderzoekingen over lijnenmeetkunde which discussed a topic in classical differential geometry.
    • Using the Study method of dual vectors, the author develops the line geometry in a Euclidean three-space (without using point coordinates).
    • This enables him to state a "translation rule," which converts theorems in spherical or two-dimensional elliptic geometry to line geometry theorems in Euclidean three-dimensional space.
    • In 1959 he published the textbook Analytische meetkunde (verklaard met lineaire algebra) [Analytic geometry (interpreted by linear algebra)].
    • An English translation Linear algebra and geometry was published in 1962:- .
    • differential geometry, differential topology, and algebraic topology, and nurturing a number of doctoral students and post-doctoral visitors.
    • The book Tight and taut submanifolds contains a paper based on the Roever Lectures in Geometry Kuiper gave at Washington University in St Louis, USA, from 20 January to 24 January 1986.

  109. Pythagoras biography
    • Anaximander certainly was interested in geometry and cosmology and many of his ideas would influence Pythagoras's own views.
    • ',13)">13] says that Pythagoras learnt geometry from the Egyptians but it is likely that he was already acquainted with geometry, certainly after teachings from Thales and Anaximander.
    • Of course today we particularly remember Pythagoras for his famous geometry theorem.
    • After [Thales, etc.] Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.
    • Again Proclus, writing of geometry, said:- .
    • I emulate the Pythagoreans who even had a conventional phrase to express what I mean "a figure and a platform, not a figure and a sixpence", by which they implied that the geometry which is deserving of study is that which, at each new theorem, sets up a platform to ascend by, and lifts the soul on high instead of allowing it to go down among the sensible objects and so become subservient to the common needs of this mortal life.
    • In addition to his beliefs about numbers, geometry and astronomy described above, he held [Encyclopaedia Britannica.',2)">2]:- .

  110. Edge biography
    • After graduating, he continued working for his doctorate at Trinity on projective geometry.
    • Cambridge was at that time a centre for geometry research with Baker's school flourishing there.
    • After studying classical geometry, Edge moved towards the topic which is most associated with him, namely finite geometry.
    • Other topics Edge worked on, all of which exhibit his mastery of the subject, include nets of quadric surfaces, the geometry of the Veronese surface, Klein's quartic, Maschke's quartic surfaces, Kummer's quartic, the Kummer surface, Weddle surfaces, Fricke's octavic curve, the geometry of certain groups, finite planes and permutation representations of groups arising from geometry.
    • Edge taught the algebra courses at Edinburgh at this time but he taught algebra with a strong geometric flavour reflecting his deep knowledge, feel and love for geometry.

  111. Mori biography
    • Mori works on algebraic geometry.
    • to classify algebraic varieties has always been a fundamental problem of algebraic geometry and even an ultimate dream of algebraic geometers.
    • Mori's work achieved a remarkable continuation of classification efforts in algebraic geometry and in many ways provides a fitting chapter in the progress of algebraic geometry through the 20th century.
    • The most profound and exciting development in algebraic geometry during the last decade or so was the Minimal Model Program or Mori's Program in connection with th classification problems of algebraic varieties of dimension three.
    • In 1998 Mori published the monograph Birational geometry of algebraic varieties which he coauthored with Janos Kollar.
    • The minimal model program, or Mori's program, was one of the great successes of algebraic geometry in the 1980s.
    • The basic goal was to understand the birational geometry of threefolds in a way analogous to the birational theory of surfaces.

  112. Lax Anneli biography
    • Her interest in mathematics was awakened by Euclidean geometry, which she first studied in the lyceum in Berlin around 1935.
    • She was attracted to the constructions and logic of geometry, completing advanced problems with ease [Humanistic Mathematics Network Journal 21 (California State University Press, 1999).',6)">6]:- .
    • On her arrival in the United States, she studied geometry with a teacher she described as 'a lovely old lady, Miss Eaton' at a high school in Queens.
    • These two significant encounters with geometry convinced her that logic was what made mathematics satisfying and pleasing.
    • Because she could not see its underlying logical structure, the algebra class that followed on Miss Eaton's geometry class was a disappointment [Humanistic Mathematics Network Journal 21 (California State University Press, 1999).',6)">6]:- .
    • 'The Geometry of Numbers' was derived essentially from a raw manuscript left incomplete by C D Olds (1912 - 1979) who was a professor of mathematics at San Jose State University.
    • In the end I'm sure Anneli would have been pleased with the final result: a fine introduction to the geometry of numbers ..
    • Lax's view on mathematics deeply influenced Marchisotto's approach to the subject, particularly in the areas of analysis and geometry.

  113. Kagan biography
    • In 1922 he went to Moscow when the Department of Differential Geometry was founded at Moscow State University.
    • Kagan was the first Head of Department and he founded an important School of Differential Geometry.
    • He founded a publication associated with this seminar Transactions of the seminar on Vector and Tensor Analysis with its applications to Geometry, Mechanics and Physics in 1933.
    • In 1934 Kagan and other members of his School organised an International conference on differential geometry which took place at Moscow University.
    • Kagan worked on the foundations of geometry and his first work was on Lobachevsky's geometry.
    • Kagan studied tensor differential geometry after going to Moscow because of an interest in relativity.
    • Kagan wrote a history of non-euclidean geometry and also a detailed biography of Lobachevsky.

  114. Richard Jules biography
    • This paper discussed axiomatic projective geometry and built on work by Hilbert, von Staudt and Meray.
    • Then in 1908 Richard wrote Sur la nature des axiomes de la geometrie in which he looked critically at four different approaches to geometry:- .
    • Geometry is founded on arbitrarily chosen axioms - there are infinitely many equally true geometries.
    • Experience provides the axioms of geometry, the basis is experimental, the development deductive.
    • The axioms of geometry are definitions.
    • He then goes on to make his own suggestion as to how to approach geometry.
    • But of course, writes Richard, there is an ultimate goal which must be kept in mind when approaching geometry:- .
    • Richard was thinking about geometry at a time when the non-euclidean geometries had been discovered.

  115. Singer biography
    • In particular he studied group theory and differential geometry realising the importance of having a strong background in mathematics.
    • Singer is justifiably famous among mathematicians for his deep and spectacular work in geometry, analysis, and topology, culminating in the Atiyah-Singer Index theorem and its many ramifications in modern mathematics and quantum physics.
    • They have spawned many developments in differential geometry, differential topology, and analysis ..
    • However, [the Index Theorem] represents only a small part of his contributions to geometry and analysis.
    • Other significant contributions to geometry were his work with D B Ray on analytic torsion, the precursor of much modern work on "determinant" invariants in geometry, and an influential textbook joint with J A Thorpe, Lecture Notes on Elementary Topology and Geometry ..
    • for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.

  116. Tilly biography
    • In 1858 he was assigned to teach a mathematics course at the regimental school and it was from this time on that he undertook research into geometry.
    • Of course he was not in position to have contacts with other mathematicians and for a long period he was completely out of touch with modern developments in geometry which were in fact highly relevant to the research he was undertaking.
    • Tilly studied the principles of geometry, Euclid's fifth postulate and non-euclidean geometry without being aware that this had become a major topic of interest.
    • It was only in 1866 that he learnt about the work of the famous Russian mathematician on non-euclidean geometry, then in 1870 Tilly published a work Etudes de mechanique abstraite on Lobachevsky space.
    • In this work Tilly was the first to study non-euclidean mechanics, a topic he essentially invented (see [Conference on the History of Mathematics (Italian), Cetraro, 1988 (EditEl, Rende, 1991), 57-75.',2)">2] for details of his contributions to the link between geometry and "physical theories").
    • 32 (1986), 3-10; 90.',4)">4] Semenets examines some aspects of the researches on the foundations of geometry in the second half of the nineteenth century, in particular looking at axiomatic systems proposed by Tilly in his Essai sur les principes fondamentaux de la geometrie et de la mechanique (1878) [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • geometry was the mathematical physics of distances.

  117. Kiselev biography
    • Here are the most important of Kiselev's textbooks: Systematic arithmetic course for secondary schools (1884); Elementary Algebra (1888); Elementary Geometry (1892-1893); Additional topics in algebra (a course of the 7th grade real schools) (1893); Quick arithmetic for urban schools (1895); A Brief algebra for girls' schools and seminaries (1896); An elementary physics for secondary schools with a number of exercises and problems (1902); Physics (two volumes) (1908); Elements of the differential and integral calculus (1908); The initial study of derivatives for the 7th grade real schools (1911); Graphical representation of some of the features discussed in elementary algebra (1911); On topics in elementary geometry, which are usually solved by means of limits (1916); Brief algebra (1917); Brief arithmetic for urban county schools (1918); Irrational numbers, considered as infinite non-periodic fractions (1923); and Elements of algebra and analysis (2 volumes) (1930-1931).
    • However, books on other subjects such as Physics, which went through 13 editions, and Elements of the differential and integral calculus which also went through multiple editions were also popular but never came near to matching the incredible popularity of Systematic arithmetic course for secondary schools, Elementary Algebra and Elementary Geometry.
    • A review of Elementary Geometry written in 1893, the year after the book was published, states quite correctly that it was:- .
    • This was clearly the case for, in the Preface to the first edition of this book, Kiselev quotes ten geometry courses in French and German published in the previous decade.
    • In the course in geometry - Kiselev's most popular course - practically all of the assertions are grounded and proved.
    • Slowly they returned to full use after the Moscow Mathematical Society, at a meeting held on 9 April 1937, recommended that Kiselev's Geometry should be, for the time being, the book used by schools.
    • We give an idea of the style of Kiselev's Geometry by quoting from Alex Bogomolny's review of Planimetry [Mathematical Association of America (10 September 2006).',4)">4]:- .

  118. Chasles biography
    • In 1837 Chasles published his first major work Apercu historique sur l'origine et le developpement des methodes en geometrie (Historical view of the origin and development of methods in geometry) which quickly made his reputation as both a mathematician and as an historian of mathematics.
    • a philosophical examination of the different methods in modern geometry, in particular the method of reciprocal polars.
    • In Apercu historique Chasles studied the method of reciprocal polars as an application of the principle of duality in projective geometry; in the same way the principle of homography leads to a great number of properties of quadric surfaces.
    • In 1846 he was appointed to a chair of higher geometry at the Sorbonne which had been specially created for him.
    • He also wrote an extremely important text on geometry showing the power of synthetic geometry.
    • Questions of this type go back to Apollonius, but such questions had arisen while Chasles was working on geometry, in particular the Steiner "problem of five conics" was posed in 1848.
    • Chasles' developed a theory of characteristics to solve this problem and Chasles's characteristic formula is discussed in [Studies in algebraic geometry (Washington, D.C., 1980), 117-138.',5)">5].

  119. Kodaira biography
    • By the time he had completed half of the three year course, he had covered the whole syllabus of arithmetic, algebra, 2- dimensional and 3-dimensional geometry and solved all the problems in the set text.
    • He was accepted but later Suetuna wrote to him suggesting that he thought that studying geometry in Iyanaga's seminar would be more appropriate.
    • During his time at Princeton, Kodaira continued his involvement with harmonic forms, particularly in their application to algebraic geometry, the area which had also provided the motivation for Hodge's work.
    • The 1950s saw a great flowering of complex algebraic geometry, in which the new methods of sheaf theory, originating in France in the hands of Leray, Cartan and Serre, provided a whole new machinery with which to tackle global problems.
    • These papers altered the face of algebraic geometry, and provided the framework in which Hirzebruch and others of the younger generation were able to make spectacular progress.
    • This time through the influence of Hodge, he worked on harmonic integrals and later he applied this work to problem in algebraic geometry.
    • Another important area of Kodaira's work was to apply sheaves to algebraic geometry.
    • Professor Kunihikio Kodaira made a profound study of harmonic integrals with incisive, important applications to algebraic and complex geometry.

  120. Clebsch biography
    • Clebsch proved himself an outstanding teacher and combined his teaching skills with his research skills in building a school of algebraic geometry and invariant theory at Giessen which included Paul Gordan, Alexander Brill, Max Noether, Ferdinand von Lindemann and Jacob Luroth.
    • It was Otto Hesse who had advised Clebsch to investigate the algebraic geometry of Cayley, Sylvester and Salmon and he was particularly attracted to the contributions that Aronhold had made to their theories.
    • Clebsch went back to Abel's approach to algebraic geometry and, rather than the geometric approach of Riemann.
    • But he not only transplanted to German soil their theory of invariants and the interpretation of projective geometry by means of this theory; he also brought this theory into live and fruitful correlation with the fundamental ideas of Riemann's theory of functions.
    • Clebsch's work on algebraic geometry Uber die Anwendung der Abelschen Functionen in der Geometrie (1864) was published in Crelle's Journal and is described by Igor Shafarevich in [Math.
    • birth cry of modern algebraic geometry.
    • By 1868, when Clebsch accepted the chair formerly held by Riemann at Gottingen, he and his entourage of students were turning out so much new material on algebraic geometry and invariant theory that they began making plans for the inauguration of a new journal designed to give their work more visibility.
    • Two volumes of his lectures on geometry were published after his death in 1876 and 1891.

  121. Apery biography
    • Perhaps even at this stage the passion that he had for geometry throughout his life became evident.
    • He continued to be fascinated by mathematics, however, and in 1932 he met the cross-ratio as a projective invariant; his love of geometry became more specific, turning into a love of algebraic geometry.
    • He published many papers on Italian style algebraic geometry over the next few years: Sur les courbes d'ordre n ayant un point multiple O d'ordre n - 4 et n - 2 tacnodes, les tangentes tacnodales passant par O (1941) and Sur les quintiques a cinq rebroussements (1941) were followed by four papers in 1942, and another four in 1943.
    • Paul Dubreil, who spent the years of World War II in Nancy, returned to Paris in 1946 and advised Apery on submitting his thesis on algebraic geometry and ideals, which he did in 1947.
    • Returning to Apery's career, he was invited to give the prestigious Cours Peccot at the College de France in 1948; he spoke on "Algebraic geometry and ideals".
    • As we noted above, his early work was on Italian style algebraic geometry.

  122. Cech biography
    • He had little interest in physics so he choose courses which were within the area of mathematics and descriptive geometry.
    • His interest in a new area of mathematics, namely projective differential geometry, led to his first paper on the topic appearing in 1921 and, in the same year, he obtained a scholarship from the Ministry of Education to go to Italy and study with Fubini in Turin.
    • Čech studied there between 1921 and 1922 and he clearly impressed Fubini who asked him to cooperate with him in joint project to write a monograph on projective differential geometry.
    • Čech was interested in geometry but he was appointed to the chair of analysis at Masaryk University since this had been Lerch's chair.
    • There was another professor of mathematics at Masaryk University, namely Seifert, who held the geometry chair.
    • although geometry was Čech's field of research, Čech had to take over courses in analysis and algebra.
    • In the 1950s his mathematical interests turned to differential geometry and after a gap through the war years he began to publish again, writing 17 papers on that topic.

  123. Sidler biography
    • He attended lectures by J Bertrand (analysis), M Chasles (geometry), H Faye (astronomy), G Lame (mathematical physics), U J Le Verrier (popular astronomy), J Liouville (differential equations) and V Puiseux (celestial mechanics).
    • There he attended lectures on a range of mathematical topics by Dirichlet, on theoretical astronomy by Encke, on geodesy by Bremiker, on mathematical physics by Clausius and on geometry by Steiner.
    • Throughout the years he lectured on a wide variety of topics, including algebra, analysis, arithmetic, astronomy, various areas of geometry, and mathematical physics.
    • We note that Sidler had a particular interest in the geometry of triangles and collected books on this topic.
    • In addition, Sidler published a number of papers and talks on astronomical problems and phenomena as well as on problems in analysis and geometry.
    • Whilst he mainly lectured on analytic geometry, infinitesimal calculus, theory of functions and number theory, Sidler primarily gave lectures on theoretical astronomy and synthetic geometry.

  124. Tarski biography
    • Tarski made important contributions in many areas of mathematics: set theory, measure theory, topology, geometry, classical and universal algebra, algebraic logic, various branches of formal logic and metamathematics.
    • Tarski presented his paper The axiomatic method : with special reference to geometry and physics to the International Symposium held at the University of California at Berkeley from 26 December 1957 to 4 January 1958.
    • His paper, appearing in the Proceedings of the Conference in 1959, gives his axiom system for geometry.
    • His work includes Geometry (1935), Introduction to Logic and to the Methodology of Deductive Sciences (1936), A decision method for elementary algebra and geometry (1948), Cardinal Algebras (1949), Undecidable theories (1953), Logic, semantics, metamathematics (1956), and Ordinal algebras (1956).
    • In A decision method for elementary algebra and geometry Tarski showed that the first-order theory of the real numbers under addition and multiplication is decidable which is in contrast, in a way which is really surprising to non-experts, to the results of Godel and Church who showed that the first-order theory of the natural numbers under addition and multiplication is undecidable.
    • In Undecidable theories Tarski showed that group theory, lattices, abstract projective geometry, closure algebras and others mathematical systems are undecidable.

  125. Zylinski biography
    • In addition to teaching analytic geometry, set theory and higher algebra at the Polish College, he also taught at the Ukrainian State University and at the Higher Technical Institute.
    • These textbooks are (all in Polish): (with Stanislaw Ruziewicz) Algebra: handbook for the higher classes of secondary schools (1926); (with Stanislaw Ruziewicz) Algebra: handbook for the higher classes of middle schools (1926); (with Stanislaw Ruziewicz) Introduction to Mathematics, Algebra I (1927); (with Stanislaw Ruziewicz) Algebra: handbook for the higher classes of secondary schools (1928); Introduction to arithmetic theory (1932); and Analytical Geometry (1938).
    • He was much influenced by Żyliński's courses on Analytical Geometry, assisting in their teaching, and published Analytic Geometry with Particular Regard to the Textbook of Eustachy Żyliński in 1951.
    • This textbook deals with metric, affine and projective geometry of linear and quadratic varieties in the plane as well as in the three-space.
    • Besides items usually dealt with in textbooks of elementary analytic geometry the reader finds here the introduction to synthetic projective geometry, to the theory of matrices (and determinants) with the usual applications and to the (three-dimensional) elementary vector calculus.

  126. Rethy biography
    • He graduated with a degree in mathematics and descriptive geometry from the Technical University of Budapest in 1870.
    • Following his graduation, Rethy worked for two years as a teacher of mathematics and descriptive geometry at the Modern Technical School of Kormocbanya.
    • This paper throws light on Rethy's efforts to introduce and popularize absolute geometry.
    • And he went a step further: starting from the fact that in absolute geometry in infinitely small segments of space the theorems of Euclidean geometry hold true, and relying on the recognition that spherical trigonometry is independent of Euclid's 5th postulate, Rethy built up Bolyai's trigonometry independently.
    • In 1886 Mor Rethy was invited to the Technical University of Budapest, where he first lectured on geometry.
    • The atmosphere of these celebrations led me back to the absolute geometry.

  127. Lambert biography
    • He noticed that in this new geometry the sum of the angles of a triangle increases as its area decreases.
    • Of his work on geometry, Folta writes in [DVT-Dejiny Ved a Techniky 6 (1973), 189-205.',14)">14]:- .
    • Lambert tried to build up geometry from two new principles: measurement and extent, which occurred in his version as definite building blocks of a more general metatheory.
    • His axioms concerning number can hardly be compared with Euclid's arithmetical axioms; in geometry he goes beyond the previously assumed concept of space, by establishing the properties of incidence.
    • Lambert's physical erudition indicates yet another clear way in which it would be possible to eliminate the traditional myth of three-dimensional geometry through the parallels with the physical dependence of functions.
    • He also made a major contribution to philosophy and in Anlage zur Architectonic (1771) he attempted to transform philosophy into a deductive science, modelled on Euclid's approach to geometry.
    • History Topics: Non-Euclidean geometry .

  128. Chern biography
    • He was the only graduate student in mathematics to enter the university in 1930 but during his four years there he not only studied widely in projective differential geometry but he also began to publish his own papers on the topic.
    • Although Chern knew Artin well and would have liked to have worked with him, the desire to continue work on differential geometry was the deciding factor and he went to Paris.
    • From 1949 Chern worked in the USA accepting the chair of geometry at the University of Chicago after first making a short visit to Princeton.
    • His area of research was differential geometry where he studied the (now named) Chern characteristic classes in fibre spaces.
    • When Chern was working on differential geometry in the 1940s, this area of mathematics was at a low point.
    • Global differential geometry was only beginning, even Morse theory was understood and used by a very small number of people.
    • Today, differential geometry is a major subject in mathematics and a large share of the credit for this transformation goes to Professor Chern.

  129. Szekeres biography
    • Szekeres and Erdős wrote a paper in 1935 generalising this result; it became one of the cornerstones of combinatorial geometry.
    • Besides combinatorial geometry, he has also made contributions in the theory of partitions, graph theory, and other areas of combinatorics.
    • More recently, his research interests include combinatorial geometry, Hadamard determinants, and chaos theory.
    • There was the 1957 paper Spinor geometry and general field theory which Szekeres described in the introduction as follows:- .
    • The purpose of the present work is to exploit, more fully than has been done heretofore, the possibilities of spin connection from the point of view of geometrical field theories and to develop a geometry whose connection is derived exclusively from the displacement of spinors ..
    • In the same year he published On the propagation of gravitational waves and On some extremum problems in elementary geometry, this latter paper being written with Erdős.
    • He continued to publish on relativity with work such as Kinematic geometry: An axiomatic system for Minkowski space-time (1968).

  130. Thurston biography
    • Thurston has fantastic geometric insight and vision: his ideas have completely revolutionised the study of topology in 2 and 3 dimensions, and brought about a new and fruitful interplay between analysis, topology and geometry.
    • In 1976 his work on foliations led to his being awarded the Oswald Veblen Geometry Prize of the American Mathematical Society.
    • In 1997 he published Three-dimensional geometry and topology.
    • In 1978, W Thurston gave a course at Princeton University, whose subject was the geometry and topology of three-dimensional manifolds.
    • These notes created a new circle of ideas, and the expression "Thurston type geometry" has become very common.
    • The 1978 Princeton lecture notes, although written in an informal style, are self-contained and accessible to graduate students in topology or geometry.
    • On 6 January 2005, at the Joint Mathematics Meetings in Atlanta, Georgia, Thurston was awarded the American Mathematical Society Book Prize for Three-dimensional geometry and topology.

  131. Du Val biography
    • They got to know Henry Baker and he persuaded Patrick to undertake research into algebraic geometry at Cambridge; Du Val matriculated as a research student at Cambridge in 1927.
    • With Baker as his thesis advisor, Du Val wrote the thesis On certain configurations of algebraic geometry, having groups of self-transformations representable by the symmetry groups of certain polytopes for which he received a Ph.D.
    • The 100th meeting of the British Association was unquestionably one of the very greatest interest to mathematicians of almost all kinds; though by a conjunction of accidents straightforward algebraic geometry found itself almost completely left out of the programme.
    • Other branches of geometry were represented by Mr Coxeter, who gave an account of the modern analytical methods of discovering and completely enumerating the regular polytopes (a subject not as much studied in this country as one might wish), and by Mr Hodge, who explained the work which has been done towards the use of topology for the discussion of algebraic surfaces, a surface being represented by a "Riemann fourfold" in the same way as a curve may be by a Riemann surface.
    • He developed a strong interest in Byzantine culture and quickly mastered the Turkish language, in which some of his work is written, including an elementary textbook on coordinate geometry of which he was rather proud.
    • Together with Semple he led the London Geometry Seminar during the time he spent in London.
    • His doctorate was on algebraic geometry and in his thesis he generalised a result of Pieter Schoute.

  132. Hodge biography
    • His main mathematical interests were in algebraic geometry and differential geometry.
    • During this period he developed the relationship between geometry, analysis and topology and produced some of his best remembered work on the theory of harmonic integrals.
    • This work marked an important change in direction for the Cambridge school of geometry which, under Baker's leadership, had become somewhat isolated from other areas of mathematics.
    • In 1936 Hodge had been appointed as Lowndean Professor of Astronomy and Geometry, succeeding Baker, and he held this chair at Cambridge until 1970.
    • recognition of his distinguished work on algebraic geometry.
    • in recognition of his pioneering work in algebraic geometry, notably in his theory of harmonic integrals.

  133. Lueroth biography
    • It was working with Clebsch in Giessen that directed Luroth's research towards geometry and function theory.
    • From out of this capacity, which he maintained throughout his life, his actual productive activity has been developed, which stretches, with rare versatility, to geometry and mechanics, to astronomy and geodesy, to probability theory, set theory and the logical foundations of mathematics, on function theory and algebra.
    • As we have mentioned above, Luroth was taught by Hesse and Clebsch and because of their influence he continued to develop their work on geometry and invariants.
    • He published results in the areas of analytic geometry, linear geometry and continued the directions of his teachers in his publications on invariant theory.
    • Karl von Staudt's ideas of geometry interested Luroth and he further developed von Staudt's complex geometry in papers such as Uber das Rechnen mit Wurfen (1873) and Das Imaginare in der Geometrie und das Rechnen mit Wurfen.

  134. Viviani biography
    • Viviani learnt much from Galileo over this period, working with him on physics and geometry.
    • Despite his great expertise in geometry, Viviani seems to have done little original in this area, He did determine the tangent to the cycloid but he was not the first to succeed in this.
    • He believed passionately in the value of geometry, however, writing:- .
    • Speculative geometry is the unique teacher of the honest acquisition of what is useful, delightful, beautiful and good.
    • Geometry is the only true science because it produces knowledge from itself without mediation of causes.
    • Geometry alone teaches how to achieve knowledge and even reminds the human intellect - which is a spark of the divine one - that as a knower through the principles most known with the light of nature it can, if it so wishes, without deceiving itself or others, know the existence and properties of all things concerning the created universe and the order disposed by God, in number, weight, and measure.
    • This was reprinted in 1867 by Enrico Betti and Francesco Brioschi who, at that time, were trying to improve the teaching of geometry in Italy.

  135. Bose Raj biography
    • His main interests at that time were non-Euclidean geometry, n-dimensional geometry and global properties of convex curves.
    • He was guided by Shyamadas Mukherjee and studied geometry.
    • Finding time to undertake research was hard but he continued to work on geometry and published On the number of circles of curvature perfectly enclosing or perfectly enclosed by a closed convex oval in 1932.
    • Bose made important contributions to a number of areas of geometry including hyperbolic geometry and its application to statistics, multivariate statistical analysis, finite geometries, orthogonal Latin squares, experimental designs, balanced and partially balanced designs and association schemes, difference sets, orthogonal arrays, factorial designs, rotatable designs, coding theory, information theory, graph theory, projective geometry, partial geometries, characterization and embedding problems in designs and geometries, file organization, and additive number theory.

  136. Penrose biography
    • Roger's father became Director of Psychiatric Research at the Ontario Hospital in London Ontario, but he was very interested in mathematics, particularly geometry, while Roger's mother was also interested in geometry.
    • Roger, however, was set on research in mathematics and on entering St John's College he began research in algebraic geometry supervised by Hodge.
    • for his work in algebra and geometry from the University of Cambridge in 1957 but by this time he had already become interested in physics.
    • In that year he was appointed Gresham Professor of Geometry at Gresham College, London.
    • This volume covered two-spinor calculus and relativistic fields while the second volume covering spinor and twistor methods in space-time geometry appeared two years later.
    • Sir Roger Penrose, OM, FRS has been awarded the Royal Society's Copley medal the world's oldest prize for scientific achievement for his exceptional contributions to geometry and mathematical physics.

  137. Nirenberg biography
    • He published the results of his thesis in 1953 in the paper The Weyl and Minkowski problems in differential geometry in the large.
    • In Zurich he attended a lecture course by Heinz Hopf on geometry, a course by Bartel van der Waerden on Riemann surface theory, and lectures by Rolf Nevanlinna and Wolfgang Pauli.
    • The lecture course was aimed at students with a good grounding in linear operators and some familiarity with differential geometry, but limited knowledge of topology.
    • Newton's ideas were used to describe many different systems in physics and in geometry, which may depend on several variables simultaneously.
    • As an example from geometry one can mention the problem to find a surface with given curvature and from physics studies of the equations for viscose fluids and concerning existence of free streamlines.
    • The work of Louis Nirenberg has enormously influenced all areas of mathematics linked one way or another with partial differential equations: real and complex analysis, calculus of variations, differential geometry, continuum and fluid mechanics.
    • His range of interest is very broad: differential equations, harmonic analysis, differential geometry, functional analysis, complex analysis, etc.

  138. Rohn biography
    • In 1884 Rohn was promoted to extraordinary professor at Leipzig, then in 1887 he became a full professor at Technische Hochschule in Dresden where he held the chair of descriptive geometry.
    • In the following years, Rohn further developed these capacities and became an acknowledged master in all questions concerning the algebraic geometry of the real P2 and P3, where it is possible to overlook the different figures.
    • His love of geometry is also illustrated by his beautiful thread models which were especially produced to excite the curiosity of the uninitiated.
    • Dr Rohn was himself a pupil of Professor Klein, and the latter, in his appreciative introduction, speaks highly of his skill in the field of geometry.
    • The work sets forth in succinct form the essential features of modern projective geometry with respect to solids, thus extending the ordinary treatment of the projective properties of figures in a plane to those of three dimensions.
    • It begins with a review of plane geometry (50 pages) and then considers the sphere, cylinder, and cone, proceeding later to the properties of conic sections and other plane figures in space.
    • The work shows a return to the better type of German bookmaking of pre-war days and will be welcomed by students of modern geometry as an aid to their advanced work in this field.

  139. Rey Pastor biography
    • He graduated with a PhD in algebraic geometry from Madrid University in 1910.
    • This resulted in two major publications on geometry in 1912 and 1916.
    • The 1916 monograph was on synthetic geometry in n-dimensions and introduced [Dictionary of Scientific Biography (New York 1970-1990).
    • His lectures there on n-dimensional geometry and conformal mappings, developing the work of Schwarz, was written up by Esteban Terrades who attended the lectures, and the course was published in Catalan.
    • In this course, Rey Pastor presented his students with the concept of geometry based on group theory, using methods of establishing invariants of each group, with topological methods being the most general.
    • His second course, given in 1921, was a specialised one for engineering students and included the following topics: functions of a complex variable, conformal mapping, advanced geometry (non-euclidean), mathematical analysis and mathematical methodology.
    • In 1927, he was given a permanent appointment at the University of Buenos Aires and held two chairs: one of Mathematical Analysis and the other of Higher Geometry.

  140. Archimedes biography
    • These machines [Archimedes] had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with King Hiero's desire and request, some little time before, that he should reduce to practice some part of his admirable speculation in science, and by accommodating the theoretic truth to sensation and ordinary use, bring it more within the appreciation of the people in general.
    • Again Plutarch describes beautifully Archimedes attitude, yet we shall see later that Archimedes did in fact use some very practical methods to discover results from pure geometry:- .
    • His fascination with geometry is beautifully described by Plutarch:- .
    • And while they were anointing of him with oils and sweet savours, with his fingers he drew lines upon his naked body, so far was he taken from himself, and brought into ecstasy or trance, with the delight he had in the study of geometry.
    • The treatise On plane equilibriums sets out the fundamental principles of mechanics, using the methods of geometry.
    • certain things first became clear to me by a mechanical method, although they had to be proved by geometry afterwards because their investigation by the said method did not furnish an actual proof.
    • It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations.

  141. Fergola biography
    • Fergola studied Latin literature at the Dominican school of Thomas Aquinas and here, for the first time, he came across geometry taught by an excellent teacher.
    • The University of Naples taught some courses on geometry and arithmetic but the topics that were at the forefront were medicine and law.
    • He also took a particular interest in a whole range of geometry topics, being interested in both the synthetic and the analytic approach to the subject.
    • 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.
    • In the following years, his interest was to be directed almost exclusively towards pure geometry and synthetic methods.
    • In addition to purely algebraic means, and with the rules of Cartesian geometry, I propose to develop the most useful and notable properties, relative to the diameters of these curves, the tangents, the secants, the foci and their dimensions.
    • A vigorous argument between supporters of the synthetic method and those of the analytic method broke out in 1810 when Ottavio Colecchi (1773-1847), who taught differential and integral calculus at the Scuola di Applicazione in Naples, criticised Fergola for putting too much emphasis on pure geometry and not enough emphasis on the new methods of analysis.

  142. Golab biography
    • Hoborski was quick to spot the mathematical talents in his young pupil, and he guided him through his secondary school career, giving Golab a deep love of geometry.
    • In 1949 he was appointed as Head of the Department of Differential Geometry at the Mathematics Institute of the Polish Academy of Sciences.
    • In 1955 he became a full professor at the Jagiellonian University where he was Head of Geometry.
    • Professor Golab dealt with different fields of mathematics such as geometry, topology, algebra, analysis, logic, functional and differential equations, the theory of numerical methods and various applications of mathematics.
    • However, the most important results he obtained were in the field of geometry.
    • One can divide them into three almost equal parts; papers on the theory of geometric objects (40), papers on classical differential geometry under weak regularity assumptions (43) and papers belonging to various other domains in geometry (50) mainly connected with some special spaces such as spaces with linear or projective connection, Riemann, Minkowski and Finsler spaces, general metric spaces, etc.

  143. Barrow biography
    • Barrow studied arithmetic, geometry and optics and, like all students of the time, was encouraged not to specialise in a subject such as mathematics before graduating.
    • His study of church history led him to astronomy which in turn led him to study geometry.
    • He taught himself geometry, writing a simplified edition of Euclid's Elements which was printed in 1655 and remained the standard textbook for half a century.
    • Barrow's interest in mathematics and his small income made the position of Professor of Geometry at Gresham College, London appear very attractive when it became vacant in 1662.
    • At Gresham he taught geometry for two hours a week, one hour in English and the other in Latin.
    • The university re-opened for the second time in Easter 1667 when he gave further geometry lectures before delivering his optics lectures in the 1668-9 session.
    • Geometry is the basic mathematical science, for it includes arithmetic, and mathematical numbers are simply the signs of geometrical magnitude.

  144. Saunderson biography
    • Of course, he still required friends to read the advanced mathematics texts to him but, with West's help, he made rapid progress in the study of algebra and geometry.
    • Much of what Saunderson studied was geometry.
    • We have already indicated that one of his most successful lecture courses was on optics, which is basically a study of geometry.
    • Geometry requires geometrical figures to be considered and one might again reasonably ask how Saunderson coped with this problem.
    • Other arrangements allowed him to consider 3-dimensional geometry.
    • Saunderson then presents applications of algebra to geometry, in particular studying ratio and proportion from Book 5 of Euclid's Elements.
    • He goes on to consider solid geometry giving results on prisms, cylinders, and spheres.

  145. Riemann biography
    • He prepared three lectures, two on electricity and one on geometry.
    • Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry.
    • Riemann's lecture Uber die Hypothesen welche der Geometrie zu Grunde liegen (On the hypotheses that lie at the foundations of geometry), delivered on 10 June 1854, became a classic of mathematics.
    • The second part of Riemann's lecture posed deep questions about the relationship of geometry to the world we live in.
    • He asked what the dimension of real space was and what geometry described real space.
    • History Topics: A History of Fractal Geometry .
    • History Topics: Non-Euclidean geometry .

  146. Lyndon biography
    • Lyndon's last book was Groups and geometry (1985).
    • This book is a very readable introduction to group theory, geometry, and the connections between them.
    • The geometries studied include Euclidean geometry, affine geometry, projective geometry, inversive geometry, and hyperbolic geometry.

  147. Klugel biography
    • There are few truths which can be demonstrated in geometry without the help of the theorem of the parallels, the fewer there are, which may be necessary, to prove that.
    • It is, however, not the geometry which appears a disgrace, when a proposition established by its principles will certainly be known, whose truth is not shown by precise observation, but from the clear concept we have of the straight line.
    • His work is cited by almost all later contributors to non-euclidean geometry.
    • Things here are just as with the geometry of the ancients, and of the Englishmen imitating them, according to whom negative quantities will not occur in any proposition, since it is determined in any case what is a sum, or what is a difference, and since it can never be demanded, for a given difference, to subtract the whole from the part.
    • The work includes articles such as "Abacus", "Algebra", "Analysis", "Geometry", "Logarithms", "Parallels" and a host of similar ones.
    • It is known that the theory of parallels makes difficulties which is strange, in fact, because it is one of the first elements of geometry.
    • So many attempts are made to make the justification of the geometry quite perfect in consideration of the parallels, but it is also something to remember is present.

  148. Gauss biography
    • From the early 1800s Gauss had an interest in the question of the possible existence of a non-Euclidean geometry.
    • In a book review in 1816 he discussed proofs which deduced the axiom of parallels from the other Euclidean axioms, suggesting that he believed in the existence of non-Euclidean geometry, although he was rather vague.
    • Gauss confided in Schumacher, telling him that he believed his reputation would suffer if he admitted in public that he believed in the existence of such a geometry.
    • indicating that he had known of the existence of a non-Euclidean geometry since he was 15 years of age (this seems unlikely).
    • Gauss had a major interest in differential geometry, and published many papers on the subject.
    • A letter from Gauss to Taurinus discussing the possibility of non-Euclidean geometry.
    • History Topics: Non-Euclidean geometry .

  149. Halsted biography
    • He communicated his enthusiasm, particularly for non-euclidean geometry, to a small group of undergraduates at Princeton and in particular to Fine, who had a natural bent for questions of a logical order.
    • Student reminiscences picture Halsted as one of the more colourful professors in the university's early history and a popular speaker on and off campus whether talking about his travels to Germany, Mexico, Japan, and Hungary or about religion or science or about geometry.
    • His main interests were the foundations of geometry and he introduced non-euclidean geometry into the United States, both through his own research and writings as well as by his many important translations.
    • His work on the foundations of geometry led him to publish Demonstration of Descartes's theorem and Euler's theorem in the Annals of Mathematics in 1885, the year after he arrived at Austin, and then, in the same journal, Klein's Evanston lectures in 1893.
    • In a classroom of freshmen one of his main purposes was to challenge what he regarded as the ill founded notions that pervaded the teaching of geometry.
    • His criticism of such a definition could be the starting point for a discussion of the fundamentals of geometry.

  150. Desargues biography
    • In his later years, these seem to have included designing an elaborate spiral staircase, and an ingenious new form of pump, but the most important of Desargues' interests was Geometry.
    • He invented a new, non-Greek way of doing geometry, now called 'projective' or 'modern' geometry.
    • Desargues' most important work, the one in which he invented his new form of geometry, has the title Rough draft for an essay on the results of taking plane sections of a cone (Brouillon project d'une atteinte aux evenemens des rencontres du Cone avec un Plan).
    • When projective geometry was reinvented, by the pupils of Gaspard Monge (1746 -1818), the reinvention was from descriptive geometry, a technique that has much in common with perspective.
    • University of Minnesota (Desargues theorem and its relationship to one of Monge's geometry theorems) .

  151. Simson biography
    • 1712), Savilian Professor of Geometry at Oxford, William Jones, and finally Humphrey Ditton (1675-1715), Mathematical Master at Christ's Hospital, with whom Simson was particularly friendly.
    • after which he gave a satisfactory specimen of his skill in mathematicks and dexterity in teaching geometry and algebra, he also produced sufficient testimonials from Mr Caswell the Professor of astronomy at Oxford and from others in London well skilled in the mathematicks, upon all which the faculty resolve he shall be admitted the nineteenth day of this instant November.
    • Simson also set himself the task of preparing an edition of Euclid's Elements in as perfect a form as possible, and his edition of Euclid's books 1-6, 11 and 12 was for many years the standard text and formed the basis of textbooks on geometry written by other authors.
    • For Simson the best vehicle for presenting a mathematical argument was geometry and, although he was familiar with the recent developments in algebra and the infinitesimal calculus, he preferred to express himself in geometrical terms wherever possible.
    • That Simson's work was not restricted to Greek geometry is illustrated by Tweddle's paper [Arch.
    • the Restorer of Grecian Geometry, and by his Works the Great Promoter of its Study in the Schools.
    • Simson also made many discoveries of his own in geometry and the Simson line is named after him.

  152. Mazur Barry biography
    • Mazur received four prizes from the American Mathematical Society, namely the Veblen Prize for geometry in 1966, the Cole Prize for number theory in 1982, the Chauvenet Prize for exposition in 1994, and the Steele Prize for seminal contribution to research in 2000.
    • Mazur began his research career in geometric topology but has become one of the world's leading experts in number theory after working in algebraic geometry.
    • I came to number theory through the route of algebraic geometry and before that, topology.
    • 47 (4) (2000), 477-480.',1)">1] how he moved from topology to algebraic geometry:- .
    • the question provided quite an incentive for a topologist to look at algebraic geometry.
    • I began to learn the elements of algebraic geometry working with Mike Artin.
    • This paper determined the possible torsion of the rational points of elliptic curves defined over Q and also laid the foundation for many of the most important results in arithmetic algebraic geometry over the last 20 years, including (but not limited to): .

  153. Schlafli biography
    • This treatise, which I have the honour of presenting to the Imperial Academy of Science, is an attempt to found and to develop a new branch of analysis that would, as it were, be a geometry of n dimensions, containing the geometry of the plane and space as special cases for n = 2, 3.
    • I call this the "theory of multiple continuity" in the same sense in which one can call the geometry of space that of three-fold continuity.
    • This treatise surpasses in scientific value a good portion of everything that has been published up to the present day in the field of multidimensional geometry.
    • Most of Schlafli's work was in geometry, arithmetic and function theory.
    • Schlafli made an important contribution to non-Euclidean (elliptic) geometry when he proposed that spherical three-dimensional space could be regarded as the surface of a hypersphere in Euclidean four-dimensional space.
    • Other papers which he published investigate a variety of topics such as partial differential equations, the motion of a pendulum, the general quintic equation, elliptic modular functions, orthogonal systems of surfaces, Riemannian geometry, the general cubic surface, multiply periodic functions, and the conformal mapping of a polygon on a half-plane.

  154. Savile biography
    • geometry is almost totally unknown and abandoned in England.
    • He gave the first geometry lecture himself to a large number of students with the first holders of the chairs in his audience, and again we shall spend a moment looking at its contents.
    • The Savilian chair of Geometry was first occupied by Briggs and Savile ended his lecture with the words (see for example [J Fauvel, R Flodd and R Wilson (eds.), Oxford figures : 800 years of the mathematical sciences (Oxford, 2000), 51-56.',4)">4]):- .
    • I hand on the lamp to my successor, a most learned man, who will lead you to the innermost mysteries of geometry.
    • Many famous mathematicians have held this chair, see the list of those who have occupied the Savilian Chair of Geometry.
    • The professor of geometry was required to teach the whole of Euclid's Elements, Apollonius's Conics and the complete works of Archimedes having first provided all the necessary mathematical background for an understanding of the texts.
    • Perhaps more unusual, especially to those thinking in terms of mathematics taught in universities today, was the requirement that field work was to be undertaken in the country when the weather allowed such activities, and the students would there study practical geometry.

  155. Janovskaja biography
    • There she studied mathematics under Timchenko, who we mentioned above, and also Samuil Osipovich Shatunovsky who was interested in a wide variety of mathematical topics including group theory, the theory of numbers, and geometry.
    • He used the axiomatic method to lay the logical foundations of geometry, algebraic fields, Galois theory and analysis and his areas of interest had a large influence on his student Neimark.
    • The history of mathematics was another topic which attracted Janovskaja and she published work on Egyptian mathematics On the theory of Egyptian fractions (1947), Zeno of Elea's paradoxes, Rolle's criticisms of the calculus in Michel Rolle as a critic of the infinitesimal analysis (1947), Descartes's geometry (see below), and Lobachevsky's work on non-euclidean geometry in papers such as The leading ideas of N I Lobachevsky - a combat weapon against idealism in mathematics (1950), On the philosophy of N I Lobachevsky (1950), and On the Weltanschauung of N I Lobachevsky (1951).
    • In 1966 she published On the role of mathematical rigour in the creative development of mathematics and especially on Descartes' 'Geometry'.
    • Construction tools of Euclidean geometry are described as the ruler and the compass.
    • The author then traces the widening of the reserves of the means of construction to include the methods of cartesian geometry.

  156. Keen biography
    • At this High School she was turned on to mathematics by her study of geometry.
    • Keen, working with Nikola Lakic, wrote the book Hyperbolic geometry from a local viewpoint which was published in 2007 by Cambridge University Press in the London Mathematical Society Student Texts series.
    • Written for graduate students, and accessible to upper-level undergraduates, this book presents topics in two-dimensional hyperbolic geometry.
    • The authors begin with rigid motions in the plane, which are used as motivation for a full development of hyperbolic geometry in the unit disk.
    • The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups.
    • The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains.
    • We have quoted above from the summary of her Emmy Noether Lecture Hyperbolic Geometry and Spaces of Riemann Surfaces given at San Antonio, Texas in 1993.

  157. Tate biography
    • Tate's deep insights have had a crucial impact on the development of arithmetic algebraic geometry from the sixties onwards.
    • Through his discovery of rigid analytic spaces, he established new foundations for p-adic global analysis which have wide applicability in number theory, algebraic geometry and representation theory.
    • He has been deeply influential in many of the important developments in algebra, algebraic geometry, and number theory during this time.
    • For over a quarter of a century, Professor John Tate's ideas have dominated the development of arithmetic algebraic geometry.
    • The most obvious drawback to a text for undergraduates in a field such as this is that it is not possible to be entirely rigorous, and so, as the authors declare, "much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than rigorously prove." An appendix does develop the necessary algebraic geometry, but throughout the book the approach to the underlying geometry is informal, allowing a more rapid and intuitive access to the number theory.
    • According to the Abel committee, "Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate.

  158. Koszul biography
    • Apart from the more or less standard theorems on symmetric spaces, the author discusses the geometry of geodesics, the Bergmann metric, and finally investigates the bounded domains in considerable detail.
    • In the mid 1960s Kosul lectured at the Tata Institute of Fundamental Research in Bombay On groups of transformations and On fibre bundles and differential geometry.
    • After a number of further important publications which appeared in the proceedings of various conferences that Koszul attended such as Convegno sui Gruppi Topologici e Gruppi di Lie in Rome (1974), Symplectic geometry in Toulouse (1981), an international meeting on geometry and physics in Florence (1982), he published Introduction to symplectic geometry in Chinese in 1986.
    • During the past eighteen years there has been considerable growth in the research on symplectic geometry.
    • Many new ideas have also been derived with the help of a great variety of notions from modern algebra, differential geometry, Lie groups, functional analysis, differentiable manifolds and representation theory.

  159. Donaldson biography
    • for his fundamental investigations in four-dimensional geometry through application of instantons, in particular his discovery of new differential invariants ..
    • Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered.
    • On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry.
    • (1) Differential geometry of holomorphic vector bundles.
    • He has created an entirely new and exciting area of research through which much of mathematics passes and which continues to yield mysterious and unexpected phenomena about the topology and geometry of smooth 4-manifolds.
    • groundbreaking work in four-dimensional topology, symplectic geometry and gauge theory, and for his remarkable use of ideas from physics to advance pure mathematics.
    • Donaldson's breakthrough work developed new techniques in the geometry of four-manifolds and the study of their smooth structures.

  160. Klingenberg biography
    • Klingenberg obtained a doctorate in 1950 with a thesis on affine differential geometry.
    • from 1950 to 1952 he was a research assistant at Kiel where F Bachmann interested him in the foundations of geometry.
    • His interests had turned away from affine and projective differential geometry and turned towards Riemannian geometry.
    • Not without some pain and struggle, I finally accepted the change and concentrated my activities on my own differential geometry group.
    • His major books include A course in differential geometry (1978), Lectures on closed geodesics (1978) and Riemannian geometry (1982).

  161. Atiyah biography
    • Atiyah was soon to fill the highly prestigious Savilian Chair of Geometry at Oxford from 1963, holding this chair until 1969 when he was appointed professor of mathematics at the Institute for Advanced Study in Princeton.
    • Michael Atiyah has contributed to a wide range of topics in mathematics centring around the interaction between geometry and analysis.
    • This 'index theorem' had antecedents in algebraic geometry and led to important new links between differential geometry, topology and analysis.
    • Atiyah initiated much of the early work in this field and his student Simon Donaldson went on to make spectacular use of these ideas in 4-dimensional geometry.
    • He gave the Royal Society's Bakerian Lecture on Global geometry in 1975 and was President of the Royal Society from 1990 to 1995.
    • Savilian Geometry Professor1963 .

  162. Lichnerowicz biography
    • is devoted mainly to the discussion of two propositions concerning differential geometry in the large, the truth of which is held to be of fundamental importance for the relativistic theory of gravitation.
    • He published Elements de Calcul Tensoriel, which is an introduction to Riemannian geometry and its applications, in 1950.
    • The approach is that of Elie Cartan, whose method of the "repere mobile," satisfying in its naturalness, is applicable to more general types of geometry.
    • In 1957 he published an important text on differential geometry, Theorie globale des connexions et des groupes d'holonomie.
    • He did important work on Riemannian geometry, in particular his results on the equivalence of various definitions of a Kahler manifold are now part of the standard theory.
    • In 1964 he posed a famous conjecture on differential geometry in his paper Sur les transformations conformes d'une variete riemannienne compacte.
    • The papers are divided into three sections: relativity theory, differential geometry, and infinite-dimensional Lie algebras.

  163. Roth biography
    • He then undertook research gaining great inspiration from H F Baker's Principles of geometry.
    • an able student of algebraic geometry of great industry and strength who is likely if the opportunity is afforded him, of doing very important work in the subject.
    • Almost all his work was on geometry where he extended work begun in the Italian school.
    • He also took an approach to geometry which produced large amounts of experimental material and this is highlighted in the review of his most famous book from which we quote below.
    • The famous book we referred to is Introduction to Algebraic Geometry which Roth wrote with Jack Semple, and it was published in 1949.
    • Since it is obviously impossible to write a treatise in one volume on the whole of algebraic geometry (even if the transcendental and topological theories are excluded), one would be tempted to conclude a priori that the present book must be something in the nature of an encyclopaedia article.
    • In addition we should mention Roth's other books: Elements of probability (1936), written with Hyman Levy, and Modern elementary geometry (1948).

  164. Delone biography
    • There he continued his interest in number theory, particularly the geometry of numbers, but he also worked on polyhedra, and on crystallography with important papers such as On the question of the uniqueness of the determinations of the foundations of a parallelopiped crystal structure by the method of Debye (1926).
    • From 1932 to 1960 Delone was Head of the Department of Algebra in Steklov Mathematical Institute and from 1960 to 1980 he was Head of the Department of Geometry.
    • He was professor of mathematics at Moscow State University from 1935 to 1943, being Head of the Department of Higher Geometry in the Faculty of Mechanics and Mathematics.
    • In 1943, remaining at Moscow State University, he was made Professor of Higher Geometry and Topology, holding this position until 1960.
    • He also published a number of texts aimed at school pupils including (with O K Zhitomirski) Problems with solutions for a revision course in elementary mathematics (1928), (with O K Zhitomirski) Problems in geometry (1935), Analytical geometry I (1948), and (with D A Raikov) Analytical geometry II (1949).

  165. Lang biography
    • Lang's mathematical research ranged over a wide range of topics such as algebraic geometry, Diophantine geometry (a term Lang invented), transcendental number theory, Diophantine approximation, analytic number theory and its connections to representation theory, modular curves and their applications in number theory, L-series, hyperbolic geometry, Arakelov theory, and differential geometry.
    • Other books by Lang include Introduction to algebraic geometry (1958), Abelian varieties (1959), Diophantine geometry (1962), Introduction to differentiable manifolds (1962), Algebraic numbers (1964), Linear algebra (1966), Introduction to diophantine approximations (1966), Introduction to transcendental numbers (1966), Algebraic structures (1967), Algebraic number theory (1970), Introduction to algebraic and abelian functions (1972), Differential manifolds (1972), Elliptic functions (1973), SL(R) (1975), Introduction to modular forms (1976), Complex analysis (1977), Cyclotomic fields (1978), Elliptic curves: Diophantine analysis (1978), Undergraduate analysis (1983), Complex multiplication (1983), Riemann-Roch algebra (1985), The beauty of doing mathematics.
    • Three public dialogues (1985), Introduction to complex hyperbolic spaces (1987), Introduction to Arakelov theory (1988), Topics in Nevanlinna theory (1990), Basic analysis of regularized series and products (1993), Fundamentals of differential geometry (1999), and Math talks for undergraduates (1999).

  166. Moufang biography
    • Both Ruth and Erica Moufang had considerable artistic skills and this was put to good use in their work on Schwan's geometry book.
    • Let us note that Wilhelm Schwan, in addition to the geometry text, edited a volume of mathematics lectures.
    • in 1931 for her thesis Zur Struktur der projektiven Geometrie der Ebene on projective geometry [Encyclopedia of World Scientists (Infobase Publishing, 2007), 327-328.',4)">4]:- .
    • In it, she transported the axioms David Hilbert had postulated for the realm of plane geometry into the field of projective geometry.
    • In [The History of Combinatorial Group Theory: A Case Study in the History of Ideas (New York - Heidelberg - Berlin, 1982), 123; 136-137.',2)">2] Chandler and Magnus describe her contributions to geometry, putting them into context as follows:- .
    • A large part of her work is dedicated to the foundations of geometry.
    • Reversing a development going from Euclid to Descartes in which geometry is replaced by algebra as a fundamental discipline of mathematics, Hilbert had shown that a subset of his axioms for plane geometry (essentially the incidence axioms) together with the incidence theorem of Desargues permits the introduction of coordinates on a straight line which are elements of a skew field.
    • In this paper, which was motivated by the two papers of Hilbert on geometry mentioned above (published in 1901 and 1930), she examines the group M = F/F'', the free metabelian group on two generators.
    • Moufang also gives applications of the result to number theory, knot theory and the foundations of geometry.

  167. Koenigs biography
    • Koenigs was greatly influenced by Darboux and his first work was on geometry following work of Plucker and Klein.
    • Further, the method should have the necessary breadth and unity, and be possessed of the clearness and directness of geometry and the power and generality of analysis.
    • This multiplicity of demands Professor Koenigs has met admirably by basing his exposition on the geometry of the straight line and employing the mobile trieder of reference; the latter in the hands of Ribaucour and Darboux has proved itself to be the most certain and powerful implement yet used in infinitesimal geometry and it naturally lends itself with equal facility and elegance to the geometry of displacement.
    • The Bordin Prize of 1890 was to be awarded for work on differential geometry; Koenigs won for his essay on geodesics.
    • He also received the Poncelet Prize in 1913 for his contributions to geometry and mechanics.

  168. Cherubino biography
    • Considered an exceptional student, he was appointed as an assistant to the professor of analytic geometry at the University of Naples for the year 1910-1911.
    • These were Giuseppe Veronese, who held the Chair of Algebraic Geometry, Tullio Levi-Civita, who had been appointed to the Chair of Rational Mechanics in 1898, a post which he held for 20 years, and Francesco Severi, who had been appointed to the Chair of Projective and Descriptive Geometry in 1905.
    • Cherubino was awarded a government scholarship for advanced study which allowed him to attend the courses of Levi-Civita and Veronese while, advised by Severi, he devoted himself to the study of algebraic geometry.
    • Continuing to work in that city, he taught analysis and descriptive, analytic and projective geometry at the Air Force Academy in 1926.
    • In 1933, approaching the age of fifty, Cherubino won the competition for the Chair of Analytical, Projective and Descriptive Geometry and Drawing at the University of Messina.
    • He only held this chair for two years before moving to Pisa in 1935 to an ordinary professorship in geometry.

  169. Coxeter biography
    • Coxeter's work was mainly in geometry.
    • In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics.
    • Among his most famous geometry books are The real projective plane (1955), Introduction to geometry (1961), Regular polytopes (1963), Non-euclidean geometry (1965) and, written jointly with S L Greitzer, Geometry revisited (1967).

  170. Ghetaldi biography
    • These contain early application of algebra to geometry.
    • It is reasonable to ask: what is the most impressive ideas contained in Ghetaldi's work? Without doubt, it is his application of algebraic methods to the solution of problems in geometry.
    • We now think of Descartes as founding the application of algebra to geometry, and although Ghetaldi never quite managed to achieve this breakthrough (nowhere in his work are there algebraic equations for geometric objects) nevertheless he came very close.
    • Perhaps it is just as well since somehow 'Ghetaldian geometry' does not quite have the same ring as 'Cartesian geometry'.
    • He certainly used such algebraic geometry in Variorum problematum collectio but his main contributions in this area are contained in his book De resolutione et de compositione mathematica, libri quinque published in 1630, four years after his death.

  171. Eudemus biography
    • We know of three works on the history of mathematics by Eudemus, namely History of Arithmetic (two or more books), History of Geometry (two or more books), and History of Astronomy (two or more books).
    • The History of Geometry is the most important of the three mathematical histories of Eudemus.
    • To illustrate with one example, the work of Hippocrates on the quadrature of lunes is only known to us through Eudemus's History of Geometry.
    • It is unclear exactly when the History of Geometry was lost.
    • Paul Tannery (see for example [Annales de la Fuculte des Lettres de Bordeaux 4 (1882), 70-76.',7)">7]) believed that it was lost before the time of Pappus while others such as J L Heiberg have argued that Pappus and Eutocius both wrote with an open copy of Eudemus's History of Geometry in front of them.
    • The History of Astronomy again was heavily used by later writers and in exactly the same way as his geometry text, much information has survived in the works of others despite the loss of the original text.

  172. Egorov biography
    • Fedor Ivanovich taught algebra and geometry as part of the three-year course to train secondary school teachers at the Moscow Teachers' Institute.
    • Due to his outstanding mathematical talents, his excellent general education and his real inclination towards scientific knowledge, Mr Egorov, in his remarkably conscientious and completely successful passage through all subjects of the Mathematical Department, found the opportunity of considerably enlarging his knowledge by a very extensive acquaintance with mathematical literature and a special study of many branches of modern geometry .
    • Egorov worked on triply orthogonal systems and potential surfaces, making a major contribution to differential geometry.
    • Since the appearance of the classic memoir of Gauss (in 1827) and the no less famous treatise of Lame (in 1850), curvilinear coordinates have been an indispensable tool for the investigation of almost all branches of differential geometry and for many fields of applied mathematics.
    • In fact, the theory of curvilinear coordinates is, essentially, nothing other than the theory of lines on a surface and of systems of surfaces in space, and from this point of view this theory is evidently one of the most important constituents of differential geometry.
    • After he was appointed as a professor, he taught courses on differential geometry, the integration of differential equations, integral equations, the calculus of variations, number theory, and the theory of surfaces.

  173. West biography
    • [Leslie] had the advantage of receiving the instruction of Mr West, the author of an elementary course of mathematics, a man of original and inventive genius, and, after Dr Matthew Stewart, one of the greatest masters of the ancient geometry, whom Scotland has produced.
    • The first thing to say about this work is that it is a geometry book.
    • My original intention was not to include the First Elements of Geometry ..
    • My design was only to build upon the foundation which that illustrious author had laid, and, under the several heads of 'Conic Sections', Mensuration', and 'Spherics', to complete a system of Geometry for the purpose of youth.
    • [but] considering the improved state of mathematics, Euclid's Geometry is now inadequate and defective, as an elementary work ..
    • [This] determined me to attempt a new theory of proportion, and to introduce a new system of the Elements of Geometry as the first part of my work.

  174. Cohn-Vossen biography
    • The book will make a very wide appeal, not only to experts in all branches of pure mathematics, providing as it does a genial connecting link between almost all the maze-like ramifications of the subject; but also to many others who at school or later have felt something of the fascination that geometry ever exercises over the human mind.
    • We should comment that the title of the English translation was Geometry and the Imagination which is not really a translation of the German title.
    • A more accurate translation of Anschauliche Geometrie would be 'Intuitive Geometry'.
    • brilliant intuitive and concrete approach to geometry provided by Hilbert and his collaborator Cohn-Vossen ..
    • Later, still in 1934, he emigrated to Russia and under his influence a school of "geometry in the large" was set up in Moscow and Leningrad.
    • The importance of this work to the Russian school is seen from the fact that in 1959 a 303-page Russian book Some problems of differential geometry in the large (Russian) was published containing Russian translation of seven of Cohn-Vossen's papers.

  175. De Vries Hendrik biography
    • After four years of study, de Vries graduated from the Eidgenossische Polytechnicum in 1890 and was appointed as an assistant to D Fiedler, who had taught him as an undergraduate, to work on descriptive and projective geometry.
    • At Amsterdam, he supervised the doctoral studies of a number of students who went on the make important contributions to mathematics, the most famous of these being Bartel van der Waerden who was awarded a doctorate in 1926 for a thesis on the foundations of algebraic geometry.
    • Although initially de Vries' interest involved research in geometry, and in particular projective geometry, he became interested in the history of mathematics through reading the works of Monge, Plucker and Mobius.
    • His lectures took in algebra and analysis, but from 1921-22 onwards, he focussed increasingly on his preferred field, giving public lectures on the development of geometry.
    • He continued to publish Historical studies, and as examples we give the title of a small number of these later articles: On the contact and intersection of circles and conic sections (1946), How analytic geometry became a science (1948), On the infinite and the imaginary, or "surrealism" in mathematics (1949), and On relations and transformations (1949).

  176. Titchmarsh biography
    • He sent in an application on a single sheet saying that he wished to apply for the geometry Chair but could not undertake to lecture on geometry as Hardy had done.
    • Two days later he was appointed and the statutes were changed to say that the Savilian Professor of Geometry no longer had to lecture on geometry.
    • Titchmarsh held the Savilian Chair of Geometry at Oxford for 30 years.
    • Savilian Geometry Professor1932 .

  177. Haack biography
    • Haack did not spend long in Berlin for in 1937 he was appointed to a lectureship in Mathematics and Geometry at the Technische Hochschule of Karlsruhe.
    • Up to this time Haack had produced many excellent papers on geometry, in particular on differential geometry.
    • He explained in later writings and interviews that, seeing the approaching war and realising that differential geometry was "not very much tuned to war purposes", he changed areas.
    • There was a pure mathematician (Haack) who made use of his special expertise (differential geometry) ..
    • As well as these books on geometry, he also continued his work on gas dynamics; for example in 1958 he published the paper (published jointly with his doctoral student Gerhard Bruhn) Ein Charakteristikenverfahren fur dreidimensionale instationare Gasstromungen in which he derived the characteristic equations for the unsteady three-dimensional motion of inviscid perfect gas.

  178. Lefebure biography
    • He became a repetiteur in descriptive geometry in 1815 and, from 1817 to 1820, he was an examiner in descriptive geometry and the 'graphic arts'.
    • a detailed plan, but a list of books by which it is taught, and the authors for the mathematics texts are Euler, Lagrange and Lacroix in algebra, Lacroix or Biot for applications of algebra to geometry, Monge and Poinsot for statics, and Delambre or Biot for astronomy.
    • Traite de geometrie descriptive was essentially translated into English by Thomas Grainger Hall, of King's College, London, when he wrote his book The elements of descriptive geometry (1841).
    • In the Preface, Hall explains the position of descriptive geometry:- .
    • The treatise on Descriptive geometry, by Mr Lefebure de Fourcy has, therefore, been selected, and the following pages are, for the most part, translated from it.

  179. Berzolari biography
    • Then, remaining at Pavia, he became an assistant to Ferdinando Aschieri, who held the chair of projective and descriptive geometry.
    • In 1892 he received his 'libero docente', which is similar to the habilitation and gives the right to lecture in universities, and became a lecturer in analytic and projective geometry at Pavia having, by this time, published ten papers including the major work Ricerche sulle trasformazioni piane, univoche, involutorie, e loro applicazioni alla determinazione delle involuzioni di quinta classe in 1889 and two papers with the title Intorno alla rappresentazione delle forme binarie cubiche e biquadratiche sulla cubica gobba in 1891.
    • Berzolari entered the competition for the chair of analytic and projective geometry at the University of Rome in 1891.
    • Two years later, in 1893, Berzolari entered the competition for the chair of projective geometry at the University of Naples.
    • Later in the same year of 1893 there was a second chair competition, this time for the chair of projective and descriptive geometry at the University of Turin.
    • To give an idea of his contributions to geometry, we mention a few of his papers: Sugli invarianti differenziali proiettivi delle curve di un iperspazio (1897), Sulle coniche appoggiate in pi punti a date curve algebriche (1900), Sul significato geometrico di alcune identita lineari fra quadrati di forme algebriche (1918), and Sul complesso di covarianti di tre complessi lineari a due a due in involuzione (1922).

  180. Berwald biography
    • Berwald's scientific work was mainly in the area of differential geometry.
    • A portion of his work set up the basic theory of Finsler geometry and Spray geometry (i.e., differential geometry of path spaces).
    • Many people working in Finsler geometry consider that Ludwig Berwald is the founder of Finsler geometry.

  181. Bolyai Farkas biography
    • All his life Bolyai was interested in the foundations of geometry and the parallel axiom.
    • His main work, the Tentamen, was an attempt at a rigorous and systematic foundation of geometry, arithmetic, algebra and analysis.
    • The Tentamen is built on Bolyai's belief that mathematics consists of arithmetic and geometry with arithmetic as the mathematics of time and geometry as the mathematics of space.
    • However, in 1825 Bolyai's son Janos showed him his discovery of non-euclidean geometry.
    • History Topics: Non-Euclidean geometry .

  182. Avicenna biography
    • The first is a scientific encyclopaedia covering logic, natural sciences, psychology, geometry, astronomy, arithmetic and music.
    • In fact he divided mathematics into four branches, geometry, astronomy, arithmetic, and music, and he then subdivided each of these topics.
    • Geometry he subdivided into geodesy, statics, kinematics, hydrostatics, and optics; astronomy he subdivided into astronomical and geographical tables, and the calendar; arithmetic he subdivided into algebra, and Indian addition and subtraction; music he subdivided into musical instruments.
    • In fact ibn Sina does not present geometry as a deductive system from axioms in this work.
    • In other writings on geometry he, like many Muslim scientists, attempted to give a proof of Euclid's fifth postulate.
    • The topics dealt with in the geometry section of the encyclopaedia are: lines, angles, and planes; parallels; triangles; constructions with ruler and compass; areas of parallelograms and triangles; geometric algebra; properties of circles; proportions without mentioning irrational numbers; proportions relating to areas of polygons; areas of circles; regular polygons; and volumes of polyhedra and the sphere.

  183. Helmholtz biography
    • In the second half of the 19th century, scientists and philosophers were involved in a heated discussion on the principles of geometry and on the validity of so-called non-Euclidean geometry.
    • Moving from the observation that our geometric faculties depend on the existence, in nature, of rigid bodies, he presumed he had given a proof that Euclidean geometry was the only one compatible with these bodies, maintaining, at the same time, the empirical, not a priori, origin of geometry.
    • he realized he had made a mistake: the empirical concept of a rigid body and mathematics alone were not enough to characterize Euclidean geometry.
    • The following year, fully sharing the mathematical itinerary that, through Gauss, Riemann, Lobachevsky and Beltrami, led to the creation of the new geometry, he proposed to spread this knowledge among philosophers while at the same time criticizing the Kantian system.

  184. Codazzi biography
    • His research led him into deep results in geometry and he began to think that submitting an entry for the Grand Prix of the Paris Academy of Sciences would let his research become known to the top mathematicians.
    • and this was exactly in the right area for Codazzi's research in differential geometry.
    • All three pieces of work are important contributions to differential geometry but, although the manuscripts of Bour and Bonnet were published in Comptes-Rendus des seances de l'Academie des sciences fairly soon after the prize was awarded, Codazzi's entry was not published until 1883 (10 years after his death).
    • Certainly Codazzi became well known for his contributions to differential geometry and this led to his appointment to a chair of algebra and analytic geometry at the University of Pavia in 1865.
    • He remained at the University of Pavia for the rest of his life, although he only lived for a further eight years after his appointment to the chair of algebra and analytic geometry.

  185. Andreotti biography
    • Severi was, by this time, involved in many non-mathematical pursuits, but these years, first as a research student at the National Institute of Advanced Mathematics (INdAM), and later as an assistant in geometry, were important years during which Andreotti became an extremely innovative mathematician.
    • His interest in algebraic geometry, which had begun while he was in Pisa, developed greatly under Severi's guidance.
    • He returned to Italy, entered the national competition for the selection of a full Professor of Geometry at the University of Turin and was interviewed in November 1951.
    • All Italian mathematicians were a bit ignorant, but especially the researchers of algebraic geometry; the old venerated master Severi was often an obstacle to updating because he didn't encourage young people enough to look at other schools which he considered competitive to the Italian one which he defended at all costs.
    • It was absurd to continue to teach descriptive geometry in the first two years of the mathematics course and Andreotti replaced this by several topics in algebra such as ideal theory, polynomials, resultants etc.
    • His early work was on algebraic geometry but he went on to make major contributions to the theory of several complex variables and to partial differential operators.

  186. Gowers biography
    • However, it was not until he took Bela Bollobas' course on the geometry of Banach spaces while studying for Part III of the Mathematical Tripos that Gowers found an area of mathematics which he felt was the right one for him to begin research [Newsletter of the European Mathematical Society 33 (1999), 8-9.',10)">10]:- .
    • His first paper Symmetric block bases in finite-dimensional normed spaces was published in 1989 and in the same year he gave a survey lecture Symmetric sequences in finite-dimensional normed spaces to the conference 'Geometry of Banach spaces' held in Strobl, Austria.
    • Over the past five years, Gowers has made the geometry of Banach spaces look completely different.
    • William Timothy Gowers' work has made the geometry of Banach spaces look completely different.
    • 26-dimensional) space?; What's the deal with non-Euclidean geometry?; How can mathematics address questions that cannot be answered exactly, but only approximately?; Is it true that mathematicians burn out at the age of 25? and other sociological questions about the mathematical community.
    • An encyclopedia might focus primarily on definitions, a survey article might focus on history or on the latest research, and a popular introduction might focus on analogies, personalities or entertaining narrative; in contrast, this book is intended to answer (or at least address) basic questions about mathematics, such as "What is arithmetic geometry?", "Why do we care about function spaces?", "How is mathematics used today in biology?", "What is the significance of the Riemann hypothesis?", "Why are there so many number systems?'", or "Is mathematical research all about proving theorems rigorously?".

  187. Schoute biography
    • Schoute studied various topics in geometry such as quadrics and algebraic curves.
    • In his early work he investigated quadrics, algebraic curves, complexes, and congruences in the spirit of nineteenth-century projective, metrical, and enumerative geometry.
    • From 1891 Schoute studied Euclidean geometry of more than 3 dimensions, writing 28 papers, some jointly with Alica Boole Stott, the daughter of George Boole, such as On the sections of a block of eightcells by a space rotating about a plane.
    • Schlafli's work of the 1850's was brought to the Netherlands by Pieter Hendrik Schoute (1846-1913) who, in three papers beginning in 1893 and in his elegant two-volume textbook on many-dimensional geometry 'Mehrdimensionale Geometrie' (2 volumes 1902, 1905), studied the sections and projections of regular polytopes and compound polyhedra.
    • The subject-matter corresponds roughly to the work given, for ordinary space, in such books as Nixon's 'Geometry in Space'.
    • The book contains extensions of Euler's theorem, of the theory of regular polyhedra, of spherical geometry, and so on; the work is well illustrated by numerous carefully drawn figures.

  188. Ingarden biography
    • concerned geometrical optics with a pioneering application of differential geometry methods, in particular those of Finsler geometry.
    • The research conducted by his Torun group resulted in a number of important papers on the aforementioned applications of Finsler geometry in statistical physics, the dynamics of open quantum systems and the geometry of quantum state spaces, quantum generalisations of entropy, among others.
    • He published a number of textbooks aimed at students at Polish universities as well as monographs (we give English titles to these Polish books): (with Andrzej Jamiolkowski) Classical Electrodynamics (1979); (with Andrzej Jamiolkowski) Classical Mechanics (1980); (with Lech Gorniewicz) Mathematical analysis for physicists (1981); (with Marian Grabowski) Quantum mechanics: in a Hilbert space setting (1989); (with Andrzej Jamiolkowski) Statistical physics and thermodynamics (1990); and (with Lech Gorniewicz) Algebra and Geometry for physicists (1993).
    • At that time Professor Ingarden was the leading researcher in the fields of quantum information and Finsler geometry.

  189. Kasner biography
    • in 1899 for his thesis The Invariant Theory of the Inversion Group: Geometry upon a Quadric Surface.
    • Kasner gave the lecture Present Problems of Geometry and was delighted to have Henri Poincare, another of the speakers at the Congress, in the audience.
    • Kasner's principal mathematical contributions were in the field of geometry, chiefly differential geometry.
    • For his graduate students, who included Joseph Ritt, Jesse Douglas and John De Cicco, Kasner put on his famous Seminar in Differential Geometry [Biographical Memoirs of the National Academy of Sciences (1958), 179-209.',4)">4]:- .
    • I think it may truly be said that nearly every mathematician who arose in the New York area during the first half of this century - whether his main interest was differential geometry or not - attended this seminar at some time during his course of study, and derived from it illumination and inspiration.

  190. Grossmann biography
    • Fiedler (1832-1912), a student of August Mobius, had been professor of geometry at the Polytechnic Institute of Prague before taking up a similar position at the Polytechnikum at Zurich.
    • He had taught geometry to Grossmann, Einstein and Maric during their undergraduate studies and it was Grossmann who had excelled in the examinations of the geometry course.
    • He had continued to undertake research in geometry and he published Die fundamentalen Konstruktionen der nichteuklidischen Geometrie in Frauenfeld in 1904.
    • In 1907 Grossmann became professor of descriptive geometry at the Eidgenossische Technische Hochschule in Zurich.
    • In 1910, Grossmann co-founded the Swiss Mathematical Society along with Rudolf Fueter, professor of mathematics at the University of Basel, and Henri Fehr (1870-1954), professor for algebra and higher geometry at the University of Geneva.

  191. Killing biography
    • In particularly the study of geometry at the Gymnasium convinced Killing that he should become a mathematician.
    • The lecturer in mathematics and astronomy at the Academy was Eduard Heis but he did not teach mathematics to a high level and Killing learnt his mathematics from studying books on his own: in particular he read Plucker's works on geometry and tried to extend the results which Plucker proved.
    • Despite this he published his first paper Uber zwei Raumformen mit konstanter positiver Krummung in 1879 in Crelle's Journal and two further papers, also in Crelle's Journal, on non-euclidean geometry in n-dimensions: Die Rechnung in den Nicht-Euklidischen Raumformen (1880) and Die Mechanik in den Nicht-Euklidischen Raumformeni (1885).
    • He published the book Die nichteuklidischen Raumformen in analytischer Behandlung on non-euclidean geometry in Leipzig in 1885.
    • Killing introduced them independently with quite a different purpose since his interest was in non-euclidean geometry.
    • His students loved and admired Killing because he gave himself unsparingly of time and energy to them, never being satisfied for them to become narrow specialists, so he spread his lectures over many topics beyond geometry and groups.

  192. Grieve biography
    • Grieve then undertook research in geometry.
    • For example he wrote: A B Grieve, Some Points in the Geometry of Cubic Surfaces, Proc.
    • In 1925 he published a book of 314 pages with title Analytical Geometry of Conic Sections and Elementary Solid Figures.
    • This book claims to contain the substance of the plane and solid analytical geometry (except straight line and circle) required for Pass and Engineering students at Universities, and for the more advanced pupils of Secondary Schools.
    • It is refreshing to find plane and solid geometry in one volume, but the plan possesses the inherent objection that not very much ground in either can be covered in one book.
    • The book therefore consists in the main of what may be called numerical analytical geometry of the conics and conicoids.

  193. Morley biography
    • He wrote papers mainly on geometry but also on algebra.
    • Many years later, in 1933, he published Inversive geometry written jointly with his son Frank V Morley.
    • Morley's own favourite among his geometry papers was On the Luroth quartic curve which he published in 1919.
    • (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.
    • Geometry without pictures I found it hard to approve; indeed, I prefer it with pictures to this present day.) .

  194. Halphen biography
    • the theory of differential invariants is to the theory of curvature as projective geometry is to elementary geometry.
    • He examined problems in the areas of systems of lines, classification of space curves, enumerative geometry of plane conics, singular points of plane curves, projective geometry and differential equations, elliptic functions, and assorted questions in analysis.
    • he worked in analytic and differential geometry, a subject so unfashionable today as to be almost extinct.
    • Perhaps with its inevitable revival, analytic geometry will restore Halphen to the eminence he earned.

  195. Bellavitis biography
    • Bellavitis believed that algebra had to be founded on geometry, and that number systems could only be defined through geometric concepts.
    • He was appointed professor of geometry at Padua on 4 January 1845 after a competitive examination to find the best candidate for the chair.
    • In 1867, Bellavitis moved from the chair of geometry at Padua to take the chair of complementary algebra and analytic geometry there.
    • In addition to the work described above, Bellavitis made significant contributions to algebraic geometry, where he classified curves in particular completing Newton's classification of cubic curves, and descriptive geometry with an important textbook on the topic.

  196. Rokhlin biography
    • Rokhlin's interest in mathematics is so strong that he independently studied the beginnings of calculus, analytical geometry and higher algebra.
    • He became a private in the 995th Artillery Regiment and by the beginning of October 1941 he was with a unit in Vyazma, defending Moscow [Topology, ergodic theory, real algebraic geometry: Rokhlin\'s memorial (AMS Bookstore, 2001).',1)">1]:- .
    • Aleksandr Danilovic Aleksandrov was professor of geometry at Leningrad University from 1944 and, in 1952, he became the rector of the University.
    • He wanted to build up all the scientific departments and, in 1961, he invited Rokhlin to the chair of geometry at Leningrad.
    • Rokhlin's scientific legacy is comparatively small in volume and can be approximately divided into four parts: topology (basically four-dimensional topology and the algebraic apparatus of topology); real algebraic geometry (in his last years); ergodic theory (basically the spectral theory, entropy theory and algebraic aspects); and, lastly, works on history, teaching and the methodology of mathematics.
    • In the last years of his life after his retirement Rokhlin continued to work mainly in real algebraic geometry and four-dimensional topology.

  197. Wiener Christian biography
    • In 1847 he took the state examinations to qualify him to teach in secondary schools in Germany and in the following year of 1848 he became a teacher of physics, mechanics, hydraulics and descriptive geometry at the Technische Hochschule in Darmstadt.
    • In 1852, however, he was appointed to the chair of descriptive geometry at the Technische Hochschule (Technical University) of Karlsruhe.
    • We shall say a little about his contributions in each of these areas in turn beginning with his mathematical contributions which were mainly on descriptive geometry.
    • His chief work is a two volume book on geometry Lehrbuch der darstellenden Geometrie which supplements Chasles's work and contains important historical information [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
    • Wiener treated the basic problems of descriptive geometry by a single method: a varied use of the principal lines of a plane.
    • Wiener extended work on descriptive geometry to physics and calculated the amount of solar radiation received at different latitudes during the varying lengths of days in the course of the year.

  198. Radon biography
    • In mathematics he took lecture courses by Hans Hahn (one on Theoretical arithmetic and one on the Foundations of geometry), Wilhelm Wirtinger (Ordinary differential equations) and Franz Mertens (one on Algebra and one on Number theory) among others.
    • Radon applied the calculus of variations to differential geometry which led to applications in number theory.
    • It was while he was studying applications of the calculus of variations to differential geometry that he discovered curves which are now named Radon curves.
    • During 1918-19 he worked on affine differential geometry, then in 1926 he considered conformal differential geometry.
    • His wide interests led him to study Riemannian geometry and geometrical problems which arose in the study of relativity.

  199. Schooten biography
    • May we suggest that this was not poor judgement on the part of Descartes, rather it was because Descartes was trying to push the thesis that only by using his 'method' could discoveries such as Cartesian geometry be made so he played down Fermat's work.
    • Van Schooten was one of the main people to promote the spread of Cartesian geometry and this is a more important contribution than the results of his own researches.
    • Van Schooten established a vigorous research school in Leiden which included his private pupils Christiaan Huygens, Henrik van Heuraet and Johannes Hudde, and this school was one of the main reasons for the rapid development of Cartesian geometry in the mid 17th century.
    • [Van Schooten] was very interested in and well aware of the important new developments in mathematics which combined geometry and algebra into analytic geometry.
    • Book I gives a review of arithmetic and basic geometry.

  200. Douglas biography
    • He took part in Kasner's seminar on differential geometry and it was there that Douglas developed a love of geometry.
    • He submitted his doctoral thesis On Certain Two-Point Properties of General Families of Curves; The Geometry of Variations in 1920.
    • Douglas continued to undertake research in differential geometry while teaching at Columbia College from 1920 to 1926.
    • In a series of papers from 1927 onwards Douglas worked towards the complete solution: Extremals and transversality of the general calculus of variations problem of the first order in space (1927), The general geometry of paths (1927-28), and A method of numerical solution of the problem of Plateau (1927-28).
    • Another five papers by Douglas appeared in 1940: Theorems in the inverse problem of the calculus of variations; Geometry of polygons in the complex plane; On linear polygon transformations; A converse theorem concerning the diametral locus of an algebraic curve and A new special form of the linear element of a surface.

  201. Zorawski biography
    • He was awarded his doctorate in 1891 from the University of Leipzig for his thesis on the applications of Lie groups to differential geometry.
    • After returning to Krakow, Zorawski continued to teach courses on analytical and synthetic geometry, differential geometry, the formal theory of the differential equations, the theory of the forms, and the theory of the Lie groups.
    • The main topics of his research were invariants of differential forms, integral invariants of Lie groups, differential geometry, and fluid mechanics.
    • Retirement meant that his formal responsibilities at the university had ended but this meant that he had more time to now undertake the writing of a major mathematical work on analytical geometry.
    • His house and the three-quarters finished multi-volume work he had been writing on analytical geometry were destroyed.

  202. Mandelbrot biography
    • Benoit Mandelbrot was largely responsible for the present interest in fractal geometry.
    • His work was first put elaborated in his book Les objets fractals, forn, hasard et dimension (1975) and more fully in The fractal geometry of nature in 1982.
    • The problem was both one of geometry concerning the nature of the line thought of as built up of points and of arithmetic thought of as the theory of the real numbers.
    • That is until our honorary graduand created out of them an entirely new science, the theory of fractal geometry: it was his insight and vision which saw in those objects and the many new ones he discovered, some of which now bear his name, not mathematical curiosities, but signposts to a new mathematical universe, a new geometry with as much system and generality as that of Euclid and a new physical science.
    • History Topics: A History of Fractal Geometry .

  203. Hachette biography
    • Monge had set up a descriptive geometry course at the Ecole Royale du Genie which Ferry was teaching when Hachette was appointed to the staff.
    • Hachette assisted Ferry in teaching the descriptive geometry course.
    • He was appointed as an assistant professor in descriptive geometry at the School in November 1794.
    • Hachette worked on descriptive geometry, collected work by Monge and edited Monge's Geometrie descriptive which was published in 1799.
    • He also published on a wide range of topics from his own major works on geometry, to works on applied mechanics including the theory of machines.
    • Although not a scientist of the first rank, Hachette nevertheless contributed to the progress of French science at the beginning of the nineteenth century by his efforts to increase the prestige of the Ecole Polytechnique and by making Monge's work widely known, especially in descriptive and analytic geometry and in the theory of machines.

  204. Hirst biography
    • However he did make friends with many of the mathematicians in Berlin and he continued to study mathematics concentrating on geometry.
    • In particular he attended lectures by Dirichlet and Steiner, being strongly influenced by Steiner to undertake further research on geometry.
    • Hirst began to attend lectures again in Paris and his own researches into geometry progressed well.
    • In 1869 he gave a course of twenty-four lectures on the Elements of Geometry to the Ladies educational Association of London.
    • This fitted well with his long held belief that Euclid's Elements should be replaced as the main geometry teaching text in schools.
    • His research had been mostly in geometry, in particular on Cremona transformations, and it was for this work that he was awarded the Royal Medal from the Royal Society in 1883.

  205. Chow biography
    • It was van der Waerden who introduced Chow to algebraic geometry at this time, pointing him towards the work of Severi, Bertini and Enriques.
    • In Zur algebraische geometry IX (published in Mathematische Annalen in 1937) he introduced the notion now known as Chow coordinates.
    • In 1955 Chow proved the so-called "Chow's moving lemma" in algebraic geometry, providing an intersection theory for algebraic cycles based on ideas and results of Severi, later also developed by van der Waerden, Hodge and Pedoe.
    • The "Chow ring" is just as fundamental in algebraic geometry as its topological counterpart.
    • It is a historical fact that this school of algebraic geometry [at Johns Hopkins University] was 'created' by Chow.
    • In addition to his research and leadership of the algebraic geometry group, Chow played an important role as editor-in-chief of the American Journal of Mathematics from 1953 to 1977.

  206. Voevodsky biography
    • He continued to work on ideas coming from Grothendieck and in 1991 published Galois representations connected with hyperbolic curves (Russian) which gave partial solutions to conjectures of Grothendieck, made on nonabelian algebraic geometry, contained in his 1983 letter to Faltings and also in his unpublished 'Esquisse d'un programme' mentioned above.
    • In one of his most influential papers, A Weil (1949), proved the "Riemann hypothesis for curves over functions fields", an analogue in positive characteristic algebraic geometry of the classical Riemann hypothesis.
    • To this outside observer, one of the most significant strands in the recent history of algebraic geometry has been the search for good cohomology theories of schemes.
    • The paper at hand is an almost elementary introduction to these ideas, mostly presenting the formal structure without getting into any proofs that require deep algebraic geometry.
    • Vladimir Voevodsky made one of the most outstanding advances in algebraic geometry in the past few decades by developing new cohomology theories for algebraic varieties.
    • We start with geometry, the category of topological spaces.

  207. Adler biography
    • After taking undergraduate courses at the University of Vienna, Adler undertook research in descriptive geometry graduating in 1884.
    • In 1901 he submitted his habilitation thesis on descriptive geometry to the University of Vienna.
    • As well as his interest in descriptive geometry, Adler was also interested in mathematical education, particularly in teaching mathematics in secondary schools.
    • His publications on this topic began around 1901 and by the end of his career he was publishing more on mathematical education than on geometry.
    • Most of his papers on mathematical education were directed towards teaching geometry in schools, but in 1907 he wrote on modern methods in mathematical instruction in Austrian middle schools.
    • He produced various teaching materials for teaching geometry in the sixth-form in Austrian schools such as an exercise book which he published in 1908.

  208. Mihoc biography
    • At Bucharest University, he was taught geometry by Gheorghe Titeica, the professor of Analytical Geometry.
    • Titeica had just published the monograph Geometrie differentielle projective des reseaux (1923) and, when he taught Mihoc, was writing another important monograph The projective differential geometry of lattices.
    • Titeica's geometry lectures gave the spirit of current research and inspired Mihoc.
    • Octav Onicescu (1892-1983) had studied geometry under Tullio Levi-Civita in Rome before spending some time on a visit to Paris.
    • There were many outstanding mathematicians in the Faculty including Francesco Cantelli, one of the most prominent Italian contributors to the mathematical theory of probability, and Guido Castelnuovo who, although a specialist in algebraic geometry, had written the probability text Calcolo della probabilita (1919) and was teaching courses on probability.

  209. Drinfeld biography
    • He worked on differential geometry, particularly on measure and integration.
    • [Drinfeld's] vision of mathematics was, to a great extent, influenced by Yu I Manin, his advisor, and by the Algebraic Geometry Seminar (Manin's Seminar) that functioned with regularity at Moscow State University for about two decades.
    • Not only do they span work in algebraic geometry and number theory, but his most recent ideas have taken a strikingly different direction: he has been doing significant work on mathematical questions motivated by physics, including the relatively new theory of quantum groups.
    • Drinfeld and Manin worked on the construction of instantons using ideas from algebraic geometry.
    • This book presents a comprehensive approach to the theory of chiral algebras from the point of view of algebraic geometry.
    • One of Drinfeld's most recent articles is Infinite-dimensional vector bundles in algebraic geometry: an introduction.

  210. Shimura biography
    • He felt that the course he took on analytical geometry was taught by a teacher who did not fully understand the subject.
    • The second event that Shimura considers began his mathematical career was his attendance at a conference on algebraic geometry and number theory in March 1953 organised by Yasuo Akizuki at Kyoto University.
    • Akizuki was building a strong School of Algebraic Geometry in Kyoto and he asked Shimura to talk at the conference, which was also attended by Yutaka Taniyama.
    • The progress of algebraic geometry has had a strong influence on number theory.
    • By means of the language of algebraic geometry we can now add new knowledge in that direction.
    • To Goro Shimura for his important and extensive work on arithmetical geometry and automorphic forms; concepts introduced by him were often seminal, and fertile ground for new developments, as witnessed by the many notations in number theory that carry his name and that have long been familiar to workers in the field.

  211. Choquet-Bruhat biography
    • This is an extremely elegant account of the methods of differential geometry and exterior differential systems, which as Andre Lichnerowicz says in his preface, remains faithful to the spirit of Elie Cartan.
    • [It] is to be most enthusiastically recommended to both pure and applied mathematicians, and it will be of particular value to theoretical physicists who desire a glimpse into the fascinating field of modern differential geometry.
    • They have succeeded in gathering in one volume the mathematical infrastructure of modern mathematical physics, which includes the theories of differentiable manifolds and global analysis, Riemannian and Kahlerian geometry, Lie groups, fibre bundles and their connections, characteristic classes and index theorems, distributions, and partial differential equations.
    • These lecture notes give a pedagogical account of the basic concepts in the differential geometry of graded manifolds and supermanifolds.
    • Both graded manifolds and supermanifolds extend classical geometry to include anticommuting variables, an approach which is motivated by the desire to extend the geometric techniques used in theoretical physics to systems which include fermions.

  212. Perelman biography
    • He had already published a number of papers: Realization of abstract k-skeletons as k-skeletons of intersections of convex polyhedra in R2k-1 (Russian) (1985); (with I V Polikanova) A remark on Helly's theorem (Russian) (1986); a supplement to A D Aleksandrov's, On the foundations of geometry (Russian) (1987) in which Perelman discussed the equivalence of a Pasch-style axiom of Aleksandrov and some of its consequences; and On the k-radii of a convex body (Russian) (1987).
    • It recognizes that the home of various important theorems of Riemannian geometry is the theory of Aleksandrov spaces, that both statements and proofs become more satisfactory (but not necessarily easier) in this context, and other theorems emerge naturally to complete the picture.
    • After visiting the IHES near Paris, Perelman returned to the Steklov Mathematics Institute in Leningrad but, thanks to Gromov, Perelman was invited to the United States to talk at the 1991 Geometry Festival held at Duke University in Durham, North Carolina.
    • Thurston proposed that, in a way analogous to the case of 2-manifolds, 3-manifolds can be classified using geometry.
    • For his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow.

  213. Bonnycastle biography
    • This was not the only book Bonnycastle published in 1782 for in the same years Introduction to Mensuration and Practical Geometry appeared in print.
    • Bonnycastle's Elements of geometry (1789) contains the propositions from Euclid's Elements books, 1-6, 11 and 12 with "critical and explanatory notes" by Bonnycastle.
    • He believed that the approach to geometry exhibited by Euclid provided an outstanding method for teaching young people logical and precise thinking.
    • (There may be further editions of this, and other works, by Bonnycastle and we have simply given the last edition we could find for each book.) He published another geometry book nearly 20 years later, A Treatise on Plane and Spherical Geometry (1806).

  214. De Beaune biography
    • De Beaune did, however, publish some work for he produced the first important introduction to Descartes' cartesian geometry.
    • The treatise is of interest chiefly because it is a work of pure synthetic geometry consisting of some 125 propositions composed in the style of Euclid by one of the ablest practitioners of the new analytical art typified by the works of Descartes and Fermat.
    • And while he built on results found in works on trigonometry by Snell, Girard, and Briggs, de Beaune avoided the numerical methods employed by these and other mathematical practitioners of the period in favour of pure geometry, indicating that he distinguished clearly between two genres of mathematics.
    • I do not think that one could acquire any solid knowledge of nature in physics without geometry, and the best of geometry consists of analysis, of such kind that without the latter it is quite imperfect.

  215. Schroeter biography
    • Richelot was a worthy successor to Jacobi and continued to work in his spirit while Hesse continued Jacobi's algebraic work making fundamental contributions to the algebraic treatment of analytic geometry.
    • Schroter created at Breslau the leading centre for synthetic geometry.
    • He edited his transcripts of Steiner's lectures on synthetic geometry which he interweaved with other material by Steiner, together with his own improvements, to produce Jacob Steiner's Vorlesungen uber synthetische Geometrie (1867).
    • This important work became the textbook for the synthetic geometry of conics with a second edition appearing in 1876 and a third edition in 1898 after Schroter's death.
    • Schroter was honoured for his contributions to synthetic geometry.

  216. Kaczmarz biography
    • He was taught by Stanislaw Zaremba, Ivan Sleszynski, Marian Smoluchowski, Kazimierz Zorawski (1866-1953), who worked on differential geometry and fluid mechanics, and Antoni Hoborski (1879-1940), whose main interest was also in differential geometry.
    • From 1923 to 1939 Kaczmarz taught many university level courses at Lwow such as: Analytical Geometry, Higher Analysis, Integral Equations, Algebraic Curves, Trigonometric Series, Non-Euclidean Geometry and the Theory of Groups, and Differential Geometry.

  217. Lonie biography
    • Their programme exhibited an extensive course of teaching in geometry, practical mathematics, algebra, and geography; and in all these branches they sustained an active examination.
    • An advertisement appearing in 1859 lists the topics taught in the Mathematics Department of Madras College: Theoretical mathematics - Geometry and algebra; Practical mathematics - Trigonometry, surveying, navigation, etc.; Natural philosophy; Private Class geography.
    • Topics taught in Lonie's Mathematics Department in 1879 are: Mathematics - Euclid, Elementary Modern Geometry and Conic Sections, Plane and Spherical Trigonometry, with practice, Elementary Algebra with Higher Equations, Mensuration, and Mechanics; Physics - after Balfour Stewart and Modern Views of Natural Forces including Energy, Sound, Heat, Light; Geography - Modern Geography.
    • ',3)">3] that Lonie altered the timetable moving his Geometry class from 12 noon to 6 am.
    • He was really a capital teacher for boys, gifted with a faculty of lucid explanation, the still rarer faculty of making geometry and algebra nearly as interesting as a game of chance, and, rarest of all, the faculty of being able to secure attention and maintain complete order in large classes without the tawse, birch, twigs, or any other instrument of physical force.

  218. Salmon biography
    • Although the main topic of interest was synthetic geometry, Salmon only worked in this area for a short time before moving into the area of algebraic geometry.
    • Salmon became interested in the algebraic approach to geometry taken by Cayley, Sylvester, Hermite and later by Clebsch.
    • These four treatises on conic sections, higher plane curves, modern higher algebra, and the geometry of three dimensions not only gave a comprehensive treatment of their respective fields but also were written with a clarity of expression and an elegance of style that made them models of what a textbook should be.
    • The famous four textbooks referred to in this quote are A treatise on conic sections (1848), A treatise on higher plane curves: Intended as a sequel to a treatise on conic sections (1852), Lessons introductory to the modern higher algebra (1859), and A treatise on the analytic geometry of three dimensions (1862).

  219. Allman biography
    • However, Allman's most significant contribution was Greek geometry from Thales to Euclid published in Dublin in 1889.
    • In this work I propose to give some account of the progress of geometry during the first of these periods, and also to notice briefly the chief organs of its development.
    • In this he traced the rise and progress of geometry and arithmetic, and threw new light on the history of the early development of mathematics.
    • Nothing so painstaking, so lucid, and so satisfactory has been written on the history of geometry during the period selected, even in laborious Germany.
    • 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.

  220. Fiedler biography
    • Bydzovsky's interests were mainly in geometry and he passed this love to Fiedler who he advised while he wrote his undergraduate thesis Hyperosculating points of algebraic plane curves and their generalization in Sr.
    • Fiedler was already undertaking research supervised by Eduard Čech when the new degree was introduced and he became one of the first to be awarded the degree in 1955 for his thesis Geometry of the simplex.
    • As well as geometry, he began to study numerical methods, matrix theory and graph theory.
    • Since the early days of his research career, his favourite subjects have been geometry, graph theory, linear algebra, and their applications to numerical computations.
    • In 2001 he published a book which mentions three of the topics which made up his special interest in its title: Matrices and graphs in Euclidean geometry (2001).

  221. Dupin biography
    • Dupin was educated at the Ecole Polytechnique in Paris, where he learnt geometry from Monge.
    • He was appointed as secretary to the Ionian Academy which had been founded only a short time before and he undertook deep research on mathematical topic, in particular studying the differential geometry of surfaces, and applied mechanics where he investigated the resistance of materials.
    • contains many contributions to differential geometry, notably the introduction of conjugate and asymptotic lines on a surface ..
    • Other contributions to differential geometry which occur in this work include his invention of the 'Dupin indicatrix' which gives an indication of the local behaviour of a surface up to the terms of degree two.
    • But he not only had an academic life, publishing further important works on the applications of differential geometry to industry and the arts, but he also took a major part in politics from 1828.

  222. Plucker biography
    • For example, much of his mathematics followed the French style of geometry as developed by Monge.
    • His next move was to go to France in 1823 where he attended courses on geometry at the University of Paris.
    • Steiner was the leader of the German school of synthetic geometry, while Plucker followed the analytical approach.
    • In each volume he discussed the plane analytic geometry of the line, circle, and conic sections; and many facts and theorems - either discovered or known by Plucker - were demonstrated in a more elegant manner.
    • The characteristic features of Plucker's analytic geometry were already present in this work, namely, the elegant operations with algebraic symbols occurring in the equations of conic sections and their pencils.

  223. Moore Eliakim biography
    • Moore's doctoral dissertation was entitled Extensions of Certain Theorems of Clifford and Cayley in the Geometry of n Dimensions and this led to the award of his doctorate in 1885.
    • Moore was a fiery enthusiast, brilliant, and keenly interested in the popular mathematical research movements of the day; Bolza, a product of the meticulous German school of analysis led by Weierstrass, was an able, and widely read research scholar; Maschke was more deliberate than the other two, sagacious, brilliant in research, and a most delightful lecturer in geometry.
    • In his work on the foundations of geometry begun around 1900 Moore examined the independence of Hilbert's axioms.
    • His 1902 paper On the projective axioms of geometry showed that Hilbert's axiom system contained redundant axioms.
    • Other topics he worked on include algebraic geometry, number theory and integral equations.

  224. Paman biography
    • Paman wrote a paper on this treatise which he called The Harmony of the Ancient and Modern Geometry Asserted.
    • Before setting out on the voyage, Paman had given his paper, The Harmony of the Ancient and Modern Geometry Asserted, to his friend Dr Hartley, and when he returned, in February 1742, Paman sent it to the Royal Society.
    • The book was titled, as the paper, The Harmony of the Ancient and Modern Geometry asserted.
    • The harmony of the ancient and modern geometry asserted; in answer to the Analyst, etc.
    • of introducing them first into Geometry, and that whilst I aimed at the Rigour of the Ancients, I might avoid the Tedium and Perplexity of their Demonstrations ad absurdum.

  225. Chang biography
    • She also studies the related extremal inequalities and problems in isospectral geometry.
    • The Ruth Lyttle Satter Prize is awarded to Sun-Yung Alice Chang for her deep contributions to the study of partial differential equations on Riemannian manifolds and in particular for her work on extremal problems in spectral geometry and the compactness of isospectral metrics within a fixed conformal class on a compact 3-manifold.
    • Such functions play an important role in the study of the blow-up phenomenon in a number of problems in geometry.
    • We have also applied this approach in conformal geometry to the isospectral compactness problem on 3-manifolds when the metrics are restricted in any given conformal class.
    • The course was entitled "Geometric PDE" and described using analytic tools like that of partial differential equations to solve problems in geometry.

  226. Schonflies biography
    • Shortly after this letter was written Arthur Schonflies was appointed ausserordentlicher Professor in Gottingen, where for the next seven years he attracted droves of students to his classes in descriptive geometry.
    • Schonflies worked first on geometry and kinematics but became best known for his work on set theory and crystallography.
    • Schonflies also wrote on kinematics and projective geometry.
    • He wrote textbooks on descriptive geometry and analytic geometry, and a calculus textbook Einfuhrung in die mathematische Behandlung der Naturwissenschaft (1895) written jointly with Walter Nernst.

  227. Bloch biography
    • Bloch worked on a large range of mathematical topics; for example, function theory, geometry, number theory, algebraic equations and kinematics.
    • The name of Andre Bloch is attached to magnificent works on the theory of analytic functions of a complex variable and on many different areas of geometry, in particular geometrical inversion and non-Euclidean geometry for which he had a particular fondness.
    • The integral geometry of the closed contour has already been the subject of several important pieces of research.
    • Suffice it to say in particular work by Gabriel Koenigs and his students on the determination of the volume generated by any movement of a closed contour, those of Jacques Hadamard on the generalization of the theorem of Paul Guldin and those of M A Buhl on the geometry and the analysis of multiple integrals.

  228. Weatherburn biography
    • He took me through the topics in his two books on vector analysis, and perhaps also some differential geometry..
    • At about this time his research interests changed from vector analysis to differential geometry.
    • He wrote two major volumes Differential geometry of three dimensions (1927, 1930) as well as nearly 30 papers on this topic.
    • Other topics are however included, with the result that the two volumes together give an account of most of the principal branches of classical differential geometry.
    • He published An Introduction to Riemannian Geometry and the Tensor Calculus in 1938 and it was reissued in 1966.

  229. Bertini biography
    • Cremona recommended him to teach descriptive and projective geometry as a lecturer at the University of Rome.
    • In 1875 he was appointed professor of geometry at the University of Pisa, accepting the offer of a chair for which he had been proposed by Betti.
    • His work in algebraic geometry extended Cremona's work.
    • These two fundamental theorems are among the ones most used in algebraic geometry.
    • We should note that Bertini had a number of outstanding students and their work continued the Italian tradition of outstanding contributions to geometry.

  230. Dehn biography
    • In this thesis he proved the Saccheri-Legendre theorem which states that in absolute geometry the sum of the angles in a triangle is at most 180°.
    • By absolute geometry, we mean geometry satisfying the axioms of Euclidean geometry except for the parallel postulate.
    • In Mathematics his courses included History of Mathematics and Projective Geometry.

  231. Fubini biography
    • There he was taught by Dini and Bianchi who quickly influenced Fubini to undertake research in geometry.
    • However, Fubini was lucky for his teacher Bianchi was about to publish an important work on differential geometry and he discussed the results of Fubini's thesis in his treatise which appeared in 1902.
    • Fubini's interests were exceptionally wide moving from his early work on differential geometry towards analysis.
    • His most important work was on differential projective geometry where he used the absolute differential calculus.
    • His contributions opened new paths for research in several areas of analysis, geometry, and mathematical physics.

  232. Scheffers biography
    • Scheffers' favourite field of study was geometry and, more specifically, the differential geometry of intuitive space.
    • The first volume of an intended two volume collaboration between Lie and Scheffers was published in 1896 entitled Geometrie der Beruhrungstransformationen (Geometry of Contact Transformations).
    • In 1908 and 1909, Eduard Study, the Professor of Mathematics at Bonn, published two papers in which he criticised the standard treatment of differential geometry in general, and Scheffers' treatment in particular.
    • Over three hundred of the twelve hundred pages are devoted to applications to geometry.

  233. Bottasso biography
    • After graduating, he was appointed assistant professor in projective geometry at the University of Turin where he taught for three years.
    • Arriving back in Italy Bottasso was appointed as an assistant professor of projective geometry at the University of Bologna.
    • Following this series of appointments, he became a lecturer in algebra and analytic geometry at the University of Parvia (the city is 35 km south of Milan).
    • Bottasso studied differential geometry and mechanics but also made contributions to actuarial and financial mathematics.
    • He used the vector calculus in studying problems in geometry, mechanics and physics.

  234. Savage biography
    • His grades began to improve: C in analytic geometry; B in calculus; B in differential equations; A in Raymond Wilder's foundations of mathematics; and A in Raymond Wilder's point set topology course.
    • In 1941 Savage received his PhD with a thesis was on metric and differential geometry.
    • degree at the University of Michigan was on applications of vectorial methods to metric geometry (in the sense of the Menger school), especially with a view to the merging of metric geometry in that sense with differential geometry.

  235. 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.
    • These topics are discussed in connection with one or another of the major mathematical creations which have been introduced since ancient times and hence accompany some presentation of the concepts of Euclidean geometry, trigonometry, projective geometry, coordinate geometry, statistics, the theory of probability, transfinite numbers, non-Euclidean geometry, and other mathematical subjects.

  236. Pacioli biography
    • The work studies arithmetic, algebra, geometry and trigonometry and, despite the lack of originality, was to provide a basis for the major progress in mathematics which took place in Europe shortly after this time.
    • An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid's geometry.
    • Pacioli was appointed to teach geometry at the University of Pisa in Florence in 1500.
    • He remained in Florence, teaching geometry at the university, until 1506.
    • This biography accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry.

  237. Descartes biography
    • Rene Descartes was a philosopher whose work, La geometrie, includes his application of algebra to geometry from which we now have Cartesian geometry.
    • Algebra makes it possible to recognise the typical problems in geometry and to bring together problems which in geometrical dress would not appear to be related at all.
    • Algebra imports into geometry the most natural principles of division and the most natural hierarchy of method.
    • Some ideas in La Geometrie may have come from earlier work of Oresme but in Oresme's work there is no evidence of linking algebra and geometry.

  238. Bruno biography
    • From 1852 to 1858 he taught algebra and geometry, while from 1860 to 1862 he taught differential and integral calculus.
    • In 1863 he taught descriptive geometry at the University and, in 1863, he became Professor of Descriptive Geometry.
    • His title was changed to Professor of Projective Geometry with Design in 1875 and in 1881 he became Chairman of the Faculty of Science.
    • All Bruno's publications deal with research into geometry.

  239. Kaestner biography
    • However Kastner was quite unenthusiastic about logic, but this is not surprising for a mathematician of this period who was interested in geometry.
    • Folta writes in [DVT - Dejiny Ved a Techniky 6 (1973), 189-205.',3)">3] about Kastner's work on geometry:- .
    • Kastner [is] among the mathematicians of the 18th century whose broad interests compelled [him] to concern [himself] with the principal problems of geometry.
    • [His] results included new features that more precisely formulated the traditional interpretation of elementary geometry.
    • Kastner, in spite of his rather great inclination for Euclid's Elements, based his version of the axiomatics of geometry in his Kompendium on other principles (e.g., on motions) and attempted both to seize on other fundamental properties (continuity, ordering) and to determine the selection of the parallel axiom as a foundation.

  240. Grandi biography
    • In Florence he met with Vincenzo Viviani and learned from him and his students the methods of classical geometry and also the infinitesimal methods of Bonaventura Cavalieri.
    • Viviani had answered the question using geometry but had not given a proof.
    • a work which contains more than its title would lead us to expect, and in which the author remarks many other curiosities in geometry of the same kind, and among others, a portion of the surface of a right cone which can be squared.
    • Having learnt much geometry, between 1699 and 1700 Grandi began to look at applications to optics, mechanics, astronomy.
    • However, he did not neglect mathematics, publishing an Italian version of Euclid's Elements in 1731 and a series of works on mechanics Instituzioni meccaniche (1739), arithmetic Instituzioni di aritmetica pratica (1740) and geometry Instituzioni geometriche (1741).

  241. Turner biography
    • In 1620 he succeeded Briggs first to the chair of geometry at Gresham College in London, then, in 1630, to the Savilian chair of Geometry at Oxford [Dictionary of National Biography (Oxford, 2004).',2)">2]:- .
    • On 9 November 1648 representatives of Parliament removed Turner from his fellowship at Merton and from the Savilian chair of Geometry.
    • Through Laud, Turner gained the appointment to the Savilian chair of Geometry at Oxford.
    • Savilian Geometry Professor1631 .

  242. Backlund biography
    • Despite writing a thesis on astronomy, Backlund published a paper on geometry Några satser om plana algebraiska kurvor, som gå genom samma skarningspunkter (Some comments on planar algebraic curves, which go through the same intersection points) in 1868.
    • After being appointed as an assistant at the astronomical observatory in Lund on 19 January 1869, he was appointed as a docent in geometry later in the same year.
    • Backlund was the youngest of the candidates and did not stand much chance of being appointed despite being able to present five papers in algebraic geometry.
    • Having failed to win the chair of mathematics with his contributions to algebraic geometry, he now changed topic feeling that this would give him a better chance of promotion.
    • All Backlund's papers on algebraic geometry are in Swedish and published in Swedish journals.

  243. Aleksandrov biography
    • But his interest was mainly directed towards fundamental problems of mathematics: the foundations of geometry and non-euclidean geometry.
    • Today the Department of General Topology and Geometry of Moscow State University is Russia's leading centre of research in set-theoretic topology.
    • After Aleksandrov's death in November 1982, his colleagues from the Department of Higher Geometry and Topology, in which he had held the chair, sent a letter to Moscow University's rector A A Logunov proposing that one of Aleksandrov's former students should become Head of the Department, to preserve Aleksandrov's scientific school.
    • On 28 December 1982 the rector issued a circular creating the Department of general topology and Geometry.

  244. Morin Ugo biography
    • In 1935 he became a lecturer in descriptive geometry at the University of Padua and held this position until December 1942.
    • Morin competed for the Chair of Analytic Geometry at the University of Florence in which he was successful, taking up the appointment in December 1942.
    • Unirationality and rationality problems are, in the field of algebraic geometry, among the most significant topics of the scientific legacy of Ugo Morin.
    • Some of the profound themes of twentieth century geometry are intertwined with some of the beautiful geometric ideas that Ugo Morin has developed and nurtured over the course of his life.
    • Morin also edited some textbooks for teaching geometry in secondary schools during the 1950s and in the 1960s worked on texts for teachers of mathematics in secondary schools prepared for the Ministry of Education.

  245. Lie biography
    • The type of mathematics that Lie would study became more clearly defined during 1868 when he avidly read papers on geometry by Plucker and Poncelet.
    • Despite the common link through Plucker's line geometry, Lie and Klein were rather different in character as Freudenthal points out in [Dictionary of Scientific Biography (New York 1970-1990).
    • Jordan seems to have succeeded in a way that Sylow did not, for Jordan made Lie realise how important group theory was for the study of geometry.
    • He began to discuss with Klein these new ideas on groups and geometry and he would collaborate later with Klein in publishing several papers.
    • This joint work had as one of its outcomes Klein's characterisation of geometry in his Erlangen Program of 1872 as properties invariant under a group action.

  246. Mathews biography
    • Most of Mathews' research was on number theory but he also wrote texts on Bessel functions and on projective geometry.
    • Mathews also wrote Algebraic equations (1907) which is a clear exposition of Galois theory, and Projective geometry (1914).
    • This latter book develops the subject of projective geometry without using the concept of distance and it bases projective geometry on a minimal set of axioms.
    • The book contains a wealth of information concerning the projective geometry of conics and quadrics.

  247. Minkowski biography
    • This lecture is particularly interesting, for it contains the first example of the method which Minkowski would develop some years later in his famous "geometry of numbers".
    • In a paper published in 1908 Minkowski reformulated Einstein's 1905 paper by introducing the four-dimensional (space-time) non-Euclidean geometry, a step which Einstein did not think much of at the time.
    • His most original achievement, however, was his 'geometry of numbers' which he initiated in 1890.
    • It gave an elementary account of his work on the geometry of numbers and of its applications to the theories of Diophantine approximation and of algebraic numbers.
    • Work on the geometry of numbers led on to work on convex bodies and to questions about packing problems, the ways in which figures of a given shape can be placed within another given figure.

  248. Roomen biography
    • There is a science common to geometry and arithmetic which considers quantity generally as measurable.
    • Surely there is a certain science common to arithmetic and geometry to which properties common to all quantities pertain: since a proportion is common to all quantities, not only abstract ones such as numbers and magnitudes, but also concrete ones such as times, sounds, voices, places, motions, and forces (for all these and many others are said to have a proportion if their relation is considered from the viewpoint of quantity).
    • Van Roomen proposes unifying geometry and arithmetic under his concept of 'mathesis universalis'.
    • Those who have written about algebra considered it to be a part of arithmetic, although it might as well be considered to be part of geometry.
    • Algebraic propositions usually are demonstrated by geometrical constructions, so that algebra should perhaps better be considered as a part of geometry.

  249. Schouten biography
    • At first Barrau advised Schouten but, after he left Delft for Groningen in 1913 to become Schoute's successor as professor of geometry, Jacob Cardinaal became Schouten's advisor.
    • Klein's Erlanger Programm of 1872 looked at geometry as properties invariant under the action of a group.
    • For example in 1924 he published Uber die Geometrie der halb-symmetrischen Ubertragungen jointly with Alexander Friedmann, and in 1926 he published two papers written jointly with Elie Cartan: On Riemaniann geometries admitting an absolute parallelism, and On the Geometry of the Group-manifold of Simple and Semi-simple Groups.
    • He attended the international conference on differential geometry organised by Benjamin Fedorovich Kagan which took place at Moscow University in 1934.
    • Since the publication of the author's 'Der Ricci-Kalkul' [1924] the subject of tensor calculus as applied to differential geometry has grown very considerably, and the present edition is more than a mere translation of the first.

  250. Kahler biography
    • This paper was the starting point for Kahler geometry with its notions of Kahler manifolds and Kahler groups which are today fundamental ideas in string theory and the study of space-time.
    • [He] gave a five semester course with sometimes more than 10 hours of classes per week, in which he expostulated on algebra, algebraic geometry, function theory and arithmetic.
    • The subject combines algebra and geometry, with some arithmetical flavouring; but the author, instead of following in his terminology the accepted usage in either one of those subjects, or adapting it to his purposes, has chosen to borrow his vocabulary from philosophy, so that rings, homomorphisms, factor-rings, ideals, complete local rings appear as "objects", "perceptions", "subjects", "perspectives", "individualities".
    • His speculative considerations are illustrated by suggestive examples from set theory, mathematical logic, abstract algebra and differential, algebraic and analytic geometry.
    • The main thesis of the paper is that algebraic geometry is a prolegomenon to a mathematical theory of monades.

  251. Hunyadi biography
    • In a paper published in Crelle's Journal in 1880 he explained his view on the relation of algebra and geometry (see for example [Internat.
    • considering that analytic geometry is principally a geometrical discipline, it is the author's modest view that we have to make every effort not to delegate the main role to algebra and analysis, confusing the means with the end, but to give the main aim 'geometry' its due importance.
    • This position is further supported by the argument that if we do approach analytic geometry from the opposite end, we might commit the error of reducing the analytic teaching of geometry to a collection of exercises in algebra and analysis which would certainly go against the spirit of the science.

  252. Casey biography
    • He published several research papers on geometry while studying for his degree, and the influence of the group of geometers working at Trinity College, including W R Hamilton, A S Hart, Salmon and Townsend, is apparent in this work.
    • A little later, he was offered a specially created professorship in geometry at Trinity College.
    • 1910); A treatise on the analytical geometry of the point, line, circle and conic sections (1885; 2nd ed.
    • It was in his Sequel to Euclid that Casey presented for the first time in a textbook those extensions of the theorems of Euclid that became known as the newer geometry of the triangle; indeed, he and the French mathematician Emile Lemoine (1840-1912) are held to be the founders of the so-called Modern Geometry of the circle and triangle.

  253. Gergonne biography
    • Gergonne's mathematical interests were in geometry so it is not surprising that it was this topic which figured most prominently in his journal.
    • Perhaps surprisingly, since Gergonne was himself a geometer, he suggests that algebra is a more important topic than geometry.
    • We will examine Gergonne's contributions to geometry later in this article, but for the moment it is worth noting that he did publish on other topics.
    • Gergonne introduced the word polar and the principle of duality in projective geometry was one of his main contributions.
    • He noticed the fact that certain forms of geometry yielded theorems which appeared in related pairs, and this led him to a more detailed analysis of why this was so.

  254. Ceva Giovanni biography
    • But he was also interested in undertaking scientific activities and, in addition to the financial dealing he was doing, he studied geometry and hydraulics.
    • For most of his life Giovanni Ceva worked on geometry.
    • He discovered one of the most important results on the synthetic geometry of the triangle between Greek times and the 19th Century.
    • In addition to his work on geometry, Ceva studied applications of mechanics and statics to geometric systems.
    • It investigates questions of pure geometry as well as applications of mathematics, particularly to hydrodynamics.

  255. Chisini biography
    • Geometry teaches you how to carry out the correct reasoning on the wrong picture.
    • According to Eugenio Giuseppe Togliatti [Discorso commemorativo pronunciato nella seduta ordinaria del 19 aprile 1969 : Accademia Nazionale dei Lincei, Celebrazioni Lince No 26 (Accademia Nazionale dei Lincei, Rome, 1969).',2)">2], the activity of Oscar Chisini in the mathematical sciences was threefold: scientific research in the field of algebraic geometry, high level original reconstruction of mathematical theories, and active involvement in the teaching of mathematics at secondary school level.
    • For example he published in this journal: Sul principio di continuita (1956) which is an expository lecture on the principle of continuity in algebraic geometry, beginning with the ideas of Kepler; La superficie cubica (1957) which gives a clear and original treatment of the principal properties of cubic surfaces, presenting it as a preliminary introduction to the study of algebraic geometry; and Isoperimetri (1960) which contains elementary thoughts on the plane isoperimetric problem.
    • As an algebraic geometry researcher, Oscar Chisini is to be considered part of the so-called Italian school, featuring among others Luigi Cremona, Corrado Segre, Guido Castelnuovo, Francesco Severi, Beniamino Segre and, of course, Federigo Enriques.

  256. Henrici biography
    • introduced projective geometry, vector analysis, and graphical statics into the University College mathematics syllabus - a radical departure from the analytically biased Cambridge-style course previously taught.
    • Henrici wrote some excellent little books to introduce undergraduates to mathematical ideas such as projective geometry in Congruent Figures (1878), and vector methods in Vectors and Rotors (1903).
    • He was also a major contributor to the eleventh edition of Encyclopaedia Britannica (published in 1910 and 1911) contributing articles on 'calculating machines', 'Euclidean geometry', 'projective geometry', 'projection', 'descriptive geometry', and 'perspective'.

  257. Whitehead Henry biography
    • In pure geometry he has not been over diligent ..
    • He attended a seminar which Veblen gave on differential geometry and it must have been a very fine talk for it persuaded Whitehead that he would undertake research in that topic.
    • He worked mainly on differential geometry although towards the end of his three years there he became interested in topology.
    • Whitehead's joint work with his doctoral supervisor Veblen led to The Foundations of Differential Geometry (1932), now considered a classic.
    • Soon after his return to England, Whitehead wrote another major work on differential geometry On the Covering of a Complete Space by the Geodesics Through a Point (1935).

  258. Wald biography
    • He worked under Menger's supervision on geometry and was awarded his doctorate in 1931.
    • Between 1931 and 1937 Wald published 21 papers on geometry which Menger describes in [D L Sills (ed.), International Encyclopedia of Social Sciences 16 (1968), 435-438.',5)">5] as:- .
    • However his work with Schlesinger did not only give him financial security and hence the opportunity to undertake research in geometry.
    • As we mentioned above, in Vienna Wald worked on pure mathematics, mostly geometry, and on econometrics.
    • His first pure mathematical work was on metric spaces, an extension of Steinitz's work to infinite dimensional vector spaces, and some beautiful results on differential geometry.

  259. Houel biography
    • Houel published a work on geometry in 1863.
    • At this stage he did not know of the published work on non-euclidean geometry but he clearly was working his way towards the idea.
    • Since long, the scientific researches of mathematicians on the fundamental principles of elementary geometry have concentrated themselves almost exclusively on the theory of parallels, and if, hitherto, the efforts of so many eminent minds have produced no satisfactory result, it is perhaps permitted to conclude thence that in pursuing these researches they have followed a false path and attacked an insoluble problem, of which the importance has been exaggerated in consequence of inexact ideas on the nature and origin of the primordial truths of the science of space.
    • Houel became interested in non-euclidean geometry once he had been made aware of the work of Bolyai and Lobachevsky.
    • He corresponded with Tilly on non-euclidean geometry.

  260. James biography
    • In 1970 James was appointed to the famous chair of Savilian Professor of Geometry.
    • James was elected a Fellow of New College Oxford in 1970 when he became Savilian Professor of Geometry and he continues to hold the fellowship.
    • He retired from the Savilian Professor of Geometry in 1995 and became professor emeritus.
    • These volumes covered Henry Whitehead's work in differential geometry, complexes and manifolds, homotopy theory, and algebraic and classical topology.
    • Savilian Geometry Professor1969 .

  261. Julia biography
    • This appointment to a professorship at the Sorbonne came without a specific chair, but in 1925 he was named to the Chair of Applications of Analysis to Geometry at the Sorbonne.
    • In 1931 he was appointed to the Chair of Differential and Integral Calculus, then in 1937 he was appointed to the Chair of Geometry and Algebra at the Ecole Polytechnique when Maurice d'Ocagne retired.
    • Volume 5 contains works on (i) Number theory; and (ii) Geometry, mechanics, and electricity.
    • About two-thirds of the first volume is devoted to the applications of analysis to geometry.
    • History Topics: A History of Fractal Geometry .

  262. Theon of Smyrna biography
    • It is, rather, a handbook for philosophy students, written to illustrate how arithmetic, geometry, stereometry, music, and astronomy are interrelated.
    • One who had become skilled in all geometry and all music and astronomy would be reckoned most happy on making acquaintance with the writings of Plato, but this cannot be come by easily or readily, for it calls for a very great deal of application from youth upwards.
    • The work begins with a collection of theorems which Theon says will be useful for the study of arithmetic, music, geometry, and astronomy in Plato.
    • However his coverage of geometry is none too good and later in the book he makes an excuse for this saying that anyone who reads his book, or the works of Plato, will have already studied elementary geometry.

  263. Schubert biography
    • He then entered the University of Berlin where he took his first degree in 1867, then moved to the University of Halle where he was awarded a doctorate in 1870 for his thesis on enumerative geometry Zur Theorie der Charakteristiken.
    • He had published two papers on enumerative geometry before submitting his doctoral dissertation, these being on the system of sixteen spheres that touch four given spheres.
    • Schubert is famed for his work on enumerative geometry which considers those parts of algebraic geometry that involves a finite number of solutions.
    • Algebraically, the solution of the problems of enumerative geometry amounts to finding the number of solutions for certain systems of algebraic equations with finitely many solutions.

  264. Menger biography
    • Personally, I am occupied by geometry of all kinds, furthermore by epistemology.
    • In 1927 Menger was invited by Hahn to accept the chair of geometry at the University of Vienna when Kurt Reidemeister left for Konigsberg.
    • This book reprints all the articles (in German) along with chapters (in English) surveying the important developments in economics, logic, topology and geometry that were reported in the 'Ergebnisse'.
    • There is an important phase in the development of modern point set theoretical geometry which has been closely associated with the concept of dimensionality, - we refer to the attempt to create precise mathematical meaning for the simple geometric spaces of our intuition in terms of primitive non-arithmetical concepts.
    • Around this time Menger's interests in mathematics broadened and he began to work on hyperbolic geometry, probabilistic geometry and the algebra of functions.
    • Menger's work on geometry failed to have the impact that his work on dimension theory had.
    • This is almost certainly because geometry, at this time, was a rather neglected area of mathematics, particularly in the United States.
    • This led to his interest in mathematical education and, during the 1950s and 1960s, he wrote articles on mathematical education and published books with new ideas on teaching calculus, geometry and other branches of mathematics.

  265. Poinsot biography
    • He had published a number of works on geometry, mechanics and statics beginning with Elements de statique in 1803 and following this with [Dictionary of Scientific Biography (New York 1970-1990).
    • His research in geometry, statics and dynamics is important.
    • However he is best known for his dedication to geometry and, together with Monge, he contributed to the topic regaining its leading role in mathematical research in France in the eighteenth century.
    • As well as his research in geometry, Poinsot contributed to its increasing importance by creating a chair of advanced geometry at the Sorbonne in 1846.

  266. Sturm biography
    • He used his time well and began to write articles on geometry which were published in Gergonne's Annales de mathematiques pures et appliquees.
    • The author describes how Tarski showed in 1940 that Sturm's method of proof could be used in mathematical logic to prove the completeness of elementary algebra and geometry.
    • His time for research was now limited but he still made important contributions undertaking research on infinitesimal geometry, projective geometry and the differential geometry of curves and surfaces.

  267. Reidemeister biography
    • Immediately he had written his doctoral thesis, Reidemeister became interested in geometry.
    • It was Wilhelm Blaschke who came up with the particular problems in differential geometry on which Reidemeister began to work [Mathematicians under the Nazis (Princeton University Press, Princeton, 2003).',3)">3]:- .
    • In Hamburg, he met Wilhelm Blaschke who turned him toward an interest in geometry, and Blaschke entrusted the brilliant student with cooperation on the second volume of his 'Differential Geometry'.
    • For example his reviews of Principles of Geometry by Henry Baker and Grundzuge der mehrdimensionalen Differentialgeometrie by Dirk Struik appeared in 1923.
    • On Hans Hahn's recommendation, despite having never habilitated, Reidemeister was appointed as associate professor of geometry at the University of Vienna in October 1923.
    • Reidemeister worked on the foundations of geometry and he wrote an important book on knot theory Knoten und Gruppen (1926).
    • He established a geometry and topology based on group theory without the concept of a limit.
    • In the second part, a purely axiomatic development of geometry is based on the concept of a 3-web, and its completeness and consistency are demonstrated by forming from it a number system satisfying the postulates embodied in the first part.
    • Both as an original composition and for its collection of interesting developments, this work is a distinct addition to geometrical literature, and should appeal strongly to anyone deeply interested in the logical foundations of geometry.
    • In the section of the encyclopaedia referred to above, he showed that combinatory topology was one of the most primitive branches of geometry, in which the concept of limiting values had as yet no place.
    • The chapter on analytic geometry shows how translations, rotations and dilatations are represented by linear transformations of a complex variable, and how the scope is extended by considering linear fractional transformations.
    • The fourth chapter gives a system of axioms for Euclidean geometry, using distance as a primitive concept.
    • The sixth (on geometry and logic) begins with Hjelmslev's idea of representing the points and lines of the Euclidean (or non-Euclidean) plane by involutory transformations that leave them invariant ..
    • This is followed by remarks about Russell's paradox and about affine geometry over an arbitrary field.
    • The ninth (on geometry and number theory) develops the theory of algebraic numbers, leading to rigorous proofs of the impossibility, by Euclidean constructions, of duplicating the cube and trisecting an angle of 600.

  268. Hirzebruch biography
    • He also studied algebraic topology and algebraic geometry with Heinz Hopf at the Eidgenossische Technische Hochschule in Zurich from 1949 to 1950.
    • We give some details of those which followed his 1956 masterpiece mentioned above which was translated as Topological methods in algebraic geometry and appeared in several updated editions over many years.
    • for outstanding work combining topology, algebraic and differential geometry, and algebraic number theory; and for his stimulation of mathematical cooperation and research.
    • For the past three and a half decades, the name of Professor Friedrich Hirzebruch has been connected with famous results in the areas of topology, algebraic geometry, and global differential geometry, results which all mark the beginning of important theories and which have had an enormous influence on the development of modern mathematics.

  269. Weyr biography
    • Then he studied at the Prague Polytechnic from 1865 to 1868 where he was taught geometry by Otto Wilhelm Fiedler (1832-1911).
    • Fiedler, who had obtained his doctorate from the University of Leipzig in 1858 advised by August Ferdinand Mobius, became professor of descriptive geometry at the Technical University of Prague in 1864 but left three years later to become a professor in Zurich.
    • The two brothers collaborated on the first volume of their three volume book on projective geometry entitled Foundations of higher geometry (Czech).
    • He acquainted himself with the latest achievements in projective and synthetic geometry and wrote several treatises that helped him to gain the attention of European geometers.
    • They were interested in descriptive geometry, then in projective geometry and their interests turned towards algebraic and synthetic methods in geometry.
    • Emil Weyr led the geometry school in Vienna throughout the 1880's up until his death.
    • His approach to synthetic geometry was in the style of Chasles and Cremona while the other geometric approach at the time, namely that of Karl von Staudt and Theodor Reye, had little influence on him.
    • If among Austrian middle school teachers geometric knowledge and understanding of the methods of geometry are widely used, that is mostly due to Weyr.

  270. Freudenthal biography
    • A particular interest that Freudenthal had in the history of mathematics was geometry.
    • Bos, in [The legacy of Hans Freudenthal (Dordrecht, 1993), 51-58.',4)">4], discusses his contributions to the history of geometry around 1900:- .
    • In the late 1950s Freudenthal published several articles on the history of geometry around 1900, in particular on Hilbert's innovative approach to the foundations of geometry.
    • In particular, his essay-review of the eighth edition of Hilbert's Grundlagen der Geometrie has become a standard reference in historical studies of geometry.

  271. Seidenberg biography
    • Seidenberg contributed important research to commutative algebra, algebraic geometry, differential algebra, and the history of mathematics.
    • An example of one of his papers on algebraic geometry is The hyperplane sections of normal varieties (1950) which has proved fundamental in later advances.
    • 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 Geometry and Algebra in Ancient Civilizations Van der Waerden puts forward similar views for which he gives credit to Seidenberg, saying that Seidenberg made him look at the history of mathematics a new way.

  272. Monte biography
    • This treatise represented a major step forward in understanding the geometry of perspective and it was a major contribution towards the development of projective geometry.
    • It goes beyond most other works of the period on perspective in using three dimensional geometry based on the more advanced Book XI of Euclid's Elements.
    • References to Euclid's work on solid geometry clearly no longer looked intimidating.

  273. Gallarati biography
    • These reflect the major area of his research which was mostly in algebraic geometry.
    • Three papers given in [Dionisio Gallarati : Collected papers of Dionisio Gallarati (Kingston, ON, 2000).',1)">1] relate to Gallarati's contributions to Grassmannian geometry.
    • introduces the modern reader to the almost forgotten world of projective algebraic geometry, Italian style.
    • Gallarati taught algebraic geometry for many years and notes from these courses still exist [Dionisio Gallarati : Collected papers of Dionisio Gallarati (Kingston, ON, 2000).',1)">1]:- .

  274. Padoa biography
    • He returned to the University of Turin for session 1894-95 where he attended two courses given by Giuseppe Peano, one on the infinitesimal calculus and the other on higher geometry which made particular study of the geometric contributions of Hermann Grassmann.
    • This was certainly not the first time that Padoa had attempted to leave school teaching and become a university lecturer; he had applied unsuccessfully for a lectureship in Mathematical Logic in 1901, a lectureship in Descriptive Geometry in 1909, and a lectureship in Theoretical Philosophy in 1912.
    • After he was appointed at the staff of the University of Genoa, he taught courses on 'Mathematical Logic' (1932-34), 'Ideographic Logic' (1934-37), and 'Descriptive Geometry' (1935-36).
    • Padoa spoke on A new system of definitions for Euclidean geometry but began with a summary of his lecture at the Philosophy Congress.

  275. Vitali biography
    • His main teachers at Bologna were Cesare Arzela, who had held the chair of Higher Analysis, and Federigo Enriques who taught at Bologna on a temporary basis from 1894, but held the chair of projective and descriptive geometry from 1896.
    • There he was strongly influenced by Luigi Bianchi, who taught him analytic geometry, and Ulisse Dini who taught him infinitesimal calculus.
    • Vitali published three papers in 1900; two of them (Sulle funzioni analitiche sopra le superficie di Riemann and Sui limiti per n = infinity delle derivate nme delle funzioni analitiche) were on the material that he had produced for his thesis, while the other (Sulle applicazioni del Postulato della continuita nella geometria elementare) was an essay on the applications of the postulate of continuity in elementary geometry.
    • In his last years he worked on a new absolute differential calculus and a geometry of Hilbert spaces.

  276. Kostrikin biography
    • The necessity of giving a unified presentation of standard course material in algebra, linear algebra and geometry has been felt for a long time.
    • The natural evolution of the standard programs, both on behalf of unifying the courses in linear algebra and higher-dimensional analytic geometry, and on behalf of separating them and sprinkling elements of number theory into a course in algebra, are reflected in the pages of the present book, which has been based on the previously mentioned textbook of the same name, although considerably extended and divided into three parts for the convenience of readers.
    • In 1980 Kostrikin co-authored another famous textbook Linear algebra and geometry with Yuri Ivanovich Manin.
    • Affine and projective geometry.

  277. Euler biography
    • He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc.
    • He made decisive and formative contributions to geometry, calculus and number theory.
    • Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces.
    • In this latter topic he had to solve various problems of differential geometry and geodesics.

  278. Banu Musa biography
    • Jafar Muhammad worked mainly on geometry and astronomy while Ahmad worked mainly on mechanics and al-Hasan worked mainly on geometry.
    • The brothers were given the best education in Baghdad, studying geometry, mechanics, music, mathematics and astronomy.
    • In the text areas as described as products of linear magnitudes, so the terminology of arithmetic is perhaps for the first time applied to the operations of geometry.

  279. Wallis biography
    • There, to avoid being diverted to other discourses and for some other reasons, we barred all discussion of Divinity, of State Affairs, and of news (other than what concerned our business of philosophy) confining ourselves to philosophical inquiries, and related topics; as medicine, anatomy, geometry, astronomy, navigation, statics, mechanics, and natural experiments.
    • He was appointed to the Savilian Chair of geometry at Oxford in 1649 by Cromwell mainly because of his support for the Parliamentarians.
    • Wallis replied with the pamphlet Due Correction for Mr Hobbes, or School Discipline for not saying his Lessons Aright to which Hobbes wrote the pamphlet The Marks of the Absurd Geometry, Rural Language etc.
    • History Topics: Non-Euclidean geometry .

  280. Jordanus biography
    • He wrote several books on arithmetic, algebra, geometry and astronomy.
    • Geometry is developed in the work Liber phylotegni de triangulis which is an excellent example of a Middle Ages Latin geometry text.
    • The Demonstratio de plana spera is a specialised work on geometry which studies stereographic projection.

  281. Hermann biography
    • In fact, while in Padua, Hermann lectured on the standard topics of the day, namely classical geometry, mechanics, optics, hydraulics, and gnomonics.
    • He also worked on the textbook Abrege des mathematiques (1728-1730), writing the first volume, Arithmetic and Geometry, and the third volume on Military Fortifications.
    • However, he also leaned towards Newton and, on many occasions, preferred to deal with dynamical problems in terms of geometry.
    • In his work on curves in space, Hermann discusses the spherical epicycloid; the problem of finding the shortest distance between two points on a given surface; and the equations and properties of various surfaces from the point of view of analytic geometry of three dimensions.

  282. McMullen biography
    • In this monograph, the author presents a comprehensive study of a theory which brings into parallel two recent and very deep theorems, involving geometry and dynamics.
    • He has made important contributions to various branches of the theory of dynamical systems, such as the algorithmic study of polynomial equations, the study of the distribution of the points of a lattice of a Lie group, hyperbolic geometry, holomorphic dynamics and the renormalization of maps of the interval.
    • As the title of the talk suggests, there were many different areas of mathematics touched on by McMullen, including: Fermat's Last Theorem, Zeno's Paradoxes, hyperbolic and spherical geometry, the harmonic series, and tiling.
    • More recently he gave a lecture The Geometry of 3-Manifolds to the annual meeting of the American Association for the Advancement of Science in Boston in February 2008:- .

  283. Freedman biography
    • In his paper, The topology of four-dimensional manifolds, published in the Journal of Differential Geometry (1982), Freedman solved this problem, and in particular, the four-dimensional Poincare conjecture.
    • My primary interest in geometry is for the light it sheds on the topology of manifolds.
    • Here it seems important to be open to the entire spectrum of geometry, from formal to concrete.
    • In the nineteenth century there was a movement, of which Steiner was a principal exponent, to keep geometry pure and ward off the depredations of algebra.

  284. Libermann biography
    • She was appointed Professor at the University of Rennes where she continued her research on differential geometry.
    • Perhaps she is best known for her monograph Symplectic geometry and analytical mechanics written jointly with Charles-Michel Marle.
    • She continued to help to run the Geometry and Mechanics seminar in Paris until the end of 2006.
    • Her final publication Charles Ehresmann's concepts in differential geometry was published in 2007.

  285. Gudermann biography
    • Gudermann worked almost exclusively on spherical geometry and special functions but he is not remembered for any original mathematical results in these areas.
    • He did write a book on spherical geometry and [Dictionary of Scientific Biography (New York 1970-1990).
    • For this reason and because of its constant curvature there exist many similarities between spherical geometry and plane geometry; yet at the same time Gudermann considered more interesting the study of cases where the similarity no longer holds.

  286. Stark biography
    • He published the 629-page textbook Analytic Geometry (Polish) in 1951.
    • This textbook deals with metric, affine and projective geometry of linear and quadratic varieties in the plane as well as in the three-space.
    • Besides items usually dealt with in textbooks of elementary analytic geometry the reader finds here the introduction to synthetic projective geometry, to the theory of matrices (and determinants) with the usual applications and to the (three-dimensional) elementary vector calculus.

  287. Scherk biography
    • At Halle, Scherk taught a wide range of courses such as: analytic geometry of lines and the conic sections; analytic geometry of lines and surfaces of the first and second degree; algebra and algebraic geometry; plane and spherical trigonometry; integral calculus; and differential calculus and its application to algebra, analysis and geometry.

  288. Floer biography
    • There he worked for his doctorate with Clifford Taubes on gauge theory and with Alan Weinstein on symplectic geometry but, before completing his thesis with Taubes on monopoles on 3-manifolds, he returned to Germany in the summer of 1984 to undertake military service.
    • Floer developed a new method for "counting" the solutions of maximum-minimum problems arising in geometry.
    • The value of his work was grasped immediately by specialists in differential geometry, topology, and mathematical physics, for whom "Floer homology" has become an essential part of their problem-solving toolkit.
    • After reviewing Morse theory in finite dimensions, Floer went on to outline applications to symplectic geometry, working on the loop space on a symplectic manifold.

  289. Peano biography
    • Among Peano's teachers in his first year at the University of Turin was D'Ovidio who taught him analytic geometry and algebra.
    • In his second year he was taught calculus by Angelo Genocchi and descriptive geometry by Giuseppe Bruno.
    • In his third year Francesco Faa di Bruno taught him analysis and D'Ovidio taught geometry.
    • Among his teachers in his final year were again D'Ovidio with a further geometry course and Francesco Siacci with a mechanics course.

  290. Moore Robert biography
    • Halsted had suggested a problem in one of his classes which had led Moore to prove that one of Hilbert's geometry axioms was redundant.
    • Eliakim Moore, who was the head of mathematics at Chicago University, heard of this contribution and, since his research interests at the time were precisely on the foundations of geometry, Eliakim Moore organised the award of a scholarship that would allow Robert Moore to study for his doctorate in Chicago.
    • at University of Chicago and the degree was awarded in 1905 for a dissertation entitled Sets of Metrical Hypotheses for Geometry.
    • It was at the University of Pennsylvania that Moore first tried out his teaching methods in a Foundations of Geometry course he taught there.

  291. McDuff biography
    • McDuff is best known for her work in the geometry of multi-dimensional structures.
    • Her work in symplectic geometry, functional analysis and diffeomorphism groups has provided understanding and unexpected results in a whole range of areas of great importance.
    • She is the Helen Lyttle Kimmel '42 Professor of Mathematics at Barnard College where she teaches "Introduction to Higher Mathematics" and courses in geometry and topology.
    • In the early eighties, shortly before Gromov's work on pseudo-holomorphic curves began to move the subject in new directions, McDuff began her study of symplectic topology and geometry.

  292. Fine biography
    • The second part covers geometry and is in two volumes.
    • It begins with setting geometry up in a similar axiomatic way to Euclid's Elements, but then it goes on to more practical considerations of measuring length, height, surface area, and volumes.
    • The second of the geometry volumes covers topics in trigonometry but only at an elementary level.
    • As we have noted, Fine gave the value of π to be 22/7 in the geometry part of the Protomathesis (which is dated 1530).

  293. Gerbaldi biography
    • After the award of his degree, he became an assistant to Enrico D'Ovidio who occupied the Chair of Algebra and Analytic Geometry at the University of Turin.
    • In 1881 Gerbaldi published La superficie di Steiner studiata sulla sua rappresentazione analitica mediante le forme ternarie quadratiche which contained his work on conic sections, projective geometry and projective planes.
    • He was appointed as an assistant at the University of Rome and, while he was there, he entered a competition for the Chair of Analytic and Projective Geometry at the University of Palermo.
    • The goal was to stimulate the study of higher mathematics by means of original communications presented by the members of the society on the different branches of analysis and geometry, as well as on rational mechanics, mathematical physics, geodesy, and astronomy.

  294. Mei Wending biography
    • For example, the gougu theorem (referred to in the West as the Pythagorean Theorem) was a well-known and important focus of ancient Chinese geometry, but since the time of Liu Hui and Zhao Shang, two brilliant mathematicians of the 3rd century, no proof of the gougu theorem had been given in any mathematical books.
    • Mei used traditional Chinese methods in Jihe bubian (Complements of Geometry) Mei to calculate the volumes and relative dimensions of regular and semi-regular polyhedrons.
    • The Jihe tongjie (Complete Explanation of Geometry) contains Mei's approach to Euclidean geometry.

  295. Andreev biography
    • He taught at Kharkov University as a privatdocent from January 1874, starting to teach his first course on analytical geometry at that time.
    • Andreev is best known for his work on geometry, although he also made contributions to analysis.
    • In the area of geometry he did major pieces of work on projective geometry.

  296. Schramm biography
    • (The spheres must have disjoint interiors, but they don't have to be the same size.) It's a standard theorem in classical geometry, also related to important work in hyperbolic geometry and complex analysis, that you can realize any planar simple graph by kissing circles in R2, i.e., the circles are the vertices and the kissing pairs are the edges.
    • Schramm was recognized for his development of stochastic Loewner equations and for his contributions to the geometry of Brownian curves in the plane.
    • For his contributions to discrete conformal geometry, where he discovered new classes of circle patterns described by integrable systems and proved the ultimate results on convergence to the corresponding conformal mappings, and for the discovery of the Stochastic Loewner Process as a candidate for scaling limits in two dimensional statistical mechanics.

  297. Walker Arthur biography
    • He received a First Class degree from Oxford in 1931, having specialised in differential geometry for which he won a special distinction.
    • Walker worked on geometry, in particular differential geometry, relativity, and cosmology.
    • This is an account of a course of 12 lectures given at the University of Arizona on the geometry of cosmology.

  298. Ramanujam biography
    • The Director refused his resignation but later in the year he again resigned and went to the 1970-71 Algebraic geometry year at the University of Warwick in England.
    • His excitement and enthusiasm was one of the main factors that made "Algebraic geometry year" a success.
    • We discussed many topics involving topology and algebraic geometry at that time, and especially Kodaira's Vanishing Theorem.
    • As a result of his work with Shafarevich and Mumford, Ramanujam went on to make contributions to algebraic geometry which Mumford describes in [C P Ramanujam - a tribute (Berlin-New York, 1978), 8-10.',2)">2].

  299. Birman biography
    • She attended the Julia Richmond High School, an all-girls school in New York, where she developed a love for geometry [Notices Amer.
    • We had a course in Euclidean geometry, and every single night we would have telephone conversations and argue over the solutions to the geometry problems.
    • She has served on the editorial boards of several journals and was among the founding editors of two journals, 'Geometry and Topology' and 'Algebraic and Geometric Topology'.

  300. Magnitsky biography
    • In February, Magnitskii was appointed to the school and simultaneously ordered to compile a book "in the Slavonic dialect, selected from arithmetic, geometry and navigation." The 'Arithmetic' was therefore specifically commissioned to be the textbook of the Moscow School.
    • It used the methods of algebra, geometry, and trigonometry [History of Education Quarterly 13 (4) (1973), 325-345.',9)">9]:- .
    • Geometry and trigonometry were not abstract entities, but solution methods for navigational problems, just as contemporary English and American texts relied on examples of commercial transactions to induce students to do their calculations.
    • arithmetic, geometry, trigonometry, even as far as navigation, and having finished these sciences, the students are ordered to be sent to the other school, to the foreigner Henry Farquharson and his colleague Gwyn.

  301. Mansion biography
    • In November 1865 he began lecturing on advanced algebra, analytic geometry and descriptive geometry at the School of Civil Engineering, attached to the University of Ghent.
    • He taught advanced algebra, analytic geometry, astronomy and mathematical methodology.
    • Author of many works on mathematical analysis, the calculus of probabilities, non-Euclidean geometry, the history and philosophy of science, [Mansion] held a prominent place in the Belgian scientific world.

  302. Hippocrates biography
    • One of the Pythagoreans [Hippocrates] lost his property, and when this misfortune befell him he was allowed to make money by teaching geometry.
    • He then came to Athens to persecute the offenders and, during a long stay, attended lectures, finally attaining such proficiency in geometry that he tried to square the circle.
    • He was the first to write an Elements of Geometry and although his work is now lost it must have contained much of what Euclid later included in Books 1 and 2 of the Elements.
    • Eudemus of Rhodes, who was a pupil of Aristotle, wrote History of Geometry in which he described the contribution of Hippocrates on lunes.

  303. Archytas biography
    • He claimed that mathematics was composed of four branches, namely geometry, arithmetic, astronomy and music.
    • Indeed, they have transmitted to us a keen discernment about the velocities of the stars and their risings and settings, and about geometry, arithmetic, astronomy, and, not least of all, music.
    • One interesting innovation which Archytas brought into his solution of finding two mean proportionals between two line segments was to introduce movement into geometry.
    • In these Eutocius claims to quote the description given in History of geometry by Eudemus of Rhodes but the accuracy of the quotation is doubted by the authors of [Centaurus 18 (1973/74), 1-5.',10)">10].

  304. Bliss Nathaniel biography
    • Halley had been appointed to the Savilian chair of geometry at Oxford in 1704 and had been appointed Astronomer Royal in 1720.
    • Following the death of Halley in January 1742, Bradley applied for the position of Astronomer Royal and Bliss applied for the Savilian chair of geometry.
    • Bliss, of course, was Savilian professor of geometry at Oxford so, although his research interests were mainly in astronomy, he also taught mathematics at Oxford.
    • It seems slightly ironical, but Bliss seems to have been more productive in astronomy research when he was the professor of geometry than when he was Astronomer Royal.

  305. Cassels biography
    • His mathematical publications started in about 1947 with a series of papers on the geometry of numbers, in particular papers on theorems of Khinchin and of Davenport, and on a problem of Mahler.
    • Then in 1959 he published another book, An introduction to the geometry of numbers.
    • His work includes numerous papers on Diophantine Approximation and the Geometry of Numbers, and seminal contributions to the theory of quadratic forms and sums of squares.
    • He has written excellent books on Diophantine Approximation, Geometry of Numbers, Algebraic Number Theory and Rational Quadratic Forms.

  306. Legendre biography
    • Legendre's work replaced Euclid's "Elements" as a textbook in most of Europe and, in succeeding translations, in the United States and became the prototype of later geometry texts.
    • In 1803 Napoleon reorganised the Institut and a geometry section was created and Legendre was put into this section.
    • In 1832 (the year Bolyai published his work on non-euclidean geometry) Legendre confirmed his absolute belief in Euclidean space when he wrote:- .
    • History Topics: Non-Euclidean geometry .

  307. Privalov biography
    • Of the mathematicians, Konstantin Alekseevich Andreev was best known for his work on geometry and was Dean of the Faculty during Privalov's undergraduate years, Dimitri Fedorovich Egorov was a leading researcher in differential geometry and integral equations, Leonid Kuzmich Lakhtin was interested in analysis and probability, and Boleslav Kornelievich Mlodzeevskii had been the first to give lectures at Moscow University on set theory and the theory of functions.
    • A course of lectures that he gave in Saratov gave rise to Analytic geometry in the plane (1918).
    • His textbook Analytical geometry (1927) was another that proved extremely popular with a 12th edition appearing in 1939 - editions had appeared at the rate of one per year for twelve years.

  308. Bradwardine biography
    • Another interesting, but fallacious, argument was produced by Bradwardine when he tried to disprove atomism using geometry.
    • However, he questioned the logic of his own arguments as he felt perhaps the existence of geometry already assumes that atomism is false.
    • Speculative geometry contains elementary geometry which is not all based on Euclid.

  309. Keller biography
    • He had published a number of papers on geometry and algebraic geometry between 1933 and 1939, but only one publication Eine Bemerkung zu den Pluckerschen Formeln (1943) between 1940 and 1948.
    • Looking at Keller's work one notices immediately that he made contributions to several, not only all neighbouring, areas of mathematics, the most important being to geometry, to algebraic geometry and to topology; in addition he studied number theoretic and analytic topics as well as those of a more philosophical character.

  310. Walsh biography
    • Memoir on the Invention of Partial Equations; The Theory of Partial Functions; Irish Manufactures: A New Method of Tangents; An Introduction to the Geometry of the Sphere, Pyramid and Solid Angles; General Principles of the Theory of Sound; The Normal Diameter in Curves; The Problem of Double Tangency; The Geometric Base; The Theoretic Solution of Algebraic Equations of the Higher Orders.
    • The Elements of Geometry, by John Walsh (Folio).
    • Discovered the general solution of numerical equations of the fifth degree at 114 Evergreen Street, at the Cross of Evergreen, Cork, at nine o'clock in the forenoon of July 7th, 1844; exactly twenty-two years after the invention of the Geometry of Partial Equations, and the expulsion of the differential calculus from Mathematical Science.
    • And the falsehood of the offspring of that method, namely, the no less celebrated doctrine of fluxions, differentials, limits, etc., the boast and glory of England, France and Germany, demonstrated by the great invention of the geometry of partial equations which has superseded them, at least in my hands, and indefinitely surpassed the old system in power.

  311. Besicovitch biography
    • At Cambridge Besicovitch lectured on analysis in most years but he also gave an advanced course on a topic which was directly connected with his research interests such as almost periodic functions, Hausdorff measure, or the geometry of plane sets.
    • His work on sets of non-integer dimension was an early contribution to fractal geometry.
    • Kenneth Falconer, one of the leading experts on fractal geometry, seeing me [EFR] writing this article commented:- .
    • History Topics: A History of Fractal Geometry .

  312. Burali-Forti biography
    • Burali-Forti taught analytic projective geometry at the Military Academy where he continued to teach for the rest of his life.
    • Burali-Forti attended the Congress and presented a paper The postulates for the geometry of Euclid and of Lobachevsky to the Geometry section of the Congress.
    • As well as set theory and vector analysis, Burali-Forti also worked on linear transformations and their applications to differential geometry.

  313. Hudson biography
    • Charlotte Angas Scott, who had studied under Cayley and shared Hudson's interests in algebraic geometry, was Head of the Mathematics department there.
    • in which [Hudson] included a lot of elegant geometry in an exposition of the range and limitations of ruler and compass constructions.
    • However she essentially gave up publishing mathematics after her treatise appeared in print, except for one notable exception which was an article on Analytic geometry, curve and surface in the 14th edition of Encyclopaedia Britannica published in 1929.
    • She will long be remembered by the mathematical world for her contributions to geometry and by Newnham and Cambridge as one of their distinguished alumni.

  314. Peschl biography
    • this work lies on the common boundary between differential geometry, function theory (of one and several variables) and partial differential equations.
    • The book, the result of lectures given at the University of Bonn, is a valuable contribution to that approach to analytic geometry in which is stressed, at the example of Schreier and Sperner, the necessity of basing the traditional material on the strict concepts of modern algebra.
    • The titles of the chapters are: Algebra and geometry of complex numbers; Fundamental topological concepts, sets, sequences of complex numbers and infinite series; Functions, real and complex differentiability and holomorphy; Integral theorems and their consequences; Winding number and curves homologous to zero; Taylor development of holomorphic functions; Elementary transcendental functions; Laurent series, isolated singularities and residue calculus; Holomorphic and meromorphic functions obtained by limiting processes; Analytic continuation; and Conformal mappings.
    • Partielle Differentialgleichungen erster Ordnung (1973) provides an elementary introduction to first order partial differential equations while Differential-geometrie (1973) provides a clear, elementary and concisely presented introduction to local differential geometry in Euclidean and Riemannian spaces.

  315. Wilson Edwin biography
    • He became interested in the foundations of geometry, particularly in projective and differential geometry and he published a paper The so-called foundations of geometry in 1903 which criticised Hilbert's approach to geometry.

  316. Todd biography
    • After graduation Todd remained at Trinity to study for his doctorate in geometry under H F Baker's supervision.
    • Among his fellow students at Cambridge at this time were a number of others studying geometry including P du Val, H S M Coxeter and W L Edge.
    • Todd and Hodge began to change the geometry at Cambridge to areas that were then of great interest internationally.
    • Todd was a superb technician and manipulator of formulae but he also brought to bear a keen appreciation of the underlying geometry.

  317. Borel Armand biography
    • This was an opportunity for Borel to learn a great deal about algebraic geometry and number theory from Weil.
    • They focus on Lie groups, and their actions, as well as on algebraic and arithmetic groups, and broach core questions regarding many different areas: algebraic topology, differential geometry, analytic geometry, analytic and algebraic geometry, number theory etc.

  318. Levi Eugenio biography
    • There he was taught by Luigi Bianchi who had been promoted to professor of analytic geometry at the Scuola Normale Superiore in 1890.
    • He had published two important works before Levi began to study with him, namely Lectures on differential geometry (1894) and Lectures on the theory of groups of substitutions (1900).
    • The other mathematician in Pisa who had a major influence on Levi's research was Ulisse Dini who held two chairs at the University of Pisa, the chair of analysis and higher geometry, and the chair of infinitesimal analysis.
    • However, he also wrote on issues relating to: differential geometry, Lie groups, partial differential equations and the minimum principle.

  319. Gromoll biography
    • The first five chapters comprise an introduction to Riemannian geometry, accessible to students with a background in real analysis, linear algebra and first concepts of general topology.
    • Among many other lectures he gave to international meetings we mention the his address to the 4th Geometry Festival, UNC Chapel Hill (1988), to the 38th AMS Summer Inst., Los Angeles (1990), his Plenary Lecture at the CMS Meeting, St John (1998); and his Plenary Lecture at the 50th Anniversary of IMPA, Rio de Janeiro (2002).
    • Differential geometry: Riemannian geometry and gave an overview of what was known at the time concerning manifolds of nonnegative curvature.

  320. Darboux biography
    • Then, in 1878 he became suppleant to Chasles in the chair of higher geometry, also at the Sorbonne.
    • Two years later Chasles died and Darboux succeeded him to the chair of higher geometry, holding this chair until his death.
    • Darboux made important contributions to differential geometry and analysis.
    • In 1873 Darboux wrote a paper on cyclides and between 1887 and 1896 he produced four volumes on infinitesimal geometry which included most of his earlier work it was titled Lecons sur la theorie general des surfaces et les applications geometriques du calcul infinitesimal.

  321. Kotelnikov biography
    • The thesis he presented for the Master's Degree was The Cross-Product Calculus and Certain of its Applications in Geometry and Mechanics.
    • Much of his career is spent working on physics and non-euclidean geometry.
    • In 1927 he published one of his most important works, The Principle of Relativity and Lobachevsky's Geometry.
    • He also worked on quaternions and applied them to mechanics and geometry.

  322. Clairaut biography
    • Alexis used Euclid's Elements while learning to read and by the age of nine he had mastered the excellent mathematics textbook of Guisnee Application de l'algebre a la geometrie which provided a good introduction to the differential and integral calculus as well as analytical geometry.
    • A geometry book Elements de geometrie was published in 1741 and a book on algebra Elements d'algebre was published in 1749.
    • I intended to go back to what might have given rise to geometry; and I attempted to develop its principles by a method natural enough so that one might assume it to be the same as that of geometry's first inventors, attempting only to avoid any false steps that they might have had to take..

  323. Francesca biography
    • One has the impression that Piero retained everything - the saints' legends read in school, the mathematics of abbaco school, his creative explorations into ancient geometry - and that all his learning enriched his painting.
    • His labour in the artisan culture remained formative in Piero's life and art, not withstanding his great later achievements in painting, mathematics, and geometry that required access to and participation in an elite culture.
    • Also in Borgo San Sepolcro around this time he painted the Baptism of Christ, which uses geometry to great effect, for example in placing the dove at the centre of the circle forming the top of the painting.
    • It deals with arithmetic, starting with the use of fractions, and works through series of standard problems, then it turns to algebra, and works through similarly standard problems, then it turns to geometry and works through rather more problems than is standard before (without warning) coming up with some entirely original three-dimensional problems involving two of the 'Archimedean polyhedra' (those now known as the truncated tetrahedron and the cuboctahedron).

  324. Menelaus biography
    • Three books on the "Elements of Geometry", edited by Thabit ibn Qurra, and "The Book on the Triangle".
    • Book 2 applies spherical geometry to astronomy.
    • We gave a quotation above from the 10th century Arab register which records a book called Elements of Geometry which was in three volumes and was translated into Arabic by Thabit ibn Qurra.
    • Another Arab reference to Menelaus suggests that his Elements of Geometry contained Archytas's solution of the problem of duplicating the cube.

  325. Pasch biography
    • He worked on the foundations of geometry.
    • His main interests were the foundations of projective geometry and of analysis.
    • a watershed in the development of the foundations of geometry.
    • Drawing on the work of his predecessors, Pasch was the first to explicitly state all the basic concepts and axioms necessary for his construction of projective geometry.

  326. Mannheim biography
    • In the following year Mannheim was appointed as Professor of Descriptive Geometry at the Ecole Polytechnique.
    • He made numerous contributions to geometry and for his outstanding contributions to the subject he was awarded the Poncelet Prize of the Academie des Sciences in 1872.
    • He studied the polar reciprocal transformation introduced by Chasles and applied his results to kinetic geometry.
    • Mannheim's own definition of kinetic geometry considered it to be the study of motion of a figure without reference to any forces, time or other properties external to the figure.

  327. Sullivan biography
    • It was this work which led to Sullivan receiving the Oswald Veblen Prize in Geometry in 1971 from the American Mathematical Society.
    • During the 1980s the resources of the [Albert Einstein] Chair allowed the founding of a regular seminar in geometry and chaos theory that brought first-rank international scholars to CUNY [the City University of New York] and New York City.
    • He has contributed to several diverse areas of mathematics including topology, geometry and dynamics and complex analysis.
    • In addition to the 1971 Oswald Veblen Prize in Geometry mentioned above, he received the 1981 Prix Elie Cartan from the French Academy of Sciences, the 1994 King Faisal International Prize for Science (mathematics), and the Ordem Scientifico Nacional by the Brazilian Academy of Sciences in 1998.

  328. Steiner biography
    • This connection and transition is the real source of all the remaining individual propositions of geometry.
    • He was appointed to a new extraordinary professorship of geometry at the University of Berlin on 8 October 1834.
    • He was one of the greatest contributors to projective geometry.
    • Steiner disliked algebra and analysis and believed that calculation replaces thinking while geometry stimulates thinking.

  329. Picard Emile biography
    • He himself wrote that he hated geometry but he:- .
    • Picard made his most important contributions in the fields of analysis, function theory, differential equations, and analytic geometry.
    • Building on work by Abel and Riemann, Picard's study of the integrals attached to algebraic surfaces and related topological questions developed into an important part of algebraic geometry.
    • But you also escaped, you introduced us not only to hydrodynamics and turbulence, but to many other theories of mathematical physics and even of infinitesimal geometry; all this in lectures, the most masterly I have heard in my opinion, where there was not one word too many nor one word too little, and where the essence of the problem and the means used to overcome it appeared crystal clear, with all secondary details treated thoroughly and at the same time consigned to their right place.

  330. Novikov Sergi biography
    • In 1963 Novikov had been appointed to the staff of the Steklov Institute of Mathematics and, the following year, he was also appointed to the Department of Differential Geometry at Moscow University.
    • Novikov also became head of the Department of Higher Geometry and Topology of Moscow University in 1983 and, the following year he became head of the Department of Geometry and Topology of the Mathematical Institute of the USSR Academy of Sciences.
    • Since 1996 he has been working at the University of Maryland in the United States but retains close links with Russia with a research appointments in Moscow University, in the Landau Institute for Theoretical Physics, and as Head of the Geometry and Topology research groups at the Steklov Institute.

  331. Malgrange biography
    • Malgrange took Cartan's courses on differential geometry and Lie groups in his second year.
    • Grothendieck's second thesis was on sheaf theory, and this work may have planted the seeds for his interest in algebraic geometry, where he was to do his greatest work.
    • We have mentioned his important work on linear partial differential equations above, but he has made numerous other very significant contributions to differential geometry, non-linear differential equations, and singularities of functions and mappings.
    • The modern algebraic theory of differential systems, or "theory of D-modules'', brings us new relations between two mathematical areas traditionally far apart: the theory of systems of linear partial differential equations and algebraic geometry.

  332. Pincherle biography
    • His childhood education, however, gave him a love for music and literature, and these interests continued throughout his life [The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).',5)">5]:- .
    • Betti was a strong influence on Pincherle as was Ulisse Dini who, after working on differential geometry was, by this time, studying and teaching the foundations for the theory of functions of a real variable.
    • In the spring of 1880, following a competition, he was appointed to the chair of algebraic analysis and analytic geometry at the University of Palermo.
    • Although Pincherle referred to Grassmann and Peano, his approach went far beyond the framework of geometry and placed itself in quite a general context using, in particular, infinite-dimensional linear spaces.

  333. Finsler biography
    • In fact in 1934 Cartan wrote a book Les espaces de Finsler which established Finsler's name in differential geometry.
    • A Finsler space is a generalisation of a Riemannian space where the length function is defined differently and Minkowski's geometry holds locally.
    • Differential geometry was not Finsler's research topic for long since he moved to take up set theory.
    • At Zurich, in addition to his work on set theory he also worked on differential geometry, number theory, probability theory and the foundations of mathematics.

  334. Playfair biography
    • In the eighteenth century geometry was systematically studied from Euclid's Elements in the universities, while the schools were generally content to accept the theorems and constructions without proof.
    • To these books, which specifically deal with plane geometry, Playfair added three more books intended to supplement the preceding six; On the Quadrature of the Circle and the Geometry of Solids, Elements of Plane and Spherical Trigonometry and The Arithmetic of Sines.
    • History Topics: Non-Euclidean geometry .

  335. Davies biography
    • for a thesis on n-dimensional geometry in 1926.
    • His steady stream of publications is testimonial to his authority in the fields of Riemannian geometry and the calculus of variations.
    • His steady stream of publications in differential geometry and the calculus of variations attests to his authority in this field.
    • He extended the ideas in these papers to generalisations of Riemannian manifolds such as Finsler manifolds and Cartan manifolds in later papers, for example: Lie derivation in generalized metric spaces (1939), Subspaces of a Finsler space (1945), Motions in a metric space based on the notion of area (1945), and The theory of surfaces in a geometry based on the notion of area (1947).

  336. Bombieri biography
    • He has significantly influenced number theory, algebraic geometry, partial differential equations, several complex variables, and the theory of finite groups.
    • The award was made for the book Heights in Diophantine Geometry jointly authored by Bombieri and Gubler.
    • The book is a research monograph on all aspects of Diophantine geometry, both from the perspective of arithmetic geometry and of transcendental number theory.

  337. Hironaka biography
    • He had already published three papers before submitting his thesis, On the arithmetic genera and the effective genera of algebraic curves (1957), A note on algebraic geometry over ground rings.
    • Classical algebraic geometry studies properties of varieties which are invariant under birational transformations.
    • Some fundamental theorems in the theory of several complex variables and of the geometry of complex manifolds are proved in a simple but rigorous form.
    • Hironaka was invited to give one of the featured lectures on recent research advances and he spoke on Resolution of Singularities in Algebraic Geometry.

  338. Wilczynski biography
    • It is characteristic of the man, however, that during the long illness which followed he never lost his interest in geometry and never gave up hope and the belief that he would some day be able to return to his academic duties.
    • However, Wilczynski's main work was in projective differential geometry and ruler surfaces.
    • It has often been stated that Wilczynski was the founder, or inventor, of projective differential geometry.
    • But Wilczynski was the first ever to appreciate, demonstrate and exploit the utility of completely integrable systems of linear homogeneous differential equations for projective differential geometry.

  339. Hooke biography
    • His rapidly gained understanding of geometry was soon applied to his real love of mechanics and he began to invent possible flying machines.
    • Wilkins gave him a copy of his book Mathematical Magick, or the wonders that may be performed by mechanical geometry which he had published five years before Hooke arrived in Oxford.
    • He did however secure another appointment, namely that of Professor of Geometry at Gresham College, London, being appointed there in 1665.
    • In addition to his post as Professor of Geometry at Gresham College, Hooke held the post of City Surveyor.

  340. Gerbert biography
    • It is noteworthy that arithmetic and music are only very briefly mentioned at the beginning and that geometry is described in a short paragraph at the end, while the rest of the account is devoted to a description of astronomical tools fabricated by Gerbert in order to introduce his disciples to astronomy.
    • With their minds well trained in these exercises his pupils advanced to the higher arts of the quadrivium - arithmetic, music, astronomy, and geometry.
    • It is to be noticed that Gerbert was the first to introduce into the schools instruments as an assistance to the study of arithmetic, astronomy, and geometry.
    • Procure the 'Historia' of Julius Caesar from Lord Adso, abbot of Montier-en-Der, to be copied again for us in order that you may have whichever books are ours at Rheims, and may expect ones that we have since discovered at Bobbio, namely eight volumes: Boethius 'On astrology', also some beautiful figures of geometry; and others no less worthy of being admired.

  341. Hippias biography
    • He was a master of the science of calculation, geometry, astronomy, 'rhythms and harmonies and correct writing'.
    • the Spartans could not endure lectures on astronomy or geometry or calculation; it was only a small minority of them who could even count; what they liked was history and archaeology.
    • Perhaps the highest compliment that we can pay to Hippias is to report on the arguments of certain historians of mathematics who have claimed that the Hippias who discovered the quadratrix cannot be Hippias of Elis since geometry was not far enough advanced at this time to have allowed him to make these discoveries.
    • Pappus wrote his major work on geometry Synagoge in 340.

  342. Heegaard biography
    • Klein had me give two lectures in the 'Mathematische Gesellschaft' with a summary of Zeuthen's work on enumerative geometry.
    • He was a successful lecturer, teaching Geometry, Rational Mechanics, Elementary Mathematics, History of Mathematics and General Mathematics for Actuaries, but he found many difficulties dealing with his colleagues.
    • He reformed the teaching of geometry and gave lectures on Mathematical Education.
    • He felt it a good omen that this hope may be fulfilled by the fact that Professor Cartan was present to give the historical lecture (interspersed with personal reminiscences) entitled: "The role of Sophus Lie's theory of groups in the development of modern geometry".

  343. Witten biography
    • Speaking at the American Mathematical Society Centennial Symposium in 1988, Witten explained the relation between geometry and theoretical physics:- .
    • It used to be that when one thought of geometry in physics, one thought chiefly of classical physics - and in particular general relativity - rather than quantum physics.
    • One of Witten's subsequent works was a paper which Atiyah singles out for special mention in [Proceedings of the International Congress of Mathematicians, Kyoto, 1990 I (Tokyo, 1991), 31-35.',3)">3], namely Supersymmetry and Morse theory which appeared in the Journal of differential geometry in 1984.
    • Since this highly influential paper, the ideas in it have become of central importance in the study of differential geometry.

  344. Schwerdtfeger biography
    • His early papers include On generalized Hermitian matrices (1942), On contact transformations associated with the symplectic group (1942), Skew-symmetric matrices and projective geometry (1944), On the representation of rigid rotations (1945), and The Isoperimetric Problem (1945).
    • In 1962 he published Geometry of complex numbers : Circle Geometry, Mobius Transformations, Non-Euclidean Geometry which:- .

  345. Kontsevich biography
    • Also in 1994 he published Gromov-Witten classes, quantum cohomology, and enumerative geometry which was written jointly with Yuri Manin.
    • In 1993 he published Formal (non)commutative symplectic geometry which was reviewed by Alexander Voronov.
    • The paper places emphasis on the three fundamental types of algebras - Lie, associative and commutative - as functional models of three hypothetical versions of noncommutative symplectic geometry (in fact, the usual commutative one in the third case).
    • Calculus of differential forms, symplectic forms, Hamiltonian vector fields and Poisson brackets in noncommutative geometry are sketched.

  346. Bocher biography
    • At Gottingen he also attended lecture courses by Klein on the potential function, on partial differential equations of mathematical physics and on non-euclidean geometry.
    • It required for its treatment not so much a specific knowledge of the theory of the potential, although Bocher was thoroughly equipped on that side, even familiarity with the geometry of inversion, of which he made himself a master, but rather the power to carry through a piece of detailed analytic investigation with accuracy and skill ..
    • Because of the clarity and care with which his elementary texts on analytic geometry and trigonometry were written they are still in demand.
    • He also wrote elementary texts such as Trigonometry (written jointly with Gaylord) and Analytic geometry.

  347. Jungius biography
    • Empiricus remained verifiable through experience, Epistemonicus is grounded in principles and rules - as are the axioms of Euclid's geometry - and Heureticus reveals new methods for the solution of problems previously insoluble.
    • In [Jungius's] opinion a distinct science of nature required above all a finite number of principles, just as Euclidean geometry relies upon a small number of basic entities such as the point, the line, and the angle.
    • Jungius's attempt to rebuild the system of physical knowledge belongs to the widespread quest for making both philosophy and natural science as axiomatically structured as geometry.
    • In 1633, commenting on Sennert's 'Epitome scientiae naturalis' of 1618, in which Sennert had shown that arguments from geometry about divisibility and continuity must not be applied to the physical sciences, Jungius remarked that until then no physical body had ever been proved to be entirely homogeneous.

  348. Eells biography
    • In it the whole geometry or topology of the spaces involved play a role, rather than just the equations describing the behaviour or motion in small areas.
    • This he did with "Global Analysis" in 1971-72, "Geometry of the Laplace Operator" in 1976-77, and "Partial Differential Equations in Differential Geometry", in 1989-90.
    • harmonic maps pervade differential geometry and mathematical physics: they include geodesics, minimal surfaces, harmonic functions, Abelian integrals, Riemannian fibrations with minimal fibres, holomorphic maps between Kahler manifolds, chiral models, and strings.

  349. Dieudonne biography
    • He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups.
    • Here the term "classical group" is used as in the author's monograph, Sur les groupes classiques (1948) and the "elementary theory" refers roughly to results which involve subgroup and homomorphisms as opposed to results concerned for example with topology, differential geometry, etc.
    • In writing Algebre lineaire et geometrie elementaire Dieudonne aims to provide teachers in the lycees of France with sufficient background in geometry so that they can prepare their pupils properly for entry to university study.
    • He published texts such as History of functional analysis (1981), History of algebraic geometry (1985), Pour l'honneur de l'esprit humain (1987), A history of algebraic and differential topology 1900-1960 (1989), and L'ecole mathematique francaise du XXe siecle (2000).

  350. Wallace biography
    • Leslie and he are said to be on the eve of battle - for the "Elements of Geometry" and curves of the second order are to be discarded for Playfair's Euclid! Love me, love my dog - the saw says; still more should it say: love me, love my book.
    • Wallace's work was on geometry and Simson's line (which is definitely not due to Simson!) appears first in a paper of Wallace in 1799.
    • in explaining the foundations of the method, we have endeavoured to show that it rests upon purely analytical, namely the theory of limiting ratios, and this being the case, the subject may be treated as a branch of pure mathematics, without having occasion to introduce any ideas foreign to geometry ..
    • Though unable to walk, and almost to stand, I never ceased to think [and write books on geometry and conic sections].

  351. Pisier biography
    • My main research field is functional analysis, taken in a broad sense, ranging from the geometry of Banach spaces to the theory of star algebras or von Neumann algebras, through single operator theory on a Hilbert space.
    • Other areas of Pisier's research appear in the books Factorization of linear operators and geometry of Banach spaces (1986) and The volume of convex bodies and Banach space geometry (1989).
    • and the geometry of the corresponding finite-dimensional normed spaces.

  352. Sylvester biography
    • Sylvester tried hard to return to being a professional mathematician and he applied for a lectureship in geometry at Gresham College, London in 1854 but he was not appointed.
    • In particular he used matrix theory to study higher dimensional geometry.
    • When Smith died in 1883 Sylvester, although 68 years old at this time, was appointed to the Savilian chair of Geometry at Oxford.
    • Savilian Geometry Professor1883 .

  353. Fiorentini biography
    • However [Commutative algebra and algebraic geometry, Ferrara (Dekker, New York, 1999), x-xiii.
    • Other examples of his work from a slightly later period are (with L Badescu) Criteri di semifattorialita e di fattorialita per gli anelli locali con applicazioni geometriche (1975) and (with Alexandru T Lascu) A formula of enumerative geometry (1981).
    • regular sequences and refinements of this notion, leading to special classes of rings, important in algebraic geometry (complete intersection, Gorenstein, Cohen-Macaulay, Buchsbaum, etc.); .
    • classical (Italian) geometry in the perspective of the modern methods.

  354. Bott biography
    • Samelson was a real master of geometry and Lie group theory.
    • The main themes of the papers included in [Volume 4] are the geometry and topology of the Yang-Mills equations and the rigidity phenomena of vector bundles.
    • These works were physics-inspired and very much contributed to the convergence between geometry and high energy physics we are witnessing today.
    • his many fundamental contributions in topology and differential geometry and their application to Lie groups, differential operators and mathematical physics.

  355. Almgren biography
    • In addition he had written the short book Plateau's problem: An invitation to varifold geometry which was published in 1966.
    • Among his service we should mention he was an editor of three journals: the Journal of Experimental Mathematics, the Journal of Geometric Algebra, and Differential Geometry and its Applications.
    • He administered the Geometry Supercomputer Project for the Geometry Computing Group and served on the American Mathematical Society Committee on Applications of Mathematics.

  356. Dandelin biography
    • Following the publication of two papers which solved problems in elementary geometry, published in the Correspondance sur l'Ecole Polytechnique, he presented the memoir Sur quelque parties de la geometrie to the Royal Belgium Academy of Science in 1817.
    • This was not the ideal appointment, for he would have loved to have had a chair of geometry, but it was the best that could be arranged.
    • Dandelin, having lost out on his university education, received most of his early mathematical influence from Quetelet, who was two years younger than him, and his early interests were in geometry.
    • His mathematical contributions were not restricted to geometry, however, and he wrote papers on statics, algebra and probability.

  357. Samuel biography
    • The nature of the applications of the various theorems to algebraic geometry is in general rapidly explained.
    • In 1955 Samuel published Methodes d'algebre abstraite en geometrie algebrique which aimed to give a complete foundation for algebraic geometry:- .
    • Unlike the first, however, the second volume is concerned in large measure with those parts of commutative algebra that are the fruits of its union with algebraic geometry ..

  358. Pitt biography
    • Experienced staff were qualified to teach the major areas of analysis, algebra, geometry, statistics, and the mechanics of rigid and deformable bodies, fluids and electromagnetism.
    • This compact account develops the theory as it applies to abstract spaces, describes its importance to differential and integral calculus, and shows how the theory can be applied to geometry, harmonic analysis, and probability.
    • The second part also has three chapters: Each one of them considers an area of application: geometry, harmonic analysis and probability.

  359. Diaconis biography
    • The techniques used are a combination of tools from geometry, PDE, group theory and probability.
    • In 2001 he was a main speaker at Groups St Andrews 2001 in Oxford giving a series of lectures on Random walks on groups: characters and geometry.
    • This involves the character theory of the group and the geometry of the group in various generating sets.

  360. 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.
    • In particular, using a method resembling descriptive geometry, he maps circles on the sphere into the equatorial plane.

  361. Al-Kindi biography
    • Al-Kindi "was the most leaned of his age, unique among his contemporaries in the knowledge of the totality of ancient scientists, embracing logic, philosophy, geometry, mathematics, music and astrology.
    • In geometry al-Kindi wrote, among other works, on the theory of parallels.
    • Also related to geometry was the two works he wrote on optics, although he followed the usual practice of the time and confused the theory of light and the theory of vision.

  362. Siegel biography
    • Geometry of numbers and its applications to algebraic number theory.
    • Serge Lang published Diophantine geometry in 1962 and Mordell wrote a critical review of it two years later.
    • When I first saw [Lang's Diophantine geometry], about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible.

  363. Roberval biography
    • This method of drawing tangents makes Roberval the founder of kinematic geometry.
    • Roberval wrote a treatise on algebra and one of analytic geometry which appeared in his posthumous 1693 publication.
    • He certainly introduced algebraic methods into solving geometric problems before Rene Descartes did, but although he deserves credit for this nevertheless he did not produce Cartesian geometry.

  364. Wirtinger biography
    • Not only did he write beautiful papers on function theory, he also wrote on geometry, algebra, number theory, plane geometry and the theory of invariants.
    • When Reidemeister was appointed as associate professor of geometry at the University of Vienna in 1923 he became a colleague of Wirtinger.

  365. Frattini biography
    • Frattini was taught by some outstanding mathematicians at the University of Rome, being tutored by the geometers Guiseppe Battaglini, Eugenio Beltrami (who had just published his masterpiece on non-euclidean geometry).
    • He taught at the Technical Institute there, becoming Head of Mathematics and Descriptive Geometry in the year following his appointment.
    • His work on differential geometry is important as is his papers on the analysis of second degree indeterminates.

  366. MacMahon biography
    • In fact he had suffered a great disappointment in the previous year when he was a candidate for the Savilian Chair of Geometry at Oxford.
    • Sylvester, who held the Savilian Chair of Geometry, died in March 1897.
    • Before he was a candidate for the Savilian Chair of Geometry, MacMahon had been elected a Fellow of the Royal Society in 1890.

  367. Nash biography
    • In September 1948 Nash entered Princeton where he showed an interest in a broad range of pure mathematics: topology, algebraic geometry, game theory and logic were among his interests but he seems to have avoided attending lectures.
    • He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well.
    • His research on the theory of real algebraic varieties, Riemannian geometry, parabolic and elliptic equations was, however, extremely deep and significant in the development of all these topics.

  368. Bartholin biography
    • From 1656 he was professor of geometry at the University of Copenhagen but transferred to become an extraordinary professor of medicine in the following year.
    • Bartholin edited Introduction to the geometry of Descartes by van Schooten and also translated Optics of Larissa into Latin.
    • He wrote his text Experimenta crystalli Islandici disdiaclastici quibus mira & insolita refractio detegitur on the geometry of crystals in 1669.

  369. Weise biography
    • Weise's mathematical work was mainly on questions from differential geometry and topology.
    • The field of higher geodesy and cartography is one specialized branch of geometry and analysis which has been the subject of many particular investigations.
    • Weise acted as supervisor of PhD students from a wide range of mathematical fields, a dozen of them went on to become professors, among them Wolfgang Gaschutz (finite groups), Wolfgang Haken (knot theory and the solution of the four-colour-problem), Wilhelm Klingenberg (differential geometry) and Jens Mennicke (topology).

  370. Zeno of Sidon biography
    • Some modern authors have suggested that these claims give Zeno of Sidon some justification to be considered as having been the first person to consider the possibility of non-Euclidean geometry.
    • Zeno argued generally that, even if we admit the fundamental principles of geometry, the deductions from them cannot be proved without the admission of something else as well which has not been included in the said principles, and he intended by means of these criticisms to destroy the whole of geometry.

  371. Grunewald biography
    • over infinite group theory, finite group theory, Diophantine decision problems, arithmetic groups, automorphic forms and algebraic geometry.
    • His 60th birthday had been celebrated a year earlier with a well-attended conference 'Group theory, number theory and geometry' held at the University of Oxford.
    • A special issue of the journal Groups, Geometry, and Dynamics, to mark his 60th birthday was edited by Martin Bridson and Dan Segal with the cooperation of Alex Lubotzky and Peter Sarnak.

  372. Schwarz biography
    • Through him Schwarz became interested in geometry.
    • His interest in geometry was soon combined with Weierstrass's ideas of analysis.
    • That he found such general methods says much for his great intuition which was perhaps based on a deep feeling for geometry.

  373. Bronowski biography
    • He continued with mathematical research at Cambridge after the award of his first degree working on problems in geometry.
    • Bronowski then returned to Cambridge where he received his doctorate in mathematics for a thesis which looked at problems in geometry and topology.
    • He continued his research in geometry publishing a series of papers On triple planes and a paper The figure of six points in space of four dimensions.

  374. Pick biography
    • His mathematical work was extremely broad and his 67 papers range across many topics such as linear algebra, invariant theory, integral calculus, potential theory, functional analysis, and geometry.
    • However more than half of his papers were on functions of a complex variable, differential equations, and differential geometry.
    • Pick's theorem is on reticular geometry.

  375. Davenport biography
    • There he was influenced by Mordell to become interested in both the geometry of numbers and Diophantine approximation.
    • At this time Davenport worked mainly on the geometry of numbers and on Diophantine approximation; he also acquired a lasting interest in problems of packing and covering.
    • Davenport worked on number theory, in particular the geometry of numbers, Diophantine approximation and the analytic theory of numbers.

  376. Coble biography
    • He undertook research into algebraic geometry and was awarded his Ph.D.
    • His early papers, written while he was at Johns Hopkins University, include: On the relation between the three-parameter groups of a cubic space curve and a quadric surface (1906); An application of the form-problems associated with certain Cremona groups to the solution of equations of higher degree (1908); An application of Moore's cross-ratio group to the solution of the sextic equation (1911); An application of finite geometry to the characteristic theory of the odd and even theta functions (1913); and Point sets and allied Cremona groups (1915).
    • In 1929 Coble published the monograph Algebraic geometry and theta functions in the American Mathematical Society Colloquium Publications, being the tenth such volume.

  377. Suss biography
    • He returned to Frankfurt to complete his studies after this three year break, where his research in geometry was supervised by Bieberbach.
    • In 1921 Bieberbach moved from Frankfurt to the University of Berlin where he was appointed to the chair of geometry.
    • In the same year he published Eichflachenprinzipien in der projektiven Flachentheorie which aims to put in place the foundations of a general projective theory of surfaces in a manner roughly corresponding to Berwald's treatment of the Euclidean and affine case but strongly employing the methods of relative differential geometry.

  378. Hilbert biography
    • Hilbert's work in geometry had the greatest influence in that area after Euclid.
    • A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance.
    • He published Grundlagen der Geometrie in 1899 putting geometry in a formal axiomatic setting.

  379. Brisson biography
    • He specialised in the design and construction of ship canals; in particular he applied descriptive geometry to problems of canals.
    • In this work they applied method of geometry they had learnt from Monge to problems of geography, in particular systems of canal routing [ Essai sur le systeme general de navigation interieure de la France (Paris, 1829).',2)">2]:- .
    • [They] applied descriptive geometry to the actual geography of France [discovering] a way to find the lowest points in the watersheds between basins by a simple examination of existing topographical maps, which lacked contour lines and all but a few isolated points of altitude.

  380. Peterson biography
    • Peterson wrote an unpublished "candidate's" thesis, Uber die Biegung der Flachen (On the bending of surfaces) (Dorpat, 1853) on differential geometry.
    • His main work is in differential geometry but he obtained an honorary doctorate for his work on partial differential equations.
    • Peterson's paper 'On curves on surfaces' (1867) and the book 'Uber Curven und Flachen' (1868) were devoted to differential geometry.

  381. McCrea biography
    • He published his second book in 1942 which was Analytical Geometry of Three Dimensions.
    • This book is based upon a short course of lectures to first-year Honours students who have just completed a course on Algebra approximating to that covered by Dr Aitken's 'Determinants and Matrices' in these Texts, and who will have later in their curriculum a course of modern projective geometry.
    • For instance, homographic correspondence is not introduced, largely on the ground that it plays its fundamental part in non-metrical geometry which the student will normally encounter at a later stage.

  382. Antoine biography
    • However, it was impossible to draw the figures which were required in geometry.
    • These figures were made with such dazzling precision that one day one of his students ventured to ask him in a geometry course suited for cartography, "But Sir, how do you make them?" He got this humorous response: "You know, I'm blind because of the war, a bullet has deprived me of my sight, and the last thing I saw was a military map".
    • He added, "In geometry, I have an advantage over you; you, you see the figure, but the whole figure.

  383. Theodosius biography
    • So Theodosius was the author of Sphaerics, a book on the geometry of the sphere, written to provide a mathematical background for astronomy.
    • Sphaerics was written to supplement Euclid's Elements in particular to make up for the lack of results on the geometry of the sphere in Euclid's work.
    • It then goes on to consider geometry results which are relevant to astronomy and these continue to be studied through Book III.

  384. Rado Ferenc biography
    • After this Rado's interests turned towards the algebraic foundations of geometry.
    • He continued with non-injective collineations of two Desarguesian projective planes, which led him to ring geometry.
    • Finally we note Rado's contributions as on the editorial board of the Journal of Geometry and Aequationes Mathematicae.

  385. Lebesgue biography
    • Lebesgue held his post at the Sorbonne until 1918 when he was promoted to Professor of the Application of Geometry to Analysis.
    • After 1922 he remained active, but his contributions were directed towards pedagogical issues, historical work, and elementary geometry.
    • History Topics: A History of Fractal Geometry .

  386. Bateman biography
    • He published eight problems in the Educational Times, between 1902 and 1910, and of these seven were on geometry.
    • Some of Bateman's early work was on geometry and the influence of geometry on all his work is evident.

  387. Levy Paul biography
    • His grandfather was a professor of mathematics while Paul's father, Lucien Levy, was an examiner with the Ecole Polytechnique and wrote papers on geometry.
    • He also studied geometry.
    • History Topics: A History of Fractal Geometry .

  388. Stolz biography
    • in 1864 for a thesis on geometry.
    • Topics covered are those of interest to Klein and to Stolz and include Klein's Erlangen program, nowhere differentiable continuous functions, geometry, and the topology of the line.
    • Stolz's earliest papers were concerned with analytic or algebraic geometry, including spherical trigonometry.

  389. Frege biography
    • He received his doctorate in 1873 from Gottingen for a dissertation Uber eine geometrische Darstellung der imaginaren Gebilde in der Ebene, in which he tried to lay down foundations for a portion of geometry.
    • He lectured on all branches of mathematics, in particular analytic geometry, calculus, differential equations, and mechanics, although his mathematical publications outside the field of logic are few.
    • He decided instead that one had to base the whole of mathematics on geometry.

  390. Eratosthenes biography
    • This work was heavily used by Theon of Smyrna when he wrote Expositio rerum mathematicarum and, although Platonicus is now lost, Theon of Smyrna tells us that Eratosthenes' work studied the basic definitions of geometry and arithmetic, as well as covering such topics as music.
    • 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.
    • Another book written by Eratosthenes was On means and, although it is now lost, it is mentioned by Pappus as one of the great books of geometry.

  391. Boscovich biography
    • Boscovich's implicit, or working, philosophy of mathematics centred on a geometry that was axiomatically Euclidean, abstracted from phenomenal experience, and able to describe, in an approximate manner, phenomena on a macroscopic level.
    • This geometry admitted extension by elements, such as ideal points, with no correspondent in phenomenal experience.
    • In appropriate circumstances, the geometry of the continuum can describe physical reality that, on the microscopic level, is discontinuous and finite.

  392. Perron biography
    • Geometry became the topic of Perron's doctoral thesis directed by Lindemann and Perron went on to complete his habilitation at Munich and was appointed a lecturer there in 1906.
    • However he also worked on differential equations, matrices and other topics in algebra, continued fractions, geometry and number theory.
    • Perhaps most remarkable of all was his text on non-euclidean geometry which he published at the age of 82.

  393. Faedo biography
    • After graduating, he was appointed as a substitute lecturer in geometry at the Scuola Normale in Pisa.
    • In 1937 he was appointed as an assistant to Federigo Enriques who held the chair of higher geometry in Rome.
    • Enriques' choice fell on Faedo, who moved to Rome in March 1937 after overcoming certain perplexities associated with the relocation and to the discipline itself (geometry) in which he had not earned a degree.

  394. Lexell biography
    • Lexell's work in mathematics is mainly in the area of analysis and geometry.
    • Lexell made major contributions to spherical geometry and trigonometry.
    • Spherical geometry was a major tool in his astronomical studies.

  395. Lovasz biography
    • In 1975 he moved to the Jozsef Attila University in Szeged where he was appointed as a Docent but, three years later, he was promoted to professor filling the Chair of Geometry.
    • Combinatorics has grown a lot in the last decade, especially in those fields interacting with other branches of mathematics, like polyhedral combinatorics, algebraic combinatorics, combinatorial geometry, random structures and, most significantly, algorithmic combinatorics and complexity theory.
    • His lecture Geometric algorithms and algorithmic geometry appeared in the conference proceeding and also as a video.

  396. Weyr Eduard biography
    • Solin taught the course on the 'Geometry of Position' and the course on 'Graphical Statics' which Weyr attended.
    • The two brothers collaborated on a book on Projective Geometry.
    • During the spring of 1875, Weyr also served as an assistant to professor Karl Josef Kupper (1828-1900), the professor of descriptive geometry at the Technical University of Prague.
    • Also in 1878 the third volume of Emil and Eduard Weyr's projective geometry book was published.
    • Eduard Weyr wrote geometrical papers and books mainly in projective geometry and differential geometry.
    • Topics he studied include conic sections and quadrics, projective and synthetic geometry, geometrical affinities, differential geometry of curves, differential geometry of surfaces, and algebraic curves.

  397. Kelly biography
    • Let us look briefly at some of the Kelly's mathematical papers, which were mainly on topics in geometry and graph theory, and at his books.
    • In 1953 he wrote Projective geometry and projective metrics jointly with Herbert Busemann.
    • Kelly's 1979 text Geometry and convexity was written jointly with Max L Weiss.

  398. Bollobas biography
    • Bollobas attended Budapest University where he joined the research group headed by Laszlo Fejes Toth who worked on discrete geometry.
    • A while after he had returned to Budapest from Moscow, he received the offer of a scholarship from Oxford where Atiyah held the Savilian Chair of Geometry.
    • Throughout this book the reader will discover connections with various other branches of mathematics, including optimization theory, linear algebra, group theory, projective geometry, representation theory, probability theory, analysis, knot theory and ring theory.

  399. Rees David biography
    • Douglas Northcott, who had become interested in algebra while attending Emil Artin and Claude Chevalley's seminar in Princeton in 1946-47, organised a seminar at Cambridge to study Andre Weil's book The Foundations of Algebraic Geometry (1946).
    • Secondly, through Northcott's seminar, he met another of the participants, the young mathematician Joan Sybil Cushen (25 August 1924 to August 2013) who had written her thesis on algebraic geometry ("before Grothendieck got his hands on it" as she described it) in 1951 in London and returned to Cambridge to teach at Girton College.
    • It grew up on a tough block, living between algebraic number theory and algebraic geometry.

  400. Macaulay biography
    • He wrote 14 papers on algebraic geometry and polynomial ideals.
    • It is important pioneering work in the development of algebraic geometry.
    • He also contributed a number of articles: Bolyai's science of absolute space (1900), On continued fractions (1900), Projective geometry (1906), On the axioms and postulates employed in the elementary plane constructions (1906), On a problem in mechanics and the number of its solutions (1906), and Some inequalities connected with a method of representing positive integers (1930).

  401. Ibrahim biography
    • Ibrahim ibn Sinan was a son of Sinan ibn Thabit and a grandson of Thabit ibn Qurra and studied geometry and in particular tangents to circles.
    • He also studied the apparent motion of the Sun and the geometry of shadows.
    • We know of Ibrahim's works through his own work Letter on the description of the notions Ibrahim derived in geometry and astronomy in which Ibrahim lists his own works.

  402. Smith biography
    • In 1860 he was appointed Savilian professor of geometry despite a strong field of applicants including George Boole.
    • His first two papers were on geometry and, in 1868, he wrote Certain cubic and biquadratic problems which won him the Steiner prize of the Royal Academy of Berlin.
    • Savilian Geometry Professor1861 .

  403. Lusztig biography
    • His first two papers were A model of plane affine geometry over a finite field (Romanian) (1965) and Construction of a universal bundle over arbitrary polyhedra (Romanian) (1966).
    • However, by its very nature, since it draws from various deep and rich theories such as algebraic geometry, intersection cohomology and enveloping algebras, it requires a great deal of effort and commitment on the part of the reader.
    • Extensive connections have already been found between quantum groups and various mathematical areas such as Lie theory, low-dimensional topology, group theory, noncommutative geometry and so on.

  404. Heinonen biography
    • It is a well-written and enjoyable book containing something for both beginners and experts and deserves to be on the shelf of anyone interested in the interplay of analysis, geometry and topology.
    • The purpose of this course was to introduce the students to several aspects of metric geometry.
    • He also gave invited addresses at the XVIIth Geometry Festival at the Courant Institute in New York in 2002 and at the American Mathematical Society meeting in Boulder, Colorado in 2003.

  405. Plato biography
    • Let no one unversed in geometry enter here.
    • The first students of conic sections, and possibly Theaetetus, the creator of solid geometry, were members of the Academy.
    • the exact sciences - arithmetic, plane and solid geometry, astronomy, and harmonics - would first be studied for ten years to familiarise the mind with relations that can only be apprehended by thought.

  406. Bass biography
    • So I tried to learn what others were doing, attending many of the graduate courses: number theory and algebraic geometry from Lang, Lie groups and class field theory from Harish-Chandra, differential algebra from Kolchin, category theory from Eilenberg, and fiber bundles from Albrecht Dold.
    • We have mentioned some areas of Bass's work above, but let us note that he himself gives his research interests as algebraic K-theory; number theory; group theory (geometric methods); and algebraic geometry.
    • These informal reminiscences, presented at the ICTP 2002 Conference on algebraic K-theory, recount the trajectory in the author's early research, from work on the Serre conjecture (on projective modules over polynomial algebras), via ideas from algebraic geometry and topology, to the ideas and constructions that eventually contributed to the founding of algebraic K-theory.

  407. Halley biography
    • Halley was appointed Savilian professor of geometry at Oxford in 1704 following the death of Wallis.
    • After some compliments to the university, he proceeded to the original and progress of geometry, and gave an account of the most celebrated of the ancient and modern geometricians.
    • Savilian Geometry Professor1704 .

  408. Shatunovsky biography
    • As early as at the end of the 19th century, i.e., before the appearance of Gilbert's book "Foundations of Geometry," Shatunovskii gave, in his course of lectures, "Introduction to Analysis," an axiomatic definition of the notion "magnitude" and verified the independence of these axioms by using models.
    • In particular he produced excellent work in group theory, the theory of numbers, and geometry.
    • He used the axiomatic method to lay the logical foundations of geometry, algebraic fields, Galois theory and analysis.

  409. Oresme biography
    • Oresme invented a type of coordinate geometry before Descartes, finding the logical equivalence between tabulating values and graphing them in De configurationibus qualitatum et motuum.
    • Clagett writes in [Nicole Oresme and the Medieval Geometry of Quantities and Motions (Madison, 1968).',3)">3]:- .
    • the invention of analytic geometry before Descartes, with propounding structural theories of compounds before nineteenth century organic chemists, with discovering the law of free fall before Galileo, and with advocating the rotation of the Earth before Copernicus.

  410. Study biography
    • Reye had published a two volume work on synthetic geometry Geometrie der Lage in 1866 and 1868.
    • Study became a leader in the geometry of complex numbers.
    • He reformulated, independently of Francesco Severi, the fundamental principles of enumerative geometry due to Hermann Schubert.
    • With Corrado Segre, Study was one of the leading pioneers in the geometry of complex numbers.

  411. Hesse biography
    • Subsequently this concept has been widely applied in algebraic geometry.
    • Another result by Hesse which has proved particularly influential is the 'principle of transfer' which he gave in 1866 during his work on projective geometry.
    • In his long years as a lecturer, he continually showed his enthusiasm for mathematics, and his textbooks on analytical geometry must be seen in this context.

  412. Abu Kamil biography
    • The Fihrist includes a reference to Abu Kamil and among his works listed there are: (i) Book of fortune, (ii) Book of the key to fortune, (iii) Book on algebra, (vi) Book on surveying and geometry, (v) Book of the adequate, (vi) Book on omens, (vii) Book of the kernel, (viii) Book of the two errors, and (ix) Book on augmentation and diminution.
    • Works by Abu Kamil which have survived, and will be discussed below, include Book on algebra, Book of rare things in the art of calculation, and Book on surveying and geometry.
    • The Book on surveying and geometry is studied in detail in [Centaurus 38 (1996), 1-21.',9)">9].

  413. MacCullagh biography
    • Also in 1843 MacCullagh published his most important work on geometry, namely On surfaces of the second order which described how surfaces such as the ellipsoid could be generated.
    • Although he produced much less work on geometry than on light, it is his work on geometry which has survived and proved in the end the more important.

  414. Blichfeldt biography
    • Some of the many topics that he covered were diophantine approximations, orders of linear homogeneous groups, theory of geometry of numbers, approximate solutions of the integers of a set of linear equations, low-velocity fire angle, finite collineation groups, and characteristic roots.
    • Blichfeldt wrote papers on the geometry of numbers and he has an important book Finite Collineation Groups.
    • In [Yearbook : surveys of mathematics 1980 (Mannheim, 1980), 9-41.',3)">3] Hlawka looks at Blichfeldt's contributions to the geometry of numbers, in particular looking at Blichfeldt's principle.

  415. Crofton biography
    • He did discuss mathematical questions there with John Casey, for when Casey published his book A treatise on the analytic geometry of the point, line, circle and conic section he both acknowledged Crofton's help with the work and also included a number of exercises which he explicitly acknowledged were due to Crofton.
    • Crofton wrote most of his papers on pure mathematics, publishing on geometry and the operator calculus.
    • He is perhaps best known, however, for his contributions to geometric probability, integral geometry and Crofton's formula (see for example [Archimede 35 (3) (1983), 110-126.',9)">9] and [Arch.

  416. Olivier biography
    • In his role as professor Olivier lectured on descriptive geometry and mechanics.
    • His greatest fame, however, is as a result of the mathematical models which he created to assist in his teaching of geometry.
    • Initially the style of the academy was modelled on the Royal Military Academy at Woolwich but later it was modelled in the French style with descriptive geometry becoming a major topic.

  417. Grothendieck biography
    • However it was during this period that his research interests changed and they moved towards topology and geometry.
    • Grothendieck's Seminaire de Geometrie Algebrique established the IHES as a world centre of algebraic geometry, and him as its driving force.
    • During this period Grothendieck's work provided unifying themes in geometry, number theory, topology and complex analysis.

  418. Petersen biography
    • His research was on a wide variety of topics from algebra and number theory to geometry, analysis, differential equations and mechanics.
    • He wrote a series of textbooks based on courses he had given at the College of Technology: one on plane geometry in 1877; one on statics in 1881; one on kinematics in 1884; and one on dynamics in 1887.
    • His most important work however was in geometry with innovative ideas on regular graphs.

  419. Tao biography
    • Covering number theory, algebra, analysis, Euclidean geometry, and analytic geometry, Solving mathematical problems includes numerous exercises and model solutions throughout.
    • Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics as diverse as partial differential equations, analytic number theory, the geometry of 3-manifolds, nonstandard analysis, group theory, model theory, quantum mechanics, probability, ergodic theory, combinatorics, harmonic analysis, image processing, functional analysis, and many others.

  420. Fuss biography
    • Most of Fuss's papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations.
    • He made contributions to differential geometry and won a prize from the French Academy in 1778 for a paper on the motion of comets near some planet Recherche sur le derangement d'une comete qui passe pres d'une planete (see [Istor.-Astronom.

  421. Takagi biography
    • Takagi studied Algebra for beginners by Todhunter and Geometry by Wilson.
    • At Tokyo University Takagi took courses on calculus and analytic geometry.
    • Hilbert had left this topic immediately after writing the Zahlbericht and by the time Takagi reached Gottingen he was engaged in studying the foundations of geometry and then integral equations.

  422. Zeeman biography
    • At this point he became Principal of Hertford College, Oxford, and Gresham professor of geometry at Gresham College London.
    • Among the books which Zeeman has published are the texts Catastrophe theory (1977), Geometry and perspective (1987) and Gyroscopes and boomerangs (1989).
    • He was Gresham Professor of Geometry from 1988 to 1994, delivering an annual series of public lectures.

  423. Dedekind biography
    • There he was to receive a good understanding of basic mathematics studying differential and integral calculus, analytic geometry and the foundations of analysis.
    • Dedekind was then qualified as a university teacher and he began teaching at Gottingen giving courses on probability and geometry.
    • presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space.

  424. Lehmer Derrick N biography
    • In 1917 Lehmer published An Elementary Course in Synthetic Projective Geometry.
    • The following course is intended to give, in as simple a way as possible, the essentials of synthetic projective geometry.
    • The writer has not followed the usual practice of inserting historical notes at the foot of the page, and has tried instead, in the last chapter, to give a consecutive account of the history of pure geometry, or, at least, of as much of it as the student will be able to appreciate who has mastered the course as given in the preceding chapters.

  425. Jonquieres biography
    • In 1846 Chasles had been appointed to a chair of higher geometry at the Sorbonne which had been specially created for him.
    • After his introduction to advanced mathematics by Chasles it is not surprising that his main interests were geometry throughout his life.
    • In addition de Jonquieres discovered results in the area of Schubert's Abzahlende Geometrie (Enumerative geometry).

  426. Pascal biography
    • Blaise however, his curiosity raised by this, started to work on geometry himself at the age of 12.
    • It contained a number of projective geometry theorems, including Pascal's mystic hexagon.
    • He worked on conic sections and produced important theorems in projective geometry.

  427. Mahler biography
    • In particular, there was an elementary book on geometry which he would not then understand, but of which he liked to copy the figures.
    • Already, from the summer vacation of 1917, he began teaching himself logarithms (the arithmetic properties of which turned out to be one of his abiding interests in transcendental number theory) plane and spherical trigonometry, analytic geometry and calculus.
    • Other major themes of his work were rational approximations of algebraic numbers, p-adic numbers, p-adic Diophantine approximation, geometry of numbers (a term coined by Minkowski to describe the mathematics of packings and coverings) and measure on polynomials.

  428. Meyer biography
    • This impressive work extended apolarity theory as introduced by Reye to projective geometry in several dimensions using the theory of rational curves.
    • As indicated by the thesis which he submitted for his habilitation, the title of which we quoted above, Meyer's interests lie in the study of algebraic geometry, algebraic curves and projective invariant theory.
    • He also co-authored an article on potential theory with H Burkhardt, and an article on the geometry of the triangle with G Berkhan.

  429. Hermite biography
    • Hadamard, who unlike his teacher Hermite worked in all areas of mathematics, spoke of Hermite's dislike for geometry:- .
    • [Hermite] had a kind of positive hatred of geometry and once curiously reproached me with having made a geometrical memoir.
    • History Topics: A History of Fractal Geometry .

  430. Grauert biography
    • At Gottingen he supervised the doctoral studies of 44 students, see [I Bauer, F Catanese, Y Kawamata, T Peternell and Y-T Siu (eds), Complex geometry : Collection of papers dedicated to Hans Grauert (Springer, Berlin, 2002), xii-xiii.
    • Note that on one hand we place great importance on geometry, particularly on the interface between geometric visualization and mathematical-logical formulation; on the other hand, we also treat in this book a large part of the basic theory that is needed, say, by students of mathematical economics or physics, even though it only reflects the contents of the first part of the two-semester course 'Analytical geometry and linear algebra'.

  431. Cimmino biography
    • Cimmino was only nineteen years old when he graduated with his thesis on approximate methods of solution for the heat equation in 2-dimensions, but he was appointed as an assistant to Picone who held the chair of analytical geometry at the University of Naples.
    • Then he was in charge of the courses of Analytic Geometry from 1935 to 1938 [Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322, USA (5 January 2004).',3)">3]:- .
    • However, he also made important contributions to many other areas of mathematics, for example the calculus of variations; differential geometry; conformal and quasi-conformal mappings; topological vector spaces; and the theory of distributions.

  432. Bartels biography
    • Bartels took up his post at professor of mathematics at Kazan in 1808 and, during the following twelve years, he lectured on the History of Mathematics, Higher Arithmetic, Differential and Integral Calculus, Analytical Geometry and Trigonometry, Spherical Trigonometry, Analytical Mechanics and Astronomy.
    • It was this course which made Lobachevsky think about non-euclidean geometry.
    • Bartels founded the Centre for Differential Geometry at Dorpat, and remained there until his death in 1836.

  433. Hadamard biography
    • His work was therefore a major contribution to both geometry and to dynamics.
    • This volume on two dimensional geometry appeared in 1898, and was followed by a second volume on three dimensional geometry in 1901.

  434. Lefschetz biography
    • in mathematics in 1911 with a thesis on algebraic geometry entitled On the existence of loci with given singularities.
    • It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry.
    • for indomitable leadership in developing mathematics and training mathematicians, for fundamental publications in algebraic geometry and topology, and for stimulating needed research in nonlinear control processes.

  435. Durer biography
    • It was not only the mathematical theory of proportion which influenced Durer's art at this period, but also his mastery of perspective through his study of geometry.
    • Descriptive geometry originated with Durer in this work although it was only put on a sound mathematical basis in later work of Monge.
    • One of the methods of overcoming the problems of projection, and describing the movement of bodies in space, is descriptive geometry.

  436. De Vries biography
    • He was professor of geometry at the University of Utrecht from 1897 to 1928.
    • In early 1908 de Vries submitted a manuscript to Nieuw Archief voor Wiskunde and, in April of that year, the editor of the journal, Jan Cornelis Kluyver (1860-1932), who was the professor at the University of Leiden working on analysis, differential geometry and number theory, replied to Korteweg (see [6]):- .
    • In each of the three years he taught for seven hours a week using his own textbook which he had published in 1907 and also Leerboek der vlakke meetkunde (Textbook of plane geometry) written jointly by his brother Jan de Vries and Willem Henrik Leonard Janssen van Raaij (1862-1937).

  437. Burckhardt biography
    • The teacher should present only those features of plane geometry which are necessary for the complete understanding of the theorems which form the foundation of surveying.
    • Spherical trigonometry, which should be preceded by certain necessary theorems of geometry which were not taken up in the earlier course.
    • I have not mentioned solid geometry, but it is easy to determine a geographic position [referring to M Barbie Du Bocage's standing as a leading geographer] even if ignorant that the cone is a third of a cylinder, &c, &c.

  438. Doppler biography
    • He applied to schools in Linz, Salzburg, Gorizia and Ljubljana and for the chair of higher mathematics at Vienna Polytechnic and on 23 March 1833 for the professorship of arithmetic, algebra, theoretical geometry and accountancy at the Technical Secondary School in Prague.
    • Doppler did get another chance of a post at the Polytechnic, however, and at the end of 1837 the professorship in practical geometry and elementary mathematics became vacant.
    • The same number of students sat for the examination in "theoretical geometry" in June and July of the same year in a twelve day examination.

  439. Mobius biography
    • Mobius's 1827 work Der barycentrische Calcul, on analytical geometry, became a classic and includes many of his results on projective and affine geometry.
    • He introduced a configuration now called a Mobius net, which was to play an important role in the development of projective geometry.

  440. Bell Robert biography
    • by the University in 1911 for his treatise Coordinate Geometry of Three Dimensions.
    • Professor Robert J T Bell, M.A., D.Sc., LL.D., Sc.D., author of "An Elementary Treatise on Coordinate Geometry of Three Dimensions" (1910; third edition 1944), chapters I-IX of which have been issued separately as "Coordinate Solid Geometry" since 1938, died at Dunedin, New Zealand, on September 8 at the age of 86, leaving a son and a daughter.

  441. Iacob biography
    • At Cluj he worked in the departments of Analytic Geometry, descriptive Geometry, Analysis, and Complex Functions.
    • The author's declared goal is to stimulate both the teacher and his disciples to go beyond the material usually offered in courses on geometry and mechanics.

  442. Sperry biography
    • Wilczynski had begun his research career as a mathematical astronomer but his study of the dynamics of astronomical objects had turned his interests towards differential equations and then to projective differential geometry and ruler surfaces.
    • Given Wilczynski's interests, it is not surprising that Sperry worked on geometry and astronomy for her doctorate which was awarded in 1916 for the thesis Properties of a certain projectively defined two-parameter family of curves on a general surface in 1916.
    • Her distinguished teaching career, during which time she published two excellent texts, one on plane geometry and the other on spherical trigonometry, came to a premature end in 1950.

  443. Kubilius biography
    • Kubilius was awarded a Candidate's degree (equivalent to a Ph.D.) in 1951 for his thesis Geometry of Prime Numbers.
    • He published results from his thesis in On some problems of geometry of prime numbers (1952).
    • In the same year he published On some problems of the geometry of prime numbers (Russian).

  444. Guccia biography
    • The goal was to stimulate the study of higher mathematics by means of original communications presented by the members of the society on the different branches of analysis and geometry, as well as on rational mechanics, mathematical physics, geodesy, and astronomy.
    • In 1889 Guccia was appointed to the chair of geometry at Palermo which he held for the rest of his life.
    • As we have indicated above, Guccia's work was on geometry, in particular Cremona transformations, classification of curves and projective properties of curves.

  445. Osipovsky biography
    • In On space and time Osipovsky criticised Kant's doctrine of the a priori nature of geometric notions (quotation from [, A History of Non-Euclidean Geometry : Evolution of the Concept of a Geometric Space (Springer, 1988).',1)">1]):- .
    • Osipovsky in On space and time examines Kant's argument [, A History of Non-Euclidean Geometry : Evolution of the Concept of a Geometric Space (Springer, 1988).',1)">1]:- .
    • In the same work Osipovsky sums up his ideas on space and time [, A History of Non-Euclidean Geometry : Evolution of the Concept of a Geometric Space (Springer, 1988).',1)">1]:- .

  446. Antiphon biography
    • Aristotle claims that a geometer only needs to show that false arguments are false if they are based on geometry, otherwise he can ignore them.
    • Antiphon therefore deserves an honourable place in the history of geometry as having originated the idea of exhausting an area by means of inscribed regular polygons with an ever increasing number of sides, an idea upon which ..
    • In modern times it has often been supposed that Antiphon was simply making a bad mistake in geometry by supposing that any approximation could ever amount to coincidence between a polygon with however many sides and a continuously curved circumference of a curved circle.

  447. Grassmann biography
    • Grassmann also realised that once geometry is put into this algebraic form then the apparent restrictions of 3-dimensional space vanish.
    • Clifford algebras appear together with Grassmann's exterior algebra in differential geometry.
    • Grassmann's mathematical methods were slow to be adopted but eventually they inspired the work of Elie Cartan and have since been used in studying differential forms and their application to analysis and geometry.

  448. Dodgson biography
    • He was the author of a fair number of mathematics books including: A syllabus of plane algebraical geometry (1860), Two Books of Euclid (1860), The Formulae of Plane Trigonometry (1861), Condensation of Determinants (1866), Elementary Treatise on Determinants (1867), Examples in Arithmetic (1874), Euclid and his modern rivals (1879), Curiosa Mathematica, Part I: A New Theory of Parallels (1888), and Curiosa Mathematica, Part II: Pillow Problems thought out during Sleepless Nights (1893).
    • Dodgson wrote it to defend using Euclid's Elements as a means of teaching geometry.
    • It is a serious work with well argued cases on all sides regarding the teaching of geometry.

  449. Gregory biography
    • James learnt mathematics first from his mother who taught him geometry.
    • In this work Gregory lays down exact foundations for the infinitesimal geometry then coming into existence.
    • For his reluctance to publish his "several universal methods in geometry and analysis" when he heard through Collins of Newton's own advances in calculus and infinite series, he postumously paid a heavy price ..

  450. Laguerre biography
    • His most important work was in the areas of analysis and geometry.
    • His work in geometry was important at the time but has been overtaken by Lie group theory, Cayley's work and Klein's work.
    • He was one of the most penetrating geometers of our age: his discoveries in geometry place him in the first rank among the successors of Chasles and Poncelet.

  451. Bonnet biography
    • However between these two papers on series, Bonnet had begun his work on differential geometry in 1844.
    • Bonnet did important work on differential geometry.
    • Between 1844 and 1867 he published a series of papers on the differential geometry of surfaces.

  452. Franklin biography
    • While this "back-tracking" is frequently brought out in analysis, it is seldom emphasised in connection with plane and solid geometry.
    • I remember some half-baked ideas about the geometry of Pfaffians ..
    • He was well-versed in many fields of geometry, algebra and analysis.

  453. Guarini biography
    • He was familiar with contemporary mathematics especially geometry, but did not adopt algebra - his encyclopedic Euclides adauctus et methodicus mathematicaeque universalis (Turin, 1671) was aimed at clerical students unable to command such a newfangled technique, or maybe he had his own reservations about it.
    • Guarini was adept at most applications of advanced curvature and projective techniques, if not indulging outright in the projective geometry of Desargues as his modern admirers have alleged, confusing 'projection' with 'projective geometry'.

  454. Bagnera biography
    • Francesco Gerbaldi was appointed to the chair of analytic and projective geometry at the University of Palermo in 1890.
    • He was appointed as an ordinary professor of algebra and analytic geometry at the University of Messina in 1905.
    • During these years Michele de Franchis, who was also a Sicilian, was professor of projective and descriptive geometry at the University of Parma, and Bagnera and de Franchis collaborated on an outstanding series of papers on the theory of hyperelliptic surfaces.

  455. White biography
    • White began teaching advanced courses in his area of research interest, in particular on algebraic geometry, projective geometry, and invariant theory.
    • He worked on invariant theory, the geometry of curves and surfaces, algebraic curves and twisted curves.

  456. Forsyth biography
    • After his 1893 treatise he published many other texts, the most important of which are Lectures on the differential geometry of curves and surfaces (1912), Lectures introductory to the theory of functions of two complex variables (1914), Calculus of variations (1927), Geometry of four dimensions which was in two volumes and published in 1930, and Intrinsic geometry of ideal space also in two volumes, published in 1935.

  457. Scott biography
    • Scott continued research at Girton on algebraic geometry under Cayley's supervision, receiving her doctorate in 1885.
    • In 1894 Scott published an important textbook An Introductory Account of Certain Modern Ideas and Methods in Plane Analytical Geometry.
    • Miss Scott was a geometer who whenever possible brought to analytical geometry the full resources of pure geometrical reasoning.

  458. Lowenheim biography
    • He managed to get a post teaching eurythmy (harmony of bodily movement developed with the aid of music into an educational system) and geometry at the Anthroposophic School of Eurythmy in Berlin.
    • In fact he lost unpublished manuscripts on logic, geometry, music and the history of art.
    • The result implies that no uncountable mathematical system, such as those involved in analysis, geometry, and set theory, can be characterised up to isomorphism using only first-order sentences.

  459. Macdonald William biography
    • He was a pioneer in the introduction of modern geometry into the mathematical curriculum, and his book A Higher Geometry was widely used in schools and colleges.
    • He is author of "A Higher Geometry." He was President of the Edinburgh Mathematical Society 1888 - 9, and President of the Scottish Secondary School Teachers' Association 1898 - 9.

  460. Grave biography
    • At St Petersburg he taught analytic geometry, algebra, calculus I and a special course on the theory of surfaces.
    • He also began publishing books based on his lecture courses, for example: Course of analytical geometry (1893); and A course in differential calculus (1895).
    • Among the courses he taught at Kiev we mention: "Group theory"; "Elementary course in the theory of numbers"; "Elements of the theory of elliptic functions"; "Fundamentals of analytical geometry"; "Mathematics of insurance "; and "The elements of algebra".

  461. Okounkov biography
    • The new techniques of working with random partitions invented and successfully developed by Okounkov lead to a striking array of applications in a wide variety of fields: topology of module spaces, ergodic theory, the theory of random surfaces and algebraic geometry.
    • For his contributions bridging probability, representation theory and algebraic geometry.
    • This combination has proven powerful in attacking problems from algebraic geometry and statistical mechanics.

  462. Rolle biography
    • He worked on Diophantine analysis, algebra (using methods of Claude Gaspar Bachet de Meziriac involving the use of the Euclidean algorithm) and, to a lesser extent, on geometry.
    • Geometry has always been considered as an exact science, and indeed as the source of the exactness which is widespread among other parts of mathematics.
    • But it seems that this feature of exactness doe not reign anymore in geometry since the new system of infinitely small quantities has been mixed to it.

  463. Servois biography
    • Servois worked in projective geometry, functional equations and complex numbers.
    • He introduced the word pole in projective geometry.
    • Considered as a leading expert by many mathematicians of his day, he was consulted on many occasions by Poncelet while he was writing his book on projective geometry Traite des proprietes projective.

  464. Gregory David biography
    • He lectured at Edinburgh University on optics, geometry, mechanics and hydrostatics.
    • His lecture notes on geometry were to form the basis of Maclaurin's Treatise of Practical Geometry which was published in 1745.

  465. Gorenstein biography
    • There he worked under Saunders Mac Lane and became interested in finite groups, the subject he would return to after a few years studying algebraic geometry to make it his life's major work.
    • This work led to a thesis on algebraic geometry in which he introduced rings which are now named after him.
    • He returned from algebraic geometry to his early research topic of finite groups in 1957, stimulated by a collaboration with Herstein.

  466. Campbell biography
    • His research turned later towards differential geometry.
    • The theory in which Mr Campbell was so specially interested underlies most of his more recent work on differential geometry in general, and on that particular branch of it connected with Einstein's gravitational theory.
    • Campbell on Differential Geometry .

  467. 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.
    • among the most beautiful, as well as most general, propositions known in the whole compass of geometry.
    • Michel Chasles ranked [Simson and Stewart] among the most important contributors to the progress of geometry.

  468. Betti biography
    • Back in Pisa he moved in 1859 to the chair of analysis and higher geometry.
    • His final move was to substitute the chair of celestial mechanics for his chair of analysis and geometry in 1870.
    • Dini, who Betti had taught earlier, was appointed to fill his chair of analysis and higher geometry.

  469. Catalan biography
    • He remained a pupil at the Ecole Gratuite de Dessin until 1831 but, in 1829, after a competition, he was appointed to teach geometry to his fellow pupils at the school.
    • In the same year, on 20 November, he was appointed assistant tutor (repetiteur) in descriptive geometry at the Ecole Polytechnique, assisting Charles Francois Antoine Leroy who was teaching this topic.
    • Catalan lectured on part of the descriptive geometry course in 1840, the year in which his second daughter Fanny was born.

  470. Minding biography
    • At Dorpat Minding taught algebra, analysis, geometry, probability, mechanics and physics.
    • His work, which continued Gauss's study of 1828 on the differential geometry of surfaces, greatly influenced Peterson.
    • Lobachevsky had published, also in Crelle's Journal, related results three years earlier and these results by Lobachevsky and Minding formed the basis of Beltrami's interpretation of hyperbolic geometry in 1868.

  471. Bunyakovsky biography
    • Bunyakovskii worked on number theory, geometry and applied mathematics.
    • Bunyakovskii also worked on geometry.
    • He then attempted his own proof, unaware that Lobachevsky had invented non-euclidean geometry 25 years before and, although it was published, it had been rejected by Ostrogradski when it had been submitted for publication in the St Petersburg Academy of Sciences.

  472. Bernoulli Jacob biography
    • Jacob Bernoulli also studied the work of Wallis and Barrow and through these he became interested in infinitesimal geometry.
    • Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687.
    • His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines.

  473. Leonardo biography
    • During his time in Milan, Leonardo became interested in geometry.
    • He illustrated Pacioli's Divina proportione and he continued to work with Pacioli and is reported to have neglected his painting because he became so engrossed in geometry.
    • Leonardo studied Euclid and Pacioli's Suma and began his own geometry research, sometimes giving mechanical solutions.

  474. Kac biography
    • One of the topics he tutored was geometry and Mark, although only five years old, became fascinated by what his father was teaching and he persuaded his father to teach him some geometry.
    • That one could know how to prove theorems of elementary geometry without knowing how much seven times nine was seemed more than slightly strange.

  475. Stiefel biography
    • After spending time in Hamburg and Gottingen during 1932, he returned to ETH Zurich where he was appointed as an assistant to Walter Saxer who worked in geometry.
    • At ETH Zurich, Stiefel taught a course on descriptive geometry and geometrical mappings for ten years starting in about 1936.
    • The first chapter discusses orthographic projection and its use in solving graphically problems of solid geometry.

  476. Al-Khwarizmi biography
    • Their tasks there involved the translation of Greek scientific manuscripts and they also studied, and wrote on, algebra, geometry and astronomy.
    • However, Gandz in [The geometry of al-Khwarizmi (Berlin, 1932).',6)">6] (see also [23]), argues for a very different view:- .
    • because his treatment of practical geometry so closely followed that of the Hebrew text, Mishnat ha Middot, which dated from around 150 AD, the evidence of Semitic ancestry exists.

  477. Brouwer biography
    • revealed the twin interests in mathematics that dominated his entire career; his fundamental concern with critically assessing the foundations of mathematics, which led to his creation of intuitionism, and his deep interest in geometry, which led to his seminal work in topology ..
    • He gave his inaugural lecture on 12 October 1909 on 'The nature of geometry' in which he outlined his research programme.
    • Point set theory was widely applied in analysis and somewhat less widely applied in geometry, but it did not have the character of a unified theory.

  478. Lindemann biography
    • Later Lindemann was able to make use of the lecture notes he had taken attending Clebsch's geometry lectures when he edited and revised these note for publication in 1876.
    • At Erlangen he studied for his doctorate and, under Klein's direction, he wrote a dissertation on non-Euclidean line geometry and its connection with non-Euclidean kinematics and statics.
    • Lindemann's main work was in geometry and analysis.

  479. 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.

  480. Reichardt biography
    • Reichardt's mathematical interests turned towards differential geometry and later towards the history of mathematics.
    • Students may find it an adequate introduction, and draw educational benefit from the models of (pseudo-) Euclidean geometry given in the last chapter.
    • This book contains a reprint of Reichardt's 1976 book Gauss und die nicht-euklidische Geometrie but to this had been added reprints of papers on non-euclidean geometry by Janos Bolyai, Nikolai Ivanovich Lobachevsky and Felix Klein.

  481. Cauchy biography
    • He also failed to be appointed to the geometry section of the Institute, the position going to Poinsot.
    • An example of how Cauchy treated colleagues is given by Poncelet whose work on projective geometry had, in 1820, been criticised by Cauchy:- .
    • When questioned by Cauchy on a problem in descriptive geometry, the prince was confused and hesitant.

  482. Belanger biography
    • The mathematical knowledge required to present for admission to the Ecole Centrale des Arts et Manufactures is merely arithmetic, elementary geometry and a part of algebra.
    • So the Ecole Centrale des Arts et Manufactures devised a teaching plan, attempting to satisfy the condition without sacrificing in proofs the tight logic without which mathematics becomes an often misleading semi-science, not compromising clarity by excessive brevity, but by choosing those parts of analytic geometry and the infinitesimal calculus which every engineer must know, and especially those which are necessary for the study of mechanics viewed from the point of view of its practical application to industrial work.

  483. Puissant biography
    • Louis trained as a land-surveyor but felt that he did not know enough mathematics to carry out this work to the standard that he wished, so decided that he must study geometry.
    • The map was produced with considerable detail, the projection used spherical trigonometry, truncated power series and differential geometry.

  484. Reeb biography
    • In the following year Charles Ehresmann and Andre Lichnerowicz organised the conference "Differential geometry" at the University of Strasbourg.
    • For example, at the Fourth International Colloquium on Differential Geometry at Santiago de Compostela in 1978 he gave the talk Equations differentielles et analyse non classique which surveyed results on the perturbation of dynamical systems obtained using methods of nonstandard analysis.

  485. Trail biography
    • By this time the earlier system of appointing post graduate teaching bursars to help with arithmetic and elementary geometry had been abandoned as no longer necessary.
    • His widely used Elements of Algebra, which he published for his students in 1770, ranged from first principles to equations of all orders and included applications to problem solving, physics and geometry.

  486. Kneser Hellmuth biography
    • Kneser published on sums of squares in fields, on groups, on non-Euclidean geometry, on Harald Bohr's almost periodic functions, on iteration of analytic functions, on the differential geometry of manifolds, on local uniformisation and boundary values.

  487. Weierstrass biography
    • The topics of his lectures included:- the application of Fourier series and integrals to mathematical physics (1856/57), an introduction to the theory of analytic functions (where he set out results he had obtained in 1841 but never published), the theory of elliptic functions (his main research topic), and applications to problems in geometry and mechanics.
    • History Topics: A History of Fractal Geometry .

  488. Chung biography
    • Her interests are wide and among her nearly 200 publications there are contributions to spectral graph theory, extremal graphs, graph labelling, graph decompositions, random graphs, graph algorithms, parallel structures and various applications of graph theory in Internet computing, communication networks, software reliability, and discrete geometry.
    • Chung studies this topic from the point of view of spectral geometry in this book drawing the analogy to the spectrum on Riemannian manifolds.

  489. Bennett biography
    • He therefore treated a curve defined in the area of kinematics by the methods of algebraic geometry.
    • He was a close friend of James Bennet Peace, at that time teaching in the Engineering Laboratory of the university (afterwards Secretary to the Syndics of the University Press); it is natural to suppose that their association fostered Bennett's instinct for the geometry of mechanism; his historical paper on Sarrut's mechanism gives interesting glimpses of his search in the records of the Engineering School.

  490. Braikenridge biography
    • William Braikenridge was an Anglican clergyman who worked on geometry and discovered independently many of the same results as Maclaurin.
    • A further work on geometry was Sections of a solid hitherto not considered by geometers.

  491. Steinitz biography
    • Steinitz is no longer a very young experienced lecturer of great versatility and who has worked on numbers theory, set theory, polyhedron geometry and analysis situs; he has recently been on the recommended list almost everywhere but has not been appointed due to adverse circumstances.
    • Steinitz took up his appointment on 30 April 1920 and during his years at Kiel he taught a wide-ranging collection of courses including courses on algebra, polyhedra, number theory, complex analysis, topology, geometry, vector analysis and mechanics.

  492. Cardan biography
    • His father was a lawyer in Milan but his expertise in mathematics was such that he was consulted by Leonardo da Vinci on questions of geometry.
    • In addition to his law practice, Fazio lectured on geometry, both at the University of Pavia and, for a longer spell, at the Piatti foundation in Milan.

  493. Arnauld biography
    • In 1667 Arnauld published New Elements of Geometry.
    • This work was based on Euclid's Elements and was intended to give a new approach to teaching geometry rather than new geometrical theorems.

  494. Emerson biography
    • It properly follows these two fundamental branches, Arithmetic and Geometry, but is vastly superior in nature to both, as it can solve questions quite beyond the reach of either of them.
    • As it contained a new system of philosophy, built upon the most sublime geometry, the greatest mathematicians were obliged to study it with great care and attention, and few became masters of the subject; so that for a long time it was little read.

  495. Ward Seth biography
    • Arithmetic and geometry are sincerely and profoundly taught, analytical algebra, the solution and application of equations, containing the whole mystery of both those sciences, being faithfully expounded in the Schools by the Professor of Geometry, and in several Colleges by particular tutors.

  496. Anaxagoras biography
    • After [Pythagoras] Anaxagoras of Clazomenae dealt with many questions in geometry..
    • There is also other evidence to suggest that Anaxagoras had applied geometry to the study of astronomy.

  497. Branges biography
    • George Thomas was writing a text on the calculus and analytic geometry which was tested on the incoming freshman class.
    • In the summer break I read the recently published Lectures on Classical Differential Geometry by Dirk Struik.

  498. Leimanis biography
    • After the award of his Master's Degree Leimanis was appointed as an assistant in the Department of Descriptive Geometry at the University, but he supplemented his income by teaching mathematics at the Riga School of Commerce during session 1930-31.
    • Returning to Latvia, he was awarded a doctorate for a thesis on algebraic geometry and appointed as a dozent in the Department of Pure Mathematics at the University of Latvia.

  499. Chen biography
    • The authors of [Contemporary trends in algebraic geometry and algebraic topology, Tianjin (2000, World Sci.
    • Richard Hain, who was Chen's last doctoral student graduating in 1980 with his thesis Iterated Integrals, Minimal Models and Rational Homotopy Theory, is the coauthor of the articles [Contemporary trends in algebraic geometry and algebraic topology, Tianjin (2000, World Sci.

  500. Tonelli biography
    • His lecturers at Bologna included Salvatore Pincherle, who taught him analysis, Federigo Enriques, who taught him projective and descriptive geometry, and Cesare Arzela who taught function theory.
    • After the award of his doctorate, Tonelli was appointed as Salvatore Pincherle's assistant at Bologna to work in algebra and analytic geometry.

  501. Schmidt biography
    • Schmidt's ideas were to lead to the geometry of Hilbert spaces and he must certainly be considered as a founder of modern abstract functional analysis.
    • He was one of the earliest mathematicians to demonstrate that the ordinary experience of Euclidean concepts can be extended meaningfully beyond geometry into the idealised constructions of more complex abstract mathematics.

  502. Dee biography
    • There he studied Greek, Latin, philosophy, geometry, arithmetic and astronomy.
    • He instructed the crews on geometry and cosmography before they left for voyages to North America in 1576.

  503. Frewin biography
    • He studied for twelve years at the school obtaining passes in the Scottish Higher Leaving Certificate examinations in Latin, French, English, Mathematics, Analytical Geometry, and Geometry of Conics.

  504. Tinseau biography
    • deal with topics in the theory of surfaces and curves of double curvature: planes tangent to a surface, contact curves of circumscribed cones or cylinders, various surfaces attached to a space curve, the determination of the osculatory plane at a point of a space curve, problems of quadrature and cubature involving ruler surfaces, the study of properties of certain special ruled surfaces (particularly conoids), and various results in the analytic geometry of space.
    • Two papers were published in 1772 on infinitesimal geometry Solution de quelques problemes relatifs a la theorie des surfaces courbes et des lignes a double courbure and Sur quelques proptietes des solides renfermes par des surfaces composees des lignes droites.

  505. Cataldi biography
    • If you are still finding this a little difficult to understand, just consider a great circle on a sphere (this is the shortest distance between two points and so the straight line in this geometry).
    • This will not be a great circle, so is not the shortest distance between two points and so is not the analogue of a straight line in this geometry.

  506. Bezout biography
    • The experience of teaching non-mathematicians shaped the style of the works: Bezout treated geometry before algebra, observing that beginners were not yet familiar enough with mathematical reasoning to understand the force of algebraic demonstrations, although they did appreciate proofs in geometry.

  507. Bordoni biography
    • Four new mathematics courses were established in 1819, namely Descriptive Geometry, Static and Hydraulic Architecture, and Hydrometry and Drawing.
    • Bordoni began to study differential geometry as early as the 1820s.

  508. Maclaurin biography
    • Simson was particularly interested in the geometry of ancient Greece and his enthusiasm for the topic was to influence the young student Maclaurin.
    • Maclaurin did notable work in geometry, particularly studying higher plane curves.

  509. Hotelling biography
    • He was awarded a doctorate in 1924 and he was to make good use in his subsequent career of the topology, differential geometry, analysis and mathematical physics he had learnt at Princeton.
    • he taught his first course, "The theory of probability and statistical inference" in the year 1925-26, and it is interesting to note that in the following year he was offering "Determinants and probability" and "The mathematics of statistical inference," as well as "Analysis situs" [the older name for Topology] and "Differential geometry." .

  510. Bath biography
    • The second thing I learned from Fraser was projective geometry.
    • Bath was awarded his doctorate from Cambridge for his thesis Researches In The Geometry Of Algebraic Curves And Surfaces.

  511. Kalton biography
    • However, he was an invited participant at the meeting on 'Geometry of Banach spaces' held at the Mathematisches Forschungsinstitut in Oberwolfach in November 1973 and he was an invited principal speaker at the N.A.T.O.
    • While in the United States, Kalton was an invited speaker at the special session on 'Geometry of Banach spaces' at the American Mathematical Society regional meeting held at Ohio State University, Columbus, Ohio in April 1978.

  512. Lafforgue biography
    • He then began research in algebraic geometry and the theory of Arakelov under the direction of Christophe Soule.
    • Lafforgue became charge de recherche at the Centre National de la Recherche Scientifique (CNRS) in 1990 and worked in the Arithmetic and Algebraic Geometry team at the Universite Paris-Sud at Orsay.

  513. Ribenboim biography
    • In 1949 he was appointed as a teaching assistant in geometry in Rio de Janeiro, then later that year he was appointed assistant professor at the Centro Brasileiro de Pesquisas Fisicas, again in Rio de Janeiro.
    • During this time he attended lectures by David Mumford on algebraic geometry.

  514. Appell biography
    • Appell's first paper in 1876 was based on projective geometry continuing work of Chasles.
    • The article [Acta Mathematica 45 (1925), 161-285.',2)">2], written by Appell himself, lists 140 works in analysis, 30 works in geometry, 87 works in mechanics as well as many textbooks, addresses, lectures on the history of mathematics and lectures on mathematical education.

  515. Dougall biography
    • In 1952 he published The double six of lines and a theorem in Euclidean plane geometry in Proc.
    • By regarding Q as a 4-sphere in complex Euclidean 5-space, and making some projections, he relates this to a simple theorem of plane geometry: .

  516. Kendall biography
    • He has written on stochastic geometry and its applications, and the statistical theory of shape.
    • Kendall has been joint editor of a number of important works, including Mathematics in the Archaeological and Historical Sciences (1971), Stochastic Analysis (1973), Stochastic Geometry (1974), Analytic and Geometric Stochastics (1986).

  517. Pairman biography
    • Pairman sat the Scottish Leaving Certificate examinations passing Lower Dynamics and Lower Science, with Higher passes in English, French, Latin, Mathematics, and Analytical Geometry.
    • Geometry was a particular problem, because you really need diagrams.

  518. Engel biography
    • The year after Engel returned to Leipzig from Christiania, Klein accepted a professorship at the University of Gottingen and Lie was appointed to succeed him in Leipzig becoming professor of geometry there in February 1886.
    • Engel collaborated with Paul Stackel in studying the history of non-euclidean geometry.

  519. Hecht biography
    • These widely used school texts were on mathematics, geometry and surveying.
    • His later texts covered topics such as quadratic and cubic equations, differential and integral calculus, and arithmetic and geometry.

  520. Picken biography
    • First let us list a few of the papers that Picken published in The Mathematical Gazette: Ratio and proportion (January 1920); The complete angle and geometrical generality (December 1922); Some general principles of analytical geometry (July 1923); The complete angle (October 1923); The notation of the calculus (October 1923); Parallelism and similarity (October 1924); The approach to the logarithmic and exponential functions (December 1926); and The approach to the calculus (October 1927).
    • He lectured on Spherical Geometry and Trigonometry to the Society on 11 April 1911.

  521. Peirce Benjamin biography
    • For example An Elementary Treatise on Plane Trigonometry (1835), First Part of an Elementary Treatise on Spherical Trigonometry (1836), An Elementary Treatise on Sound (1836), An Elementary Treatise on Algebra : To which are added Exponential Equations and Logarithms (1837), An Elementary Treatise on Plane and Solid Geometry (1837), An Elementary Treatise on Plane and Spherical Trigonometry (1840), and An Elementary Treatise on Curves, Functions, and Forces Vol 1 (1841), Vol 2 (1846).
    • There is proof enough furnished by every science, but by none more than geometry, that the world to which we have been allotted is peculiarly adapted to our minds, and admirably fitted to promote our intellectual progress.

  522. Gentry biography
    • She had undertaken research at Girton College, University of Cambridge, England, on algebraic geometry under Cayley's supervision.
    • Her doctoral thesis was supervised by Charlotte Scott, not surprisingly, on geometry which was Scott's own area of expertise.

  523. Laurent Hermann biography
    • He wrote 30 books and a fair number of papers on infinite series, equations, differential equations and geometry.
    • It is divided into two parts, of two and five volumes respectively, on the differential and integral calculus, and included not only the standard treatment of the derivative and the integral and their applications to geometry but also substantial sections on the theory of functions, determinants and elliptic functions.

  524. Boethius biography
    • His understanding of mathematics was rather limited, however, and the text he wrote on arithmetic was of poor quality and although his geometry text has not survived there is little reason to believe that is was any better.
    • Boethius was one of the main sources of material for the quadrivium, an educational course introduced into monasteries consisting of four topics: arithmetic, geometry, astronomy, and the theory of music.

  525. Gillman biography
    • He had enjoyed spotting patterns with numbers from a very young age but, even at high school, there was little in the way of mathematics in the curriculum other than elementary algebra and geometry.
    • He took courses in French and Analytic Geometry to start with but, in his second year took Differential Calculus and another French course.

  526. Caratheodory biography
    • During periods when construction work had to stop due to floods, he studied mathematics from some textbooks he had with him, such as Jordan's Cours d'Analyse and Salmon's text on the analytic geometry of conic sections.
    • However, he also required a university post so he was appointed as Professor of Analytical and Higher Geometry at the University of Athens on 2 June 1920.

  527. Serenus biography
    • many persons who were students of geometry were under the erroneous impression that the oblique section of a cylinder was different from the oblique section of a cone known as an ellipse, whereas it is of course the same curve.
    • 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.

  528. Miranda biography
    • Proceedings of international Meeting dedicated to the memory of Professor Carlo Miranda (Naples, 1983).',1)">1] and divides these contributions into the following areas: (a) Integral equations, series expansions, summation methods; (b) Harmonic mappings, potential theory, holomorphic functions; (c) Calculus of variations, differential forms, elliptic systems; (d) Numerical analysis; (e) Propagation problems; (f) Differential geometry in the large; (g) General theory for elliptic equations; and (h) Functional transformations.
    • Several of these contributions are treated with some detail: results concerning normal families and approximation theory, the equivalence between Brouwer's fixed point theorem and a result for the zeroes of some systems of continuous functions, the Cauchy-Dirichlet problem for the propagation equation, the numerical integration of the Thomas-Fermi equation, integral equations (introducing the notions of pseudofunction and singular eigenvalue), problems of differential geometry "in the large", etc.

  529. Sato biography
    • It was algebra that first attracted him and soon he was reading books on his own, learning about complex numbers, projective geometry and other more advanced topics.
    • Looking back, forty years later, we realize that Sato's approach to mathematics is not so different from that of Grothendieck, that Sato did have the incredible temerity to treat analysis as algebraic geometry, and that he was also able to build the algebraic and geometric tools adapted to his problems.

  530. Drach biography
    • He published two papers on geometry, Sur les lignes d'osculation quadrique des surfaces.
    • A 33 (41) (2000), 7407-7422.',5)">5], and [CR-geometry and overdetermined systems, Osaka, 1994 (Math.

  531. Robinson Raphael biography
    • His doctoral dissertation was on complex analysis, but he also worked on logic, set theory, geometry, number theory, and combinatorics.
    • The book gives an introductory account of the methods introduced by Tarski for establishing the undecidability of several fairly simple branches of mathematics (group theory, lattices, abstract projective geometry, closure algebras and others).

  532. Li Shanlan biography
    • With Alexander Wylie, Li translated Elements of Analytical Geometry and of the Differential and Integral Calculus which had been written by Elias Loomis and published in New York in 1851.
    • Although this is the first time that the principles of algebraic geometry have been placed before the Chinese, in their own idiom, yet there is little doubt that this branch of the science will commend itself to native mathematicians, in consideration of its obvious utility ..

  533. Mascheroni biography
    • In 1786 Mascheroni became professor of algebra and geometry at the University of Pavia, mainly on the strength of his excellent work on statics Nuove ricerche su l'equilibrio delle volte which he had published one year earlier.
    • He was moved initially by a desire to make a contribution to elementary geometry.

  534. Varignon biography
    • He thus implicitly attributed to mechanics the same demonstrative perfection that Euclidean geometry had been thought to possess.
    • is organised in two parts, the first part explaining concepts of arithmetic and elementary algebra and the larger second part covering topics in Euclidean geometry.

  535. Faa di Bruno biography
    • From 1859 Angelo Genocchi held the Chair of Algebra and Complementary Geometry at Turin, then in the following year he moved to the Chair of Higher Analysis and Faa di Bruno was appointed as his deputy.
    • In 1871 he was put in charge of teaching calculus and analytic geometry and he was appointed as an extraordinary professor of higher analysis in 1876.

  536. Calugareanu biography
    • Calugareanu worked in a number of different mathematical areas such as the theory of functions of one complex variable, geometry, algebra, and topology.
    • Guided by this point of view, Calugareanu left us important works on differential geometry and topology.

  537. Whitney biography
    • A final section on other topics includes nine papers on logic, geometry, and the mathematics of physical quantities, for the last of which he received a Lester Ford Award.
    • for his fundamental work in algebraic topology, differential geometry and differential topology.

  538. Thabit biography
    • played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry.

  539. Maschke biography
    • Moore was a fiery enthusiast, brilliant, and keenly interested in the popular mathematical research movements of the day; Bolza, a product of the meticulous German school of analysis led by Weierstrass, was an able, and widely read research scholar; Maschke was more deliberate than the other two, sagacious, brilliant in research, and a most delightful lecturer in geometry.
    • Maschke's second area of work was on differential geometry in particular the theory of quadratic differential quantics.

  540. Todhunter biography
    • Among his textbooks are Analytic Statics (1853), Plane Coordinate Geometry (1855), Examples of Analytic geometry in Three Dimensions (1858).

  541. Konigsberger biography
    • They had spent the evenings reading books on differential calculus and analytical geometry, working at a table in a small sitting room which they shared.
    • Among Kummer's courses that he attended were higher number theory, the theory of surfaces, mechanics, and analytical geometry.

  542. Preston biography
    • For example, our differential geometry lectures told us about two adjacent points on a curve, or two adjacent tangents to a curve, and my immediate contemporaries at Magdalen (my college at Oxford), Michael Barrett and Victor Guggenheim, and I, read the appropriate books that did this differential geometry properly.

  543. Newton biography
    • Attempting to read a trigonometry book, he found that he lacked knowledge of geometry and so decided to read Barrow's edition of Euclid's Elements.
    • The new algebra and analytical geometry of Viete was read by Newton from Frans van Schooten's edition of Viete's collected works published in 1646.

  544. Serre biography
    • From 1956 he held the chair of Algebra and Geometry in the College de France until he retired in 1994 when he became an honorary professor.
    • It is a kind of applied mathematics: you try to use any tool in algebraic geometry and number theory that you know of, ..

  545. Rosanes biography
    • Rosanes' mathematical papers concerned the various questions of algebraic geometry and invariant theory that were current in the nineteenth century.
    • Rosanes wrote on many aspects of algebraic geometry and invariant theory (particularly between 1870 and 1890) which were in fashion at that time.

  546. Adelard biography
    • Adelard made Latin translations of Euclid's Elements from Arabic sources which were for centuries the chief geometry textbooks in the West.
    • The remaining two books of the five which compose the treatise cover geometry, which is completely Greek in style, music, and astronomy.

  547. Ramus biography
    • Given these views it is not surprising that his 1569 textbook on geometry contained strong criticisms of Euclid's Elements.
    • He wrote textbooks on arithmetic, algebra and geometry with the aim of including only theorems which could be applied to the crafts.

  548. Droz-Farny biography
    • By this time he had developed a strong preference for geometry.
    • [Droz-Farny] is known to have written four books between 1897 and 1909, two of them about geometry.

  549. Dinostratus biography
    • Amyclas of Heraclea, one of the associates of Plato, and Menaechmus, a pupil of Eudoxus who had studied with Plato, and his brother Dinostratus made the whole of geometry still more perfect.
    • Dinostratus probably did much more work on geometry but nothing is known of it.

  550. Genocchi biography
    • After a while he began teaching but he had to be tricked into entering the competition for the Chair of Algebra and Complementary Geometry at Turin.
    • From 1859 Genocchi held the Chair of Algebra and Complementary Geometry at Turin, then the following year he moved to the Chair of Higher Analysis.

  551. Bacon biography
    • He then progressed to the quadrivium, studying geometry, arithmetic, music and astronomy.
    • His most important mathematical contribution is the application of geometry to optics.

  552. Waring biography
    • We shall comment further below on this important work, covering topics in the theory of equations, number theory and geometry.
    • Proprietates algebraicarum curvarum, covering geometry, was published in 1772.

  553. Ackermann biography
    • The paper ends with remarks on geometry, as a science conveying knowledge of the external world.
    • Beyond any axiomatic treatment of geometry there are intuitive geometrical impressions which force themselves upon us with compelling power.

  554. Kempe biography
    • Most of Kempe's early contributions to mathematics were on linkages, involving applications of geometry.
    • The great geometrician Euclid, before demonstrating to us the various propositions contained in his Elements of Geometry, requires that we should be able to effect certain processes.

  555. Serrin biography
    • And this was the first evidence of his growing interest in differential geometry.
    • Mathematical analysis, calculus of variations and geometry were not sufficient for James Serrin.

  556. Cohn biography
    • His second book Linear equations was published in 1958 and another book Solid geometry was published in 1961.
    • It is Cohn's merit to provide a coherent treatment of this subject which at the same time leads the reader to a wide range of interesting and important research problems, related to questions in algebra, geometry and logic.

  557. Strong biography
    • Matthew Stewart had made some conjectures on the geometry of the circle which were proved by Glenie in 1805.
    • Another Scottish mathematician who was working on geometry was William Wallace.

  558. D'Alembert biography
    • d'Alembert was a mathematician, not a physicist, and he believed mechanics was just as much a part of mathematics as geometry or algebra.
    • What annoys me the most is the fact that geometry, which is the only occupation that truly interests me, is the one thing that I cannot do.

  559. Smale biography
    • In addition to the Fields Medal described above, he was awarded the Veblen Prize for Geometry by the American Mathematical Society in 1966:- .
    • Geometry Center (Turning a sphere inside out) .

  560. Baker Alan biography
    • Among his famous books are Transcendental number theory (1975), Transcendence theory : advances and applications (1977), A concise introduction to the theory of numbers (1984) and (with Gisbert Wustholz) Logarithmic forms and Diophantine geometry (2007).
    • This encouraged us to work out a programme that aimed to cover a large spectrum of number theory and related geometry with particular emphasis on Diophantine aspects.

  561. Remak biography
    • He had broad interests, working on mathematical economics as well as group theory and the geometry of numbers.
    • He may, without doubt, be called a leading scholar in the splendid and important field of geometry of numbers.

  562. Knopp biography
    • Volume 1 covers numbers, functions, limits, analytic geometry, algebra, set theory; volume 2 covers differential calculus, infinite series, elements of differential geometry and of function theory; and volume 3 covers integral calculus and its applications, function theory, differential equations.

  563. Vitruvius biography
    • Let him be educated, skilful with the pencil, instructed in geometry, know much history, have followed the philosophers with attention, understand music, have some knowledge of medicine, know the opinions of the jurists, and be acquainted with astronomy and the theory of the heavens.
    • Geometry is of much assistance in architecture, and in particular it teaches us the use of the rule and compasses, by which especially we acquire readiness in making plans for buildings in their grounds, and rightly apply the square, the level, and the plummet.

  564. Kirillov biography
    • We should also mention the conference Orbit method in Geometry and Physics held in his honour in Marseilles, France, in December 2000.
    • The aim of the orbit method is to unite harmonic analysis and symplectic geometry to describe the representations of a group in geometric terms.

  565. Adian biography
    • Before the spring break, as part of preparation for final examinations, another teacher of mathematics, the school headmaster, gave his students homework in solid geometry based on trigonometric formulae from the popular problem book by Rybkin.
    • dissertations in mathematical logic, algebra, number theory, geometry, and topology, first as the vice-chairman and later, after the death of Vinogradov, as the chairman.

  566. Briot biography
    • He wrote textbooks which covered most of the topics from a mathematics course: arithmetic, algebra, calculus, geometry, analytic geometry, and mechanics.

  567. Finkel biography
    • Until I was seventeen years of age I had never seen a geometry or an algebra.
    • Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions with Notes and Explanations by B F Finkel.

  568. Heuraet biography
    • Van Schooten had established a vigorous research school in Leiden which included van Heuraet, and this school was one of the main reasons for the rapid development of Cartesian geometry in the mid 17th century.
    • When van Heuraet learned that I had measured the surface of the parabolic conoid and had determined the length of the parabola equal to a given quadrature of the hyperbola (concerning both of which I wrote you previously), he found not only both of them by his own technique but, in addition, he rectified completely all other curves of those genera that we allow in geometry.

  569. Witt biography
    • Shortly after that, Witt introduced the ring of 'Witt vectors', which had a great influence on the development of modern algebraic geometry (see [Jahresber.
    • It is one of the most important theorems in mathematics, connecting such diverse areas as representation theory, differential geometry, and universal algebra.

  570. Yamabe biography
    • This was a period when his mathematical interests began to move away from Lie groups to differential equations and differential geometry.
    • Mathematicians will gather every two years at the University of Minnesota for a long weekend to hear geometry talks, discuss the latest research and interact with younger mathematicians.

  571. Boyle biography
    • he grew very well acquainted with the most useful part of arithmetic, geometry, with its subordinates, the doctrine of the sphere, that of the globe, and fortification.
    • At Oxford he joined a group of forward looking scientists, including John Wilkins, John Wallis who was the Savilian Professor of Geometry, Seth Ward who was the Savilian Professor of Astronomy, and Christopher Wren who would succeed Ward as Savilian Professor of Astronomy in 1661.

  572. Autolycus biography
    • Of these books, On the Moving Sphere is a work on the geometry of the sphere which is the same as being a mathematical astronomy text.
    • Theodosius, 200 years later, wrote Sphaerics, a similar book on the geometry of the sphere, also written to provide a mathematical background for astronomy.

  573. Wiman biography
    • Bjorling was interested in algebraic geometry so it was a natural topic for him to give to his research students.
    • His first doctoral students worked on problems with entire functions, but later students wrote theses looking at problems in group theory, irreducibility criteria for polynomials with integer coefficients, and algebraic geometry.

  574. Maurolico biography
    • Maurolico also worked on geometry, the theory of numbers (L E Dickson notes some of his results), optics, conics and mechanics, writing important books on these topics which we will discuss in more detail below.
    • Geometricarum questionum (Books 1 and 2), which is a work on trigonometry and solid geometry but also discusses the method for measuring the Earth that Maurolico proposed earlier in Cosmographia; .

  575. Margulis biography
    • Tits talks in [Proceedings of the International Congress of Mathematicians, Helsinki, 1978 (Helsinki, 1980), 57-63.',7)">7] about the range of Margulis's work in combinatorics, differential geometry, ergodic theory, dynamical systems and discrete subgroups of Lie groups.
    • The different approaches to this and related conjectures (and theorems) involve analytic number theory, the theory of Lie groups and algebraic groups, ergodic theory, representation theory, reduction theory, geometry of numbers and some other topics.

  576. Northcott biography
    • In particular he was taught to solve simultaneous equations and prove elementary theorems in Euclidean geometry which gave him a love of mathematics at this early stage in his education.
    • the book will encourage many who would not otherwise have done so to study ideal theory and algebraic geometry.

  577. Roberts biography
    • Among the subjects to which his principal papers related were plane and solid geometry, theory of numbers, and link motion.
    • His writings on geometry included several important papers on parallel curves and surfaces.

  578. Mei Juecheng biography
    • Important work of Mei Wending on mathematics published in this collection included: Bisuan (Pen Calculations), Chou suan (Napier's bones), Du suan shi li (Proportional Dividers), Shao guang shi yi (Supplement to 'What Width'), Fang cheng lun (Theory of Rectangular Arrays), Gougu ju yu (Right-angled Triangles), Jihe tong jie (Explanations in Geometry), Ping san jiao ju yao (Elements of Plane Trigonometry), Fang yuan mi ji (Squares and Circles, Cubes and Spheres), Jihe bu bian (Supplement to Geometry), Hu san jiao ju yao (Elements of Spherical Trigonometry), Huan zhong shu chi (Geodesy), and Qiandu celiang (Surveying Solids).

  579. Bhaskara II biography
    • The topics covered in the thirteen chapters of the book are: definitions; arithmetical terms; interest; arithmetical and geometrical progressions; plane geometry; solid geometry; the shadow of the gnomon; the kuttaka; combinations.

  580. Grinbergs biography
    • It was during his time in Paris that his first publication appeared; it was a work on geometry Uber die Bestimmung von zwei speziellen Klassen von Eilinien.
    • Returning to Riga, Grinbergs became a Privatdozent at the University of Latvia in 1937 and began his lecturing career in January 1938 teaching courses on geometry.

  581. Polya biography
    • He did not score particularly high marks in mathematics at the Gymnasium, his work in geometry being graded as merely "satisfactory".
    • His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics.

  582. Voronoy biography
    • Later Voronoy worked on the theory of numbers, in particular he worked on algebraic numbers and the geometry of numbers.
    • Voronoy's memoir on parallelohedra represents one of the deepest investigations in the geometry of numbers in the world's literature, and the originality of the methods employed in the purely geometrical first part stamps the memoir with the imprint of genius.

  583. Feuerbach biography
    • A study in analytic geometry by Dr Karl Wilhelm Feuerbach, Professor of Mathematics).
    • He must therefore be considered as the joint inventor of homogeneous coordinates since Mobius, in his work Der barycentrische Calcul also published in 1827, introduced homogeneous coordinates into analytic geometry.

  584. Carnot biography
    • In 1801 he published De la correlation des figures de geometrie in which he tried to put pure geometry into a universal setting.
    • In 1803 he published Geometrie de position in which sensed magnitudes were first used systematically in geometry.

  585. Stott biography
    • Alicia Boole experimented with the cubes and soon developed an amazing feel for four dimensional geometry.
    • To Ethel, and possibly Lucy too, this was a meaningless bore; but it inspired Alice (at the age of about eighteen) to an extraordinarily intimate grasp of four-dimensional geometry.
    • She then produced three-dimensional central cross-sections of all the six regular polytopes by purely Euclidean constructions and synthetic methods for the simple reason that she had never learned any analytic geometry.
    • In around 1930 she was introduced to Coxeter and they worked together on various problems [King of infinite space: Donald Coxeter, the man who saved geometry (House of Anansi Press, Totonto, 2006).',2)">2]:- .
    • He even persuaded her to talk at Henry Baker's tea party [King of infinite space: Donald Coxeter, the man who saved geometry (House of Anansi Press, Totonto, 2006).',2)">2]:- .
    • She wrote him a letter saying (see [King of infinite space: Donald Coxeter, the man who saved geometry (House of Anansi Press, Totonto, 2006).',2)">2]):- .

  586. Wiener Hermann biography
    • At the time of Hermann's birth, Christian Wiener was professor of descriptive geometry at the Technische Hochschule of Karlsruhe, having moved there five years earlier when appointed to the chair.
    • Wiener proposed that geometry be studied without using visual images, but rather by abstract axiomatic methods.

  587. Hoyle biography
    • His teaching duties were to give a geometry course and a statistical mechanics course in 1945-46.
    • He stopped teaching geometry, teaching instead courses on Electricity and Magnetism, and on Thermodynamics.

  588. Ortega biography
    • Ortega's book A Tractado subtilisimo d'arithmetica y de geometria published in Barcelona in 1512 covered commercial arithmetic and the rules of geometry.
    • In the second part of the book, devoted mostly to geometry, Ortega gives a method of extracting square roots very accurately using Pell's equation, which is surprising since a general solution to Pell's equation does not appear to have been found before Fermat over 100 years later.

  589. Elliott biography
    • His mathematical work included algebra, algebraic geometry, synthetic geometry, elliptic functions and the theory of convergence.

  590. Sinai biography
    • However the family had strong mathematical connections since Yakov Grigorevich's grandfather (Nadezda Kagan's father) was Benjamin Fedorovich Kagan, the Head of the Department of Differential Geometry at Moscow State University where he founded an important School of Differential Geometry.

  591. Wangerin biography
    • Wangerin's research was on potential theory, spherical functions and differential geometry.
    • He taught many courses at the University of Halle including: linear partial differential equations; calculus of variations; theory of elliptical functions; synthetic geometry; hydrostatics and capillarity theory; theory of space curves and surfaces; analytic mechanics; potential theory and spherical harmonics; celestial mechanics; the theory of the map projections; hydrodynamics; and the partial differential equations of mathematical physics.

  592. Steggall biography
    • His research interests were in the theory of numbers and in kinematical geometry, particularly the geometry of the triangle.

  593. Ajima biography
    • Ajima's work went towards geometry despite the strong algebraic numerical tradition in the Seki school.
    • He presented this in Kohai jutsu kai, giving a method which is the high point that traditional Japanese mathematics reached in methods of integration [Sacred Mathematics : Japanese Temple Geometry (Princeton University Press, Princeton, NJ, 2008).',2)">2]:- .

  594. Shnirelman biography
    • Shnirelman started research in algebra, geometry and topology as a student but did not consider his results sufficiently important to merit publication.
    • Shnirelman and Lyusternik also applied their "principle of the stationary point" to other problems of geometry "im Grossen".

  595. Korkin biography
    • In particular he took courses on analytic geometry, higher algebra and number theory given by Chebyshev.
    • At this time he left the First Cadet School and taught trigonometry, analytic geometry and integral calculus at the University.

  596. Cantor biography
    • Of course this had implications for geometry and the notion of dimension of a space.
    • History Topics: A History of Fractal Geometry .

  597. Benedetti biography
    • In a section on geometry, he also treats spherical triangles, circles and conic sections.
    • [However] Benedetti is not so much a mathematician as a natural philosopher employing the methods of geometry.

  598. Orlicz biography
    • monographs on Orlicz spaces: M A Krasnoselskii and Ya B Rutickii, Convex Functions and Orlicz Spaces (Groningen 1961), J Lindenstrauss and L Tzafriri, Classical Banach Spaces I, II (Springer 1977, 1979), C Wu and T Wang, Orlicz Spaces and their Applications, (Harbin 1983 - Chinese), A C Zaanen, Riesz Spaces II, (North-Holland 1983), C Wu, T Wang, S Chen and Y Wang, Theory of Geometry of Orlicz Spaces (Harbin 1986 - Chinese), L Maligranda, Orlicz Spaces and Interpolation, (Campinas 1989), M M Rao and Z D Ren, Theory of Orlicz Spaces (Marcel Dekker 1991) and S Chen, Geometry of Orlicz Spaces (Dissertationes Math.

  599. Valyi biography
    • Valyi was appointed professor of theoretical physics at Kolozsvar in 1884, and in the following year he also became professor of mathematics lecturing on analysis, geometry and number theory.
    • Valyi also made extensive studies of projective geometry, in particular studying polar reciprocity, and used elliptic functions to study third order curves.

  600. Chevalley biography
    • His papers of 1936 and 1941 where he introduced the concepts of adele and idele led to major advances in class field theory and also in algebraic geometry.
    • Chevalley also published Theory of Distributions (1951), Introduction to the theory of algebraic functions of one variable (1951), The algebraic theory of spinors (1954), Class field theory (1954), The construction and study of certain important algebras (1955), Fundamental concepts of algebra (1956) and Foundations of algebraic geometry (1958).

  601. Lansberge biography
    • The third book is devoted to the study of plane triangles while the fourth, and final, book begins with a study of spherical geometry before giving a thorough look at spherical trigonometry.
    • it is not intended for instruction in geometry and astronomy.

  602. Spanier biography
    • At Berkeley, Spanier built up a strong group working in geometry and topology by several appointments of topologists to the faculty of Berkeley and also by attracting many top topologists to spend periods as visitors at Berkeley.
    • Chern was appointed professor of geometry at Chicago in 1949, the year following Spanier's appointment, and the two began the study of homology groups of fibre spaces with their joint paper The homology structure of sphere bundles in 1950.

  603. Camus biography
    • He was then named as Professor of Geometry to the Academy of Architecture and in this capacity he lectured to students of architecture.
    • This work was to be in four parts, arithmetic, geometry, mechanics, and hydraulics.

  604. Whitehead biography
    • In 1906 he published The axioms of projective geometry and, in the following year, The axioms of descriptive geometry.

  605. Cunha biography
    • As part of his university reforms de Pombal appointed da Cunha professor of geometry in 1773.
    • The book contained the elements of geometry and algebra in addition to the calculus.

  606. Stackel biography
    • Stackel thrived during his time at Halle, publishing numerous papers, mainly on topics in analysis, mechanics and differential geometry.
    • Later, the two collaborated on the study of non-Euclidean geometry, as well as research into the history of mathematics, perhaps most notably collaborating on the publication of the complete works of Euler.

  607. Graham biography
    • His seminal work has led to the birth of at least three new branches of mathematics: Ramsey theory, computational geometry, and worst case analysis of multiprocessing algorithms (now sometimes referred to as "Graham type analysis.") Graham's characteristic quality is an indefatigable activity, both in the cause of mathematics, and on behalf of it applications.
    • He has made many important research contributions to this subject, including the development, with Fan Chung, of the theory of quasirandom combinatorial and graphical families, Ramsey theory, the theory of packing and covering, etc., as well as to the theory of numbers, and seminal contributions to approximation algorithms and computational geometry (the "Graham scan").

  608. Harish-Chandra biography
    • However Harish-Chandra made several attempts to get into algebraic geometry or number theory.
    • His scientific work, being a synthesis of analysis, algebra and geometry, is still of lasting influence.

  609. Eudoxus biography
    • The question of whether Eudoxus thought of his spheres as geometry or a physical reality is studied in the interesting paper [Studies in Hist.
    • 21 (4) (1990), 313-329.',19)">19] argues convincingly that this is not so and that the ideas which influenced Eudoxus to come up with his masterpiece of 3-dimensional geometry were Pythagorean and not from Plato.

  610. Faltings biography
    • He received the medal primarily for his proof of the Mordell Conjecture which he achieved using methods of arithmetic algebraic geometry.
    • My main interests are arithmetic geometry (diophantine equations, Shimura-varieties), p-adic cohomology (relation crystalline to etale, p-adic Hodge theory), and vector bundles on curves (Verlinde-formula, loop-groups, theta-divisors).

  611. Lemoine biography
    • Lemoine work in mathematics was mainly on geometry.
    • He produced a classification of geometry according to five operations:- .

  612. Bertrand biography
    • Joseph showed remarkable talents as a child and by the age of nine he understood algebra and elementary geometry, as well as being able to speak Latin fluently.
    • Bertrand published many works on differential geometry and on probability theory.

  613. Chrystal biography
    • Chrystal's mathematical publications cover many topics including non-euclidean geometry, line geometry, determinants, conics, optics, differential equations, and partitions of numbers.

  614. Pierpont biography
    • Von Escherich (1849-1935), an expert on analytical geometry, had been appointed to the University of Vienna in 1884.
    • In addition to the American Mathematical Society Colloquium Lectures that he gave in Buffalo in 1896, Pierpont address the International Congress of Arts and Science in St Louis in September 1904 on the History of Mathematics in the Nineteenth Century, he addressed the American Mathematical Society summer meeting at Wellesley in 1921 on Some mathematical aspects of the theory of relativity, he gave the Gibbs Lecture in Kansas City in 1925 on Some modern views of space, he addressed the annual meeting at Nashville in 1927 on Mathematical rigor, past and present, he addressed the annual meeting at New York in 1928 On the motion of a rigid body in a space of constant curvature, and the annual meeting at Berkeley in 1929 on Non-Euclidean geometry, a retrospect.

  615. Kluvanek biography
    • The book, covered Differential and Integral calculus, Analytic geometry, Differential equations and Complex variable [S Tkacik, J Guncaga, P Valihora and M Gerec (eds.), Igor Kluvanek: Prispevky zo seminara venovaneho nedozitym 75.
    • This is a monograph on the geometry of the range of a vector measure and applications to control systems governed by partial differential equations.

  616. Kelland biography
    • He wrote analytical papers on General Differentiation (1839), and Differential Equations (1853), and gave a geometrical Theory of Parallels outlining a version of non-Euclidean geometry.
    • Kelland produced a much-revised edition of John Playfair's Elements of Geometry and a successful textbook of Algebra.

  617. Viete biography
    • 6 (3) (1975), 185-208.',22)">22] where it is stated that Viete was interested purely in the geometry of the planetary theories of both Ptolemy and Copernicus, and did not consider the question of whether the theories represented the actual physical reality.
    • Viete also wrote books on trigonometry and geometry such as Supplementum geometriae (1593).

  618. Iwasawa biography
    • a general method in arithmetical algebraic geometry, known today as Iwasawa theory, whose central goal is to seek analogues for algebraic varieties defined over number field of the techniques which have been so successfully applied to varieties defined over finite fields by H Hasse, A Weil, B Dwork, A Grothendieck, P Deligne, and others.
    • today it is no exaggeration to say that Iwasawa's ideas have played a pivotal role in many of the finest achievements of modern arithmetical algebraic geometry on such questions as the conjecture of B Birch and H Swinnerton-Dyer on elliptic curve; the conjecture of B Birch, J Tate, and S Lichtenbaum on the orders of the K-groups of the rings of integers of number fields; and the work of A Wiles on the modularity of elliptic curves and Fermat's Last Theorem.

  619. Wilkins Ernest biography
    • He wrote his first papers in 1942, both on geometry, and they were published in 1943.
    • They are The first canonical pencil and A special class of surfaces in projective differential geometry both published in Duke Mathematical Journal.

  620. Segner biography
    • His publications include Elements of Arithmetic and Geometry and Nature of Liquid Surfaces.
    • The proofs of several theorems of algebra and geometry have been adopted by subsequent textbooks and some of his Latin and German technical terms ..

  621. Matsushima biography
    • Back in Osaka, Matsushima jointly began to organise the United States-Japan Seminar in Differential Geometry which was held in Kyoto in June 1965.
    • He became an editor of the new Journal of Differential Geometry in 1967, remaining on the editorial board for the rest of his life.

  622. Olech biography
    • In his first year he took three lecture courses: analysis by Tadeusz Wazewski, algebra by Andrzej Turowicz and analytic geometry by Jacek Szarski.
    • It includes a number of other branches of mathematics, and his works are classified by Mathematical Reviews as including, among others, the area of linear and multilinear algebra, measure and integration theory, calculus of variations, convex and discrete geometry, operations research and general systems theory.

  623. Balmer biography
    • From 1865 until 1890 he was also a university lecturer in mathematics at the University of Basel where his main field of interest was geometry.
    • But for his work on what amounts to a problem in physics, Balmer would be unknown today within the history of mathematics since he made no contribution to geometry of special significance despite it being the topic of interest throughout his life.

  624. Tietz biography
    • Lineare Geometrie (first edition 1967, second edition 1973) was a text based on lecture courses that Tietz had given on analytic geometry.
    • Linear geometry; 3.

  625. Van Lint biography
    • The next, written jointly with Gerard van der Geer, was Introduction to coding theory and algebraic geometry.
    • The book was based on lectures given in the seminar on "Coding Theory and Algebraic Geometry" held at Schloss Mickeln, Dusseldorf, 16-21 November 1987.

  626. Arf biography
    • Cahit Arf's interest in mathematics was stimulated during his school years in Izmir by a teacher who encouraged him to solve problems in euclidean geometry.
    • At an earlier conference on Rings and Geometry held in Istanbul in 1984, Arf had presented a paper The advantage of geometric concepts in mathematics.

  627. Anosov biography
    • In the 1970s Anosov published papers such as Existence of smooth ergodic flows on smooth manifolds (Russian) (1974) and was an invited speaker at the International Congress of Mathematicians at Vancouver in 1974 when he gave the lecture Geodesics and Finsler geometry.
    • His new approach to the problem of the existence of closed geodesies in Riemannian geometry led to him establishing important results in Some homotopies in a space of closed curves (Russian) (1980).

  628. Cavalieri biography
    • He taught Cavalieri geometry and introduced him to the ideas of Galileo.
    • In his letter, Galileo said of Cavalieri, "few, if any, since Archimedes, have delved as far and as deep into the science of geometry." In support of his application to the Bologna position, Cavalieri sent Marsili his geometry manuscript and a small treatise on conic sections and their applications in optics.
    • Cavalieri's geometry manuscript which had been a factor in his appointment to Bologna, although completed in December 1627, was not published until 1635.
    • Guldin was a "classicist" geometer, steeped in the idea of explicit construction, sceptical of considerations of infinity in the domain of geometry, and wary of the risk of ending up with an atomistic theory of the continuum.
    • I should not dare affirm that this geometry of indivisibles is actually a new discovery.
    • Whatever it was, it is certain that this geometry represents a marvellous economy of labour in the demonstrations and establishes innumerable, almost inscrutable, theorems by means of brief, direct, and affirmative demonstrations, which the doctrine of the ancients was incapable of.
    • The geometry of indivisibles was indeed, in the mathematical briar bush, the so-called royal road, and one that Cavalieri first opened and laid out for the public as a device of marvellous invention.

  629. Thomason biography
    • The author pushes the applications of stable homotopy and homotopical algebra to algebraic K-theory and algebraic geometry further than anyone else and his methods have exerted considerable influence on other workers in the field.
    • Few have had the simultaneous grasp of topology, algebraic geometry and K-theory that Thomason did.

  630. Bonferroni biography
    • After the war he was appointed to the post of assistant professor at the Turin Polytechnic, where he taught analysis, geometry and mechanics.
    • Later in the same lecture he stated that, "subjective probability is not amenable to mathematical analysis." In [Bollettino dell\'Unione Matematica Italiana 15 (1960), 570-574.',11)">11] Pagni gives a list of Bonferroni's publications under three main headings: (1) actuarial mathematics (16 articles and 1 book); (2) probability and statistical mathematics (30 articles and 1 book); and (3) analysis, geometry and rational mechanics (13 articles).

  631. Kolmogorov biography
    • His monograph on probability theory Grundbegriffe der Wahrscheinlichkeitsrechnung published in 1933 built up probability theory in a rigorous way from fundamental axioms in a way comparable with Euclid's treatment of geometry.
    • his ideas in set-theoretic topology, approximation theory, the theory of turbulent flow, functional analysis, the foundations of geometry, and the history and methodology of mathematics.

  632. Bortolotti biography
    • Bortolotti was Dean of the Faculty at Modena in 1913-19, then he was appointed professor of geometry at the University of Bologna where he remained for the rest of his life, retiring in 1936.
    • There is certainly a strong argument supporting the view that if the work had been widely known at the time it was written, the fusion of algebra and geometry which was achieved by Descartes might have occurred much earlier.

  633. Le Tenneur biography
    • Le Tenneur clearly was trying to argue against the notions current at the time on using algebra to study geometry.
    • He wished geometry to be Greek style, not in the style of Descartes and his followers.

  634. Frenet biography
    • At Toulouse Frenet undertook research in geometry and he wrote a doctoral thesis there which he submitted in 1847.
    • As this frame moves along the curve, we can look at its rate of change to determine how the curve turns and twists, two ideas that actually describe the whole geometry of the curve.

  635. Cartier biography
    • The mathematics I was taught was classical geometry, in the uncultivated, synthetic way.
    • He continued to study at the Ecole Normale for his doctorate on algebraic geometry which he defended in 1958.

  636. Rudio biography
    • It was only in the year before Rudio entered the Polytechnikum that Geiser had been appointed to a full professorship of higher mathematics and synthetic geometry, with special responsibility for teaching mathematics to engineering students and to mathematics students.
    • Rudio worked on group theory, algebra and geometry.

  637. Scholtz biography
    • Scholtz's field of research was projective geometry and theory of determinants.
    • Mertens, Pasch, Caspary, Muller, Szabo and others) who put it to good use in other questions of geometry.

  638. Srinivasan biography
    • She had a good teacher who let her appreciate the beauty of Euclidean geometry and also gave her a good grounding in understanding the concept of proof.
    • Her lecture The invasion of geometry into finite group theory was delivered at Louisville, Kentucky.

  639. Binet biography
    • He became a teacher at Ecole Polytechnique in 1807 and, one year later, he was appointed to assist the professor of applied analysis and descriptive geometry.
    • In 1814 he was appointed examiner of descriptive geometry then, in 1815, he was appointed to succeed Poisson in mechanics.

  640. Fuller biography
    • He examined a vectorial system of geometry, Energetic- Synergetic geometry, based on the tetrahedron which provides maximum strength with minimum structure.

  641. Levi biography
    • He also published two geometry books, one being a commentary and introduction to the first five books of Euclid, but not presented axiomatically.
    • The other is the Science of Geometry of which only a fragment has survived.

  642. Wussing biography
    • The existence of two additional roots of abstract group theory has been obscured mainly by the fact that the group-theoretic modes of thought in number theory and geometry remained implicit until the end of the middle third of the nineteenth century; they made no use of the term 'group' and, in the beginning, had virtually no link to the contemporary development of the theory of permutation groups.
    • His main thesis, ably defended and well documented, is that the roots of the abstract notion of group do not lie, as frequently assumed, only in the theory of algebraic equations, but that they are also to be found in the geometry and the theory of numbers of the end of the 18th and the first half of the 19th centuries.

  643. Milnor biography
    • The Todd polynomials were first studied in algebraic geometry and it is surprising that they play this fundamental role in classification of manifolds.
    • The article [Topological methods in modern mathematics (Houston, TX, 1993), 31-43.',5)">5] looks at nine papers which Milnor had written on differential geometry.

  644. Hardy biography
    • Deeply unhappy at Cambridge, Hardy took the opportunity to leave in 1919 when he was appointed as Savilian professor of geometry at Oxford.
    • Savilian Geometry Professor1920 .

  645. Marchenko biography
    • In these classes solutions are contained that can be obtained by the inverse problem method and by the methods of algebraic geometry, and also solutions that do not reduce to these methods.
    • In addition to his mathematical research we must mention his service as an editor of several journals: he was editor-in-chief of The theory of functions, functional analysis, and their applications for nearly thirty years; an honorary editor of Mathematical Physics, Analysis, Geometry; an editor of the Proceedings of the Ukrainian Academy of Sciences; and an editor of Inverse Problems.

  646. Merrifield biography
    • This was on the strength of some excellent mathematical papers on the calculation of elliptic functions, the first of which was The geometry of the elliptic equation which he published in 1858.
    • He was also active in promoting improvements in the teaching of geometry being a member of the Association for the Improvement of Geometrical Teaching.

  647. Weyl biography
    • It united analysis, geometry and topology, making rigorous the geometric function theory developed by Riemann.
    • In 1917 Weyl gave another course presenting an innovative approach to relativity through differential geometry.

  648. Hudde biography
    • Van Schooten had established a vigorous research school in Leiden which included Hudde, and this school was one of the main reasons for the rapid development of Cartesian geometry in the mid 17th century.
    • From 1654 until 1663 Hudde worked on mathematics as part of van Schooten geometry research group at Leiden.

  649. Al-Banna biography
    • He studied geometry in general, and Euclid's Elements in particular.
    • At the university in Fez Al-Banna taught all branches of mathematics, which at this time included arithmetic, algebra, geometry and astronomy.

  650. Chisholm Young biography
    • They lived for a year in Italy where they undertook research in geometry but did not find it particularly exciting.
    • Their joint work A First Book of Geometry, which was on paper folding for children, was published in 1905.

  651. Theaetetus biography
    • 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.
    • For Theaetetus had distinguished square roots commensurable in length from those which are incommensurable, and who divided the more generally known irrational lines according to the different means, assigning the medial line to geometry, the binomial to arithmetic and the apotome to harmony, as stated by Eudemus..

  652. Couturat biography
    • In fact the Couturat-Russell correspondence began in 1897, the first letter being from Couturat to Russell concerning the Russell's An essay in the foundations of geometry.
    • The topics covered in the correspondence include: the foundations of geometry, extension versus intension in logic, the Russell paradox, the axiom of choice, the controversies with Poincare, logic, Leibniz, Peano, Kant, arithmetical induction, mathematical existence, politics, international language, and some personal matters.

  653. Bartik biography
    • In other courses such as Analytic Geometry, Trigonometry and Physics, the only students besides Jean were young men undergoing naval training.
    • Special arrangements were made, two retired teachers were brought back to teach courses, and Modern Geometry and the Theory of Numbers were put on just for her.

  654. Hahn biography
    • The paper is a classic of the early set-theoretical geometry.
    • My reaction is very different: Fractal geometry demonstrates that Hahn was dead wrong.

  655. Lame biography
    • Lame was elected to the Academie des Sciences in 1843 when Louis Puissant died leaving a vacancy in the geometry section.
    • He also did important work on differential geometry and, in another contribution to number theory, he showed that the number of divisions in the Euclidean algorithm never exceeds five times the number of digits in the smaller number.

  656. Conway biography
    • In 1986 Conway left Cambridge after accepting appointment to the John von Neumann Chair of Mathematics at Princeton in the United States where much of his work has focused on geometry, in particular studying the symmetries of crystal lattices.
    • This is a beautiful and fascinating book on the geometry and arithmetic of the quaternion algebra and the octonion algebra.

  657. Guldin biography
    • Concerning pure mathematics, arithmetic is described as the science of discrete quantity, and geometry as the science of continuous quantity.
    • Guldin was a "classicist" geometer, steeped in the idea of explicit construction, sceptical of considerations of infinity in the domain of geometry, and wary of the risk of ending up with an atomistic theory of the continuum.

  658. D'Ocagne biography
    • This work is an expansion of the course in pure geometry given by the author at the Ecole des Ponts et Chaussees.
    • In 1912 he was appointed professor of geometry at the Ecole Polytechnique.

  659. Stieltjes biography
    • From September to December 1883 Stieltjes lectured on analytical geometry and on descriptive geometry.

  660. Ricci-Curbastro biography
    • He changed area somewhat to undertake research in differential geometry and was the inventor of the absolute differential calculus between 1884 and 1894.
    • In the paper, applications are given by Ricci-Curbastro and Levi-Civita to the classification of the quadratic forms of differentials and there are other analytic applications; they give applications to geometry including the theory of surfaces and groups of motions; and mechanical applications including dynamics and solutions to Lagrange's equations.

  661. Quillen biography
    • Quillen approached the Adams conjecture with two quite distinct approaches, namely using techniques from algebraic geometry and also using techniques from the modular representation theory of groups .
    • Jacek Brodzki lectured on Analysis and geometry on discrete groups, Mathai Varghese lectured on T-duality and non-commutative geometry, Joachim Cuntz lectured on K-theory for locally convex algebras, and Eric Friedlander closed the proceedings with the lecture Dan and me: looking back at some of Dan's remarkable mathematics.

  662. Brashman biography
    • For example, he published the textbook Course in Analytical Geometry (Russian) in 1838.
    • Brashman wrote one of the best analytic geometry texts of his time, for which the Russian Academy of Sciences awarded him the entire Demidov Prize for 1836.

  663. Kendall Maurice biography
    • Other monographs are A course in the geometry of n dimensions (1961) which aims to present that part of the theory of n-dimensional geometry which has statistical applications, and to sketch very briefly what those applications are.

  664. Tucker Albert biography
    • A few weeks into my Euclidean geometry course, the Principal decided to give us a test.
    • However, his first research paper came about because he found an error in Luther Eisenhart's course on Riemannian Geometry which he attended during his year as an Instructor.

  665. Feigl biography
    • Feigl worked on geometry, in particular the foundations of geometry and topology.

  666. Somov biography
    • Among his works (all written in Russian) were Analytic theory of the undulatory motion of the ether (1847), Foundations of the theory of elliptical functions (1850), Course in differential calculus (1852), Analytic geometry (1857), Elementary algebra (1860), Descriptive geometry (1862) and the two volume treatise Rational mechanics (1872-74).

  667. Ruffini biography
    • Among his teachers of mathematics at Modena were Luigi Fantini, who taught Ruffini geometry, and Paolo Cassiani, who taught him calculus.
    • Fantini, who had taught Ruffini geometry when he was an undergraduate, found his eyesight deteriorating and in 1791 he had to resign his post at Modena.

  668. Dantzig George biography
    • This continued at Central High School where he became fascinated by geometry.
    • gave me thousands of geometry problems while I was still in high school.

  669. Ball Robert biography
    • In 1892 John Couch Adams, the Lowndean Professor of Astronomy and Geometry at Cambridge and the director of the Cambridge Observatory, died.
    • Ball applied for the vacant position and was appointed as Lowndean Professor of Astronomy and Geometry but disputes with the university meant that he had to wait a year before he was appointed director of the Cambridge Observatory.

  670. Carlitz biography
    • But his publications extend beyond these areas to include algebraic geometry, commutative rings and algebras, finite differences, geometry, linear algebra, and special functions.

  671. Boruvka biography
    • Boruvka then became Čech's research assistant and Čech interested Boruvka in differential geometry.
    • Under the influence of Eduard Čech and Elie Cartan he worked on differential geometry, then he became interested in algebra, and undertook research on groups and groupoids (algebraic systems in which the associative law does not hold).

  672. Tannery Paul biography
    • 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.
    • 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.

  673. Ollerenshaw biography
    • When she finally sat the entrance exams, Kathleen found the algebra-geometry paper set by Oxford to be very easy [1]:- .
    • It was moreover a matter of geometry -- pure mathematics -- a nice problem that had a neat and successful solution.

  674. Cosserat biography
    • In mathematics, we have already noted his early work on geometry.
    • In particular in 1905 he published works by Karl Mikhailovich Peterson on curves, surfaces and their deformations, and work on differential geometry related to questions from the theory of elasticity.

  675. Picone biography
    • In his third year at the Institute he was taught mathematics by his father for a short while, but in his third and fourth year his main mathematics teacher was Michele de Franchis (1875-1946) who taught him algebra and analytic geometry, taking him to the second year university level.
    • three different topics: (i) boundary value problems for second order linear ordinary differential equations, for which Picone developed his well-known "identity", and the subsequent extension of these results to second order linear partial differential equations of elliptic and parabolic types; (ii) partial differential equations of hyperbolic type (in two independent variables), for which Picone studied problems generalizing Goursat's problem; (iii) research on differential geometry in the direction set by L Bianchi, with particular attention to the characterization of the ds2 of a ruling and to W congruences.

  676. Wilf biography
    • The title of the thesis was "The transmission of neutrons in multilayered slab geometry." It solved the transport equation in multilayered geometry by regarding each homogeneous layer as a little black box with prescribed inputs and outputs (which point of view was Jerry's hallmark), and it wired them together by representing each by a matrix.

  677. Hay biography
    • I found the logical aspects of mathematics much more congenial than the numerical aspects, and when I showed aptitude for this, Mr Rosenbaum suggested I read up on non-Euclidean geometry, to put the subject in a new perspective.
    • He had me get Wolfe's book on non-Euclidean geometry, which I found fascinating and which ultimately was the basis of the project I wrote as a senior for the Westinghouse Science Talent Search, in which I won third prize.

  678. Davidov biography
    • Of these, the geometry and algebra textbooks enjoyed special success and were republished many times.
    • Through the next half century the geometry text underwent thirty-nine editions and the algebra text twenty-four.

  679. Robins biography
    • Pemberton soon had Robins reading, in English translations, the classic Greek texts on geometry by Apollonius, Archimedes and by Pappus.
    • Robins loved this geometrical approach to mathematics and retained a preference for geometry over algebra or analysis throughout his life.

  680. Prufer biography
    • After his death his lectures notes on projective geometry were collected and published as the 314 page book Projektive Geometrie (1933).

  681. Keldysh Lyudmila biography
    • In August 2004 the conference Geometric Topology, Discrete Geometry and Set Theory was held in Moscow to mark the centenary of her birth.

  682. Thompson Abigail biography
    • Despite the recent influence of algebraic and geometric techniques such as quantum groups, hyperbolic geometry, and algebraic varieties in the study of 3-manifolds, most of the fundamental arguments involve or can be reduced to cutting and pasting surfaces and manifolds and studying their possible combinatorial configurations.

  683. Dechales biography
    • Topics covered in this wide ranging work included practical geometry, mechanics, statics, magnetism and optics as well as topics outwith the usual topics of mathematics such as geography, architecture, astronomy, natural philosophy and music.

  684. Wilkins biography
    • Wilkins also wrote on mechanical devices, publishing Mathematical Magick, or the wonders that may be performed by mechanical geometry in 1648.

  685. Vallee Poussin biography
    • Volume 2 covered multiple integrals, differential equations, and differential geometry.

  686. Carslaw biography
    • Other topics to interest Carslaw throughout his career, which we have not touched on above, included an interest in non-euclidean geometry, Green's functions and the history of Napier's logarithms.

  687. Tacquet biography
    • This major contribution contained works on astronomy, spherical trigonometry, practical geometry, and fortifications.

  688. Gutzmer biography
    • Highly gifted as a teacher, showing infectious enthusiasm for his topic, Gutzmer taught courses on a wide variety of topics including differential and integral calculus, and analytic geometry at lower level.

  689. Bryant biography
    • In 1863, when Sophie was thirteen years old, her father was appointed to the Chair of Geometry in the University of London, and the family moved to London.

  690. Dinghas biography
    • His work is in many areas of mathematics including differential equations, functions of a complex variable, functions of several complex variables, measure theory and differential geometry.

  691. Liouville biography
    • Liouville contributed to differential geometry studying conformal transformations.

  692. Chatelet biography
    • She was laid on a quarto book of geometry.

  693. Diocles biography
    • On burning mirrors is a collection of sixteen propositions in geometry mostly proving results on conics.

  694. Borcherds biography
    • The first such formulae were found in the one-dimensional case by Euler and Jacobi, and the conventional wisdom in algebraic geometry was that such product formulae could not exist in higher dimensions.

  695. Kalman biography
    • Not only have they led to eminently useful developments, such as the Kalman-Bucy filter, but they have contributed to the rapid progress of systems theory, which today draws upon mathematics ranging from differential equations to algebraic geometry.

  696. Mayer Adolph biography
    • In the following years he taught Differential and Integral Calculus, Theory of Definite Integrals, Some chapters from mechanics and the calculus of variations, Higher Algebra, Differential Equation of Mechanics and the Calculus of Variations, Analytic Geometry, and many more courses of a similar type.

  697. Hartogs biography
    • Daniele Struppa writes in [Geometry Seminars, 1987-1988, Bologna, 1987-1988 (Univ.

  698. Rogosinski biography
    • There was little science and the mathematics course contained no calculus but plenty of geometry.

  699. Cocker biography
    • And after that, Geometry command.

  700. Dixon Arthur biography
    • His mathematics, very much in the English tradition of Cayley, studied applications of algebra to geometry, elliptic functions and hyperelliptic functions.

  701. Guinand biography
    • Guinand worked on summation formulae and prime numbers, the Riemann zeta function, general Fourier type transformations, geometry and some papers on a variety of topics such as computing, air navigation, calculus of variations, the binomial theorem, determinants and special functions.

  702. Huygens biography
    • Tutored at home by private teachers until he was 16 years old, Christiaan learned geometry, how to make mechanical models and social skills such as playing the lute.

  703. Mordell biography
    • Together with Davenport and Mahler, Mordell initiated great advances in the geometry of numbers while he held the Chair of Pure Mathematics at Manchester.

  704. Vladimirov biography
    • Hermann Minkowski had initiated a study of the geometry of numbers in 1890 and, over the next twenty years, he studied many problems including packing problems for convex bodies.

  705. Iyanaga biography
    • He did publish a number of papers, however, which arose through the various courses such as algebraic topology, functional analysis, and geometry, which he taught.

  706. Delaunay biography
    • From 1845 to 1850 he taught courses at the Ecole des Mines; these were descriptive geometry, stereotomy, mechanical drawing, analytical mechanics, and elementary physics.

  707. Insolera biography
    • Following this, he won a scholarship to enable him to continue undertaking research at the University of Rome, where he was supervised by Guido Castelnuovo, who held the Chair of Analytic and Projective Geometry, and Vito Volterra who held the Chair of Mathematical Physics.

  708. Jarnik biography
    • During the decade 1939-49 he wrote a series of papers dealing with the geometry of numbers, in particular dealing with Minkowski's inequality for convex bodies.

  709. Escher biography
    • When, later, in stereometry [solid geometry], an appeal was made to my imagination, it went a bit better, but in school I never excelled in that subject.

  710. Rouche biography
    • Another of his famous geometry texts was written jointly with Ch De Comberousse, namely Traite de geometrie.

  711. Korteweg biography
    • Korteweg showed a similar versatility in his teaching, with his usual courses being analytic and projective geometry, mechanics, astronomy and probability theory.

  712. Hua biography
    • Thus Hua became interested in matrix algebra and wrote several substantial papers on the geometry of matrices.

  713. Cusa biography
    • He was interested in geometry and logic and had clearly made a study of at least parts of Euclid's Elements and works of Thomas Bradwardine and Campanus of Novara.

  714. Laplace biography
    • Imparting geometry, trigonometry, elementary analysis, and statics to adolescent cadets of good family, average attainment, and no commitment to the subjects afforded little stimulus, but the post did permit Laplace to stay in Paris.

  715. Khayyam biography
    • In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-euclidean geometry, although this was not his intention.

  716. Herschel Caroline biography
    • Slowly Caroline turned more and more towards helping William with his astronomical activities while he continued to teach her algebra, geometry and trigonometry.

  717. Borda biography
    • In 1753, at the age of twenty, Borda produced his first memoir on geometry and sent it to d'Alembert.

  718. Lorgna biography
    • We indicate below the titles of some of his works but let us record here that, among the pure mathematical topics he worked on, was geometry, convergence of series and algebraic equations.

  719. Aida biography
    • It is a book of geometry problems, developing formulae for ellipses, spheres, circles etc.

  720. Nalli biography
    • She was awarded her laurea on 10 January 1910 after submitting a thesis on algebraic geometry.

  721. Whiteside biography
    • This first volume appeared in 1964 with the second volume, containing English translations of Newton's Lucasian lectures on algebra and analytical geometry, his enumeration of cubics, and his tract on finite differences, appearing in 1967.

  722. Mackenzie biography
    • At Honours level: Natural Philosophy, Mathematics, Final Natural Philosophy, Final Mathematics, Calculus, General Analysis, Heat, Electricity I and II, General Physics, Higher Algebra and Geometry.

  723. Fibonacci biography
    • It contains a large collection of geometry problems arranged into eight chapters with theorems based on Euclid's Elements and Euclid's On Divisions.

  724. Heawood biography
    • He also wrote five papers and 23 notes for the Mathematical Gazette on a variety of mathematical topics but perhaps more on geometry than any other topic.

  725. Luzin biography
    • In 1931 Luzin himself turned to a new area when he began to study differential equations and their application to geometry and to control theory.

  726. Al-Biruni biography
    • These include: theoretical and practical arithmetic, summation of series, combinatorial analysis, the rule of three, irrational numbers, ratio theory, algebraic definitions, method of solving algebraic equations, geometry, Archimedes' theorems, trisection of the angle and other problems which cannot be solved with ruler and compass alone, conic sections, stereometry, stereographic projection, trigonometry, the sine theorem in the plane, and solving spherical triangles.

  727. Turner John biography
    • For example he contributed the paper On the teaching of geometry to the meeting of the Society on Friday 8 December 1905.

  728. Taniyama biography
    • During his undergraduate years he read Claude Chevalley's Theory of Lie groups and Andre Weil's Foundations of algebraic geometry as well as two other books by Weil on algebraic curves and abelian varieties.

  729. Kolchin biography
    • Although the articles in this volume are in the main devoted to commutative algebra, linear algebraic group theory, and differential algebra, the diversity of subjects covered - complex analysis, algebraic K-theory, logic, stochastic matrices, differential geometry, ..

  730. Wiltheiss biography
    • Wiltheiss predominantly lectured to students who were beginning their studies, on topics such as differential and integral calculus, geometry, and algebra.

  731. Redei biography
    • In 1941 Redei was appointed to the Chair of Geometry in Szeged but later he was appointed to the Chair of Algebra and Number Theory.

  732. Al-Nayrizi biography
    • A later work, written in the 13th century, described al-Nayrizi as both a distinguished astronomer and as a leading expert in geometry.

  733. Stern biography
    • During his 55 years at the University of Gottingen, Stern lectured on a wide variety of topics, including algebraic analysis, analytic geometry, differential and integral calculus, variational calculus, mechanics, popular astronomy and, of course, number theory.

  734. Watson Henry biography
    • He wrote The elements of plane and solid geometry (1871), Treatise on the kinetic theory of gases (1876).

  735. Olds biography
    • Most regular work emphasized calculus or geometry; the discrete mathematics I learned with the outside readings was what was required.

  736. Cotes biography
    • a new sort of construction in geometry which appear to me very easy, simple and general.

  737. Strassen biography
    • He is considered the founding father of algebraic complexity theory with his work on the degree bound, connecting complexity to algebraic geometry.

  738. Agnesi biography
    • As Truesdell writes in [Geometry Seminars, 1988-1991 (Bologna, 1991), 145-167.',16)">16]:- .

  739. Banach biography
    • As well as continuing to produce a stream of important papers, he wrote arithmetic, geometry and algebra texts for high schools.

  740. Russell biography
    • A fourth volume on geometry was planned but never completed.

  741. Stifel biography
    • He obtained a parish at Bruck, near Wittenberg but left to go to Jena where he began lecturing at the University on mathematics, in particular on arithmetic and geometry.

  742. Banneker biography
    • and only a few semesters of elementary schooling in his childhood, Banneker taught himself the algebra, geometry, logarithms, trigonometry, and astronomy needed to become an astronomer.

  743. Krawtchouk biography
    • Other areas he wrote on included algebra (where among other topics he studied the theory of permutation matrices), geometry, mathematical and numerical analysis, probability theory and mathematical statistics.

  744. Lichtenstein biography
    • Since he presupposes on the part of his readers a familiarity with the theory of the Newtonian potential function and of integral equations, together with a good grasp of other branches of analysis, geometry, and celestial mechanics, the book is not easy to read, but the results are so significant that it should be carefully studied by everyone who is seriously interested in the mathematical treatment of the problem of figures of equilibrium of rotating fluid bodies.

  745. Kirkman biography
    • He also at this time examined certain questions in geometry.

  746. Krasnosel'skii biography
    • In August 1946 Krasnosel'skii returned to Kiev where he was able to continue his studies, but he also worked, first as an lecturer in descriptive geometry at the Kiev Highway Institute and later as a junior scientist at the Institute of Mathematics of the Ukrainian Academy of Sciences.

  747. Ruan Yuan biography
    • The school was set up with an innovative curriculum, and literature, astronomy, geometry and mathematics were taught there.

  748. Lerch biography
    • He also wrote on geometry and numerical methods.

  749. Stein biography
    • For more than a century there has been a significant and fruitful interaction between Fourier analysis, complex function theory, partial differential equations, real analysis, as well as ideas from other disciplines such as geometry and analytic number theory, etc.

  750. Cesari biography
    • Cesari lays great stress on the connection of theory to applications and devotes two chapters to illustrative examples from geometry, mechanics, aerospace science, economics and other fields.

  751. Specker biography
    • Back at ETH after the vacation ended, he attended a projective geometry course given by Kollros and also attended lectures by Paul Finsler at the University of Zurich.

  752. Piatetski-Shapiro biography
    • For almost 40 years Professor Ilya Piatetski-Shapiro has been making major contributions in mathematics by solving outstanding open problems and by introducing new ideas in the theory of automorphic functions and its connections with number theory, algebraic geometry and infinite dimensional representations of Lie groups.

  753. Galileo biography
    • At Padua his duties were mainly to teach Euclid's geometry and standard (geocentric) astronomy to medical students, who would need to know some astronomy in order to make use of astrology in their medical practice.

  754. Luchins biography
    • Despite having to interrupt her doctoral studies, Luchins began to publish a series of papers with her husband including: Towards Intrinsic Methods in Testing (1946), A Structural Approach to the Teaching of the Concept of Area in Intuitive Geometry (1947), The Satiation Theory of Figural After-Effects and Gestalt Principles of Perception (1953), and Variables and Functions (1954).

  755. Nicomedes biography
    • prided himself inordinately on his discovery of this curve, contrasting it with Eratosthenes's mechanism for finding any number of mean proportionals, to which he objected formally and at length on the ground that it was impracticable and entirely outside the spirit of geometry.

  756. Brill biography
    • He was taught mathematics by his uncle Christian Wiener, attending his course on descriptive geometry, and by Alfred Clebsch who taught the course on mechanics.
    • Their joint work was the first systematic use of algebraic techniques, which today form part of commutative algebra, in the study of geometry.
    • He contributed to the study of algebraic geometry, trying to bring the rigour of algebra into the study of curves.
    • As is well known Brill together with Max Noether are the twin-stars of German geometricians who did the important pioneer work concerning the geometry on algebraic curves.
    • One may therefore expect that a treatise on the subject by such a man should contain much that is of value and of fundamental importance for the student of geometry and, in a wider sense, for the mathematician in general.
    • In other words, the student will have a fairly good start in algebraic geometry after he has mastered Brill's lectures.
    • Brill, one of the remaining representatives of an important period of geometric development, has given enough of an algebraic, function theoretic treatment of algebraic geometry to stimulate the student for further reading and research in this direction.

  757. Fine Henry biography
    • Among the elementary texts he wrote are Number system of algebra treated theoretically and historically (1891), A college algebra (1905), Coordinate geometry (1909), and Calculus (1927).

  758. Stevin biography
    • In Problemata geometrica (1583) Stevin presented geometry based largely on Euclid and Archimedes but the problems which he studied show that he was also influenced by Durer.

  759. Castelli biography
    • One day he was approached by a group of literate gentlemen who wanted to be taught the principles of geometry.

  760. Al-Jayyani biography
    • There are five magnitudes that, according to al-Jayyani, are used in geometry; number, line, surface, angle, and solid.

  761. Buffon biography
    • He corresponded with Gabriel Cramer on mechanics, geometry, probability, number theory and the differential and integral calculus.

  762. Atkinson biography
    • At this time his ability in mathematics became evident and he topped his class in geometry, algebra, and arithmetic.

  763. Clavius biography
    • The second volume contains his works on geometry and algebra, while the third volume contains his commentary on the "Sphaera" of Johannes de Sacrobosco (also known as John of Holywood) from which we have quoted regarding the two solar eclipses, and his treatise on the astrolabe.

  764. Schwartz Jacob biography
    • This clear, non-technical treatment makes relativity more accessible than ever before, requiring only a background in high-school geometry.

  765. Glaisher biography
    • To this class belong the theories of magnitude and position, the former, including all that relates to quantity, whether discrete or continuous, and the latter to all branches of geometry.

  766. Lagrange biography
    • At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.

  767. Lewy biography
    • Among the first papers he published after emigrating to the United States were A priori limitations for solutions of Monge-Ampere equations (two papers, the first in 1935, the second two years later), and On differential geometry in the large : Minkowski's problem (1938).

  768. Lasker biography
    • Finally let us comment that Lasker's results on the decomposition of ideals into primary ideals was the foundation on which Emmy Noether built an abstract theory which developed ring theory into a major mathematical topic and provided the foundations of modern algebraic geometry.

  769. Rademacher biography
    • In addition to the significant contributions to real analysis and measure theory which we have briefly mentioned above, he contributed to complex analysis, geometry, and numerical analysis.

  770. Bring biography
    • There are eight volumes of his hand written mathematical work on various questions in algebra, geometry, analysis and astronomy preserved in the library at Lund.

  771. Van Vleck biography
    • He gave his retiring presidential address on The role of the point-set theory in geometry and dynamics.

  772. Panini biography
    • This may be brought out by comparing the grammar of Sanskrit with the geometry of Euclid - a particularly apposite comparison since, whereas mathematics grew out of philosophy in ancient Greece, it was ..

  773. Gnedenko biography
    • For example he even manages to discuss non-euclidean geometry and Lobachevsky's contributions without even mentioning Bolyai.

  774. Selten biography
    • During these walks I occupied my mind with problems of elementary geometry and algebra.

  775. Maskelyne biography
    • Great mathematicians have become astronomers from the facility mathematics gave them in the attainment of astronomy; but here the love of astronomy was the motive of application to mathematics without which our astronomer soon found he could not make the progress he wished in his favourite science; in a few months, without any assistance he made himself master of the elements of geometry and algebra.
    • To understand Arithmetic, Geometry, Algebra, Plane and Spherical Trigonometry, and Logarithms; to have a good eye and good ears, be well grown, and have a good constitution to enable him to apply several hours in the day to calculation, and to get up to the observations that happen at late hours in the night.
    • Although not greatly interested in original mathematical ideas, he did prove some new geometry theorems.

  776. Cohen biography
    • I never heard him lecture on set theory, but rather on algebraic geometry and p-adic fields.

  777. Infeld biography
    • In 1932 he visited Leipzig where he worked with Heisenberg and van der Waerden, with whom he wrote the paper Spinors in Riemannian geometry.

  778. Meyer Paul-Andre biography
    • Soon he was ahead of the rest of his class and was fortunate to have an excellent teacher who taught him to compute square roots and cube roots as well as giving him a liking for geometry.

  779. Schmidt F-K biography
    • Heffter, an expert on differential equations, complex analysis and analytic geometry, had been appointed to Freiburg in 1911 having previously been a full professor at RWTH Aachen and Kiel.

  780. Trahtman biography
    • At Bar-Ilan University, Trahtman taught courses in discrete mathematics, theory of sets, algebra, analytical geometry, mathematical logic, finite automata, formal languages, rings and modules, and differential equations.

  781. Bolzano biography
    • While pursuing his theological studies he prepared a doctoral thesis on geometry.

  782. Keldysh Mstislav biography
    • Sergei Novikov, in the interview [Geometry, topology, and mathematical physics (Amer.

  783. Alexiewicz biography
    • He wrote two important Polish language texts: Differential geometry (1966, second edition 1970) and Functional analysis (1969).

  784. Posidonius biography
    • 'theorem' and 'problem', and subjects belonging to elementary geometry.

  785. Adamson biography
    • The courses he took for his finals were the compulsory papers in Geometry, Algebra, Analysis, Statics, and Dynamics, along with the two Special Topic papers Fluid Mechanics and Algebra.

  786. Huntington biography
    • He gave axioms for a group, an abelian group, a boolean algebra, geometry, the real number field, and the complex numbers.

  787. Castillon biography
    • Throughout his mathematical work there is a preference for synthetic, as opposed to analytic, geometry, which is perhaps a reflection of his preoccupation with Newton's mathematics.

  788. Moore Jonas biography
    • Moore wrote the sections on arithmetic, geometry, trigonometry and cosmography while the sections on algebra, Euclid and navigation were written by Perkins.

  789. Biot biography
    • He made advances in astronomy, elasticity, electricity and magnetism, heat and optics on the applied side while, in pure mathematics, he also did important work in geometry.

  790. Forsythe biography
    • The subject matter of the books listed is mathematics, pure and applied, including tables beyond the most elementary, but excluding descriptive geometry.

  791. Peacock biography
    • In 1836 he was appointed Lowndean professor of astronomy and geometry at Cambridge and three years later was appointed dean of Ely cathedral, spending the last 20 years of his life there.

  792. Whewell biography
    • He also published The Mechanical Euclid, containing the Elements of Mechanics and Hydrostatics demonstrated after the Manner of the Elements of Geometry (1837) where he took a completely geometrical approach.

  793. Hammersley biography
    • This covered plenty of Euclidean geometry (including such topics as the nine-point circle) and algebra (Newton's identities for roots of polynomials) and trigonometry (identities governing angles of a triangle, circumcircle, incircle, etc), but no calculus.

  794. Clunie biography
    • In his final two honours years he had studied the five compulsory topics, Geometry, Algebra, Analysis, Statics, and Dynamics.

  795. Gherard biography
    • The most important, however, were on astronomy, geometry and other branches of mathematics.

  796. Bernays biography
    • In 1899 Hilbert had written Grundlagen der Geometrie and, in 1956, Bernays revised this work on the foundations of geometry.

  797. Blum biography
    • Especially striking is the interplay of various mathematical disciplines such as algebraic number theory, algebraic geometry, logic, and numerical analysis, to mention a few.

  798. Newcomb biography
    • He also wrote on non-euclidean geometry and Cayley commented on one of his theorems saying:- .

  799. Shafarevich biography
    • More recently he made important advances to algebraic geometry.

  800. Patodi biography
    • Patodi's first paper Curvature and the eigenforms of the Laplace operator was part of his thesis and the contents of this paper are described in [Geometry and analysis : papers dedicated to the memory of V K Patodi (Bangalore, 1980), i-iii.',2)">2]:- .

  801. Fefferman biography
    • Professor Charles Fefferman's contributions and ideas have had an impact on the development of modern analysis, differential equations, mathematical physics and geometry, with his most recent work including his sharp (computable) solution of the Whitney extension problem.

  802. Allardice biography
    • For example at the meeting held on Friday 14 March 1884 he read a paper on the geometry of the spherical surface; at the meeting on Friday 8 January 1886 he discussed a problem of symmetry in an algebraical function; on 11 February 1887 he communicated a note on a theorem in algebra; on 11 January 1889 he contributed a note on a formula in quaternions; on 13 December 1889 he discussed some theorems in the theory of numbers; on 13 November 1891 his paper Barycentric Calculus of Mobius was read by John Alison; on 14 December 1901 his paper Four Circles Touching a Common Circle was communicated to the meeting by Mr George Duthie; and on 13 January 1911 his paper On the envelope of the directrices of a system of similar conics through three points was communicated by E D Williamson.

  803. Dini biography
    • Dini progressed quickly in his career at the University of Pisa, being appointed to Betti's chair of analysis and higher geometry in 1871.

  804. Lyapin biography
    • Analysis, algebra, geometry, and topology being rich in examples of the latter, their abstract theory deserves recognition.

  805. Dionis biography
    • He published a treatise on the analytic geometry of plane curves in 1756 Traite des courbes algebrique.

  806. Neumann Hanna biography
    • Her introduction to higher mathematics was a course given by Georg Feigl and, in addition, she was taught analytic and projective geometry by Ludwig Bieberbach, differential and integral calculus by Erhard Schmidt, and number theory by Issai Schur.

  807. De L'Hopital biography
    • He studied geometry even in his tent.

  808. Behnke biography
    • Also Die Autonomie der Geometrie (1971) which considers the way that geometry is taught in schools.

  809. Cherry biography
    • My love of camping and mountaineering connects in one direction with 'do it yourself' and in another direction - via the shapes of hills - with geometry and mathematics.

  810. Konig Denes biography
    • Even if a collection in the Hungarian language exists, it is much more elementary than the work here, and only includes rather commonplace arithmetic and geometry puzzles.

  811. Christoffel biography
    • Christoffel published papers on function theory including conformal mappings, geometry and tensor analysis, Riemann's o-function, the theory of invariants, orthogonal polynomials and continued fractions, differential equations and potential theory, light, and shock waves.

  812. Jordan biography
    • Jordan's use of the group concept in geometry in 1869 was motivated by studies of crystal structure.

  813. Leslie biography
    • Leslie was a successful professor of mathematics, attracting large classes of students and publishing his lectures in popular textbooks such as the three part work Elements of Geometry, Geometrical Analysis, and Plane Trigonometry (1809).

  814. Fricke biography
    • The present volume is confined to analysis and theory of functions, a second is announced as in preparation, to treat of advanced portions of algebra and geometry.

  815. John biography
    • He applied this in his study of general properties of linear partial differential equations, convex geometry and the mathematical theory of water waves.

  816. Schwarzschild biography
    • Schwarzschild's relativity papers give the first exact solution of Einstein's general gravitational equations, giving an understanding of the geometry of space near a point mass.

  817. Wright biography
    • These topics are: prime numbers; congruences and the quadratic reciprocity law; continued fractions; irrational, algebraic and transcendental numbers; quadratic fields; arithmetical functions, their order of magnitude and the Dirichlet or power series which generate them; partitions and representations of numbers as sums of squares, cubes and higher powers; Diophantine approximation; and the geometry of numbers.

  818. Magenes biography
    • This feeling was clear in the excellent and interesting speech given by Francesco Severi, and also in the lack of a section dedicated to algebra, while the tradition of the great Italian school of algebraic geometry was confirmed.

  819. Weisbach biography
    • His interests were always wide and this is reflected in the range of courses that Weisbach was teaching around this time: descriptive geometry, crystallography, optics, mechanics and machine design.

  820. Tits biography
    • The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classification of algebraic and Lie groups as well as finite simple groups, to Kac-Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces.

  821. Grosseteste biography
    • Grosseteste worked on geometry, optics and astronomy.

  822. Zenodorus biography
    • The treatise contains three-dimensional geometry results as well as two-dimensional.

  823. Roch biography
    • These included: Differential and Integral Calculus; Analytic Geometry; and Elliptic and Abelian Functions.

  824. Schafer biography
    • Under Lane she studied projective differential geometry and submitted her dissertation on Singularities of Space Curves.

  825. Jeffery Ralph biography
    • He took a degree in economics at Acadia University, but while studying for this degree he took two mathematics courses, one in calculus and one in analytic geometry.

  826. Riesz biography
    • His doctoral dissertation was on geometry.

  827. Brown Alexander biography
    • He contributed papers to meetings of the Society such as On the Ratio of Incommensurables in Geometry to the meeting on Friday 9 June 1905 and Relation between the distances of a point from three vertices of a regular polygon, at the meeting on Friday 11 June 1909, communicated by D C McIntosh.

  828. Zygmund biography
    • He attended lectures by Sierpinski on set theory, Mazurkiewicz on analytic functions, Samuel Dickstein on algebra and history of mathematics, and Stefan Kwietniewski on projective geometry.

  829. Montmort biography
    • Montmort went on to study the latest mathematics, in particular studying algebra and geometry.

  830. Puiseux biography
    • He wrote on geometry, where he discovered new properties of evolutes and involutes and mechanics where he studied the conical pendulum, the tautochrone and similar topics.

  831. Clarke Joan biography
    • During the time that Joan Clarke was an undergraduate at Cambridge, Gordon Welchman had supervised her in Geometry during Part II and, aware of her mathematical ability, he was responsible for recruiting Clarke to join the 'Government Code and Cypher School' (GCCS) at Bletchley Park.

  832. Lipschitz biography
    • In the paper [The history of modern mathematics III (Boston, MA, 1994), 113-138.',4)">4] the author shows convincingly how Lipschitz mechanical interpretation of Riemann's differential geometry would prove to be a vital step in the road towards Einstein's special theory of relativity.

  833. Shen Kua biography
    • This requires at least some understanding of spherical geometry and trigonometry.

  834. Burkhardt biography
    • Other topics on which Burkhardt published papers included groups, differential equations, differential geometry and mathematical physics.

  835. Leibniz biography
    • He began to study the geometry of infinitesimals and wrote to Oldenburg at the Royal Society in 1674.

  836. Hall Marshall biography
    • He also became interested in projective geometry and began a major study of projective planes.

  837. Francais Jacques biography
    • He then appears to have lost interest in mathematics until, under the command of Malus, he was encouraged to prepare his work on analytic geometry for publication.

  838. Tibbon biography
    • And here is geometry, the basis for all mathematical sciences, and this book is the basis, the root and the beginning for all later books on this science.

  839. Prager biography
    • He also wrote textbooks in Turkish for his students, one on descriptive geometry and another on elementary mechanics.

  840. Copernicus biography
    • Geometry Net (A list of about 100 links) .

  841. 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.

  842. Galois biography
    • Geometry.net (Links to other Galois references) .

  843. Feigenbaum biography
    • Using fractal geometry to describe natural forms such as coastlines, mathematical physicist Mitchell Feigenbaum developed software capable reconfiguring coastlines, borders, and mountain ranges to fit a multitide of map scales and projections.

  844. Sharp biography
    • In 1717 Sharp published Geometry Improved.

  845. Rogers biography
    • However, this was only the first of a remarkable number of papers that Rogers published while a research student: A note on a theorem of Blichfeldt (1946); (with Harold Davenport) Hlawka's theorem in the geometry of numbers (1947); A note on a problem of Mahler (1947); A note on irreducible star bodies (1947); Existence theorems in the geometry of numbers (1947); (with J H H Chalk) The critical determinant of a convex cylinder (1948); A problem of Hirsch (1948); The product of the minima and the determinant of a set (1949); The product of n homogeneous linear forms (1949); The successive minima of measurable sets (1949); On the critical determinant of a certain nonconvex cylinder (1949); (with Harold Davenport) A note on the geometry of numbers (1949); and (with J H H Chalk) The successive minima of a convex cylinder (1949).
    • Many results in the geometry of numbers assert, in effect, that inequalities of a certain type are soluble in integers, the constant on the right of the inequality being the best possible.
    • In collaboration with Geoffrey Shephard and James Taylor during that period his interest in convex geometry and Hausdorff Measure Theory widened.
    • As related above, his early work was on number theory, and he wrote on Diophantine inequalities and the geometry of numbers.
    • His later work covered a wide range of different topics in geometry and analysis including Borel functions, Hausdorff measure and local measure, topological properties of Banach spaces and upper semicontinuous functions.
    • In the period after the Second World War, Rogers rapidly emerged in the forefront of the renaissance of the geometry of numbers.

  846. Scheiner biography
    • Lantz had moved to Munich and Scheiner succeeded him in Ingolstadt as Professor of Mathematics and Hebrew, teaching geometry, astronomy and, in addition, his specialist subjects of sundials and telescopes.

  847. Mydorge biography
    • Mydorge studied geometry and physics.

  848. Riccati Vincenzo biography
    • It probably came too late, at the end of the period of construction of the curves, when geometry has given way to algebra, and when series became the tool of choice to represent the solutions of differential equations.

  849. Hirsch biography
    • III (1969); B I Plotkin, Groups of automorphisms of algebraic systems (1972); I R Shafarevich, Basic algebraic geometry (1974); D A Suprunenko, Matrix groups (1976).

  850. Kramp biography
    • Kramp was elected to the geometry section of the Academie des Sciences in 1817.

  851. Von Dyck biography
    • In 1880 Klein left Munich to take up the chair of geometry at Leipzig.

  852. Wolf biography
    • Wolf wrote on prime number theory and geometry, then later on probability and statistics - a series of papers discussed Buffon's needle experiment in which he estimated π by Monte Carlo methods.

  853. Dowker biography
    • The lectures on coherent sheaves are dealt with in the same analytical spirit, and there is no attempt to go far into the applications to algebraic geometry or complex manifolds.

  854. Seress biography
    • on Symbolic and Algebraic Manipulation) and was hailed as "a groundbreaking work" that "marks a turning point in Majorana Theory." His most recent work, with Harald Helfgott, under publication in the Annals of Mathematics, gives a long-sought bound on the diameter of the alternating and symmetric groups and represents a tour de force in the study of the geometry of finite simple groups.

  855. Aristotle biography
    • Notice that Euclid and his axiom system for geometry came after Aristotle.

  856. Adams Frank biography
    • In 1970 Adams succeeded Hodge as Lowndean Professor of Astronomy and Geometry at Cambridge, and at this time he returned to Trinity College.

  857. Ree biography
    • He also published On projective geometry over full matrix rings (1955).

  858. Springer biography
    • lies in the treatment of the prerequisites from algebraic geometry and commutative algebra.

  859. Holder biography
    • From 1900 Holder became interested in the geometry of the projective line and philosophical questions, which had interested him throughout his career, began to play a prominent role.

  860. Sinan biography
    • ',1)">1] the author points out that only two of the four could have beed written by Sinan, one on Archimedes work On triangles and one On the elements of geometry.

  861. Rittenhouse biography
    • He was fascinated with mathematics from his early years but, with little opportunity for schooling, was largely self-taught from books on elementary arithmetic and geometry and a box of tools inherited from an uncle, David Williams.

  862. Ozanam biography
    • The geometry of these sundials, which could be adjusted to work at any latitude, is studied in [Centaurus 29 (3) (1986), 165-177.',5)">5] particularly those types with hour-lines which were rectilinear, parabolic, elliptical, and hyperbolical.

  863. Banu Musa Muhammad biography
    • In addition to making perhaps the major contribution to the geometry text described above, Jafar Muhammad also wrote Premises of the book of conics which was a critical revision of Apollonius's Conics.

  864. Simplicius biography
    • In his commentary on Aristotle's Physics Simplicius quotes at length from Eudemus's History of Geometry which is now lost.

  865. De Witt biography
    • His most important work Elementa curvarum linearum (1659-61) was finished in 1649, and was the first systematic development of the analytic geometry of the straight line and conic.

  866. Arzela biography
    • He also studied at the University of Pisa and his thesis was directed by Enrico Betti who held the chair of analysis and higher geometry at the University of Pisa.

  867. Frisi biography
    • A letter by Frisi written in 1753 on mechanics and geometry is given and discussed in [Ist.

  868. Takebe biography
    • By this study, the author draws the conclusion that Takebe (whose work is still unpublished, and partly lost) had a role in the history of scientific thought, as his aim was to improve, by the assistance of geometry and mathematics, calculation techniques so that astronomy and calendar science could become, in Japan too, exact sciences.

  869. Faddeev biography
    • His primary area was algebra, but he made significant contributions to other areas such as number theory, function theory, geometry and probability.

  870. Al-Khalili biography
    • Of course, giving tables for timekeeping using astronomical events, requires a thorough understanding of geometry on the sphere and the work by al-Khalili can be seen as the end-product of the work of the Arabs on this mathematical topic.

  871. Ferrari biography
    • When Cardan generously resigned his post at the Piatti Foundation in Milan to make way for him in 1541, Ferrari easily defeated Zuanne da Coi, his only rival for the post, in a debate and, at the age of twenty, became a public lecturer in geometry.

  872. Walsh Joseph biography
    • He continued to publish a steady stream of papers with On the location of the roots of the derivative of a polynomial appearing in 1920 and then two papers A generalization of the Fourier cosine series and A theorem on cross-ratios in the geometry of inversion in 1921.

  873. Heisenberg biography
    • He avoided courses by Lindemann, however, so his mathematical interests moved from number theory to geometry.

  874. Lamy biography
    • In previous publications [Lamy] had addressed the fields of rhetoric, mechanics, mathematics and geometry, and his discussion of perspective gains force and depth from his interest in the study of optics.

  875. Nicomachus biography
    • Nicomachus wrote Arithmetike eisagoge (Introduction to Arithmetic) which was the first work to treat arithmetic as a separate topic from geometry.

  876. Lawson biography
    • In a meeting held on Friday 9 March 1923 a discussion on the teaching of elementary geometry was opened by Mr A J Tressland, M.A., F.R.S.E., of the Edinburgh Academy, who advocated the adoption of the sequence in geometrical teaching contained in the schedule recently issued by a special committee of the Assistant-Masters' Association.

  877. Gregory Duncan biography
    • Two other important works by Duncan Gregory are Examples of the Processes of the Differential and Integral Calculus and A Treatise on the Application of Analysis to Solid Geometry.

  878. Schwartz biography
    • He fell in love with geometry, taught by an inspiring teacher, but was disappointed by the philosophy course where the teaching was much less good.

  879. Paoli biography
    • His research was on analytic geometry, calculus, partial derivatives, and differential equations.
    • Among all those who in Italy are given to the study of mathematics, if we except a few sublime geniuses, who, with their strength of spirit have triumphed over all obstacles and reached a place at the highest level of geometry, there are few others that come to mediocrity.

  880. Atwood biography
    • In the summer of 1781 Atwood sent the Royal Society details of his applications of geometry to the problems of correcting sightings through sextant mirrors.

  881. Friedrichs biography
    • Knowledge of elementary Euclidean geometry is presupposed, and some familiarity with the basic notions of physics will be helpful.

  882. Mumford biography
    • I turned from algebraic geometry to an old love - is there a mathematical approach to understanding thought and the brain? This is applied mathematics and I have to say that I don't think theorems are very important here.

  883. Smullyan biography
    • it must be wondered at that the whole fascinating subject of Infinity is so little known to the general public! Why isn't it taught in high schools? It is no harder to understand than algebra or geometry, and it is so rewarding! .

  884. Spencer biography
    • He was a member of Editorial Board of the Transactions of the American Mathematical Society (1950-1955), the Annals of Mathematics (1958-1962), the Proceedings of the National Academy of Sciences (1965-1967), the American Journal of Mathematics (1967-1975), and the Journal of Differential Geometry (1967-1981).

  885. Volterra biography
    • His interest in mathematics started at the age of 11 when he began to study Legendre's Geometry.

  886. Nikodym biography
    • Some of his other books were: Introduction to differential calculus, (Warsaw, 1936) (jointly with his wife), Theory of tensors with applications to geometry and mathematical physics, I, (Warsaw, 1938), Differential Equations, (Poznan, 1949).

  887. De Rham biography
    • He gave a reducibility theorem for Riemann spaces which is fundamental in the development of Riemannian geometry.

  888. Foulis biography
    • These are Fundamental Concepts of Mathematics (1962), (with Mustafa A Munem) Calculus (1978), (with Mustafa A Munem) Calculus: With Analytic Geometry (1984), (with Mustafa A Munem) After Calculus: Algebra (1988), (with Mustafa A Munem) After Calculus: Analysis (1989), (with Mustafa A Munem) Algebra and Trigonometry with Applications (1991), and (with Mustafa A Munem) College Algebra with Applications (1991).

  889. Higman biography
    • After taking special topic courses on group theory and differential geometry, Higman received an MA.

  890. Zalts biography
    • Zalts continued research at the University of Latvia and in February 1944 he was awarded his doctorate for his thesis on the geometry of deformations.

  891. Krull biography
    • About thirty-five publications of fundamental importance for the development of commutative algebra and algebraic geometry date from this period.

  892. Kuczma biography
    • Fundamental notions such as existence and uniqueness of solutions of equations under consideration are treated throughout the book as well as a surprisingly wide scale of examples showing applications of the theory in dynamical systems, ergodic theory, functional analysis, functional equations in several variables, functional inequalities, geometry, iteration theory, ordinary differential equations, partial differential equations, probability theory and stochastic processes.

  893. Gordan biography
    • Gordan also worked on algebraic geometry and he gave simplified proofs of the transcendence of e and π.

  894. Bayes biography
    • This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.

  895. Alcuin biography
    • Alcuin wrote elementary texts on arithmetic, geometry and astronomy at a time when a renaissance in learning in Europe was just beginning, a renaissance mainly led by Alcuin himself.

  896. Quetelet biography
    • He also began to give public lectures at the Museum in Brussels on topics such as geometry, probability, physics, and astronomy.

  897. Pompeiu biography
    • In [\'Gheorghe Titeica and Dimitrie Pompeiu\' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Editura Acad.

  898. Magini biography
    • This mathematical work is typical of all of Magini's mathematical contributions which are all highly practical; they include treatises on the geometry of the sphere, on applications of trigonometry, and on calculating devices of his own invention.

  899. Cassels James biography
    • In 1910 he passed Lower Latin and Analytical Geometry in the Scottish Leaving Certificate examinations, and Mathematics, English, and French at Higher level.

  900. Brash biography
    • Edinburgh University bursaries: October 1905 John Welsh Mathematical Bursary (subject of examination - mathematics, especially pure geometry) William Brash, George Watson's College.

  901. Poisson biography
    • However, Poisson found that descriptive geometry, an important topic at the Ecole Polytechnique because of Monge, was impossible for him to succeed with because of his inability to draw diagrams.

  902. Stampacchia biography
    • For three years he produced outstanding examination results in a wide range of courses such as Tutorial Sessions in Analysis and in Geometry, Calculus of Variations, Theory of Functions, and Ordinary Differential Equations.

  903. Kirby biography
    • It was his outstanding work on the annulus conjecture which led to the American Mathematical Society awarding him their Veblen Prize in Geometry in 1971:- .

  904. Hardie Robert biography
    • On Friday 9 March 1923 a discussion on the teaching of elementary geometry was opened by Mr A J Tressland, M.A., F.R.S.E., of the Edinburgh Academy, who advocated the adoption of the sequence in geometrical teaching contained in the schedule recently issued by a special committee of the Assistant-Masters' Association.

  905. Kothe biography
    • The thesis was accepted on 31 January 1931 and he became a Lecturer in Geometry in 1935.

  906. Fantappie biography
    • There was another major idea introduced by Fantappie, namely a cosmological theory based on a geometry arising from a group which, in some sense, generalises the Lorentz group.

  907. Holmboe biography
    • Not everyone, of course, agreed with his ideas on teaching mathematics and between 1835 and 1838 Christopher Hansteen published textbooks for schools on geometry and mechanics.

  908. Rudolph biography
    • This is a central branch of dynamical systems with broad connections to smooth and low-dimensional dynamics, symbolic dynamics, topological dynamics, you name it, and to other branches of mathematics, functional analysis, geometry, combinatorics, number theory, you name it.

  909. Painleve biography
    • In 1900 he was elected to the geometry section of the Academie des Sciences.

  910. Heine biography
    • He also attended geometry lectures by Steiner, and astronomy lectures by J F Encke, the director of the observatory.

  911. Gardner biography
    • (Coxeter includes the dissection in his classic, 'Introduction to Geometry'.) .

  912. Wall biography
    • In 1995 Wall published The geometry of topological stability written jointly with A A du Plessis.

  913. Schroder biography
    • The extension of the power concept, originally only associated with integers, to rational fractions has been very fruitful in algebra; this suggests that we should try to do the same thing in geometry whenever the opportunity presents itself.

  914. Piaggio biography
    • Here list a few articles which Piaggio published in The Mathematical Gazette: Relativity rhymes with a mathematical commentary (January 1922); Geometry and relativity (July 1922); Mathematics for evening technical students (July 1924); Mathematical physics in university and school (October 1924); Probability and its applications (July 1931); Three Sadleirian professors: A R Forsyth, E W Hobson and G H Hardy (October 1931); Mathematics and psychology (February 1933); Lagrange's equation (May 1935); Fallacies concerning averages (December 1937); and The incompleteness of "complete" primitives of differential equations (February 1939).

  915. Snell biography
    • 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).

  916. Gruenberg biography
    • In addition to these research level texts, Gruenberg also published an undergraduate level text (written jointly with A J Weir) Linear geometry.

  917. Plessner biography
    • Then in session 1921/22 he studied in Berlin where von Mises lectured on differential and integral equations, Bieberbach on differential geometry and Schur on algebra.

  918. Werner Wendelin biography
    • for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory.

  919. Brunelleschi biography
    • Toscanelli's interest in science and mathematics was particularly significant for he taught his young pupil the principles of geometry.

  920. Aryabhata II biography
    • It discusses topics such as geometry, geography and algebra with applications to the longitudes of the planets.

  921. Beckenbach biography
    • We have mentioned one such text above, but let us give an incomplete list: College Algebra (1964), Modern Introduction to Analysis (1964), Applied Combinatorial Analysis (1964), Essential of College Algebra (1965), Integrated College Algebra and Trigonometry (1966), Modern School Mathematics (1967), Algebra (1968), Pre-algebra (1970), Modern College Algebra and Trigonometry (1969), Analysis of Elementary functions (1970), Intermediate Algebra for College Students (1971), Modern Analytic Geometry (1972), Concepts of Communications: Interpersonal, Intrapersonal and Mathematical (1972), and College Mathematics for Students of Business and the Social Sciences (1987).

  922. Remez biography
    • He gave courses at these institutions on analysis, differential equations and differential geometry while undertaking research for his doctorate.

  923. Ladyzhenskaya biography
    • He started teaching his daughters mathematics in the summer of 1930 beginning with giving explanations of the basic notions of geometry, then he formulated a theorem and in turn made his daughters prove it.

  924. Griffiths Brian biography
    • We give the titles of a few of his mathematical education article which give an overview of his interests in that topic: Pure mathematicians as teachers of applied mathematicians (1968); Mathematics Education today (1975); Successes and failures of mathematical curricula in the past two decades (1980); Simplification and complexity in mathematics education (1983); The implicit function theorem: technique versus understanding (1984); A critical analysis of university examinations in mathematics (1984); Cubic equations, or where did the examination question come from? (1994); The British Experience of Teaching Geometry since 1900 (1998); and The Divine Proportion, matrices and Fibonacci numbers (2008).

  925. Hurwitz biography
    • Schubert gave up part of every Sunday to working at geometry with the schoolboy Hurwitz, and the first of the latter's papers, written when he was still at the Andreanum, was a joint paper.

  926. Osgood biography
    • Other classic texts included Introduction to Infinite Series (1897), A First Course in the Differential and Integral Calculus (1909), Topics in the theory of functions of several complex variables published by the American Mathematical Society in 1914, Plane and Solid Analytic Geometry (with W C Graustein, 1921), Advanced Calculus (1925), and Mechanics (1937).

  927. Mittag-Leffler biography
    • Mittag-Leffler made numerous contributions to mathematical analysis particularly in areas concerned with limits and including calculus, analytic geometry and probability theory.

  928. Karlin biography
    • He was appointed to the California Institute of Technology in 1948 and began publishing papers on functional analysis such as Unconditional convergence in Banach spaces (1948), Bases in Banach spaces (1948), Orthogonal properties of independent functions (1949), and (with L S Shapley) Geometry of reduced moment spaces (1949).

  929. Green Sandy biography
    • with First Class Honours in Mathematics in 1947 having taken the compulsory courses of Geometry, Algebra, Analysis, Statics, Dynamics and the optional courses of Special Functions, and Algebra in his final year of study.

  930. Bruno Giordano biography
    • He went to Padua where he wrote Lectures on Geometry and Art of Deformation.

  931. Schlesinger biography
    • The paper introduces what today are known as the Schlesinger transformations and Schlesinger equations which have an important role in differential geometry.

  932. De Prony biography
    • Also around 1791 de Prony was working on geometry with Pierre Girard.

  933. Humbert Georges biography
    • It was as a direct consequence of his work on using abelian functions in geometry which won for him the 1892 Academie des Sciences prize for work on Kummer surfaces.

  934. Finck biography
    • His texts include books on algebra, mechanics, geometry and analysis.

  935. Frobenius biography
    • In his work in group theory, Frobenius combined results from the theory of algebraic equations, geometry, and number theory, which led him to the study of abstract groups.

  936. Dickson biography
    • Dickson studied widely within mathematics but specialised in Halsted's own subjects of euclidean and non-euclidean geometry.

  937. Casorati biography
    • After returning from the first of his journeys in 1859, Casorati was appointed as extraordinary professor of algebra and analytic geometry at the University of Pavia.

  938. Darmois biography
    • As an undergraduate, he had taken courses given by Gaston Darboux and Edouard Goursat and, in 1911, encouraged by Darboux, he began to undertake research on a topic involving both geometry and analysis.

  939. Regiomontanus biography
    • He then gives a list of the axioms he will assume, followed by 56 theorems on geometry.

  940. Hall biography
    • Among his teachers at Cambridge were Hobson, the Sadleirian professor, and Baker, the Lowndean professor of Astronomy and Geometry.

  941. Birnbaum biography
    • After arriving in Gottingen, Edmund Landau became his advisor, and he attended several lecture courses: differential equations given by Courant; calculus of variations given by Courant; power series given by Landau; higher geometry given by Herglotz; probability calculus given by Bernays; analysis of infinitely many variables given by Wegner; and attended the mathematical seminar directed by Courant and Herglotz.

  942. Blanch biography
    • Her thesis on algebraic geometry considered a transformation which first appeared in a paper by Veneroni in 1901.

  943. Browder William biography
    • The mathematics he was taught convinced him that it was a boring computational subject devoid of beauty, with the one exception that he saw Euclidean geometry as beautiful.

  944. Shelah biography
    • But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty - a system of demonstrations and theorems based on a very small number of axioms which impressed me and captivated me.

  945. Clifford biography
    • Influenced by the work of Riemann and Lobachevsky, Clifford studied non-euclidean geometry.

  946. Chauvenet biography
    • As a textbook writer we mention Chauvenet's A treatise on plane and spherical trigonometry (1850), Spherical astronomy (1863), Theory and use of astronomical instruments : Method of least squares (1863), and A treatise of elementary geometry (1870).

  947. Al-Tusi Nasir biography
    • These topics included logic, physics and metaphysics while he also studied with other teachers learning mathematics, in particular algebra and geometry.

  948. Skolem biography
    • I see Skolem as arguing that all the evidence that has been given for the existence of uncountable sets is inconclusive, and the reason why he insists on considering countable models is that axiomatisation was put forward at the time as the only way to secure set theory, and what sets are and which sets exist was claimed to be determined by the axioms and their models (much as what Euclidean geometry is about was claimed to be determined by Hilbert's axioms and their models).

  949. Fermi biography
    • This paper gave an important result about the Euclidean nature of space near a world line in the geometry of general relativity.

  950. Kruskal Martin biography
    • Methods for exact solution published in 1974 was fundamental, and the ideas developed in it were later extended to dynamical systems, inverse scattering, and symplectic geometry.

  951. Taylor biography
    • One could certainly consider this work as laying the foundations for the theory of descriptive and projective geometry.

  952. Borel biography
    • In [Enseignement mathematique 11 (1965), 1-95.',8)">8] Borel's mathematical work is divided into the following topics: Arithmetic; Numerical series; Set theory; Measure of sets; Rarefaction of a set of measure zero; Real functions of real variables; Complex functions of complex variables; Differential equations; Geometry; Calculus of probabilities; and Mathematical physics.

  953. Gould biography
    • In 1881 Henri Brocard had published Etudes d'un nouveau cercle du plan du triangle in which he introduced Brocardian geometry.

  954. Young biography
    • The other two were written jointly with his wife: The first book of geometry (1905) was an elementary work clearly written by the Youngs having the teaching of mathematics to their own children in their minds, and The theory of sets of points (1906).

  955. Rheticus biography
    • This appointment, which involved teaching arithmetic and geometry, gave Rheticus a salary of 100 gulden.

  956. Motzkin biography
    • Exceptionally broad, the range of his work included beautiful and important contributions to the theory of linear inequalities and programming, approximation theory, convexity, combinatorics, algebraic geometry, number theory, algebra, function theory, and numerical analysis.

  957. Qadi Zada biography
    • He completed his standard education in Basra and then studied geometry and astronomy with al-Fanari.

  958. Walfisz biography
    • He also served as Head of the Department of Algebra and Geometry at the Tbilisi Mathematical Institute from 1948 to 1962.

  959. Konig Julius biography
    • Konig worked on a wide range of topics in algebra, number theory, geometry, set theory, and analysis.

  960. Bolza biography
    • Moore was a fiery enthusiast, brilliant, and keenly interested in the popular mathematical research movements of the day; Bolza, a product of the meticulous German school of analysis led by Weierstrass, was an able, and widely read research scholar; Maschke was more deliberate than the other two, sagacious, brilliant in research, and a most delightful lecturer in geometry.

  961. Ince biography
    • The object of this book is to provide in a compact form an account of the methods of integrating explicitly the commoner types of ordinary differential equation, and in particular those equations that arise from problems in geometry and applied mathematics.

  962. Gopel biography
    • He wrote on Steiner's synthetic geometry and an important work, published after his death, continued the work of Jacobi on elliptic functions.

  963. Gronwall biography
    • Gronwall's work contains classical analysis (Fourier series, Gibbs phenomenon, summability theory, Laplace and Legendre series), differential and integral equations, analytic number theory (transcendental numbers, divisor function, L-function of Dirichlet), complex function theory (Dirichlet L-series, conformal mappings, univalent functions), differential geometry, mathematical physics (problems of elasticity, ballistics, induction, potential theory, kinetic theory of gases, optics), nomography, atomic physics (wave mechanics of hydrogen and helium atom, lattice theory of crystals) and physical chemistry where he is especially known as a very important contributor.

  964. Borsuk biography
    • Borsuk introduced the important concept of absolute neighbourhood retracts in his doctoral dissertation, published in 1931, which was to lead to new and fruitful ideas in metric differential geometry, see [Topol.

  965. Democritus biography
    • He wrote On numbers, On geometry, On tangencies, On mappings, On irrationals but none of these works survive.

  966. Eckmann biography
    • It was characteristic of Hopf's views on our science that this meant not only learning algebraic topology - then a very young field - but also getting acquainted with group theory, differential geometry, and algebra in the "abstract" sense of the Emmy Noether school.

  967. Hamel biography
    • At this time Klein was running a seminar which studied the theory of elasticity, descriptive geometry, and mechanics, and Hamel participated in this seminar.

  968. Bartlett biography
    • During these years he published pioneer papers on the 2 by 2 by 2 contingency table, on the geometry of the linear model, on transformations in analysis of variance, on estimation problems in factor analysis, on spatial aspects of design in field trials, as well as a remarkable group of papers on foundational problems in statistical inference and the role of the likelihood.

  969. Cantor Moritz biography
    • Chasles was an acknowledged leading expert on the history of geometry and encouraged Cantor to publish further historical material in Comptes Rendus.

  970. Weingarten biography
    • The theory of surfaces was the most important topic in differential geometry and [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .

  971. De Bruijn biography
    • He lists his interests (on his website) as: Geometry, Number Theory, Classical and Functional Analysis, Applied Mathematics, Combinatorics, Computer Science, Logic, Mathematical Language, Brain Models.

  972. Al-Khujandi biography
    • Finally, although this really proves little, the theorem appears many times in the writings of Abu Nasr Mansur: both his writings on geometry as well as those on astronomy.

  973. Lukacs biography
    • His doctoral dissertation on a geometry topic was supervised by W Meyer and, in 1930, he was awarded his doctorate.

  974. Bourgain biography
    • Bourgain's work touches on several central topics of mathematical analysis: the geometry of Banach spaces, convexity in high dimensions, harmonic analysis, ergodic theory, and finally, nonlinear partial differential equations from mathematical physics.

  975. Fawcett biography
    • Philippa Fawcett's outstanding results in algebra and geometry led to her being awarded a Gilchrist scholarship to study mathematics at Newnham College, Cambridge, the women's College that her mother had helped to found.

  976. Kummer biography
    • Kummer's Berlin lectures, always carefully prepared, covered analytic geometry, mechanics, the theory of surfaces, and number theory.

  977. Mandelbrojt biography
    • History Topics: A History of Fractal Geometry .

  978. Grimaldi biography
    • Grimaldi was well prepared to teach all branches of mathematics: geometry, optics, gnomonics, statics, geography, astronomy and celestial mechanics.

  979. Doetsch biography
    • He taught at the University of Halle from 1922 to 1924 before being appointed as Ordinary Professor of Descriptive Geometry at the Technical University of Stuttgart.

  980. Serret biography
    • Serret did important work in differential geometry.

  981. Threlfall biography
    • He was promoted to a tenured extraordinary professor of analytical geometry, calculus of variations, analysis and function theory on 1 April 1936.

  982. Montgomery biography
    • These include: Periodic one-parameter groups in three-space (1936); Translation Groups of Three-Space (1937); Compact Abelian transformation groups (1938); Non-Abelian Compact Connected Transformation Groups of Three-Space (1939); A theorem on the rotation group of the two-sphere (1940); Topological group foundations of rigid space geometry (1940); Topological transformation groups.

  983. Ohm biography
    • He worked on writing an elementary book on the teaching of geometry while remaining desperately unhappy in his job.

  984. Tucker Robert biography
    • But, like a number of schoolmaster's at this time, he also made a contribution to research in geometry.

  985. Gelfand biography
    • Gelfand's work on group representations led him to study integral geometry (a term due to Blaschke) which in turn he was led to by a study of the Radon transform.

  986. Einstein biography
    • Geometry and Experience .

  987. De Finetti biography
    • Meanwhile, against the will of his mother, who was worried about his future, he moved to the recently founded (1925) University of Milan and there, in 1927, he graduated in Applied Mathematics with a dissertation on affine geometry supervised by Giulio Vivanti, a mathematician who made some noteworthy contributions to complex analysis.

  988. Bossut biography
    • However the Ecole du Genie brought Monge into contact with Bossut who encouraged him to develop his ideas on geometry.

  989. Bernoulli Nicolaus(I) biography
    • There he worked on geometry and differential equations.

  990. Gaschutz biography
    • It had been published in 1942, and after the disastrous winter in the East the wartime economy must have deemed it beneficial to winning the war and therefore worthy of publication, despite the shortage in demand, due to its title: 'The Geometry of Fabrics'.

  991. Fraser biography
    • He spared no pains in providing for the due equipment of the Physical Laboratory, and in preparing courses of Practical Geometry and Experimental Physics suitable for boys.

  992. Zassenhaus biography
    • Zassenhaus worked on a broad range of topics and, in addition to those mentioned above, he worked on nearfields, the theory of orders, representation theory, the geometry of numbers and the history of mathematics.

  993. Koch biography
    • History Topics: A History of Fractal Geometry .

  994. Kerr Roy biography
    • Before he left, Josh and I became interested in the new methods that were entering general relativity from differential geometry at that time.

  995. Kepler biography
    • In principle this included the four mathematical sciences: arithmetic, geometry, astronomy and music.

  996. Butzer biography
    • Among his lecturers we mention Donald Coxeter, who gave courses on non-Euclidean geometry and number theory, and Bill Tutte who gave a topology course.

  997. Bieberbach biography
    • He took up this eminent chair of geometry on 1 April 1921.

  998. Bour biography
    • Bour made many significant contributions to analysis, algebra, geometry and applied mechanics despite his early death from an incurable disease.

  999. Pitman biography
    • He took me through the topics in his two books on vector analysis, and perhaps also some differential geometry ..

  1000. Tarry biography
    • Finally let us mention Tarry's contributions to geometry.

  1001. Mullikin biography
    • She displayed promise as a mathematician in her senior year by solving a problem on geometry in the 'American Mathematical Monthly'.

  1002. Mosteller biography
    • In his first year he studied engineering drawing, descriptive geometry, English and practical courses which included masonry, carpentry, sheet metal working and welding.

  1003. Wilder biography
    • an exposition of the basic theories of modern mathematics: the theory of sets, the real number system (on the basis of the Peano axioms) and the theory of groups (including some of its applications in algebra and geometry).

  1004. Bernoulli Daniel biography
    • The third part of Mathematical exercises was on the Riccati differential equation while the final part was on a geometry question concerning figures bounded by two arcs of a circle.

  1005. Toeplitz biography
    • After graduating, he continued with his studies of algebraic geometry at the University of Breslau, being awarded his doctorate in 1905.

  1006. Hausdorff biography
    • History Topics: A History of Fractal Geometry .

  1007. Wolstenholme biography
    • They were usually concerned with questions of analytical geometry, and they were marked they were marked by a peculiar analytical skill and ingenuity.

  1008. Mises biography
    • He classified his own work, not long before his death, into eight areas: practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability calculus, statistics and philosophy.

  1009. Goursat biography
    • Volume 1 covers applications to geometry, expansion in series, definite integrals, and derivatives and differentials.

  1010. Dvoretzky biography
    • Nevertheless there are remarkable theorems embodying the heuristic principle and providing powerful tools for the study of high-dimensional geometry.

  1011. Seidel Jaap biography
    • The authors of [Geometry and Combinatorics, Selected Works of J J Seidel (Academic Press, Inc., 1991).',1)">1] write:- .

  1012. Ghizzetti biography
    • Then, remaining at the Turin Polytechnic, he became an assistant teaching geometry there.

  1013. Story biography
    • in 1911 with a thesis on algebraic geometry entitled On the existence of loci with given singularities.

  1014. Synge biography
    • Professor Synge made outstanding contributions to widely varied fields: classical mechanics, geometrical mechanics and geometrical optics, gas dynamics, hydrodynamics, elasticity, electrical networks, mathematical methods, differential geometry and, above all, Einstein's theory of relativity.

  1015. Blumenthal biography
    • Blumenthal did the first pioneering work in a programme outlined by Hilbert with the aim of creating a theory of modular functions of several variables that should be just as important in number theory and geometry as the theory of modular functions of one variable was at the beginning of the [20th] century.

  1016. Dantzig biography
    • Van Dantzig studied differential geometry, electromagnetism and thermodynamics.

  1017. Mohr biography
    • It had been sent to him by Oldenburg, the secretary of the Royal Society in London, in 1675 and Leibniz replied to Oldenburg in the following year praising Mohr's skill in geometry and analysis.

  1018. Reinhardt biography
    • Bieberbach and Suss had both been at the University of Frankfurt up to 1921, and Bieberbach had assisted Reinhardt with his habilitation thesis but, shortly after Reinhardt was appointed, Bieberbach moved to take up the Chair of Geometry at the University of Berlin and Suss went with him as his assistant.

  1019. Czuber biography
    • He submitted his habilitation thesis on practical geometry (geodesy) to the Technical University at Prague in 1876 and obtained the right to lecture.

  1020. Kruskal William biography
    • Some of his later publications include When are Gauss-Markov and least squares estimators identical? A coordinate-free approach (1968), The geometry of generalized inverses (1975), and Miracles and statistics: the casual assumption of independence (1988).

  1021. Widman biography
    • The book consists of three parts: the first section is on counting with whole numbers, the second is on proportion, while the third section is on geometry.

  1022. Oenopides biography
    • [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..

  1023. Begle biography
    • This textbook was Introductory Calculus, with Analytic Geometry which Begle published in 1954.

  1024. Noether Max biography
    • Max Noether was one of the leaders of nineteenth century algebraic geometry.

  1025. Al-Karaji biography
    • So what he achieved here was defining the product of these terms without any reference to geometry.

  1026. Manfredi biography
    • Another two posthumous publications by Manfredi were Instituzioni astronomiche (1749) and his lectures on the elements of plane and solid geometry and trigonometry, Elementi della geometria piana e solida e della trigonometria (1755).

  1027. Stewart Dugald biography
    • In fact Stewart's influence on physics is especially interesting and it form the main topic of [Scottish philosophy and British Physics 1750-1880 (Princeton University Press, Princeton, 1975).',4)">4] where Olson argues that Stewart's view of mathematics put geometry at its foundations rather than algebra, and that his views on this influenced the physical thinking of Maxwell and Rankine.

  1028. Verhulst biography
    • There he gave courses on astronomy, celestial mechanics, the differential and integral calculus, the theory of probability, geometry and trigonometry.

  1029. Runge biography
    • His doctoral dissertation, submitted to the University of Berlin on 23 June 1880, dealt with differential geometry.

  1030. Narayana biography
    • In terms of geometry Narayana gave a rule for a segment of a circle.

  1031. Nakayama biography
    • Even when he was eventually confined to bed he did not give up mathematics but read Grothendieck's work on algebraic geometry.

  1032. Tietze biography
    • Topics outside topology which Tietze worked on included ruler and compass constructions, continued fractions, partitions, the distribution of prime numbers, and differential geometry.

  1033. Ferro biography
    • We know that del Ferro was appointed as a lecturer in arithmetic and geometry at the University of Bologna in 1496 and that he retained this post for the rest of his life.

  1034. Al-Battani biography
    • one of the famous observers and a leader in geometry, theoretical and practical astronomy, and astrology.

  1035. Artin biography
    • Thus, this conjecture of Artin was the origin of a wide range of activities in what is now called arithmetic geometry.

  1036. Levi-Civita biography
    • In [Italian mathematics between the two world wars (Pitagora, Bologna, 1987), 125-141.',18)">18] the authors argue that Levi-Civita was interested in the theory of stability and qualitative analysis of ordinary differential equations for three reasons: his interest in geometry and geometric models; his interest in classical mechanics and celestial mechanics, in particular, the three-body problem; and his interest in stability of movement in the domain of analytic mechanics.

  1037. Koksma biography
    • One then finds a discussion of Minkowski's analysis, his 'Geometry of Numbers' and applications to homogeneous and non-homogeneous linear forms.

  1038. Al-Haytham biography
    • The main topics on which he wrote were optics, including a theory of light and a theory of vision, astronomy, and mathematics, including geometry and number theory.

  1039. Fontenelle biography
    • Astronomy cannot be without Optics by reason of Perspective Glasses: and both, as all parts of the Mathematicks are grounded upon Geometry ..

  1040. Nightingale biography
    • Lessons included learning arithmetic, geometry and algebra and prior to Nightingale entered nursing, she spent time tutoring children in these subjects.

  1041. Hopf Eberhard biography
    • His interests and principal achievements were in the fields of partial and ordinarydifferential equations, calculus of variations, ergodic theory, topological dynamics, integral equations, differential geometry, complex function theory and functional analysis.

  1042. Apaczai biography
    • Apaczai's Hungarian Encyclopaedia contains one section on arithmetic (Part Four of the Encyclopaedia) and one section on geometry (Part Five of the Encyclopaedia).

  1043. Fagnano Giulio biography
    • he may well be considered the founder of the geometry of the triangle.

  1044. Schmid biography
    • Following his move to Berlin, the direction of Schmid's research changed somewhat and he moved away from algebraic number theory, becoming interested in topics in algebraic geometry and Lame differential equations.

  1045. Mazur biography
    • The theorem we proved - that a transformation preserving distances is linear - is now part of the standard treatment of the geometry of function spaces.

  1046. Bolibrukh biography
    • From our third year there (that is, from Autumn 1969) we were in one group containing mainly students working in higher geometry and topology.

  1047. Fatou biography
    • History Topics: A History of Fractal Geometry .

  1048. Birkhoff biography
    • Among his works, some of which we have already mentioned above, are Relativity and Modern Physics (1923), Dynamical Systems (1928), Aesthetic Measure (1933), and Basic Geometry (1941).

  1049. Haret biography
    • Despite his political involvement, Haret retained his professorship at the University of Bucharest and, in addition, he also retained his professorship of analytical geometry at the Bridges and Roads' School in Bucharest to which he had been appointed in 1882.

  1050. Ayyangar biography
    • During nearly three decades during which he taught and undertook research at Mysore he made many contributions to geometry, statistics, astronomy, the history of Indian mathematics, and other topics.

  1051. Abbott biography
    • In it Abbott tries to popularise the notion of multidimensional geometry but the book is also a clever satire on the social, moral, and religious values of the period.

  1052. Adams biography
    • It was a short tenure of the chair for, in March 1859, he succeeded Peacock as Lowndean Professor of Astronomy and Geometry at Cambridge and held the post for over 32 years.

  1053. Barbier biography
    • These cover topics such as spherical geometry and spherical trigonometry.

  1054. Geminus biography
    • geometry and arithmetic, sciences which deal with pure, the eternal and the unchangeable, but was extended by later writers to cover what we call 'mixed' or applied mathematics, which, though theoretical, has to do with sensible objects e.g.

  1055. Fermat biography
    • They asked him to divulge his methods and Fermat sent Method for determining Maxima and Minima and Tangents to Curved Lines, his restored text of Apollonius's Plane loci and his algebraic approach to geometry Introduction to Plane and Solid Loci to the Paris mathematicians.

  1056. Young Lai-Sang biography
    • Today it stands at the crossroads of several areas of mathematics, including analysis, geometry, topology, probability, and mathematical physics.

  1057. Leech biography
    • By inclination he was a pure mathematician, with a taste for number theory, geometry and combinatorial group theory, his interests tending towards the particular rather than the general.

  1058. Torricelli biography
    • Notice that we have stated this result in the modern notation of coordinate geometry which was totally unavailable to Torricelli.

  1059. Al-Tusi Sharaf biography
    • it represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.

  1060. Netto biography
    • There he taught courses on advanced algebra, the calculus of variations, mechanics, Fourier series, and synthetic geometry.

  1061. Born biography
    • However he annoyed Klein by only making irregular attendances at his lectures, so Born decided to substitute astronomy for geometry as one of his doctoral subjects.

  1062. Bourbaki biography
    • History Topics: A History of Fractal Geometry .

  1063. Ehrenfest-Afanassjewa biography
    • A discussion with E J Dijksterhuis on the intuitive approach to teaching geometry in 1924 led to the appearance of a magazine for the didactics of mathematics, published under the name 'Euclid.' The intuitive approach was characterised by contemplating obvious spatial concepts and phenomena.

  1064. Kochin biography
    • She had purchased university texts on geometry and on analysis and Kochin read these books while he was at the high school.

  1065. Bergman biography
    • He lectured first at the Massachusetts Institute of Technology in Cambridge, Massachusetts, where he gave a series of lectures on Theory of pseudo-conformal transformations and its connection with differential geometry during 1939-40.

  1066. Bouquet biography
    • Bouquet worked on differential geometry, writing on orthogonal surfaces Note sur les surfaces orthogonales (1946).

  1067. Girard Albert biography
    • The negative solution is explained in geometry by moving backward, and the minus sign moves back when the + advances.

  1068. Haar biography
    • The Department of Mathematics, consisting of the Mathematical Seminary and the Institute of Descriptive Geometry, began operating in Szeged.

  1069. Samelson biography
    • The ideas developed here lead to some of the most important developments in topology and geometry during the last 50 years.

  1070. Carcavi biography
    • For example, in the autumn of 1637 Fermat sent Carcavi Isagoge ad locos planos et solidos, an introduction to analytic geometry that he had written in the previous year before the publication of Descartes' Geometrie.

  1071. Girard Pierre biography
    • I thank you for the solution you sent me to the geometry problem..

  1072. Goldberg biography
    • Looking through the catalogue of the Lviv Regional Library I saw a card for Nevanlinna's book in the section "Analytic geometry".

  1073. Al-Mahani biography
    • It gives a full account of the Arabic literature which was available in the 10th century and in particular mentions al-Mahani, not for his work in astronomy, but rather for his work in geometry and arithmetic.

  1074. Mayer Tobias biography
    • Tobias Mayer's interest in mathematics appears to have stemmed from his reading of the two books: Christian von Wolff's 'Anfangs-Grunde aller mathematuscher Wissenschaften' and Johann Christian Sturm's 'Mathesis enucleata.' At any rate, he acknowledges his debt to these two books when, on the occasion of his eighteenth birthday, he wrote the preface to his first printed work - a treatise concerned with the application of algebraic methods to problems of elementary and higher geometry.

  1075. Beatty biography
    • There are many books dealing in an individual way with elementary aspects of Algebra, Geometry, or Analysis.

  1076. Martin biography
    • It was during these few months at the Franklin Academy that he was introduced to basic algebra, geometry and trigonometry.

  1077. Jones Vaughan biography
    • Jones has made many other contributions to the mathematical community, particularly in his editorial work as editor or associate editor for many journals: the Transactions of the American Mathematical Society, the Pacific Mathematics Journal, the Annals of Mathematics, the New Zealand Journal of Mathematics, Advances in Mathematics, the Journal of Operator Theory, Reviews in Mathematical Physics, the Russian Journal of Mathematical Physics, the Journal of Mathematical Chemistry, Geometry and Topology, and L'Enseignement Mathematique.

  1078. Vinogradov biography
    • Nauk SSSR (9) (1991), 91-103.',26)">26] gives summaries of the 10 one hour 'Vinogradov lectures' devoted to number theory and related problems in algebraic geometry.

  1079. Meders biography
    • Meders worked on differential geometry and mathematical analysis.

  1080. Al-Samawal biography
    • These teachers had covered topics including an introduction to surveying, elementary algebra, and the geometry of the first few books of Euclid's Elements.

  1081. Cramer biography
    • Cramer taught geometry and mechanics while Calandrini taught algebra and astronomy.

  1082. Straus biography
    • geometry, convexity, combinatorics, group theory and linear algebra.

  1083. Schrodinger biography
    • He also studied projective geometry, algebraic curves and continuous groups in lectures given by Gustav Kohn.

  1084. Kadets biography
    • a disciplined, keen, and concise language on a background of meticulous techniques of analysis and geometry.

  1085. Hutton biography
    • The first volume looks at topics such as: arithmetic including discussion of square and cube roots, arithmetical and geometrical progressions, compound interest, double position and permutations and combinations; logarithms; algebra including the study of quadratic equations and the Cardan-Tartaglia method for cubic equations; geometry which follows the approach in Euclid's Elements; surveying; and conic sections.

  1086. Walker John biography
    • He wrote some articles on theoretical mechanics but his more elaborate papers were on advanced algebra and geometry.

  1087. Nielsen Jakob biography
    • Eventually Fenchel wrote a book Elementary geometry in hyperbolic space which was intended to put give an approach with would make presentation of Fenchel-Nielsen theory much clearer.

  1088. Li Zhi biography
    • We should emphasise that it is not a geometry book which is why relating each problem to a single figure is possible.

  1089. Ostrowski biography
    • These are determinants, linear algebra, algebraic equations, multivariate algebra, formal algebra, number theory, geometry, topology, convergence, theory of real functions, differential equations, differential transformations, theory of complex functions, conformal mappings, numerical analysis and miscellany.

  1090. Trudinger biography
    • In recent years, members of the programme have solved major open problems in curvature flow, affine geometry and optimal transportation, using techniques from nonlinear partial differential equations.

  1091. Ampere biography
    • Mathematically he continued to produce good work, this time an interesting treatise on analytic geometry.

  1092. Heron biography
    • The mechanicians of Heron's school say that mechanics can be divided into a theoretical and a manual part; the theoretical part is composed of geometry, arithmetic, astronomy and physics, the manual of work in metals, architecture, carpentering and painting and anything involving skill with the hands.

  1093. Fenyo biography
    • The material should therefore be regarded as supplementing the methods of classical analysis, algebra and geometry which are now the bread and butter of research workers.

  1094. Greaves John biography
    • With support from John Bainbridge and Peter Turner, Greaves was appointed as Professor of Geometry at Gresham College, London, in February 1631.

  1095. Ratner biography
    • I was fascinated with mathematical reasoning in algebra and geometry, which was beautiful and exciting.

  1096. Jerabek biography
    • His main research interest was in constructive geometry.

  1097. Apollonius biography
    • While Apollonius was at Pergamum he met Eudemus of Pergamum (not to be confused with Eudemus of Rhodes who wrote the History of Geometry) and also Attalus, who many think must be King Attalus I of Pergamum.

  1098. De Moivre biography
    • De Moivre pioneered the development of analytic geometry and the theory of probability.

  1099. Schoenberg biography
    • At Princeton he began working on distance geometry, namely [J.

  1100. Heyting biography
    • His dissertation "Intuitionistische axiomatieks der projektieve meetkunde" (Intuitionistic axiomatics of projective geometry) was the first study of axiomatisation in constructive mathematics.

  1101. Wessel biography
    • He possesses a lot of theoretical knowledge of algebra, trigonometry and mathematical geometry, and as far as the last point is concerned, he has come up with some new and beautiful solutions to the most difficult problems in geographical surveying.

  1102. Riccati biography
    • Riccati also worked on cycloidal pendulums, the laws of resistance in a fluid and differential geometry.

  1103. Rado Richard biography
    • He studied inequalities, geometry and measure theory, particularly working in this area with Besicovitch.

  1104. Scholz biography
    • There is no clear indication which branches of mathematics satisfy this condition; it seems that abstract algebras and other axiomatic theories do, intuitive geometry and numerical arithmetic, called a 'kind' of calculus ..

  1105. Duarte biography
    • He was professor of geometry, algebra, analysis and mechanics at UCV (1909-1911 and 1936-1939).

  1106. Routh biography
    • The research areas which interested him most were geometry, dynamics, astronomy, waves, vibrations and harmonic analysis.

  1107. Machin biography
    • In 1706 William Jones published a work Synopsis palmariorum matheseos or, A New Introduction to the Mathematics, Containing the Principles of Arithmetic and Geometry Demonstrated in a Short and Easie Method ..

  1108. Al-Nasawi biography
    • On the one hand he says that it will act as an introduction to the Elements while on the other hand it will provide all the necessary background in geometry for anyone wanting to read Ptolemy's Almagest.

  1109. Bugaev biography
    • In this work Bugaev describes mathematics as based on the theory of functions, with geometry and the theory of probability having a minor role.

  1110. Browder Felix biography
    • The subject had its origins in the study of nonlinear ordinary and partial differential equations, but it came to encompass a wider range of questions in all branches of analysis and in differential geometry, in theoretical physics, and in economics.

  1111. Chernikov biography
    • In 1961 Chernikov was appointed as Head of the Department of Algebra and Geometry of the Sverdlovsk branch of the Steklov Institute of the USSR Academy of Sciences.

  1112. Blackwell biography
    • On the other hand he quite enjoyed geometry, having a good teacher of the suject at high school, and he applied his growing mathematical skills to games such as noughts and crosses where he began to analyse whether there was always a winning strategy for the first player.


History Topics

  1. Non-Euclidean geometry
    • Non-Euclidean geometry .
    • Geometry and topology index .
    • Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing.
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    • Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased.
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    • Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Elements de Geometrie.
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    • Elementary geometry was by this time engulfed in the problems of the parallel postulate.
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    • D'Alembert, in 1767, called it the scandal of elementary geometry.
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    • He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line.
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    • At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy.
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    • However in some sense Bolyai only assumed that the new geometry was possible.
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    • However the real breakthrough was the belief that the new geometry was possible.
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    • Nor is Bolyai's work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829.
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    • The publication of an account in French in Crelle's Journal in 1837 brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not ready to accept ideas so revolutionary.
      Go directly to this paragraph
    • In Lobachevsky's 1840 booklet he explains clearly how his non-Euclidean geometry works.
    • Lobachevsky went on to develop many trigonometric identities for triangles which held in this geometry, showing that as the triangle became small the identities tended to the usual trigonometric identities.
    • Riemann, who wrote his doctoral dissertation under Gauss's supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length.
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    • Riemann briefly discussed a 'spherical' geometry in which every line through a point P not on a line AB meets the line AB.
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    • In this geometry no parallels are possible.
      Go directly to this paragraph
    • It is important to realise that neither Bolyai's nor Lobachevsky's description of their new geometry had been proved to be consistent.
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    • In fact it was no different from Euclidean geometry in this respect although the many centuries of work with Euclidean geometry was sufficient to convince mathematicians that no contradiction would ever appear within it.
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    • The first person to put the Bolyai - Lobachevsky non-Euclidean geometry on the same footing as Euclidean geometry was Eugenio Beltrami (1835-1900).
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    • In 1868 he wrote a paper Essay on the interpretation of non-Euclidean geometry which produced a model for 2-dimensional non-Euclidean geometry within 3-dimensional Euclidean geometry.
      Go directly to this paragraph
    • It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry.
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    • Beltrami's work on a model of Bolyai - Lobachevsky's non-Euclidean geometry was completed by Klein in 1871.
      Go directly to this paragraph
    • Klein went further than this and gave models of other non-Euclidean geometries such as Riemann's spherical geometry.
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    • Klein showed that there are three basically different types of geometry.
      Go directly to this paragraph
    • In the Bolyai - Lobachevsky type of geometry, straight lines have two infinitely distant points.
      Go directly to this paragraph
    • In the Riemann type of spherical geometry, lines have no (or more precisely two imaginary) infinitely distant points.
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    • Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.
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    • Geometry and topology index .
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry.html .

  2. Fractal Geometry
    • A History of Fractal Geometry .
    • Geometry and topology index .
    • A typical student will, at various points in her mathematical career -- however long or brief that may be -- encounter the concepts of dimension, complex numbers, and "geometry".
    • If the field of mathematics does not particularly interest her, this student might see these concepts as distinct and unrelated and, in particular, she might make the mistake of thinking that the Euclidean geometry taught to her in school encompasses the whole of the field of geometry.
    • However, if she were to pursue mathematics at the university level, she might discover an exciting and relatively new field of study that links the aforementioned ideas in addition to many others: fractal geometry.
    • While the lion's share of the credit for the development of fractal geometry goes to Benoit Mandelbrot, many other mathematicians in the century preceding him had laid the foundations for his work.
    • 1972 ',5)">5] [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] Indeed, when one has only worked with curves that are differentiable almost everywhere, an obvious question when one encounters a formula for a curve that is not is, "what does it look like?" .
    • In 1883 Georg Cantor, who attended lectures by Weierstrass during his time as a student at the University of Berlin [9] and who is to set theory what Mandelbrot is to fractal geometry, [Classics on Fractals (Addison-Wesley, 1993).
    • functions that "have no tangents" in geometric parlance) could exist -- a way that involved using "elementary geometry" (reference [Classics on Fractals .
    • A E Gerald (Addison -Wesley, 1993).',6)">6]'s title translates to On a Continuous Curve without Tangent Constructible from Elementary Geometry).
    • In doing so, von Koch expressed a link between these non-differentiable "monsters" of analysis and geometry.
    • 1972 ',5)">5] Poincare, it should be noted, studied non-linear dynamics in the later 19th century, which eventually led to chaos theory, [Introduction to Fractals and Chaos (London, 1995).',2)">2] a field closely related to fractal geometry, though beyond the scope of this paper.
    • The Hausdorff dimension, d, of a self-similar set -- its connection to fractal geometry, though, as previously stated, there are many other applications of Hausdorff dimension -- which is scaled down by ratios r1 , r2 , ..
    • At nearly the same time that Hausdorff did his research, two French mathematicians, Gaston Julia and Pierre Fatou, developed results (though not together) that ended up being important to fractal geometry.
    • The boundaries of the various basins of attraction turned out to be very complicated and are known today as Julia sets, [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] an example of which can be seen in Figure 6.
    • [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] Julia published a 199-page paper in 1918 called Memoire sur l'iteration des fonctions rationelles, which discussed much of his work on iterative functions and describing the Julia set.
    • On rare occasions, they can be "dendrites" (Figure 8), where they are "made up completely of continuously sub-branching lines, which are only just connected since the removal of any point from them would split them in two," [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] at which point, they would be considered "dust".
    • [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] .
    • [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] .
    • Mandelbrot, like Helge von Koch before him, preferred visual representations of mathematical problems, as opposed to the symbolic, [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] though this may also stem from his lack of formal education, due to World War II.
    • [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] .
    • While this method was not always possible on other sections, he managed to pass [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] and after a one-day career at the Ecole Normale, Mandelbrot started at the Ecole Polytechnique, where he met another of his mentors, Paul Levy, [13] who was a professor at there from 1920 until his retirement in 1959 [12].
    • [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] .
    • Mandelbrot has managed not only to invent the discipline of fractal geometry, but has also popularized it through its applications to other areas of science.
    • As he hinted in How Long Is the Coast of Britain? fractal geometry comes in useful in representing natural phenomena; things such as coastlines, the silhouette of a tree, or the shape of snowflakes -- things are not easily represented using traditional Euclidean geometry.
    • Equally, no simple shape from Euclidean geometry comes to mind when contemplating things such as the path of a river.
    • Furthermore, fractal geometry and chaos theory have important connections to physics, medicine, and the study of population dynamics.
    • [Introducing Fractal Geometry (Cambridge, 2000).',7)">7] However, even if the field lacked these links, it would be hard for those so inclined to resist the aesthetic appeal of most fractals.
    • However, through fractal geometry, many of these seemingly abstract ideas (from mathematicians who are relatively unknown outside of their own spheres of research) develop applications that other scientists and even non-scientists can appreciate.
    • Geometry and topology index .

  3. Non-Euclidean geometry references
    • References for: Non-Euclidean geometry .
    • R Bonola, Non-Euclidean Geometry : A Critical and Historical Study of its Development (New York, 1955).
    • T R Chandrasekhar, Non-Euclidean geometry from early times to Beltrami, Indian J.
    • N Daniels,Thomas Reid's discovery of a non-Euclidean geometry, Philos.
    • J J Gray, Euclidean and non-Euclidean geometry, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 877-886.
    • J J Gray, Non-Euclidean geometry-a re-interpretation, Historia Mathematica 6 (3) (1979), 236-258.
    • J J Gray, The discovery of non-Euclidean geometry, in Studies in the history of mathematics (Washington, DC, 1987), 37-60.
    • T Hawkins, Non-Euclidean geometry and Weierstrassian mathematics : the background to Killing's work on Lie algebras, Historia Mathematica 7 (3) (1980), 289-342.
    • C Houzel, The birth of non-Euclidean geometry, in 1830-1930 : a century of geometry (Berlin, 1992), 3-21.
    • V F Kagan, The construction of non-Euclidean geometry by Lobachevskii, Gauss and Bolyai (Russian), Akad.
    • B Mayorga, Lobachevskii and non-Euclidean geometry (Spanish), Lect.
    • T Pati, The development of non-Euclidean geometry during the last 150 years, Bull.
    • B A Rosenfeld, A history of non-euclidean geometry : evolution of the concept of a geometric space (New York, 1987).
    • B A Rozenfel'd, History of non-Euclidean geometry : Development of the concept of a geometric space (Russian) (Moscow, 1976).
    • D M Y Sommerville, Bibliography of non-euclidean geometry (New York, 1970).
    • B Szenassy, Remarks on Gauss's work on non-Euclidean geometry (Hungarian), Mat.
    • I Toth, From the pre-history of non-euclidean geometry (Hungarian), Mat.
    • A Vucinich, Nikolai Ivanovich Lobachevskii : the man behind the first non-Euclidean geometry, Isis 53 (1962), 465-481.
    • http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Non-Euclidean_geometry.html .

  4. Non-Euclidean geometry references
    • References for: Non-Euclidean geometry .
    • R Bonola, Non-Euclidean Geometry : A Critical and Historical Study of its Development (New York, 1955).
    • T R Chandrasekhar, Non-Euclidean geometry from early times to Beltrami, Indian J.
    • N Daniels,Thomas Reid's discovery of a non-Euclidean geometry, Philos.
    • J J Gray, Euclidean and non-Euclidean geometry, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 877-886.
    • J J Gray, Non-Euclidean geometry-a re-interpretation, Historia Mathematica 6 (3) (1979), 236-258.
    • J J Gray, The discovery of non-Euclidean geometry, in Studies in the history of mathematics (Washington, DC, 1987), 37-60.
    • T Hawkins, Non-Euclidean geometry and Weierstrassian mathematics : the background to Killing's work on Lie algebras, Historia Mathematica 7 (3) (1980), 289-342.
    • C Houzel, The birth of non-Euclidean geometry, in 1830-1930 : a century of geometry (Berlin, 1992), 3-21.
    • V F Kagan, The construction of non-Euclidean geometry by Lobachevskii, Gauss and Bolyai (Russian), Akad.
    • B Mayorga, Lobachevskii and non-Euclidean geometry (Spanish), Lect.
    • T Pati, The development of non-Euclidean geometry during the last 150 years, Bull.
    • B A Rosenfeld, A history of non-euclidean geometry : evolution of the concept of a geometric space (New York, 1987).
    • B A Rozenfel'd, History of non-Euclidean geometry : Development of the concept of a geometric space (Russian) (Moscow, 1976).
    • D M Y Sommerville, Bibliography of non-euclidean geometry (New York, 1970).
    • B Szenassy, Remarks on Gauss's work on non-Euclidean geometry (Hungarian), Mat.
    • I Toth, From the pre-history of non-euclidean geometry (Hungarian), Mat.
    • A Vucinich, Nikolai Ivanovich Lobachevskii : the man behind the first non-Euclidean geometry, Isis 53 (1962), 465-481.
    • [http://www-history.mcs.st-andrews.ac.uk/HistTopics/References/Non-Euclidean_geometry.html] .

  5. Fractal Geometry references
    • References for: A History of Fractal Geometry .
    • Introducing Fractal Geometry (Cambridge, 2000).

  6. Fractal Geometry references
    • References for: A History of Fractal Geometry .
    • Introducing Fractal Geometry (Cambridge, 2000).

  7. Kepler's Planetary Laws
    • 5.nnEssential orthogonality of Euclid's geometry .
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    • In Kepler's day modern algebraic notation and techniques were just being developed, but for his approach to astronomy Kepler depended exclusively on the traditional geometry of Euclid in which he had been trained at the University of Tubingen, as part of the standard preparation for the ministry.
    • Thus, the distinguishing feature of the geometry of Elements was that it relied on straight lines and circles alone.
    • 129-141 AD), Kepler made use of precisely three propositions from the work of Archimedes; one of these was vital in supplying the geometrical backing for Section 6 (the other two - one cited in Section 7, one in Section 11 - were concerned with an innovative approach to 'infinitesimal' considerations which went well beyond traditional geometry).
    • Meanwhile we reiterate Kepler's belief that Euclid's Elements encapsulated the only geometry that could properly be applied to the heavens, which after all was the realm of God.
    • So he finally rejected the idea that each planet moved in a single circle, and set out to find the actual curve that was the planet's path - naturally, this had to be constructed from a combination of (arcs of) circles by the geometry of Euclid, since Kepler recognized nothing else as appropriate for the heavens.
    • Unless its focus coincides with the fixed Sun (the origin), the investigation would have been too complicated to manage by geometry.
    • Kepler was able to formulate a complete account of planetary motion using only elementary geometry, and accordingly we will highlight the two overriding reasons for his achievement, putting them in a historical context.
    • A E L Davis: 'Some plane geometry from a cone: the focal distance of an ellipse at a glance', Mathematical Gazette, forthcoming July 2007.

  8. Mathematics and Art
    • First let us state the problem: how does one represent the three-dimensional world on a two-dimensional canvass? There are two aspects to the problem, namely how does one use mathematics to make realistic paintings and secondly what is the impact of the ideas for the study of geometry.
    • In this way Brunelleschi controlled precisely the position of the spectator so that the geometry was guaranteed to be correct.
    • It is reasonable to think about how Brunelleschi came to understand the geometry which underlies perspective.
    • Certainly he was trained in the principles of geometry and surveying methods and, since he had a fascination with instruments, it is reasonable to suppose that he may have used instruments to help him survey buildings.
    • Alberti gives background on the principles of geometry, and on the science of optics.
    • In Trattato d'abaco which he probably wrote around 1450, Piero includes material on arithmetic and algebra and a long section on geometry which was very unusual for such texts at the time.
    • Perhaps it is most accurate to say that he is studying the geometry of vision which he later makes clearer:- .
    • Not only did Leonardo study the geometry of perspective but he also studied the optical principles of the eye in his attempts to create reality as seen by the eye.
    • This treatise represents a major step forward in understanding the geometry of perspective and it was a major contribution towards the development of projective geometry.
    • Three years later, in 1639, Desargues wrote his treatise on projective geometry Brouillon project d'une atteinte aux evenemens des rencontres du cone avec un plan.
    • On the other hand the algebraic approach to geometry put forward by Descartes at almost exactly the same time (1637) may have diverted attention from Desargues' projective methods.
    • In fact la Hire had treated conics from a projective point of view in his 1673 treatise New method of geometry for sections of conics and cylindrical surfaces and there he had introduced the cross ratio of four points before meeting Desargues' approach.
    • One could certainly consider this work as being an important step towards the theory of descriptive and projective geometry as developed by Monge, Chasles and Poncelet.

  9. Group theory
    • geometry at the beginning of the 19th Century, .
    • (1) Geometry has been studied for a very long time so it is reasonable to ask what happened to geometry at the beginning of the 19th Century that was to contribute to the rise of the group concept.
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    • Geometry had began to lose its 'metric' character with projective and non-euclidean geometries being studied.
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    • Also the movement to study geometry in n dimensions led to an abstraction in geometry itself.
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    • The difference between metric and incidence geometry comes from the work of Monge, his student Carnot and perhaps most importantly the work of Poncelet.
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    • Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and Janos Bolyai among others.
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    • Mobius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group.
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    • Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.
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    • Klein proposed the Erlangen Program in 1872 which was the group theoretic classification of geometry.
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  10. History overview
    • Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.
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    • The most important mathematician of the 18th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry.
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    • The period around the turn of the century saw Laplace's great work on celestial mechanics as well as major progress in synthetic geometry by Monge and Carnot.
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    • In geometry Plucker produced fundamental work on analytic geometry and Steiner in synthetic geometry.
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    • Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann.
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    • His work in differential geometry was to revolutionise the topic.
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    • Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann.
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  11. Mathematics and Architecture
    • The first mentioned type of architecture Salingaros mentions in this quote is the pyramid and here we have marked disagreement between experts on the how much geometry and number theory the architects used.
    • However, no proof exists that sophisticated geometry lies behind the construction of the pyramids.
    • tried to persuade his students that abstract geometry was historically prior to its practical applications, and that the pyramids and buildings of ancient Egypt "proved" that their architects were highly sophisticated mathematicians.
    • Geometry was the study of shapes and shapes were determined by numbers.
    • He suggests that the circle and the square are perfect figures for generating architectural designs because they approximate the geometry of the spread-eagled human body.
    • Another 17th Century mathematician was La Hire whose interests geometry arose from his study of architecture.
    • His interest in geometry arose from his study of perspective and he went on to make important contributions to conic sections.
    • He made outstanding contributions to geometry.
    • With this training he went on to become a teacher of physics, mechanics, hydraulics and descriptive geometry at the Technische Hochschule in Darmstadt.

  12. Bolzano publications.html
    • The volume contains four of Bolzano's memoirs on geometry: Betrachtungen uber einige Gegenstande der Elementargeometrie; Versuch einer objectiven Begrundung der Lehre von den drei Dimensionen des Raumes; Die drey Probleme der Rectification, der Complanation und der Cubirung; and uber Haltung, Richtung, Krummung und Schnorkelung bei Linien.
    • E Winter conjectured (in 1933) - without proof - that these folios constitute meagre fragments of Bolzano's work 'Anti-Euclid' which - according to Bolzano's own report - was lost (it is perhaps possible that the lost 'Anti-Euclid' was written "according to such a detailed plan"), and that this work contained the concept of non-Euclidean geometry.
    • The available text contains only ideas concerning the reform and improvement of Euclidean geometry.
    • This attempts an axiomatisation of geometry.
    • Most manuscripts of the present volume constitute steps toward the realization of a planned sequel to that book; their contents range from an exposition of General Mathesis, supplemented by an extensive analysis of the notion of quantity, through a theory of cause and consequence, called 'aetiology', to essays on geometry and mechanics.
    • Contains his thoughts on Euclidean geometry, manipulations of series, functions and foundations of calculus, and topics in mechanics.
    • Contains reprints of the following papers by Bolzano: Considerations on some points in elementary geometry (1804), Contributions to a better founded exposition of mathematics (1810), The binomial theorem (1816), Pure analytical proof of the intermediate value theorem (1817), and The three problems of curve length, surface area and volume (1817).
    • Covers topics such as geometry, calculus, and mechanics frequently making philosophical commnts.
    • In these entries Bolzano considers geometry at both an elementary and advanced level, mechanics, and the foundation of mathematics.

  13. Abstract linear spaces
    • Cartesian geometry, introduced by Fermat and Descartes around 1636, had a very large influence on mathematics bringing algebraic methods into geometry.
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    • methods of synthetic geometry which were coordinate free.
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    • In 1804 he published a work on the foundations of elementary geometry Betrachtungen uber einige Gegenstande der Elementargoemetrie.
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    • This is an important step in the axiomatisation of geometry and an early move towards the necessary abstraction for the concept of a linear space to arise.
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    • The move away from coordinate geometry was mainly due to the work of Poncelet and Chasles who were the founders of synthetic geometry.
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    • Another mathematician who was moving towards geometry without coordinates was Grassmann.
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  14. Arabic mathematics
    • It was a revolutionary move away from the Greek concept of mathematics which was essentially geometry.
    • Al-Khwarizmi's successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra.
    • Sharaf al-Din al-Tusi (born 1135), although almost exactly the same age as al-Samawal, does not follow the general development that came through al-Karaji's school of algebra but rather follows Khayyam's application of algebra to geometry.
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    • represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.
    • Although the Arabic mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy.
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    • 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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    • Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy and also used formulas involving sin and tan.
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  15. Euclid's definitions
    • Geometry index .
    • The next point to note is that they are very similar to the work which is ascribed to Heron called Definitions of terms in geometry.
    • Is Definitions of terms in geometry based on Euclid's Elements or have the basic definitions from this work been inserted into later versions of The Elements? .
    • Let us go back to Sextus who writes about "mathematicians describing geometrical entities" and it is interesting that the word "describing" is not used in The Elements but is used by Heron in Definitions of terms in geometry.
    • The hypothesis is that Sextus has The Elements and Definitions of terms in geometry in front of him when he is writing and he uses the word "describe" when he refers to Heron and "define" when he refers to Euclid.
    • Compare it with the definition of a straight line in Definitions of terms in geometry: .
    • Geometry index .

  16. EMS History
    • I want to speak now of the great development of geometry, for which we are indebted to Monge who is the real originator of all that is best in modern geometry.
    • Nothing in connection with our Universities is more astounding to a foreigner than the fact that there are large numbers of students enrolled every year to begin the first proposition of Euclid, and that, of all the mathematical students within the walls, by far the greater portion have confined their studies to elementary Algebra, Geometry and Trigonometry.
    • A few years later, in a less direct and poignant way, Mackay found himself forestalled by the publication of Allman's "Greek Geometry".
    • A Course of Five Lectures by D M Y Sommerville, Esq., M.A., D.Sc., Lecturer in Mathematics in the University of St Andrews, on Non-Euclidean Geometry and the Foundations of Geometry.
    • (Fellow and Lecturer of King's College, Cambridge, and University Lecturer in Mathematics), on Infinity in Geometry.

  17. Squaring the circle
    • There are three classical problems in Greek mathematics which were extremely influential in the development of geometry.
    • 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]):- .
    • There are, we say, three types of problem in geometry, the so-called 'plane', 'solid', and 'linear' problems.
    • Oenopides is thought by Heath to be the person who required a plane solution to geometry problems.
    • [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..
    • The reasoning used by the former cannot be applied to any subject other than geometry alone, whereas Bryson's argument is directed to the mass of people who do not know what is possible and what is impossible in each department, for it will fit any.

  18. Mathematics and Art references
    • A study of Taylor's role in the history of perspective geometry.
    • K Andersen, Desargues' method of perspective : its mathematical content, its connection to other perspective methods and its relation to Desargues' ideas on projective geometry, Centaurus 34 (1) (1991), 44-91.
    • J V Field, Linear perspective and the projective geometry of Girard Desargues, Nuncius Ann.
    • A Pucci, A mathematico-esthetic itinerary : from perspective to projective geometry, Quad.
    • A Sarounova, Geometry and painting : The origins of linear perspective (Czech), Pokroky Mat.

  19. Greek astronomy
    • The next development which was absolutely necessary for progress in astronomy took place in geometry.
    • Spherical geometry was developed by a number of mathematicians with an important text being written by Autolycus in Athens around 330 BC.
    • Some claim that Autolycus based his work on spherical geometry On the Moving Sphere on an earlier work by Eudoxus.
    • There Euclid worked and wrote on geometry in general but also making an important contribution to spherical geometry.

  20. Mathematics and Art references
    • A study of Taylor's role in the history of perspective geometry.
    • K Andersen, Desargues' method of perspective : its mathematical content, its connection to other perspective methods and its relation to Desargues' ideas on projective geometry, Centaurus 34 (1) (1991), 44-91.
    • J V Field, Linear perspective and the projective geometry of Girard Desargues, Nuncius Ann.
    • A Pucci, A mathematico-esthetic itinerary : from perspective to projective geometry, Quad.
    • A Sarounova, Geometry and painting : The origins of linear perspective (Czech), Pokroky Mat.

  21. Cubic surfaces
    • Geometry and topology index .
    • At the end of his 1865 treatise The Geometry of Three Dimensions Salmon described how the two had collaborated over finding the Cayley-Salmon theorem.
    • In particular Le Paige studied the geometry of algebraic curves and surfaces, building on this earlier work.
    • Fano studied with Klein in 1893 and did an Italian translation of Klein's Erlanger Program (1872), which gave his synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations.
    • Geometry and topology index .

  22. Mathematics and Architecture references
    • D J Bassegoda Nonell, Line geometry and architecture (Spanish), Mem.
    • G Necipoglu, The Topkapi scroll - geometry and ornament in Islamic architecture, Topkapi Palace Museum Library MS H.
    • M Rubin, Architecture and geometry, Structural Topology No.
    • Vienna, 1709) on Chinese painting, drawing and architecture, in Proceedings of the Third International Conference on Engineering Graphics and Descriptive Geometry 2, Vienna, 1988 (Vienna, 1988), 323-329.

  23. Christianity and Mathematics
    • I undertake to prove that God, in creating the universe and regulating the order of the cosmos, had in view the five regular bodies of geometry known since the days of Pythagoras and Plato, and that he has fixed according to those dimensions, the number of heavens, their proportions and the relations of their movements.
    • Geometry existed before the creation; is co-eternal with the mind of God; is God himself ..
    • Where there is matter there is geometry.
    • geometry provided God with a model for the Creation and was implanted into man, together with God's own likeness - and not merely conveyed to his mind through the eyes.

  24. Mathematics and Architecture references
    • D J Bassegoda Nonell, Line geometry and architecture (Spanish), Mem.
    • G Necipoglu, The Topkapi scroll - geometry and ornament in Islamic architecture, Topkapi Palace Museum Library MS H.
    • M Rubin, Architecture and geometry, Structural Topology No.
    • Vienna, 1709) on Chinese painting, drawing and architecture, in Proceedings of the Third International Conference on Engineering Graphics and Descriptive Geometry 2, Vienna, 1988 (Vienna, 1988), 323-329.

  25. Doubling the cube
    • There are three classical problems in Greek mathematics which were extremely important in the development of geometry.
    • 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.
    • We shall use some coordinate geometry in a moment to see that Archytas is correct, but first let us give the construction in the words of Eutocius, unchanged except for the names of the point which I have changed to fit the notation of our diagram and that described above (see for example [History of Mathematics : History of Problems (Paris, 1997), 89-113.',7)">7]):- .
    • In proceeding in this way, did not one lose irredeemably the best of geometry, by a regression to a level of the senses, which prevents one from creating and even perceiving the eternal and incorporeal images among which God is eternally god.

  26. Indian mathematics
    • Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita.
    • Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.
    • The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

  27. Topology history
    • Geometry and topology index .
    • In 1736 Euler published a paper on the solution of the Konigsberg bridge problem entitled Solutio problematis ad geometriam situs pertinentis which translates into English as The solution of a problem relating to the geometry of position.
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    • The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.
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    • Geometry and topology index .

  28. Egyptian Papyri references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).
    • E M Bruins, On the approximation to π/4 in Egyptian geometry (Dutch), Nederl.
    • R Lehti, Geometry of the Egyptians (Finnish), Arkhimedes (2) (1971), 15-28.

  29. Egyptian mathematics references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).
    • E M Bruins, On the approximation to π/4 in Egyptian geometry (Dutch), Nederl.
    • R Lehti, Geometry of the Egyptians (Finnish), Arkhimedes (2) (1971), 15-28.

  30. Egyptian Papyri references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).
    • E M Bruins, On the approximation to π/4 in Egyptian geometry (Dutch), Nederl.
    • R Lehti, Geometry of the Egyptians (Finnish), Arkhimedes (2) (1971), 15-28.

  31. Trisecting an angle
    • There are three classical problems in Greek mathematics which were extremely influential in the development of geometry.
    • We say that there are three kinds of problem in geometry, the so-called 'plane', 'solid', and 'linear' problems.
    • In proceeding in [a mechanical] way, did not one lose irredeemably the best of geometry..

  32. Bolzano's manuscripts
    • He had worked for many years on Grossenlehre (Theory of quantity) which was intended to be an introduction to mathematics covering many different areas of mathematics such as numbers, elementary geometry, geometry in general, function theory, methodology, and the ideas of quantity and space.
    • It contained Bolzano's ideas concerning the reform and improvement of Euclidean geometry.

  33. Egyptian mathematics references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).
    • E M Bruins, On the approximation to π/4 in Egyptian geometry (Dutch), Nederl.
    • R Lehti, Geometry of the Egyptians (Finnish), Arkhimedes (2) (1971), 15-28.

  34. Hirst's diary
    • (18 Nov 1857) I went to hear Chasles' first lecture on geometry, and was far from satisfied with it.
    • (5 June 1864) [Cremona] had a class of about 12 and lectured on the theory of the sundial in connection with his descriptive geometry.
    • The reason is clear: firstly he does not know Latin, and that among German professors is held as a necessity; secondly he is so terribly one-sided on the question of synthetic geometry that as an examiner he would not be liked.

  35. General relativity
    • If all accelerated systems are equivalent, then Euclidean geometry cannot hold in all of them.
    • Einstein then remembered that he had studied Gauss's theory of surfaces as a student and suddenly realised that the foundations of geometry have physical significance.
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    • Before that however he had written a paper in October 1914 nearly half of which is a treatise on tensor analysis and differential geometry.

  36. Special relativity references
    • M Paty, Physical geometry and special relativity, in Einstein et Poincare, 1830-1930 : a century of geometry (Berlin, 1992), 127-149.

  37. Doubling the cube references
    • W R Knorr, Textual studies in ancient and medieval geometry (Boston, 1989).
    • K Saito, Doubling the cube : a new interpretation of its significance for early Greek geometry, Historia Math.

  38. Bourbaki 1
    • A large number of subcommittees were formed, given the size of the group, and these were to cover the following topics: algebra, analytic functions, integration theory, differential equations, existence theorems for differential equations, partial differential equations, differentials and differential forms, calculus of variations, special functions, geometry, Fourier series, and representations of functions.
    • It proved impossible to retain the classical division into analysis, differential calculus, geometry, algebra, number theory, etc.

  39. Pi history
    • These are both dramatic and astonishing formulae, for the expressions on the right are completely arithmetical in character, while π arises in the first instance from geometry.
    • All of thy geometry, Herr Planck, is fairly hard..
      Go directly to this paragraph

  40. Doubling the cube references
    • W R Knorr, Textual studies in ancient and medieval geometry (Boston, 1989).
    • K Saito, Doubling the cube : a new interpretation of its significance for early Greek geometry, Historia Math.

  41. Babylonian mathematics references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).
    • E M Bruins, On Babylonian geometry, Nederl.

  42. Zero
    • 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.
    • Although Euclid's Elements contains a book on number theory, it is based on geometry.

  43. Babylonian mathematics references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).
    • E M Bruins, On Babylonian geometry, Nederl.

  44. Cartography
    • In Egypt geometry was used from very early times to help measure land.
    • Note, of course, that the use of such positional grids are an early form of Cartesian geometry.

  45. Special relativity references
    • M Paty, Physical geometry and special relativity, in Einstein et Poincare, 1830-1930 : a century of geometry (Berlin, 1992), 127-149.

  46. The four colour theorem
    • Geometry and topology index .
    • Geometry and topology index .

  47. Rose literature
    • It is preserved entire; of its thirteen books, the first six deal with plane geometry, the next three with arithmetic, Book X, the longest of all, with irrational quantities, and the remainder with solid geometry.

  48. Bourbaki 2
    • They decided on producing advanced texts on commutative algebra, algebraic geometry, Lie groups, global and functional analysis, algebraic number theory, and automorphic forms.
    • There were attempts at homotopy theory, at spectral theory of operators, at the index theorem, at symplectic geometry.

  49. Physical world
    • Euclid set up geometry in this way but there were interesting aspects of this as far as physical science was concerned.
    • More worrying as far as physical science was concerned, is the fact that the objects of Euclid's geometry can have no physical existence.

  50. function concept
    • At almost the same time that Galileo was coming up with these ideas, Descartes was introducing algebra into geometry in La Geometrie.
    • Thus began the long controversy about the nature of functions to be allowed in the initial conditions and in the integrals of partial differential equations, which continued to appear in an ever increasing number in the theory of elasticity, hydrodynamics, aerodynamics, and differential geometry.

  51. Babylonian numerals
    • Some theories are based on geometry.

  52. Babylonian and Egyptian references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

  53. Nine chapters references
    • E I Berezkina (trs.), Two texts of Liu Hui on geometry (Russian), in Studies in the history of mathematics, No.

  54. Babylonian mathematics references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

  55. Babylonian Pythagoras references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

  56. Babylonian numerals references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

  57. Topology history references
    • V L Hansen, From geometry to topology (Danish), Normat 36 (2) (1988), 48-60.

  58. Orbits references
    • N A Chernikov, Introduction of Lobachevskii geometry into the theory of gravitation, Soviet J.

  59. Greek astronomy references
    • L Wright, The astronomy of Eudoxus : geometry or physics?, Studies in Hist.

  60. Kepler's Laws references
    • Because it can be derived directly from a section of a cone in three easy steps, as demonstrated in section 6 of A E L Davis, Some plane geometry from a cone: the focal distance of an ellipse at a glance, Mathematical Gazette, forthcoming July 2007.

  61. Planetary motion references
    • A E L Davis: 'Some plane geometry from a cone ..

  62. Nine chapters references
    • E I Berezkina (trs.), Two texts of Liu Hui on geometry (Russian), in Studies in the history of mathematics, No.

  63. Babylonian Pythagoras references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

  64. Babylonian numerals references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

  65. Harriot's manuscripts
    • Three years after receiving Harriot's papers to referee for publication, Robertson was appointed Savilian Professor of Geometry and, in 1810, Savilian Professor of Astronomy.

  66. Kepler's Laws references
    • Because it can be derived directly from a section of a cone in three easy steps, as demonstrated in section 6 of A E L Davis, Some plane geometry from a cone: the focal distance of an ellipse at a glance, Mathematical Gazette, forthcoming July 2007.

  67. Bakhshali manuscript
    • It is devoted mainly to arithmetic and algebra, with just a few problems on geometry and mensuration.

  68. U of St Andrews History

  69. Perfect numbers
    • It may come as a surprise to many people to learn that there are number theory results in Euclid's Elements since it is thought of as a geometry book.

  70. Indian Sulbasutras
    • Some historians have argued that mathematics, in particular geometry, must have also existed to support astronomical work being undertaken around the same period.

  71. Sundials
    • 51-65.',5)" onmouseover="window.status='Click to see reference';return true">5] The markings on the clock indicating the four hours were very inaccurate, and were possibly not based on observation but rather some fallacy of celestial geometry.[Timing the sun in Egypt and Mesopotamia.

  72. Fair book insert
    • The next page is headed 'Practical Geometry'.

  73. Classical light
    • Grosseteste, in about 1220, stressed the significance of the properties of light to natural philosophy and in turn advocated using geometry to study light.

  74. Indian numerals
    • In this paper, however, Gupta claims that Indian numerals had reached Southern Europe by the end of the 5th century but his argument is based on the Geometry of Boethius which is now known to be a forgery dating from the first half of the 11th century.

  75. Quadratic etc equations
    • Scipione dal Ferro (1465-1526) held the Chair of Arithmetic and Geometry at the University of Bologna and certainly must have met Pacioli who lectured at Bologna in 1501-2.
      Go directly to this paragraph

  76. Egyptian mathematics references
    • B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).

  77. Greek astronomy references
    • L Wright, The astronomy of Eudoxus : geometry or physics?, Studies in Hist.

  78. Egyptian mathematics
    • Some problems involve geometry.

  79. Planetary motion
    • (It was proved, in a dynamical context, in Book I, Prop.15 of Newton's work, already cited.) However, it is possible to formulate a rational basis for the above deduction, founded on geometry - and so to produce a theoretical proof of Law III which would have been not so far beyond the conceptual understanding of a pre-Newtonian mathematician [Miscellanea Kepleriana, ed.

  80. Real numbers 2
    • In order to complete the connection presented in this section of the domains of the quantities defined [his determinate limits] with the geometry of the straight line, one must add an axiom which simple says that every numerical quantity also has a determined point on the straight line whose coordinate is equal to that quantity, indeed, equal in the sense in which this is explained in this section.

  81. Special relativity
    • These transformations, with a different scale factor, are now known as the Lorentz equations and the group of Lorentz transformations gives the geometry of special relativity.
      Go directly to this paragraph

  82. Greek sources II
    • 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.
      Go directly to this paragraph

  83. Orbits references
    • N A Chernikov, Introduction of Lobachevskii geometry into the theory of gravitation, Soviet J.

  84. Nine chapters
    • Quadratic equations are considered for the first time in Chapter 9, are solved by an analogue of division using ideas from geometry, in fact from the Chinese square-root algorithm, rather than from algebra.

  85. Topology history references
    • V L Hansen, From geometry to topology (Danish), Normat 36 (2) (1988), 48-60.

  86. Jaina mathematics
    • the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations.

  87. Elliptic functions
    • Early algebraists had to prove their formulas by geometry.

  88. Planetary motion references
    • A E L Davis: 'Some plane geometry from a cone ..


Famous Curves

  1. Devils
    • He became professor of mathematics at Geneva and wrote on work related to physics; also on geometry and the history of mathematics.

  2. Rhodonea
    • Grandi was the author of a number of works on geometry in which he considered the analogies of the circle and equilateral hyperbola.

  3. Tractrix
    • This is a surface of constant negative curvature and was used by Beltrami in 1868 in his concrete realisation of non-euclidean geometry.

  4. Nephroid
    • In 1878 he published The geometry of cycloidsin London.

  5. Circle
    • The greeks considered the Egyptians as the inventors of geometry.

  6. Watts
    • Sylvester, Kempe and Cayley further developed the geometry associated with the theory of linkages in the 1870's.

  7. Conchoidsl
    • He contributed to the geometry of spirals and the finding of geometric means.


Societies etc

  1. International Congress Speaker
    • James Pierpont, Non-Euclidean Geometry from Non-Projective Standpoint.
    • Oswald Veblen, Differential Invariants and Geometry.
    • Oswald Veblen, Spinors and Projective Geometry.
    • Shiing-shen Chern, Differential Geometry of Fiber Bundles.
    • Harold Davenport, Recent Progress in the Geometry of Numbers.
    • Andre Weil, Number Theory and Algebraic Geometry.
    • Oscar Zariski, The Fundamental Ideas of Abstract Algebraic Geometry.
    • Beniamino Segre, Geometry upon an Algebraic Variety.
    • Andre Weil, Abstract versus Classical Algebraic Geometry.
    • Shiing-shen Chern, Differential Geometry: Its Past and Its Future.
    • Israil Moiseevic Gelfand, The Cohomology of Infinite Dimensional Lie Algebras; Some Questions of Integral Geometry.
    • Phillip Augustus Griffiths, A Transcendental Method in Algebraic Geometry.
    • Roger Penrose, The Complex Geometry of the Natural World.
    • William Paul Thurston, Geometry and Topology in Dimension Three.
    • Shing-Tung Yau, The Role of Partial Differential Equations in Differential Geometry.
    • Paul Erdos, Extremal Problems in Number Theory, Combinatorics, and Geometry.
    • Yum-Tong Siu, Some Recent Developments in Complex Differential Geometry.
    • Simon Kirwan Donaldson, Geometry of Four Dimensional Manifolds.
    • Gerd Faltings, Recent Progress in Arithmetic Algebraic Geometry.
    • Mikhael Gromov, Soft and Hard Symplectic Geometry.
    • Edward Witten, String Theory and Geometry.
    • Laszlo Lovasz, Geometric Algorithms and Algorithmic Geometry.
    • Alexandre Varchenko, Multidimensional Hypergeometric Functions in Conformal Field Theory, Algebraic K-Theory, Algebraic Geometry.
    • Clifford Henry Taubes, Anti-self Dual Geometry.
    • Sun-Yung Alice Chang and Paul Chien-Ping Yang, Non-linear Partial Differential Equations in Conformal Geometry.
    • Yum-Tong Siu, Some Recent Transcendental Techniques in Algebraic and Complex Geometry.
    • Gang Tian, Geometry and Nonlinear Analysis.
    • Jean-Pierre Demailly, Kahler Manifolds and Transcendental Techniques in Algebraic Geometry.
    • Juan Luis Vazquez, Perspectives in Nonlinear Diffusion: Between Analysis, Physics, and Geometry.

  2. AMS Steele Prize
    • In 1994 the last of these three categories was put onto a five year cycle of topics: analysis, algebra, applied mathematics, geometry and topology, and discrete mathematics/logic.
    • for his paper "Algebraic geometry", and for his paper, written jointly with James B Carrell "Invariant theory, old and new".
    • for his cumulative influence on the fields of probability theory, Fourier analysis, several complex variables, and differential geometry.
    • for his expository research article "Equivalence relations on algebraic cycles and subvarieties of small codimension", and his book "Algebraic geometry".
    • for his work in algebraic geometry, especially his fundamental contributions to the algebraic foundations of this subject.
    • for the cumulative influence of his total mathematical work, high level of research over a period of time, particular influence on the development of the field of differential geometry, and influence on mathematics through Ph.D.
    • for his five-volume set "A Comprehensive Introduction to Differential Geometry".
    • for his fundamental work on geometric problems, particularly in the general theory of manifolds, in the study of differentiable functions on closed sets, in geometric integration theory, and in the geometry of the tangents to a singular analytic space.
    • for his books "Differential Geometry and Symmetric Spaces", "Differential Geometry, Lie Groups, and Symmetric Spaces", and "Groups and Geometric Analysis".
    • for having been instrumental in changing the face of geometry and topology, with his incisive contributions to characteristic classes, K-theory, index theory, and many other tools of modern mathematics.
    • for his fundamental work on global differential geometry, especially complex differential geometry.
    • for his extensive contributions in geometry and topology, the theory of Lie groups, their lattices and representations and the theory of automorphic forms, the theory of algebraic groups and their representations and extensive organizational and educational efforts to develop and disseminate modern mathematics.
    • for his numerous basic contributions to linear and nonlinear partial differential equations and their application to complex analysis and differential geometry.
    • He has been deeply influential in many of the important developments in algebra, algebraic geometry, and number theory during this time.
    • for his important and extensive work on arithmetical geometry and automorphic forms; concepts introduced by him were often seminal, and fertile ground for new developments, as witnessed by the many notations in number theory that carry his name and that have long been familiar to workers in the field.
    • for his paper "Pseudo-holomorphic curves in symplectic manifolds", which revolutionized the subject of symplectic geometry and topology and is central to much current research activity, including quantum cohomology and mirror symmetry.
    • for helping to weave the fabric of modern algebraic geometry, and to Elias Stein for making fundamental contributions to different branches of analysis.
    • for being one of the principal architects of the rapid development worldwide of discrete mathematics in recent years; and to Victor Guillemin for playing a critical role in the development of a number of important areas in analysis and geometry.
    • for his beautiful expository accounts of a host of aspects of algebraic geometry, including "The Red Book of Varieties and Schemes" (Springer, 1988).
    • for playing a pivotal role in shaping the direction of algebraic geometry, forging and strengthening ties between algebraic geometry and adjacent fields, and teaching and mentoring several generations of younger mathematicians.
    • for his book, "Commutative Algebra: With a View Toward Algebraic Geometry".

  3. European Mathematical Society Prize
    • whose work has made the geometry of Banach spaces look completely different.
    • has produced in a large variety of deep results on various aspects of arithmetic algebraic geometry.
    • He gave constructions of epsilon-nets in computational geometry, which provide tools for derandomisation of geometric algorithms.
    • He obtained the best results on several key problems in computational and combinatorial geometry and optimisation, such as linear programming algorithms and range searching.
    • whose work played a major role in the development of the theory of Alexandrov spaces of curvature bounded from below, giving new insight into to what extent the results of Riemannian geometry rely on the smoothness of the structure.
    • His results include a structure theory of these spaces, a stability theorem (new even for Riemannian manifolds), and a synthetic geometry a'la Aleksandrov.
    • contributed in a most important way to several domains of geometry and dynamical systems, in particular to symplectic geometry.
    • His most significant work is on valuations (additive functionals) on convex bodies and it has remodeled a central part of convex geometry.
    • This result is a very major advance in the subject, and provides the right formulation for the geometry of the problem.
    • whose work on the existence of metrics with special holomony is among the best in Riemannian geometry in the last decade.
    • Using a dazzling display of geometry and analysis, Joyce constructed compact examples in the exceptional cases where the holonomy is Spin7 and G2 the only remaining possibilities, the others on Berger's list had been eliminated.
    • The conjecture plays a central role in non-commutative geometry and has far-reaching connections to the Novikov conjecture on higher signatures in topology, to harmonic analysis on discrete groups and the theory of C*-algebras.
    • has created the method of dynamic diophantine approximation which has led to a series of remarkable results in complex geometry of algebraic varieties.
    • pioneered the use of measure-transportation techniques (due to Kantorovich, Brenier, Caffarelli, Mc Cann and others) in geometric inequalities of harmonic and functional analysis with striking applications to geometry of convex bodies.
    • has made fundamental and influential contributions to symplectic topology as well as to algebraic geometry and Hamiltonian systems.
    • His work is characterised by new depths in the interactions between complex algebraic geometry and symplectic topology.
    • Paul Biran not only proves deep results, he also discovers new phenomena and invents powerful techniques important for the future development of the field of symplectic geometry.
    • The new techniques of working with random partitions invented and successfully developed by Okounkov lead to a striking array of applications in a wide variety of fields: topology of module spaces, ergodic theory, the theory of random surfaces and algebraic geometry.
    • Stanislav Smirnov also made several essential contributions to complex dynamics, around the geometry of Julia sets and the thermodynamic formalism.
    • In arithmetic geometry, Iwasawa theory is the only general technique known for studying the mysterious relations between exact arithmetic formulae and special values of L-functions, as typified by the conjecture of Birch and Swinnerton-Dyer.

  4. MSJ Geometry Prize
    • Geometry Prize of the Mathematical Society of Japan .
    • MSJ Geometry Prize .

  5. Wolf Prize
    • for his inspired introduction of algebro-geometry methods to the theory of numbers.
    • creator of the modern approach to algebraic geometry, by its fusion with commutative algebra.
    • for his fundamental work in algebraic topology, differential geometry and differential topology.
    • for outstanding contributions to global differential geometry, which have profoundly influenced all mathematics.
    • for outstanding work combining topology, algebraic and differential geometry, and algebraic number theory; and for his stimulation of mathematical cooperation and research.
    • for ingenious and highly original discoveries in geometry, which have opened important new vistas in topology from the algebraic, combinatorial, and differentiable viewpoint.
    • for his revolutionary contributions to global Riemmanian and symplectic geometry, algebraic topology, geometric group theory and the theory of partial differential equations.
    • for his deep discoveries in topology and differencial geometry and their applications to Lie groups, differential operators and mathematical physics.
    • for his many fundamental contributions to topology, algebraic geometry, algebra, and number theory and his inspirational lectures and writing.
    • for his work on variations of Hodge structures; the theory of periods of abelian integrals; and for his contributions to complex differential geometry.
    • for his work in geometric analysis that has had a profound and dramatic impact on many areas of geometry and physics; .

  6. Sylvester Medal
    • for his researches in geometry and mechanics.
    • for his contributions to the topology of manifolds and related topics in algebra and geometry.
    • for his achievements in geometry, notably projective geometry, non-euclidean geometry and the analysis of spatial shapes and patterns, and for his substantial contributions to practical group-theory which pervade much modern mathematics.
    • for his important contributions to many parts of differential geometry combining this with complex geometry, integrable systems and mathematical physics interweaving the most modern ideas with the classical literature.
    • for his fundamental work in arithmetic geometry and his many contributions to the theory of ordinary differential equations.

  7. Savilian Chairs
    • Savilian Chairs of Geometry and Astronomy .
    • The Savilian Chair of Geometry was founded in 1619 at the University of Oxford by Henry Savile.

  8. Lowndean chair
    • Lowndean chair of Astronomy and Geometry .

  9. BMC Committee
    • Geometry: Differential geometry and fibre spaces.
    • It was agreed that the three days should be devoted to Analysis, Algebra and Geometry respectively, and that the organisers should aim at homogeneity in the morning lecture and talks.
    • 4) that the Geometry should be introduced by a lecture on Harmonic Integrals, and that the talks should include something on differential geometry in the large.
    • The programme for the 1952 Colloquium was discussed and it was agreed that if possible one day should be devoted to each of analysis, geometry and algebra.

  10. BMC 1983
    • Quillen, D G Infinite determinants over algebraic curves arising from problems in geometry, differential equations and number theory .
    • Batty, C J KAffine geometry, commutation and unitary equivalence in C*-algebras .
    • Kendall, W SStochastic Riemannian geometry .
    • Woodhouse, N M JSymplectic geometry and classical analogies .
    • Young, N JPower transfer and the non-Euclidean geometry of operators .

  11. Abel Prize
    • for playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory.
    • for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.
    • for his revolutionary contributions to geometry.
    • for pioneering discoveries in topology, geometry and algebra.
    • for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields .

  12. BMC 2007
    • Connes, A Recent developments on non-commutative geometry .
    • Singer, M Special metrics in Kahler geometry .
    • Wendland, K From dualities to geometry .
    • Series, CIndra's pearls: geometry and symmetry .

  13. BMC 1954
    • Northcott, D GAlgebraic geometry .
    • Scott, D BAlgebraic geometry .
    • Semple, J GAlgebraic geometry .
    • Todd, J AAlgebraic geometry .

  14. Paris Academy of Sciences
    • These in turn were each divided into three, with geometry, mechanics and astronomy being the three Mathematical Sciences, while chemistry, botany and anatomy were the three Physical Sciences.
    • Note that 'geometry' was used at this time in the sense we would use 'mathematics' today.
    • To geometry, mechanics and astronomy in the Mathematical Sciences was added 'general physics'.
    • The two categories of Mathematical Sciences and Physical Sciences were retained for the First Class with Mathematical Sciences now divided into five: geometry; mechanics; astronomy; geography and navigation; and general physics.

  15. Mathematics 2005
    • The 'BMC' had Special Sessions on Arithmetic and Algebraic Geometry (organisers E V Flynn and P E Newstead), and on Dynamical Systems (S van Strien and S M Rees); the 'BAMC' had Minisymposia on a variety of topics.
    • Algebraic and Arithmetic Geometry .
    • Geometry and Topology .

  16. Minutes for 1950
    • It was agreed that the three days should be devoted to Analysis, Algebra and Geometry respectively, and that the organisers should aim at homogeneity in the morning lecture and talks.
    • 4) that the Geometry should be introduced by a lecture on Harmonic Integrals, and that the talks should include something on differential geometry in the large.

  17. BMC 1996
    • Hitchin, N J Hyperkaler geometry .
    • Alon, NThe geometry of coin-weighing problems .
    • Special session: Differential geometry/Mathematical physicsn Organiser: N J Hitchin .

  18. BMC 1998
    • Connes, A Non-commutative geometry and the Riemann zeta function .
    • Gardener, TRigid subanalytic geometry .
    • Macintyre, A JNon-standard aspects of algebraic geometry .

  19. BMC 2003
    • Bridgeland, T Derived categories in algebraic geometry .
    • Bridson, M R Curvature and decidability in geometry and group theory .
    • Series, C M Recent developments in hyperbolic geometry .

  20. MSJ Iyanaga, Spring and Autumn Prize
    • MSJ Geometry Prize .
    • MSJ Geometry Prize .
    • MSJ Geometry Prize .

  21. Clay Award
    • for revolutionizing the field of operator algebras, for inventing modern non-commutative geometry, and for discovering that these ideas appear everywhere, including the foundations of theoretical physics; .
    • for their work in advancing our understanding of the birational geometry of algebraic varieties in dimension greater than three, in particular, for their inductive proof of the existence of flips.
    • The Langlands program is a collection of conjectures and theorems that unify the theory of automorphic forms, relating it intimately to the main stream of number theory, with close relations to harmonic analysis on algebraic groups as well as arithmetic algebraic geometry.

  22. AMS Satter Prize
    • for her outstanding work during the past five years on symplectic geometry.
    • for her deep contributions to the study of partial differential equations on Riemannian manifolds and in particular for her work on extremal problems in spectral geometry and the compactness of isospectral metrics within a fixed conformal class on a compact 3-manifold.
    • for her deep contributions to algebraic geometry, and in particular for her recent solutions to two long-standing open problems: the Kodaira problem ("On the homotopy types of compact Kahler and complex projective manifolds") and Green's Conjecture ("Green's canonical syzygy conjecture for generic curves of odd genus"; and "Green's generic syzygy conjecture for curves of even genus lying on a K3 surface".

  23. Ostrowski Prize
    • for his work on representation theory, arithmetic geometry, and modern mathematical physics.
    • for his contributions to contact and symplectic geometry.
    • for his fundamental contributions to conformal geometry, probability theory, and mathematical physics.

  24. MAA Chauvenet Prize
    • Recent Advances in the Foundations of Euclidean Plane Geometry, Amer.
    • Curves and Surfaces in Euclidean Space, Studies in Global Geometry and Analysis, MAA Stud.
    • Historical Ramblings in Algebraic Geometry and Related Algebra, Amer.

  25. AMS Veblen Prize
    • It is given in recognition for work in geometry or topology by a member of the American Mathematical Society which is published in a North American journal.
    • for his work in differential geometry and, in particular, the solution of the four-dimensional Poincare conjecture.
    • for his work in differential geometry, to Yakov Eliashberg for his work in symplectic and contact topology, and to Michael J Hopkins for his work in homotopy theory.

  26. Groups St Andrews.html
    • Geometry, Steinberg representations and complexity .
    • Group-theoretic applications of non-commutative toric geometry .
    • Random walks on groups: characters and geometry .

  27. BMC 2005
    • Fantechi, B Stacky techniques in enumerative geometry .
    • Lackenby, M The geometry anf topology of finite Cayley grphs .

  28. BMC 2006
    • Bielawski, R The geometry of monopoles .
    • Special session: Analysis and geometry on groupsn Organisers: A J Duncan and G Niblo .

  29. Kazan Physico-mathematical Society
    • Lobachevsky transformed the university journal Kazanskii Vestnik into the journal in which the scientific papers were published (in 1829 he published in this journal his first paper on his discovery of non-Euclidean geometry) and in 1834 he organized the purely scientific journal Uchenye Zapiski Kazanskogo Universiteta (Transactions of the Kazan University), where he also published his papers on non-Euclidean geometry.

  30. BMC 1989
    • Bruce, J W Singularities in geometry and geometry of singularities .

  31. Minutes for 1997
    • Prof Alain Connes (Collge de France), non-commutative geometry; .
    • Geometry/Topology *Dr B H Bowditch (Southampton), *Dr A D King (Lancaster), Dr V V Goryunov (Liverpool), Dr W J Harvey (King's), Dr J C Wood (Leeds), Dr D Singerman (Southampton), Dr A J Baker (Glasgow), *Dr D Joyce (Oxford).

  32. BMC 1999
    • Borovik, A V Combinatorial geometry in characteristic one .
    • Strickland, N P From homotopy theory to arithmetic geometry .

  33. Minutes for 1949
    • Geometry: Differential geometry and fibre spaces.

  34. BMC 1977
    • Giblin, P JJet spaces and geometry .
    • Macbeath, A MModular groups and their geometry .

  35. Hellenic Mathematical Society
    • He undertook research into differential geometry and when he became a founder member of the Hellenic Mathematical Society he had been a professor at the University of Athens since 1901.
    • Nilos Sakellariou, professor of analytical geometry at the University of Athens, then served as President of the Society from 1929.

  36. AMS Cole Prize in Algebra
    • for his work in representation theory and geometry.
    • for their groundbreaking joint work on higher dimensional birational algebraic geometry.

  37. Young Mathematician prize
    • for works on the theory of stochastic differential equations and Banach geometry.
    • for works on the geometry of Minkowski spaces.

  38. Minutes for 1997
    • Prof Alain Connes (College de France), non-commutative geometry; .
    • Geometry/Topology .

  39. Shaw Prize
    • for his initiation of the field of global differential geometry and his continued leadership of the field, resulting in beautiful developments that are at the centre of contemporary mathematics, with deep connections to topology, algebra and analysis, in short, to all major branches of mathematics of the last sixty years.
    • for their many brilliant contributions to geometry in 3 and 4 dimensions.

  40. BMC 1963
    • Cassels, J W SDiophantine analysis and algebraic geometry .
    • Schwarzenberger, R L EFibred varieties in algebraic geometry .

  41. AMS Fulkerson Prize
    • for 'The universality theorems on the classification problem of configuration varieties and convex polytope varieties', O Ya Viro (ed.), Topology and Geometry-Rohlin Seminar, Lecture Notes in Mathematics 1346 (Springer-Verlag, Berlin, 1988) 527-544.
    • for 'Upper bounds for the diameter and height of graphs of the convex polyhedra', Discrete and Computational Geometry 8 (1992) 363-372.

  42. BMC 1973
    • Atiyah, M FAnalysis and differential geometry .
    • Newstead, P EClassification problems in algebraic geometry .

  43. Fermat Prize
    • The Fermat Prize is awarded to a mathematician for decisive research in those fields to which Pierre de Fermat contributed, namely: Statements of Variational Principles; Foundations of Probability and Analytical Geometry; and Number Theory.
    • for several important contributions to the theory of variational calculus, which have consequences in Physics and Geometry.

  44. MSJ Seki-Takakazu Prize
    • It has organized 18 annual programs in widely ranging fields of mathematics, and the current program is "Recent Developments in Higher Dimensional Algebraic Geometry".
    • MSJ Geometry Prize .

  45. Scientific Committee minutes 2004
    • The Special Sessions will be Dynamical Systems and Algebraic/Arithmetic Geometry.
    • The Special Sessions are Analysis and Geometry on Groups (A G Robertson and A J Duncan) and Operator Theory (J R Partington and M A Dritschel).

  46. BMC 2008
    • Venkatesh, ADynamics and the geometry of numbers.
    • Special session: Differential Geometry and Geometric Analysisn Organisers: C Wood, and J Wood .

  47. BMC 1982
    • Yau, S T Non-linear equations in differential geometry .

  48. Rolf Schock Prize
    • for his important work in algebraic geometry and mathematical physics, in particular for those fundamental papers he has recently published about quantum groups and mirror symmetry.
    • for his fundamental contributions to combinatorics and its relationship to algebra and geometry, in particular for his important contributions to the theory of convex polytopes and his innovative work on enumerative combinatorics.

  49. BMC 1969
    • Klingenberg, W Recent developments in Riemannian geometry .
    • Duncan, JRelations between algebras and geometry in Banach algebras .

  50. BMC 1997
    • Lubotzky, A Property tau and its applications in combinatorics, geometry and group theory .
    • Penrose, R The twistor approach to space-time geometry .

  51. LMS Presidential Addresses
    • Einstein's Theory of Gravitation as an Hypothesis in Differential Geometry.
    • Harmonic spaces in differential geometry.

  52. Minutes for 1997
    • Prof Alain Connes (Collge de France), non-commutative geometry; .
    • Geometry/Topology *Dr B H Bowditch (Southampton), *Dr A D King (Lancaster), Dr V V Goryunov (Liverpool), Dr W J Harvey (King's), Dr J C Wood (Leeds), Dr D Singerman (Southampton), Dr A J Baker (Glasgow), *Dr D Joyce (Oxford).

  53. BMC 1986
    • Thurston, W P Three-dimensional geometry and topology .
    • Donaldson, S K Geometry of 4-manifolds .

  54. BMC 1993
    • Babai, L Vertex-symmetric graphs, excluded minors and hyperbolic geometry .
    • Evans, D M The geometry of algebraic closure .

  55. MSJ Analysis Prize
    • MSJ Geometry Prize .

  56. Minutes for 1977
    • 5 Analysis, 3 Topology, 4 Algebra, 2 Number Theory, 1 Geometry, 1 Combinatorics, 1 Logic, 1 Probability.

  57. Swiss Mathematical Society
    • The three mathematicians most involved in the founding of the Society were: Rudolf Fueter, who had been appointed as professor of mathematics at the University of Basel in 1908; H Fehr, who was at the University of Geneva; and Marcel Grossmann, who had been appointed as professor of descriptive geometry at the Eidgenossische Technische Hochschule in Zurich in 1907.

  58. Mathematical Circle of Palermo
    • The goal was to stimulate the study of higher mathematics by means of original communications presented by the members of the society on the different branches of analysis and geometry, as well as on rational mechanics, mathematical physics, geodesy, and astronomy.

  59. Minutes for 1998
    • Geometry/Topology .

  60. BMC 1953
    • Walker, A GDifferential geometry in the large .

  61. Minutes for 1951
    • The programme for the 1952 Colloquium was discussed and it was agreed that if possible one day should be devoted to each of analysis, geometry and algebra.

  62. Minutes for 2007
    • Special sessions: Number Theory, Differential Geometry and Geometric Analysis.

  63. BMC 1985
    • Gromov, M Differential geometry with and without infinitesimal calculus: anatomy of curvature .

  64. Minutes for 1953
    • The programme for the 1954 Colloquium was discussed and it was agreed that one day should be devoted to each of, ALGEBRA, ANALYSIS and ALGEBRAIC GEOMETRY in that order.

  65. BMC 1956
    • Edge, W LClassical groups in finite geometry .

  66. BMC 1952

  67. BMC 1960

  68. BMC 2002
    • Special session: Geometry topology and mechanics Organiser: M Robertsn .

  69. Serbian Academy of Sciences
    • In the 1980s geometry and topology moved into leading roles, while in the 1990s the original topics from the 1950s of analysis and mechanics again became among the most widely studied.

  70. AMS Bôcher Prize
    • for his work on the application of partial differential equations to differential geometry, in particular his completion of the solution to the Yamabe Problem in "Conformal deformation of a Riemannian metric to constant scalar curvature".

  71. References for Turin
    • F Fava, The Academy's contribution to the development of geometry (Italian), in The first two centuries of the Turin Academy of Sciences (Italian), Turin, 1983, Atti Accad.

  72. Minutes for 1956
    • The programme for the 1957 Colloquium was discussed and it was agreed that one day should be devoted to each of (i) Analysis (ii) Algebra and Number Theory (iii) Geometry and Topology.

  73. Minutes for 2000
    • Geometry and Topology: .

  74. BMC 2000
    • Donaldson, S KScalar curvature in Kahler geometry .

  75. Minutes for 1955
    • (c) Geometry and Topology .

  76. BMC 1988
    • Singer, I M Geometry and string theory .

  77. Scientific Committee 2007
    • Special session organisers are Jason Levesley (int) and Glyn Harman (ext) for Number Theory, and Chris Wood (int) and John Wood (ext) for Differential Geometry and Geometric Analysis.

  78. MSJ Algebra Prize
    • MSJ Geometry Prize .

  79. BMC 1958
    • Macbeath, A MTopics in integral geometry .

  80. BMC 1994
    • Falconer, K J Probabilistic methods in fractal geometry .

  81. Minutes for 1990
    • P E Newstead (0) Liverpool Geometry .

  82. BMC 1981
    • Jones, J D SThe Kervaire invariant in geometry and homotopy theory .

  83. Minutes for 1958
    • nnn(iii) Geometry and Topology; .

  84. BMC 1976
    • Hitchin, N JSpinors in differential geometry .

  85. BMC 1972
    • Armitage, J VParity questions in number theory and algebraic geometry .

  86. BMC 1991
    • Ekeland, I Symplectic geometry .

  87. Minutes for 1964
    • It was agreed to revert to separate mornings for Analysis and Functional Analysis; Algebra and Theory of Numbers; Geometry and Topology.

  88. BMC 1950
    • Scott, D BValuation theory and birational geometry .

  89. BMC 1951
    • Hodge, W V DHarmonic integrals and differential geometry in the large .

  90. Minutes for 1963
    • The claims of algebraic and differential geometry on future meetings were mentioned.

  91. Minutes for 1952
    • The programme for the 1953 Colloquium was discussed and it was agreed that if possible one day should be devoted to each of Analysis, Topology and Differential Geometry, Algebra and Number Theory.

  92. Minutes for 1957
    • (b) that one day should be devoted to each of (i) Analysis and Statistics (ii) Algebra and Logic (iii) Geometry and Topology.

  93. Bakerian Lecturer
    • Global geometry (Proc.

  94. Scientific Committee 2006
    • Professor Dominic Joyce (Oxford) (Differential Geometry) .

  95. BMC 1995
    • Reid, A W The geometry of incompressible surfaces in hyperbolic 3-manifolds .

  96. Minutes for 2006
    • Plans for York 2008: Chris Wood reported, that it is planned to have special sessions on Differential geometry and Representation of Algebras.


References

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    • R Bonola, Non-Euclidean Geometry (1955).
    • V A Bazhanov, The imaginary geometry of N I Lobachevskii and the imaginary logic of N A Vasiliev, Modern Logic 4 (2) (1994), 148-156.
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  2. References for Descartes
    • H J M Bos, Descartes and the beginning of analytic geometry (Dutch), in Summer course 1989 : mathematics in the Golden Age (Amsterdam, 1989), 79-97.
    • A V Dorofeeva, Descartes and his 'Geometry' (Russian), Mat.
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  3. References for Titeica
    • Gh Th Gheorghiu, Gh Titeica and the influence of his work on Romanian mathematicians (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Romanian) (Editura Acad.
    • Gh Gheorghiev and G Vranceanu, On the scientific work of Gheorghe Titeica (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Romanian) (Editura Acad.
    • N N Mihaileanu, Gheorghe Titeica's lectures on geometry (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Romanian) (Editura Acad.
    • O Onicescu, Gheorghe Titeica and Dimitrie Pompeiu (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Romanian) (Editura Acad.
    • I Popa, Gheorghe Titeica and the Iasi school of geometry (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Romanian) (Editura Acad.
    • N Soare, Gheorghe Titeica and affine differential geometry, Balkan J.
    • G Titeica, From the notebooks of Professor Gheorghe Titeica (Romanian), in Gheorghe Titeica and Dimitrie Pompeiu Symposium on Geometry and Global Analysis, Bucharest, 1973 (Romanian) (Editura Acad.

  4. References for Von Staudt
    • J L Coolidge, The Rise and Fall of Projective Geometry, The American Mathematical Monthly 41 (4) (1934), 217-228.
    • L W Dowling, Projective Geometry - Fields of Research, The American Mathematical Monthly 32 (10) (1925), 486-492.
    • H Freudenthal, The impact of von Staudt's foundations of geometry, in Geometry - von Staudt's point of view (Dordrecht-Boston, Mass., 1981), 401-425.
    • H Freudenthal, The impact of von Staudt's foundations of geometry, in For Dirk Struik (Dordrecht, 1974), 189-200.
    • E A C Marchisotto, The projective geometry of Mario Pieri: A legacy of Georg Karl Christian von Staudt, Historia Mathematica 33 (2006), 277-314.
    • K Reich, Karl Georg Christian von Staudt, book on Projective Geometry (1847), in I Grattan-Guinness and R Cooke (eds.), Landmark writings in Western mathematics 1640-1940 (Elsevier, 2005), 441-447.
    • M Wilson, The Royal Road from Geometry, Nous 26 (2) (J1992), 149-180.

  5. References for Poincare
    • R Torretti, Philosophy of geometry from Riemann to Poincare (Dordrecht-Boston, Mass., 1984).
    • S Bagce, Poincare's philosophy of geometry and its relevance to his philosophy of science, in Henri Poincare : science et philosophie, Nancy, 1994 (Berlin, 1996), 299-314; 582.
    • L C Biedenharn and Y Dothan, Poincare's work on the magnetic monopole and its generalization in present day theoretical physics, in Differential topology-geometry and related fields, and their applications to the physical sciences and engineering (Leipzig, 1985), 39-50.
    • J J Gray, Poincare and Klein - groups and geometries, in 1830-1930 : a century of geometry, Paris, 1989 (Berlin, 1992), 35-44.
    • P A Griffiths, Poincare and algebraic geometry, Bull.
    • G Heinzmann, Helmholtz and Poincare's considerations on the genesis of geometry, in 1830-1930 : a century of geometry, Paris, 1989 (Berlin, 1992), 245-249.

  6. References for Pompeiu
    • C Calude, The Pompeiu distance between closed sets (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Editura Acad.
    • C Iacob, Dimitrie Pompeiu's lectures on mechanics (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Editura Acad.
    • D V Ionescu, The connections of Dimitrie Pompeiu with the University of Cluj (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Editura Acad.
    • O Onicescu, Gheorghe Titeica and Dimitrie Pompeiu (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Editura Acad.
    • E Popoviciu, Quelques recherches de D Pompeiu liees a la theorie de l'interpolation (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Editura Acad.
    • M Rosculet, Dimitrie Pompeiu (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Editura Acad.
    • N Teodorescu, Dimitrie Pompeiu and his work (Romanian), in 'Gheorghe Titeica and Dimitrie Pompeiu' Symposium on Geometry and Global Analysis, Bucharest, 1973 (Editura Acad.

  7. References for Kiselev
    • A Bogomolny, Review: Kiselev's Geometry.
    • A Bogomolny, Review: Kiselev's Geometry.
    • http://www.maa.org/publications/maa-reviews/kiselevs-geometry-book-i-planimetry .
    • A Bogomolny, Review: Kiselev's Geometry.
    • http://www.maa.org/publications/maa-reviews/kiselevs-geometry-book-ii-stereometry .
    • A Givental, Translator's Preface, in Kiselev's Geometry.
    • A Karp and A Werner, On the teaching of geometry in Russia, in A Karp and B Vogeli (eds.), Russian mathematics education: Programs and practices (World Scientific, London, New Jersey, Singapore, 2011), 81-128.

  8. References for Pieri
    • E A Marchisotto and J T Smith, The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).
    • M Avellone, A Brigaglia and C Zappulla, The foundations of projective geometry in Italy from De Paolis to Pieri, Arch.
    • E A Marchisotto, The projective geometry of Mario Pieri: a legacy of Georg Karl Christian von Staudt, Historia Math.
    • E A Marchisotto, Mario Pieri : His contributions to the foundations and teaching of geometry, Historia Math.
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    • J T Smith, Definitions and nondefinability in geometry, Amer.

  9. References for Grassmann
    • A Brigaglia, The influence of H Grassmann on Italian projective n-dimensional geometry, in Hermann Gunther Grassmann (1809-1877) : visionary mathematician, scientist and neohumanist scholar (Dordrecht, 1996), 155-163.
    • J Browne, Grassmann, geometry and mechanics, in From Past to Future: Grassmann's Work in Context (Basel, 2010), 287-302 .
    • G Chatelet, La capture de l'extension comme dialectique geometrique: dimension et puissance selon l'Ausdehnung de Grassmann (1844), in 1830-1930 : a century of geometry (Berlin, 1992), 222-244.
    • D Flament, La 'Lineale Ausdehnungslehre' (1844) de Hermann Gunther Grassmann, in 1830-1930 : a century of geometry (Berlin, 1992), 205-221.
    • J Pfalzgraf, A short note on Grassmann manifolds with a view to noncommutative geometry, in From Past to Future: Grassmann's Work in Context (Basel, 2010), 333-341 .
    • A E Tokmakidis, Der Begriff der Determinante in Hermann Gunther Grassmanns (1809-1877) Ausdehnungslehre, Proceedings of the 4th International Congress of Geometry, Thessaloniki, 1996 (Thessaloniki, 1997), 409-416.

  10. References for Efimov
    • V B Gurevich, More about the textbook of N V Efimov, 'Short-course of analytical geometry' (review) (Russian), Uspekhi Mat.
    • L A Lyusternik, Review: Higher geometry by N V Efimov, Uspekhi Mat.
    • E J F Primrose, Review: An Elementary Course in Analytical Geometry.
    • Part I-Analytical geometry in the plane; Part II-Three-dimensional geometry by N V Efimov, The Mathematical Gazette 52 (380) (1968), 204-205.
    • N Ya Sysoeva, E E Brenev and L E Sadovskii, Review: Short-course of analytical geometry by N V Efimov (Russian), Uspekhi Mat.

  11. References for Weyl
    • M Friedman, Carnap and Weyl on the foundations of geometry and relativity theory, Erkenntnis 42 (2) (1995), 247-260.
    • M I Monastyrskii, On H Weyl's paper 'Riemann's ideas on geometry, their influence and connections with group theory' (Russian), Istor.-Mat.
    • R Penrose, Hermann Weyl, space-time and conformal geometry, Hermann Weyl, 1885-1985 (Eidgenossische Tech.
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    • E Scholz, Hermann Weyl's 'Purely infinitesimal geometry', in Proceedings of the International Congress of Mathematicians, Vol.

  12. References for Riemann
    • H Grauert, Bernhard Riemann and his ideas in philosophy of nature, in Analysis, geometry and groups: a Riemann legacy volume (Palm Harbor, FL, 1993), 124-132.
    • E Portnoy, Riemann's contribution to differential geometry, Historia Math.
    • E Scholz, Riemann's vision of a new approach to geometry, in 1830-1930: a century of geometry (Berlin, 1992), 22-34.
    • J D Zund, Some comments on Riemann's contributions to differential geometry, Historia Math.

  13. References for Vranceanu
    • Gheorghe Vranceanu : founder of modern geometry in Romania (Romanian), Gaz.
    • S Ianus and I Popovici, Gheorghe Vranceanu : a precursor of global differential geometry, An.
    • I D Teodorescu, Gheorghe Vranceanu-le fondateur de l'ecole de geometrie differentielle a Bucarest, in Differential geometry and topology applications in physics and technics, Bucharest, 1991, Polytech.
    • I D Teodorescu, Recent results in the geometry seminar of G Vranceanu at the Faculty of Mathematics in Bucharest (Romanian), in Proceedings of the National Colloquium on Geometry and Topology, Busteni, 1981, Univ.

  14. References for Enriques
    • E A Marchisotto and J T Smith, The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).
    • A Conte, The discovery of and first attempts at classifying Enriques surfaces in the unpublished correspondence of Federigo Enriques to Guido Castelnuovo (Italian), Algebra and geometry (1860-1940): the Italian contribution, Rend.
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    • G Israel, Poincare et Enriques: deux points de vue differents sur les relations entre geometrie, mecanique et physique, in 1830-1930: a century of geometry, Paris, 1989 (Springer, Berlin, 1992),107-126.
    • G Israel and M Menghini, The 'Essential Tension' at Work in Qualitative Analysis: A Case Study of the Opposite Points of View of Poincare and Enriques on the Relationships between Analysis and Geometry, Historia Mathematica 25 (1998), 379-411.

  15. References for Bolyai
    • A C Albu, Janos Bolyai and the foundations of geometry (Romanian), in Proceedings of Symposium in Geometry (Cluj-Napoca, 1993), 7-23.
    • V F Kagan, The construction of non-Euclidean geometry by Lobachevsky, Gauss and Bolyai (Russian), Proc.
    • O Mayer, Janos Bolyai's life and work, in Proceedings of the national colloquium on geometry and topology (Cluj-Napoca, 1982), 12-26.

  16. References for Desargues
    • K Andersen, Desargues' method of perspective : its mathematical content, its connection to other perspective methods and its relation to Desargues' ideas on projective geometry, Centaurus 34 (1) (1991), 44-91.
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  17. References for Euclid
    • E Filloy, Geometry and the axiomatic method.
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    • S R Palmquist, Kant on Euclid : geometry in perspective, Philos.
    • A Seidenberg, Did Euclid's 'Elements, Book I,' develop geometry axiomatically?, Arch.

  18. References for Veronese
    • L Boi, The influence of the Erlangen Program on Italian geometry, 1880-1890 : n-dimensional geometry in the works of D'Ovidio, Veronese, Segre and Fano, Arch.
    • P Freguglia, The foundations of higher-dimensional geometry according to Giuseppe Veronese (Italian), in Geometry Seminars, 1996-1997 (Bologna, 1998), 253-277.

  19. References for Griffiths
    • Part 1 Analytic geometry (American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2003).
    • Part 2 Algebraic geometry (American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2003).
    • Part 1 Analytic geometry (American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2003), xv-xvi.
    • Part 1 Analytic geometry (American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2003), xiii-xiv.

  20. References for Chow
    • S S Chern, Wei-Liang Chow, 1911-1995, in Contemporary trends in algebraic geometry and algebraic topology, Tianjin, 2000 (World Sci.
    • Chow's bibliography, Birational algebraic geometry, Baltimore, MD, 1996 (Amer.
    • J Igusa, Remarks on Chow, Birational algebraic geometry, Baltimore, MD, 1996 (Amer.
    • S Lang, Comments on Chow's work, in Contemporary trends in algebraic geometry and algebraic topology, Tianjin, 2000 (World Sci.

  21. References for Helmholtz
    • G Heinzmann, Helmholtz and Poincare's considerations on the genesis of geometry, in 1830-1930: a century of geometry (Berlin, 1992), 245-249.
    • J L Richards, The evolution of empiricism: Hermann von Helmholtz and the foundations of geometry, British J.
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  22. References for Fano
    • L Boi, The influence of the Erlangen Program on Italian geometry, 1880-1890 : n-dimensional geometry in the works of D'Ovidio, Veronese, Segre and Fano, Arch.
    • J P Murre, On the work of Gino Fano on three-dimensional algebraic varieties, Algebra and geometry (1860-1940) : the Italian contribution, Rend.

  23. References for Pincherle
    • E A Marchisotto and J T Smith, The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).
    • U Bottazzini, Pincherle and the theory of analytic functions (Italian), Geometry and complex variables, Lecture Notes in Pure and Appl.
    • S Francesconi, The teaching of mathematics at the University of Bologna from 1860 to 1940 (Italian), in Geometry and complex variables, Bologna 1988-90 (Dekker, New York, 1991), 415-474.

  24. References for Singer
    • R V Kadison, Which Singer is that?, in Surveys in differential geometry, .
    • G B Kolata, Isadore Singer and differential geometry, Science 204 (4396) (1979), 933-934.
    • E Witten, Is Singer's contributions to geometry and physics, in S-T Yau (ed.), The founders of index theory: reminiscences of Atiyah, Bott, Hirzebruch, and Singer (International Press, Somerville, MA, 2003).

  25. References for Cohn-Vossen
    • T A A B, Review: Geometry and the Imagination by D Hilbert and S Cohn-Vossen, The Mathematical Gazette 36 (317) (1952), 231-232.
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  26. References for Einstein
    • Gh Gheorghiev, Albert Einstein and the development of differential geometry (Romanian), An.
    • M Paty, Physical geometry and special relativity.
    • Einstein et Poincare, 1830-1930 : a century of geometry (Berlin, 1992), 127-149.

  27. References for Lie
    • S-S Chern, Sophus Lie and differential geometry, in The Sophus Lie Memorial Conference, Oslo, 1992 (Oslo, 1994), 129-137.
    • D E Rowe, Klein, Lie, and the 'Erlanger Programm' 1830-1930 : a century of geometry, Paris, 1989, Lecture Notes in Phys.
    • A Weinstein, Sophus Lie and symplectic geometry, Exposition.

  28. References for Monge
    • B Kvetonova, Gaspard Monge and descriptive geometry (Czech), Pokroky Mat.
    • K M Liu and S Z Yang, The history and contemporary significance of descriptive geometry : commemorating the 200th anniversary of the publication of Monge's 'Descriptive geometry' (Chinese), Math.

  29. References for Grauert
    • I Bauer, F Catanese, Y Kawamata, T Peternell and Y-T Siu (eds), Complex geometry : Collection of papers dedicated to Hans Grauert (Springer, Berlin, 2002).
    • List of doctoral students of Hans Grauert, in I Bauer, F Catanese, Y Kawamata, T Peternell and Y-T Siu (eds), Complex geometry : Collection of papers dedicated to Hans Grauert (Springer, Berlin, 2002), xii-xiii.
    • List of research publications of Hans Grauert, in I Bauer, F Catanese, Y Kawamata, T Peternell and Y-T Siu (eds), Complex geometry : Collection of papers dedicated to Hans Grauert (Springer, Berlin, 2002), vii-xi.

  30. References for D'Ovidio
    • E A Marchisotto and J T Smith, The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).
    • L Boi, The influence of the Erlangen Program on Italian geometry, 1880-1890: n-dimensional geometry in the works of D'Ovidio, Veronese, Segre and Fano, Arch.

  31. References for Poncelet
    • A N Kolmogorov, Mathematics of the 19th Century: Geometry, Analytic Function Theory (Birkhauser, 1996).
    • A N Bogolyubov, Geometry and mechanics in the works of J-V Poncelet (Russian), Studies in the history of physics and mechanics 271 'Nauka' (Moscow, 1986), 178-191.
    • J L Coolidge, The Rise and Fall of Projective Geometry, Amer.

  32. References for Rokhlin
    • V G Turaev, A M Vershik and V A Rokhlin, Topology, ergodic theory, real algebraic geometry: Rokhlin's memorial (AMS Bookstore, 2001).
    • A M Vershik, Vladimir Abramovich Rokhlin [1919-1984], in Topology, ergodic theory, real algebraic geometry (Amer.
    • A M Vershik, V A Rokhlin and the modern theory of measurable partitions, in Topology, ergodic theory, real algebraic geometry (Amer.

  33. References for Comessatti
    • C Ciliberto and C Pedrini, Annibale Comessatti and real algebraic geometry, in Algebra and geometry (1860-1940): the Italian contribution, Cortona, 1992, Rend.
    • F Russo, The antibirational involutions of the plane and the classification of real del Pezzo surfaces, in M C Beltrametti, F Catanese, C Ciliberto, A Lanteri and C Pedrini (eds.), Algebraic Geometry: A Volume in Memory of Paolo Francia (Walter de Gruyter, Berlin, 2002), 289-312.

  34. References for Bartels
    • U Lumiste, Martin Bartels as researcher: his contribution to analytical methods in geometry, Historia Math.
    • U Lumiste, Differential geometry in Estonia: history and recent developments (Finnish), Arkhimedes (4) 1996 (1996), 31-34.
    • Yu G Lumiste, Bartels - the investigator, and his achievements in the analytic methods of geometry (Russian), in Dedicated to the memory of Lobachevskii No 1 (Russian) (Kazan.

  35. References for Archimedes
    • D C Gazis and R Herman, Square roots geometry and Archimedes, Scripta Math.
    • E Kreyszig, Archimedes and the invention of burning mirrors : an investigation of work by Buffon, in Geometry, analysis and mechanics (River Edge, NJ, 1994), 139-148.
    • J M Rassias, Archimedes, in Geometry, analysis and mechanics (River Edge, NJ, 1994), 1-4.

  36. References for Viete
    • P Freguglia, Algebra and geometry in the work of Viete (Italian), Boll.
    • E Giusti, Algebra and geometry in Bombelli and Viete, Boll.
    • B A Rozenfel'd, Viete's vectors and pseudovectors and their role in the creation of analytic geometry (Russian), Istor.-Mat.

  37. References for Levi Beppo
    • E A Marchisotto and J T Smith, The Legacy of Mario Pieri in Geometry and Arithmetic (Springer, 2007).
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    • E Knobloch, Leibniz and Euler : problems and solutions concerning infinitesimal geometry and calculus, Conference on the History of Mathematics (Rende, 1991), 293-313.

  182. References for Lacroix
    • C B Boyer, Cartesian geometry from Fermat to Lacroix, Scripta Math.

  183. References for Guarini
    • J McQuillan, Geometry and Light in the Architecture of Guarino Guarini, Ph.D.

  184. References for Zeuthen
    • S L Kleiman, Hieronymus Georg Zeuthen (1839-1920), Enumerative algebraic geometry, Contemp.

  185. References for Al-Jawhari
    • K Jaouiche, The theory of parallels in Islamic geometry (Arabic) (Tunis, 1988).

  186. References for Paramesvara
    • T A Sarasvati Amma, Geometry in ancient and medieval India (Delhi, 1979).

  187. References for Ricci Giovanni
    • M Cugiani, Commemoration of Giovanni Ricci (Italian), Geometry of Banach spaces and related topics, Milan, 1983, Rend.

  188. References for Seidel Jaap
    • D G Corneil and R Mathon, Geometry and Combinatorics, Selected Works of J J Seidel (Academic Press, Inc., 1991).

  189. References for Polya
    • D Schattschneider, Polya's geometry, Bull.

  190. References for Artin
    • R Bartolozzi and U Oliveri, Reflections on some contributions of Emil Artin to the foundations of geometry: the problem of coordinatization (Italian), Riv.

  191. References for Osipovsky
    • B A Rozenfeld, A History of Non-Euclidean Geometry : Evolution of the Concept of a Geometric Space (Springer, 1988).

  192. References for Bolibrukh
    • D V Anosov and E F Mishchenko, In Memory of Andrei Andreevich Bolibrukh (Russian), Dynamical systems and related problems of geometry, Collected papers.

  193. References for Brocard
    • L Guggenbuhl, Henri Brocard and the Geometry of the triangle, Mathematical Gazette 32 (1953), 241-243.

  194. References for Zariski
    • P Blass, The influence of Oscar Zariski on modern algebraic geometry, Austral.

  195. References for Torricelli
    • F de Gandt, L'evolution de la theorie des indivisibles et l'apport de Torricelli, in Geometry and atomism in the Galilean school (Florence, 1992), 103-118.

  196. References for Rey Pastor
    • L A Santalo, The works of Rey Pastor in geometry and topology (Spanish), Rev.

  197. References for Rado Ferenc
    • V Groze, M Tarina and A Vasiu, The life and work of the Professor Francisc Rado (1921-1990), Seminar on Geometry (Babeş-Bolyai' Univ., Cluj-Napoca, 1991), 3-18.

  198. References for Mengoli
    • E Giusti, Pietro Mengoli's first research : the sum of series (Italian), in Geometry and complex variables (New York, 1991), 195-213.

  199. References for Al-Maghribi
    • K Jaouiche, The theory of parallels in Islamic geometry (Arabic) (Tunis, 1988).

  200. References for Borelli
    • G B Halsted, Non-Euclidean Geometry: Historical and Expository, Amer.

  201. References for Drinfeld
    • V Ginzburg, A glimpse into the life and work of V Drinfeld, in Algebraic geometry and number theory (Birkhauser Boston, Boston, MA, 2006), xiii-xv.


Additional material

  1. Einstein: 'Geometry and Experience
    • Einstein: Geometry and Experience .
    • He chose as his topic Geometry and Experience.
    • Geometry and Experience .
    • Let us for a moment consider from this point of view any axiom of geometry, for instance, the following:- Through two points in space there always passes one and only one straight line.
    • The more modern interpretation:- Geometry treats of entities which are denoted by the words straight line, point, etc.
    • All other propositions of geometry are logical inferences from the axioms (which are to be taken in the nominalistic sense only).
    • The matter of which geometry treats is first defined by the axioms.
    • In axiomatic geometry the words "point," "straight line," etc., stand only for empty conceptual schemata.
    • Yet on the other hand it is certain that mathematics generally, and particularly geometry, owes its existence to the need which was felt of learning something about the relations of real things to one another.
    • The very word geometry, which, of course, means earth-measuring, proves this.
    • It is clear that the system of concepts of axiomatic geometry alone cannot make any assertions as to the relations of real objects of this kind, which we will call practically-rigid bodies.
    • To be able to make such assertions, geometry must be stripped of its merely logical-formal character by the geometry.
    • To accomplish this, we need only add the proposition:- Solid bodies are related, with respect to their possible dispositions, as are bodies in Euclidean geometry of three dimensions.
    • Geometry thus completed is evidently a natural science; we may in fact regard it as the most ancient branch of physics.
    • We will call this completed geometry "practical geometry," and shall distinguish it in what follows from "purely axiomatic geometry." The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience.
    • All linear measurement in physics is practical geometry in this sense, so too is geodetic and astronomical linear measurement, if we call to our help the law of experience that light is propagated in a straight line, and indeed in a straight line in the sense of practical geometry.
    • I attach special importance to the view of geometry which I have just set forth, because without it I should have been unable to formulate the theory of relativity.
    • Without it the following reflection would have been impossible:- In a system of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we must abandon Euclidean geometry.
    • If we deny the relation between the body of axiomatic Euclidean geometry and the practically-rigid body of reality, we readily arrive at the following view, which was entertained by that acute and profound thinker, H Poincare:- Euclidean geometry is distinguished above all other imaginable axiomatic geometries by its simplicity.
    • Now since axiomatic geometry by itself contains no assertions as to the reality which can be experienced, but can do so only in combination with physical laws, it should be possible and reasonable - whatever may be the nature of reality - to retain Euclidean geometry.
    • For if contradictions between theory and experience manifest themselves, we should rather decide to change physical laws than to change axiomatic Euclidean geometry.
    • If we deny the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclidean geometry is to be retained as the simplest.
    • Why is the equivalence of the practically-rigid body and the body of geometry - which suggests itself so readily - denied by Poincare and other investigators? Simply because under closer inspection the real solid bodies in nature are not rigid, because their geometrical behaviour, that is, their possibilities of relative disposition, depend upon temperature, external forces, etc.
    • Thus the original, immediate relation between geometry and physical reality appears destroyed, and we feel impelled toward the following more general view, which characterizes Poincare's standpoint.
    • Geometry (G) predicates nothing about the relations of real things, but only geometry together with the purport (P) of physical laws can do so.
    • Envisaged in this way, axiomatic geometry and the part of natural law which has been given a conventional status appear as epistemologically equivalent.
    • All practical geometry is based upon a principle which is accessible to experience, and which we will now try to realise.
    • Not only the practical geometry of Euclid, but also its nearest generalisation, the practical geometry of Riemann, and therewith the general theory of relativity, rest upon this assumption.
    • The existence of sharp spectral lines is a convincing experimental proof of the above-mentioned principle of practical geometry.
    • Riemann's geometry will be the right thing if the laws of disposition of practically-rigid bodies are transformable into those of the bodies of Eudid's geometry with an exactitude which increases in proportion as the dimensions of the part of space-time under consideration are diminished.
    • It is true that this proposed physical interpretation of geometry breaks down when applied immediately to spaces of sub-molecular order of magnitude.
    • Success alone can decide as to the justification of such an attempt, which postulates physical reality for the fundamental principles of Riemann's geometry outside of the domain of their physical definitions.
    • It appears less problematical to extend the ideas of practical geometry to spaces of cosmic order of magnitude.
    • Therefore the question whether the universe is spatially finite or not seems to me decidedly a pregnant question in the sense of practical geometry.
    • In accordance with Euclidean geometry we can place them above, beside, and behind one another so as to fill a part of space of any dimensions; but this construction would never be finished; we could go on adding more and more cubes without ever finding that there was no more room.
    • It would be better to say that space is infinite in relation to practically-rigid bodies, assuming that the laws of disposition for these bodies are given by Euclidean geometry.
    • The construction is never finished; we can always go on laying squares - if their laws of disposition correspond to those of plane figures of Euclidean geometry.
    • But as the construction progresses it becomes more and more patent that the disposition of the discs in the manner indicated, without interruption, is not possible, as it should be possible by Euclidean geometry of the plane surface.
    • From the latest results of the theory of relativity it is probable that our three-dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry, if only we consider parts of space which are sufficiently great.
    • For this purpose we will first give our attention once more to the geometry of two-dimensional spherical surfaces.
    • The shadow-geometry on the plane agrees with the disc-geometry on the sphere.
    • If we call the disc-shadows rigid figures, then spherical geometry holds good on the plane E with respect to these rigid figures.
    • In fact the only objective assertion that can be made about the disc-shadows is just this, that they are related in exactly the same way as are the rigid discs on the spherical surface in the sense of Euclidean geometry.
    • The representation given above of spherical geometry on the plane is important for us, because it readily allows itself to be transferred to the three-dimensional case.
    • But these spheres are not to be rigid in the sense of Euclidean geometry; their radius is to increase (in the sense of Euclidean geometry) when they are moved away from S towards infinity, and this increase is to take place in exact accordance with the same law as applies to the increase of the radii of the disc-shadows L' on the plane.
    • After having gained a vivid mental image of the geometrical behaviour of our L' spheres, let us assume that in our space there are no rigid bodies at all in the sense of Euclidean geometry, but only bodies having the behaviour of our L' spheres.
    • Then we shall have a vivid representation of three-dimensional spherical space, or, rather of three-dimensional spherical geometry.
    • In this way, by using as stepping-stones the practice in thinking and visualisation which Euclidean geometry gives us, we have acquired a mental picture of spherical geometry.
    • Nor would it be difficult to represent the case of what is called elliptical geometry in an analogous manner.
    • My only aim today has been to show that the human faculty of visualisation is by no means bound to capitulate to non-Euclidean geometry.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Einstein_geometry.html .

  2. Poincaré on non-Euclidean geometry
    • Poincare on non-Euclidean geometry .
    • These premises are either self-evident and need no demonstration, or can be established only if based on other propositions; and, as we cannot go back in this way to infinity, every deductive science, and geometry in particular, must rest upon a certain number of indemonstrable axioms.
    • All treatises of geometry begin therefore with the enunciation of these axioms.
    • Some of these, for example, "Things which are equal to the same thing are equal to one another," are not propositions in geometry but propositions in analysis.
    • But I must insist on other axioms which are special to geometry.
    • The Geometry of Lobachevsky.
    • It would be, therefore, impossible to found on those premisses a coherent geometry.
    • From these hypotheses he deduces a series of theorems between which it is impossible to find any contradiction, and he constructs a geometry as impeccable in its logic as Euclidean geometry.
    • Riemann's Geometry.
    • Let us further admit that this world is sufficiently distant from other worlds to be withdrawn from their influence, and while we are making these hypotheses it will not cost us much to endow these beings with reasoning power, and to believe them capable of making a geometry.
    • What kind of a geometry will they construct? In the first place, it is clear that they will attribute to space only two dimensions.
    • In a word, their geometry will be spherical geometry.
    • Well, Riemann's geometry is spherical geometry extended to three dimensions.
    • In the same way, in Riemann's geometry - at least in one of its forms - through two points only one straight line can in general be drawn, but there are exceptional cases in which through two points an infinite number of straight lines can be drawn.
    • For instance, the sum of the angles of a triangle is equal to two right angles in Euclid's geometry, less than two right angles in that of Lobachevsky, and greater than two right angles in that of Riemann.
    • The number of parallel lines that can be drawn through a given point to a given line is one in Euclid's geometry, none in Riemann's, and an infinite number in the geometry of Lobachevsky.
    • There is no contradiction between, the theorems of Lobachevsky and Riemann; but however numerous are the other consequences that these geometers have deduced from their hypotheses, they had to arrest their course before they exhausted them all, for the number would be infinite; and who can say that if they had carried their deductions further they would not have eventually reached some contradiction? This difficulty does not exist for Riemann's geometry, provided it is limited to two dimensions.
    • As we have seen, the two-dimensional geometry of Riemann in fact, does not differ from spherical geometry, which is only a branch of ordinary geometry, and is therefore outside all contradiction.
    • Beltrami, by showing that Lobachevsky's two-dimensional geometry was only a branch of ordinary geometry, has equally refuted the objection as far as it is concerned.
    • The geometry of these surfaces is therefore reduced to spherical geometry - namely, Riemann's.
    • Beltrami has shown that the geometry of these surfaces is identical with that of Lobachevsky.
    • Thus the two-dimensional geometries of Riemann and Lobachevsky are connected with Euclidean geometry.
    • We shall then obtain the theorems of ordinary geometry.
    • But these translations are theorems of ordinary geometry, and no one doubts that ordinary geometry is exempt from contradiction.
    • Lobachevsky's geometry being susceptible of a concrete interpretation, ceases to be a useless logical exercise, and may be applied.
    • Further, this interpretation is not unique, and several dictionaries may be constructed analogous to that above, which will enable us by a simple translation to convert Lobachevsky's theorems into the theorems of ordinary geometry.
    • Are the axioms implicitly enunciated in our text-books the only foundation of geometry? We may be assured of the contrary when we see that, when they are abandoned one after another, there are still left standing some propositions which are common to the geometries of Euclid, Lobachevsky, and Riemann.
    • Moreover, when we study the definitions and the proofs of geometry, we see that we are compelled to admit without proof not only the possibility of this motion, but also some of its properties.
    • The Fourth Geometry.
    • Among these explicit axioms there is one which seems to me to deserve some attention, because when we abandon it we can construct a fourth geometry as coherent as those of Euclid, Lobachevsky, and Riemann.
    • Now, there is an infinite number of ways of defining this length, and each of them may be the starting-point of a new geometry.
    • Most mathematicians regard Lobachevsky's geometry as a mere logical curiosity.
    • If several geometries are possible, they say, is it certain that our geometry is the one that is true? Experiment no doubt teaches us that the sum of the angles of a triangle is equal to two right angles, but this is because the triangles we deal with are too small.
    • According to Lobachevsky, the difference is proportional to the area of the triangle, and will not this become sensible when we operate on much larger triangles, and when our measurements become more accurate? Euclid's geometry would thus be a provisory geometry.
    • There would be no non-Euclidean geometry.
    • Let us next try to get rid of this, and while rejecting this proposition let us construct a false arithmetic analogous to non-Euclidean geometry.
    • Besides, to take up again our fiction of animals without thickness, we can scarcely admit that these beings, if their minds are like ours, would adopt the Euclidean geometry, which would be contradicted by all their experience.
    • Ought we, then, to conclude that the axioms of geometry are experimental truths? But we do not make experiments on ideal lines or ideal circles; we can only make them on material objects.
    • On what, therefore, would experiments serving as a foundation for geometry be based? The answer is easy.
    • What geometry would borrow from experiment would.
    • The properties of light and its propagation in a straight line have also given rise to some of the propositions of geometry, and in particular to those of projective geometry, so that from that point of view one would be tempted to say that metrical geometry is the study of solids, and projective geometry that of light.
    • If geometry were an experimental science, it would not be an exact science.
    • In other words, the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise.
    • What, then, are we to think of the question: Is Euclidean geometry true? It has no meaning.
    • One geometry cannot be more true than another; it can only be more convenient.
    • Now, Euclidean geometry is, and will remain, the most convenient: 1st, because it is the simplest, and it is not so only because of our mental habits or because of the kind of direct intuition that we have of Euclidean space; it is the simplest in itself, just as a polynomial of the first degree is simpler than a polynomial of the second degree; 2nd, because it sufficiently agrees with the properties of natural solids, those bodies which we can compare and measure by means of our senses.

  3. Semple and Kneebone: 'Algebraic Projective Geometry
    • Semple and Kneebone: Algebraic Projective Geometry .
    • J G Semple and G T Kneebone published Algebraic Projective Geometry (Oxford University Press, Oxford, 1952).
    • This book is intended primarily for the use of students reading for an honours degree in mathematics, and our aim in writing it has been to give a rigorous and systematic account of projective geometry, which will enable the reader without undue difficulty to grasp the fundamental ideas of the subject and to learn to apply them with facility.
    • Projective geometry is a subject that lends itself naturally to algebraic treatment, and we have had no hesitation in developing it in this way - both because to do so affords a simple means of giving mathematical precision to intuitive geometrical concepts and arguments, and also because the extent to which algebra is now used in almost all branches of mathematics makes it reasonable to assume that the reader already possesses a working knowledge of its methods.
    • The exception is a theorem which is fundamental in our system but is possibly not met with in quite the same form outside geometry, and this theorem we have proved in the Appendix.
    • In spite, however, of treating geometry algebraically, we have tried never to lose sight of the synthetic approach perfected by such geometers as von Staudt, Steiner, and Reye.
    • We have, therefore, tried to show that although the basis of the formal structure is algebraic, the structure itself is thoroughgoing geometry, inasmuch as its concepts, its methods, and its results are all essentially dependent on geometrical ideas.
    • Our main purpose was not just to give the above quote, but rather to quote the fascinating Introduction to Semple and Kneebone's Algebraic Projective Geometry which looks at the concept of geometry:- .
    • THE CONCEPT OF GEOMETRY .
    • Our main purpose in this book is to construct and develop a systematic theory of projective geometry, and in order to make the system both rigorous and easily comprehensible we have chosen to build it on a purely algebraic foundation.
    • Although the axiomatic form is the proper one in which to present a mathematical theory, we must not lose sight of the fact that an abstract system can only be fully appreciated when seen in relation to a more concrete background; and this is the reason why we have prefaced the formal development of projective geometry with two introductory chapters of a more informal character.
    • The present chapter is devoted to a rather general consideration of the nature of mathematics and, more specifically, of geometry, while Chapter II contains an outline of the intuitive treatment of projective geometry from which the axiomatic theory has gradually been disentangled by progressive abstraction.
    • 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.
    • 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.
    • Thus, although arithmetic is ostensibly about numbers and geometry about points and lines, the real objects of study in these branches of mathematics are the relations which exist between numbers and between geometrical entities.
    • Abstract euclidean geometry of three dimensions, for instance, has as one of its realizations the structure of ordinary space.
    • In this book we shall study the structure of projective geometry which, as is well known, is closely associated with certain simple algebraic structures, and with linear algebra particularly.
    • What we have said so far about the nature of mathematics holds quite generally, but when we limit the discussion to geometry we are able to be rather more specific.
    • 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.
    • Whenever one body can be made in this way to take the place of another, the two bodies have the same shape; and they are then equivalent as regards their geometrical properties, or, in the language of elementary geometry, 'equal in all respects'.
    • This, then, is the nature of euclidean geometry - it is the invariant-theory of the group of displacements.
    • Euclidean geometry, however, is not the whole of geometry.
    • Early in the nineteenth century it was realized that other systematic collections of geometrical properties are possible besides that of Euclid, and in 1822 Poncelet published his Traite des proprietes projectives des figures, the first systematic treatise on projective geometry.
    • Confining ourselves, for simplicity, to two-dimensional geometry, we may consider the totality of all those transformations of the plane into itself which can be resolved into finite chains of projections from one plane on to another; and it is clear that this totality of transformations is a group and that it has plane projective geometry as its invariant-theory.
    • If we were now to take any arbitrarily chosen group of transformations of the plane into itself (containing the group of displacements as a subgroup) we could use this group in order to define an associated system of geometry; and all such systems are, mathematically speaking, of equal status.
    • Some of the geometries that can be obtained in this way, such as euclidean geometry, affine geometry, and projective geometry, are very well known; others, such as inversive geometry (which arises from the group of all transformations that can be resolved into finite sequences of inversions with respect to circles) are known but not usually studied in much detail; and yet others are presumably ignored altogether.
    • In the first place, euclidean geometry is of particular interest on account of its close connexion with the space of common experience, and this alone is sufficient to single it out for special attention.
    • It so happens, however, that euclidean geometry is complicated; and we can appreciate it better when we relate it to projective geometry, where the structure is very much simpler.
    • Projective geometry is more symmetrical than euclidean, by virtue both of the existence of a principle of duality and also of the fact that it may be handled by means of homogeneous coordinates.
    • Thus the system of projective geometry is easy to work out and equally easy to comprehend when it has been worked out.
    • Furthermore, projective transformations have the property of transforming conics into conics; and this means that the conic takes its place as naturally in projective geometry as does the circle in euclidean geometry.
    • Finally, the essentials of euclidean geometry may be treated projectively by the simple artifice of introducing the line at infinity and the circular points.
    • We thus have two geometries, projective geometry and euclidean geometry, which fit naturally together and which between them include most of the classical geometrical theorems.
    • It is convenient to take in conjunction with them affine geometry, an intermediate geometry that is more general than euclidean but less so than projective; and the projective hierarchy is then complete.
    • It is customary to distinguish between two modes of reasoning in geometry, commonly referred to as synthetic and analytical.
    • Since the discussion of projective geometry which follows in Part II is to be analytical, we shall conclude this chapter by touching upon the use of coordinates; but it should be realized, nevertheless ' that we are under no logical compulsion to introduce a coordinate system at all.
    • 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.
    • A standard text-book, written in a similar spirit, is Veblen and Young's Projective Geometry (Boston, 1910).) .
    • Coordinates were first introduced into geometry by Descartes, in the seventeenth century, and the fruitfulness of the innovation soon became apparent.
    • In geometry itself, not only points but also lines and other entities can be represented by sets of coordinates; and in dynamics - to take an instance of another kind - the configuration of a system is ordinarily specified by n coordinates q1, q, ..
    • We have now seen how mathematics may be looked upon as a study of formal structure, and how geometry may be fitted into the general scheme.
    • This will be the topic of the second chapter of Part I, in which our purpose will be to recall enough of the elementary treatment of projective geometry to enable the reader to appreciate the process of abstraction which leads to the formal system of Part II.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Semple_Kneebone_geometry.html .

  4. Sommerville: 'Geometry of n dimensions
    • Sommerville: Geometry of n dimensions .
    • Duncan Sommerville published Bibliography of Non-Euclidean Geometry, including the theory of parallels, the foundations of geometry, and space of n dimensions in 1911 while a lecturer at the University of St Andrews.
    • The book contains 1832 references to n-dimensional geometry.
    • Sommerville later wrote An Introduction to the Geometry of n dimensions which he published in 1929 during his years as a Professor in Wellington, New Zealand.
    • AN INTRODUCTION TO THE GEOMETRY OF N DIMENSIONS .
    • It is scarcely necessary to apologise for writing a book on n-dimensional geometry.
    • On the other hand, it is interesting to notice that there are about an equal number in the three volumes of the journal; this seems to indicate a revival of interest.] Yet one may almost say that this country was its home of origin, for, with the exception of a few previous sporadic references, the first paper dealing explicitly with geometry of n dimensions was one by Cayley in 1843, and the importance of the subject was recognised from the first by three of our most famous pure mathematicians - Cayley, Clifford, and Sylvester.
    • The wonderful projective geometry of hyperspace has been almost entirely the product of the gifted Italian school of geometers; though this branch also was inaugurated by a British mathematician, W K Clifford, in 1878.
    • In writing this book I have not attempted to produce a complete systematic treatise, but have rather selected certain representative topics which not only illustrate the extensions of theorems of three-dimensional geometry, but reveal results which are unexpected and where analogy would be a faithless guide.
    • The first four chapters explain the fundamental ideas of incidence, parallelism, perpendicularity, and angles between linear spaces; and in Chapter I there is an excursus into enumerative geometry which may be omitted on a first reading.
    • In the latter there are given, in addition to the ordinary Cartesian formulae, some account and applications of the Plucker-Grassmann co-ordinates of a linear space, and applications to line-geometry.
    • Reference may be made to the author's Bibliography of Non-Euclidean Geometry, including the theory of parallels, the foundations of geometry, and space of n dimensions (London: Harrison, for the University of St Andrews.
    • GEOMETRY OF N DIMENSIONS .
    • Origins of Geometry.
    • Geometry for the individual begins intuitionally and develops by a co-ordination of the senses of sight and touch.
    • When the power of abstraction had proceeded to the extent of conceiving surfaces apart from solids, plane geometry arose.
    • This stage had been reached when Greek geometry started.
    • In geometry there are objects which have to be defined, and relationships between these objects which have to be deduced either from the definitions or from other simpler relationships.
    • The whole science of geometry can, thus be made to rest upon a set of definitions and axioms.
    • Thus with the ordinary ideas of point and straight line in plane geometry the axioms can still be applied when instead of a point we substitute a pair of numbers (x, y), and instead of straight line an equation of the first degree in x and y; corresponding to the incidence of a point with a straight line we have the fact that the values of x and y satisfy the equation.
    • I.2 and 5 are existence-postulates; 2 implies two-dimensional geometry, and 5 three-dimensional.
    • Projective Geometry.
    • In fact, in Euclidean geometry this is not true since parallel lines have no point in common.
    • For the present therefore we shall confine ourselves to a simpler and more symmetrical type of geometry, projective geometry, for which we add the following axiom: .
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Sommerville_Geometry.html .

  5. André Weil: 'Algebraic Geometry
    • Andre Weil: Algebraic Geometry .
    • In 1946 Andre Weil published the Foundations of Algebraic Geometry.
    • FOUNDATIONS OF ALGEBRAIC GEOMETRY .
    • Algebraic geometry, in spite of its beauty and importance, has long been held in disrepute by many mathematicians as lacking proper foundations.
    • Nor should one forget, when discussing such subjects as algebraic geometry, and in particular the work of the Italian school, that the so-called "intuition" of earlier mathematicians, reckless in their use of it may sometimes appear to us, often rested on a most painstaking study of numerous special examples, from which they gained an insight not always found among modern exponents of the axiomatic creed.
    • Thus for a time the indiscriminate use of divergent series threatened the whole of analysis; and who can say whether Abel and Cauchy acted more as "creative" or as "critical" mathematicians when they hurried to the rescue? One would be lacking in a sense of proportion, should one compare the present situation in algebraic geometry to that which these great men had to face; but there is no doubt that, in this field, the work of consolidation has so long been overdue that the delay is now seriously hampering progress in this and other branches of mathematics.
    • foundations of algebraic geometry may claim to be exhaustive unless it includes (among other topics) the definition and elementary properties of differential forms of the first and second kind, the so-called "principle of degeneration", and the method of formal power-series; but, concerning these subjects, nothing more than some cursory remarks in Chap.
    • The main purpose of the book is to present a detailed and connected treatment of the properties of intersection-multiplicities, which is to include all that is necessary and sufficient to legitimize the use made of these multiplicities in classical algebraic geometry, especially of the Italian school.
    • At the same time, this book seeks to deserve its title by being entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals.
    • But in such a subject as algebraic geometry, where earlier authors left many terms incompletely defined, and were wont to make (sometimes implicitly) assumptions from which we wish to be free, all terms have to be defined anew, and to attach precise meanings to them is a task not unworthy of our most solicitous attention.
    • How much the present book contributes to this, our readers, and future algebraic geometers, must judge; at any rate, as has been hinted above, and as will be shown in detail in a forthcoming series of papers, its language and its results have already been applied to the re-statement and extension of the theory of correspondences on algebraic curves, and of the geometry on Abelian varieties, and have successfully stood that test.
    • Our results include all that is required for a rigorous treatment of so-called "enumerative geometry", thus providing a complete solution of Hilbert's fifteenth problem.
    • They could be said, indeed, to belong to enumerative geometry, had it not become traditional to restrict the use of this phrase to a body of special problems, pertaining to the geometry of the projective spaces and of certain, rational varieties (spaces of straight lines, of conies, etc.), whereas we shall emphasize the geometry on an arbitrary variety, or at least on a variety without multiple points.
    • The theory of intersection-multiplicities, however, occupies such a central position among the topics which constitute the foundations of algebraic geometry, that a complete treatment of it necessarily supplies the tools by which many other such topics can be dealt with.
    • A history of enumerative geometry could be a fascinating chapter in the general history of mathematics during the previous and present centuries, provided it brought to light the connections with related subjects, not merely with projective geometry, but with group-theory, the theory of Abelian functions, topology, etc.; this would require another book and a more competent writer.
    • It is well known that classical algebraic geometry does not usually deal with varieties in affine spaces, but with so-called projective models; the main feature which distinguishes the latter from the former is that they are, in a certain sense, "complete", or, in the topological case (when the ground-field is the field of complex numbers), compact.
    • One main source of ambiguity, in the work of classical algebraic geometers and sometimes even in that of more modern writers, lies in their use of the word "variety", or of the word "curve" when they are dealing merely with the geometry on surfaces.
    • A point is thus reached in the systematic development of algebraic geometry, of which this volume may be regarded as the preliminary part, from which one may, with better perspective, look back on the course which has been hitherto followed, and make plans for the continuation of the voyage.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Weil_Algebraic_Geometry.html .

  6. EMS obituary
    • The striking elucidations by geometry of phenomena that sprang from other branches of mathematics, the sudden perception in a figure of some intrinsic incandescence, these had a profound effect on and a singular fascination for Baker: as though he were being led to recognise the verities of things sub specie aeternitatis.
    • Herein may well have lain the chief reason for his turning to geometry.
    • His high appreciation of analysis remained: Riemann he almost worshipped, so impressed was he with the deep insight of his ideas; the work of Weierstrass and Poincare he knew as few others could know; yet he chose geometry.
    • So he inscribed the title page of the first volume of Principles of Geometry, and so he believed.
    • The preceding sentences have indicated that a knowledge of geometry in higher space may be necessary for a proper appreciation of geometry in ordinary space, and Baker's main preoccupation in writing F was to publicise this fact; therein Segre's generation of the tetrahedral complex appears on p.
    • Cayley, as long ago as 1846, said, after remarking that Desargues' figure in a plane is a projection of the 10 edges and 10 vertices of a pentahedron, that it was only reasonable to expect, by analogy, a simplification of geometry in space by using figures in higher space.
    • Klein, in 1872, explained how geometries in spaces of different dimensions could be equivalent; a geometry does not depend primarily on the ambient space but on the group of self-transformations of the figure.
    • But it is a fascinating study, and British mathematicians may well be proud of such a splendid mine of geometrical lore as is to be found in the four volumes of Principles of Geometry.
    • Plane geometry does not demand that the absolute points I, J be either imaginary or at infinity, as they are in the Euclidean plane.
    • So any non-singular quadric S can be projected stereographically from any point N of S onto any plane n not through N; the generators i, j of S at N meet h in points I, J which can serve as absolute points in the plane geometry.
    • In inversive geometry the lines and circles form a closed family.
    • Klein showed, in the Erlanger Programm of 1872, that inversive geometry in a plane is equivalent to projective geometry on a quadric; this is because, the lines and circles in 7) answering to the plane sections of S, it is precisely the plane sections of S that must form a closed family in a geometry equivalent to the inversive geometry in h so that S must, as a surface, be unaltered and its plane sections permuted among themselves.
    • For this equivalence between inversion in a plane and geometry on a quadric in space is only an instance, for n = 3, of the equivalence between inversion in [n - 1] and geometry on a quadric in [n].
    • In 1884 Segre published a 130-page paper which is one of the landmarks of descriptive geometry and gives one to understand why Baker spoke, in 10, of Segre's power of fashioning a new world from the bare suggestions of others.
    • He was glad to have done this book and set some store by its logical framework, claiming to start absolutely from scratch with no foundations of Euclid's results or of propositions from "sequels" to Euclid, doing the geometry of circles ab initio.
    • The tract describes the geometry, in [4], of a group of 25920 linear transformations, and its first consequence was J A Todd's using the geometry to decompose the group into its conjugate classes.
    • In 1947 however he was still reading: he read with close attention Hodge and Pedoe's Method of Algebraic Geometry, noting particularly the manner in which they introduced co-ordinates.
    • I have spent some time of late in looking carefully through (B) Segre's recent "Modern Geometry, Vol.
    • This was his last paper, 15 being a brief pendant to it whereby the long procession of impressive works falls quietly to its close with a diagram depicting basic propositions of projective geometry in a plane.
    • The last word lies, after all, with "the constructive methods of the old-established geometry." In minimis maxima.
    • Principles of Geometry .
    • Plane Geometry 1922, 1930 .
    • Solid Geometry 1923 .
    • Higher Geometry 1925 .
    • Introduction to Plane Geometry 1943 .
    • On non-commutative algebra, and the foundations of projective geometry.
    • Note on the foundations of projective geometry.

  7. Proclus and the history of geometry to Euclid
    • Proclus and the history of geometry to Euclid .
    • Proclus Diadochus, in his Commentary on Euclid's Elements, relates the history of geometry up to the time of Euclid.
    • We must next speak of the origin of geometry in the present world cycle.
    • But since we must speak of the origin of the arts and sciences with reference to the present world cycle, it was, we say, among the Egyptians that geometry is generally held to have been discovered.
    • And so, just as the accurate knowledge of numbers originated with the Phoenicians through their commerce and their business transactions, so geometry was discovered by the Egyptians for the reason we have indicated.
    • After him Mamercus, the brother of the poet Stesichorus, is said to have embraced the study of geometry, and in fact Hippias of Elis writes that he achieved fame in that study.
    • After these Pythagoras changed the study of geometry, giving it the form of a liberal discipline, seeking its first principles in ultimate ideas, and investigating its theorems abstractly and in a purely intellectual way.
    • After him Anaxagoras of Clazomenae devoted himself to many of the problems of geometry, as did Oenopides of Chios, who was a little younger than Anaxagoras.
    • After them Hippocrates of Chios the discoverer of the quadrature of the lune, and Theodorus of Cyrene gained fame in geometry.
    • Plato, who lived after Hippocrates and Theodorus, stimulated to a very high degree the study of mathematics and of geometry in particular because of his zealous interest in these subjects.
    • Amyclas of Heraclea, a friend of Plato, Menaechmus, a pupil of Eudoxus who had also been associated with Plato, and Dinostratus, a brother of Menaechmus, brought the whole of geometry to an even higher degree of perfection.
    • Furthermore, Athenaeus of Cyzicus, a contemporary, distinguished himself in mathematics generally and in geometry in particular.
    • Now those who have written histories trace the development of the science of geometry down to Philippus.
    • Furthermore, there is a story that Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by the study of the Elements.
    • Whereupon Euclid answered that there was no royal road to geometry.
    • But he is most to be admired for his Elements of Geometry because of the choice and arrangement of the theorems and problems made with regard to the elements.
    • This book, therefore, is for the purification and training of the understanding, while the Elements contain the complete and irrefutable guide to the scientific study of the subject of geometry.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Proculus_history_geometry.html .

  8. Proclus and the history of geometry as far as Euclid
    • Proclus and the history of geometry as far as Euclid .
    • Proclus Diadochus, in his Commentary on Euclid's Elements, relates the history of geometry up to the time of Euclid.
    • We must next speak of the origin of geometry in the present world cycle.
    • But since we must speak of the origin of the arts and sciences with reference to the present world cycle, it was, we say, among the Egyptians that geometry is generally held to have been discovered.
    • And so, just as the accurate knowledge of numbers originated with the Phoenicians through their commerce and their business transactions, so geometry was discovered by the Egyptians for the reason we have indicated.
    • After him Mamercus, the brother of the poet Stesichorus, is said to have embraced the study of geometry, and in fact Hippias of Elis writes that he achieved fame in that study.
    • After these Pythagoras changed the study of geometry, giving it the form of a liberal discipline, seeking its first principles in ultimate ideas, and investigating its theorems abstractly and in a purely intellectual way.
    • After him Anaxagoras of Clazomenae devoted himself to many of the problems of geometry, as did Oenopides of Chios, who was a little younger than Anaxagoras.
    • After them Hippocrates of Chios the discoverer of the quadrature of the lune, and Theodorus of Cyrene gained fame in geometry.
    • Plato, who lived after Hippocrates and Theodorus, stimulated to a very high degree the study of mathematics and of geometry in particular because of his zealous interest in these subjects.
    • Amyclas of Heraclea, a friend of Plato, Menaechmus, a pupil of Eudoxus who had also been associated with Plato, and Dinostratus, a brother of Menaechmus, brought the whole of geometry to an even higher degree of perfection.
    • Furthermore, Athenaeus of Cyzicus, a contemporary, distinguished himself in mathematics generally and in geometry in particular.
    • Now those who have written histories trace the development of the science of geometry down to Philippus.
    • Furthermore, there is a story that Ptolemy once asked Euclid whether there was any shorter way to a knowledge of geometry than by the study of the Elements.
    • Whereupon Euclid answered that there was no royal road to geometry.
    • But he is most to be admired for his Elements of Geometry because of the choice and arrangement of the theorems and problems made with regard to the elements.
    • This book, therefore, is for the purification and training of the understanding, while the Elements contain the complete and irrefutable guide to the scientific study of the subject of geometry.
    • http://www-history.mcs.st-andrews.ac.uk/Extras/Proclus_history_geometry.html .

  9. Proclus and the history of geometry to Euclid

  10. Oswald Veblen Publications
    • 1902 "Hilbert's Foundation of Geometry", Monist, 13, 303-309.
    • (d) "A System of Axioms for Geometry", Trans.
    • (b) "Collineations in a Finite Projective Geometry", Trans.
    • (c) (With J W Young) "A set of Assumptions for Projective Geometry", Amer.
    • (b) (With J W Young) Projective Geometry.
    • 1918 (With J W Young) Projective Geometry.
    • 1922 (a) (with L P Eisenhart) "The Riemann Geometry and its Generalization, Proc.
    • (b) "Normal Coordinates for the Geometry of Paths", Proc.
    • (c) "Projective and Affine Geometry of Paths", Proc.
    • 1923 (a) Equiaffine Geometry of Paths", Proc.
    • (b) "Projective and Affine Geometry of Paths", Bull.
    • (d) (With T Y Thomas) "The Geometry of Paths", Trans.
    • 1925 (a) "Remarks on the Foundations of Geometry", Bull.
    • (b) (With J M Thomas) "Projective Normal Coordinates for the Geometry of Paths", Proc.
    • 1926 (With J M Thomas) "Projective Invariants of Affine Geometry of Paths", Ann.
    • (b) "Generalized Projective Geometry", J.
    • (c) "Differential Invariants and Geometry", Int.
    • 1931 (With J H C Whitehead) "A Set of Axioms for Differential Geometry", Proc.
    • 1932 (With J H C Whitehead) The Foundations of Differential Geometry, Camb.
    • 1933 (a) "Geometry of Two-component Spinors", Proc.
    • (b) "Geometry of Four-component Spinors", Proc.
    • (c) "Certain Aspects of Modern Geometry." (Three lectures delivered at the Rice Institute): .
    • The Modern Approach to Elementary Geometry.
    • Modern Differential Geometry.
    • 1935 (a) "Formalism for Conformal Geometry", Proc.
    • 1936 (With J von Neumann) Geometry of Complex Domains.
    • 1937 "Spinors and Projective Geometry", Int.

  11. Sommerville obituary.html
    • He had an original mind, and beneath his outward shyness considerable talents lay concealed his intellectual grasp of geometry was balanced by a deftness in making models, and on the aesthetic side by an undoubted talent with the brush.
    • His text-books which have appeared at regular intervals are a valuable link between the old and the new era in the teaching of geometry at College.
    • They are the Elements of Non-Euclidean Geometry (1914), Analytical Conics (1924), Introduction to the Geometry of n Dimensions (1929), and the recent Three Dimensional Geometry (1934) the appearance of which he did not live to see.
    • All are characterized by a variety of algebraic treatment and a wealth of illustrations and examples, but nowhere, does technical manipulation outrun the geometry.
    • The first, entitled Networks of the Plane in Absolute Geometry (Proc.
    • The main theme is that of combinatory geometry, exemplified by a systematic investigation of The Division of Space by Congruent Triangles and Tetrahedra (1923) in the same journal, and extended to n dimensions (Palermo, 48 (1924), 9-22).
    • Although be was conversant with the more fashionable developments of the subject his own researches are for the most part concerned with the two themes of Non-Euclidean Geometry (in the restricted sense of geometry with a projective metric) and the enumerative and other properties of configurations possessing some degree of regularity or completeness, both themes being extended to n dimensions.
    • His familiarity with Non-Euclidean Geometry must have been almost unique: he treated it as worthy of a detailed study comparable to that accorded to Euclidean Geometry.
    • There may be mentioned as examples of his researches the classification of all types of Non-Euclidean Geometry (including those usually excluded as bizarre); the extension, involving the measurement of generalized angles in higher space, of Euler's Theorem on Polyhedra; space-filling figures; the classification of polytopes (i.e.
    • His text-book Introduction to Geometry of n Dimensions gives some notion of his researches in these two directions.
    • His wide knowledge of other branches of geometry (and incidentally of European languages) are seen most clearly in his Bibliography of Non-Euclidean Geometry, whose title, bereft of its subtitles, is misleading, any work on higher space being included.
    • Networks of the Plane in Absolute Geometry.
    • Semi-Regular Networks of the Plane in Absolute Geometry.
    • The Pedal Line of the Triangle in Non-Euclidean Geometry.
    • Quadratic Systems of Circles in Non-Euclidean Geometry.
    • Metrical Co-ordinates in Non-Euclidean Geometry.
    • Bibliography of Non-Euclidean Geometry.
    • The Elements of Non-Euclidean Geometry.
    • An Introduction to the Geometry of n Dimensions.
    • Analytical Geometry of Three Dimensions.
    • Taylor's Cubics associated with a Triangle in Non-Euclidean Geometry, 33, pp.
    • Peaucellier's Cell and other Linkages in Non-Euclidean Geometry, 44, pp.

  12. St Andrews Mathematics Examinations
    • At the University of St Andrews two Mathematics papers, one on Geometry and Trigonometry, the other on Algebra and Coordinate Geometry, were set in October 1884 and two papers on the same topics in April 1885.
    • GEOMETRY AND TRIGONOMETRY.
    • ALGEBRA AND CO-ORDINATE GEOMETRY.
    • GEOMETRY AND TRIGONOMETRY.
    • ALGEBRA AND CO-ORDINATE GEOMETRY.

  13. Campbell on Differential Geometry
    • Campbell on Differential Geometry .
    • Towards the end of his career J E Campbell's interests turned from continuous groups to differential geometry.
    • He had almost completed work on a book on differential geometry at the time of his death in October 1924.
    • GEOMETRY .
    • As I had made no special study of Differential Geometry beforehand, and was entirely without expertness in the methods of which Mr.

  14. Plato on Mathematics

  15. Plato on Mathematics
    • For this, he believes, one must study the five mathematical disciplines, namely arithmetic, plane geometry, solid geometry, astronomy, and harmonics.
    • Plato argues the merits of learning plane geometry .
    • 'Do you mean geometry?,' he asked.
    • 'Clearly,' he said, 'we are concerned with that part of geometry which relates to war; for in pitching a camp, or taking up a position, or closing or extending the lines of an army, or any other military manoeuvre, whether in actual battle or on a march, it will make all the difference whether a general does or does not know geometry.' .
    • 'Yes,' I said, 'but for that purpose a very little of either geometry or arithmetic will be sufficient; the question relates rather to the greater and more advanced part of geometry - whether that tends in any degree to make more easy the vision of the idea of good; and that, as I was saying, all things tend which compel the mind to turn its attention towards that place, where is the full perfection of being, which it ought, by all means, to see.' .
    • 'Then if geometry compels us to view reality, it concerns us; if the realm of change only, it does not concern us?' .
    • 'Yet anybody who has the least acquaintance with geometry will not deny that such a conception of the science is quite the opposite to the ordinary terms of those use it.' .
    • 'They have in view practice only, and are always talking in a narrow and ridiculous manner, of "squaring" and "extending" and "applying" and the like - they confuse the ways of geometry with those of daily life; whereas knowledge is the real object of the whole science.' .
    • That the knowledge at which geometry aims is knowledge of the eternal, and not of anything transient which will decay.' .
    • 'Then, my noble friend, geometry will draw the mind towards truth, and create the spirit of philosophy, and raise up that which is now sadly allowed to fall down.' .
    • 'Then nothing should be more strongly required than that the inhabitants of your State should by all means learn geometry.
    • 'There are the military advantages of which you spoke,' I said, 'and in all departments of knowledge, as experience proves, any one who has studied geometry is infinitely quicker at learning other subjects than one who has not.' .
    • Plato argues the merits of supporting solid geometry .
    • It is clear that solid geometry was relatively undeveloped when Plato wrote The Republic so he argues strongly that its study should be encouraged.
    • 'Then take a step backward, for we have gone wrong in the order of the sciences coming after plane geometry.' .
    • 'After plane geometry,' I said, 'we proceeded at once to solids in revolution, instead of taking solids in themselves; whereas after the second dimension the third, which is concerned with cubes and dimensions of depth, ought to have followed.' .
    • First you began with geometry of plane surfaces?' .
    • 'In my hurry, the ludicrous state of solid geometry, which, in natural order, should have followed, made me pass over this branch and go on to astronomy, or the motion of solids.' .
    • Anybody understanding geometry who saw them would appreciate the exquisiteness of their workmanship, but he would never dream of thinking that in them he could learn the truth about equality or about doubling, or the truth about any other proportion.' .
    • 'Then,' I said, 'in astronomy, as in geometry, we should set problems to be solved, and leave the visible heavens alone if we want to approach the subject in the right way and so to put the natural gift of reason to a real purpose.' .

  16. EMS 1913 Colloquium
    • Professor A W Conway, of the National University of Ireland, is taking for his subject "The Theory of Relativity and the New Physical Ideas of Space and Time;" Dr Sommerville, of St Andrews University, lectures on "Non-Euclidean Geometry and the Foundations of Geometry;" and Professor Whittaker, Edinburgh University, gives a course of five lectures and demonstrations on "Practical Harmonic Analysis and Periodogram Analysis." By the courtesy of the University Court, several rooms have been set aside as reception and writing rooms, and these have been furnished for the comfort and convenience of members of the colloquium.
    • The third lecture, on the subject of "Non-Euclidean Geometry," was delivered at 2 p.m.
    • After explaining how non-Euclidean Geometry arose from attempts to prove the axiom about parallel lines, the lecturer proceeded to give an exposition of the system of geometry which was discovered by Lobachevsky, in which Playfair's axiom was directly contradicted and the sum of the angles of a triangle was always less than two right angles.
    • Dr Sommerville's second lecture on Non-Euclidean Geometry was devoted to the geometry of Riemann, in which parallel lines do not exist, and the sum of the angles of a triangle is always greater than two right angles.
    • While there are no parallel lines in this geometry, lines in space may be equidistant, and a remarkable surface is obtained by revolving one line about another to which it is equidistant.
    • This surface, discovered by W K Clifford, has the property that the geometry of shortest line upon it is the same as the geometry of Euclid.
    • In his third lecture on Non-Euclidean Geometry, Dr Sommerville elaborated the conception of the "absolute," the assemblage of points at infinity.
    • It was shown how this figure, which in Non-Euclidean Geometry was a conic, real or imaginary, degenerated in Euclidean geometry to a straight line and two imaginary points.
    • The method of determining distance and angle with reference to the absolute was explained, and it was shown how this process reduced the whole of metrical geometry to protective geometry in relation to the absolute.
    • In the second part of the lecture Dr Sommerville considered the question from the point of view of geometry on a curved surface, and showed how concrete representations of the Non-Euclidean geometries were obtained by means of certain surfaces which possessed constant measure of curvature.
    • In his fourth lecture, Dr Sommerville introduced the subject of the foundations of geometry.
    • The problem was to establish a system of axioms, or assumptions, satisfying the tests of consistency, independence, and categoricalness, and such that the whole of geometry can be developed from these by pure logical deduction.
    • The lecturer confined the discussion to projective geometry, and showed how the necessary assumptions were analysed into their primary constituents.
    • When the method of denial was applied to these as to the parallel-postulate, new forms of non-Euclidean geometry emerged.
    • In his fifth and concluding lecture, Dr Sommerville continued the subject of the foundations of geometry.
    • It was shown how the complete proof of the fundamental theorem of projective geometry requires an assumption of continuity, which in a curious way implies the theorem of Pascal and the commutative law of multiplication.

  17. Finlay Freundlich's Inaugural Address, Part 2
    • It concerns a problem which formerly was thought to be the domain of pure mathematics - I am thinking of geometry.
    • Until only about 40 years ago it was believed that the laws according to which distances in space are measured were given by Euclid's laws of geometry, the first mathematical system systematically and consistently built up from axioms, i.e.
    • However, full agreement whether Euclid's fundamental axioms were really all indisputable, that means whether the omission of one of them would necessarily lead to an inconsistency in the resulting Geometry, was never reached.
    • It was still taken for granted that the laws, according to which distances in physical space have to be measured, remained those given by Euclid's geometry.
    • It is, however, usually not realized, that when proving the congruency of triangles in geometry very definite assumptions as to the free mobility of rigid bodies in space have to be made.
    • In fact the laws of Geometry of triangles drawn on the surface of a sphere differ systematically from the laws of Euclid's Geometry in the plane.
    • The spherical geometry is the most simple case of a non-Euclidean geometry.
    • When, however, the true distances of celestial bodies in space are considered, it was still taken for granted that the laws of Euclidean geometry have to be applied.
    • geometries which give metric laws for spaces of positive or negative curvature, it was the mathematician Riemann, who, in 1854, for the first time clearly stated that in physical space, that means in the space in which all phenomena of natural science proceed, the laws of geometry must be determined by the forces which act between the celestial bodies.
    • The geometry of physical space is not given a priori.
    • The Euclidean geometry would be strictly applicable only if no forces were acting and if rigid bodies were absolutely freely movable in space.
    • That this assumption is of really general and fundamental importance has been investigated and discussed with all rigour, when the foundations of geometry were in the forefront of discussion during the last century.
    • But due to the changed geometrical conditions of space, arising from the gravitational field produced by the Sun, this shortest connection is no longer a straight line of Euclid's geometry, but the arc of a Kepler orbit.
    • The existence of gravitation manifests itself in changed laws of geometry.
    • The geometry of physical space is not given a priori, but depends on the distribution of masses in space.
    • The changed geometry determines the mobility of the bodies in space and thus the orbits along which they have to move.
    • Again we experience that it is astronomical research that made it possible to investigate this profound problem concerning the foundations of the geometry in physical space.

  18. 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 thus indirectly expounds the point of view which outside England led, already many years ago, to the revolution in the study of Geometry above referred to.
    • The Italian original, which appeared in 1900, consisted of two parts, the first deals with the Axioms of Geometry, the second, translated into German and enlarged, constitutes the volume before us.
    • The first part of that work is concerned with the more general problems of the logic of science; the second part discusses the concepts of geometry, mechanics, physics, and biology.
    • The able work, 'Problemi della Scienza', from the pen of F Enriques, Professor of Geometry in the University of Bologna, well deserved an English translation.
    • After an introduction dealing with the presuppositions and limitations of the scientific concept of reality, the author proceeds to review the basic principles of logic, geometry, mechanics, and the mechanical theory of life.
    • Among mathematicians Enriques, who is professor of projective and descriptive geometry in the University of Bologna, has long been favourably known for his contributions to geometry, especially for his admirable treatise on "Projective Geometry" and for his penetrating essays on "The Foundations of Geometry." In the work before us the distinguished geometrician addresses a far wider circle of students and thinkers: not only mathematicians, but psychologists, logicians, philosophers, astronomers, mechanicians, physicists, chemists, biologists and others.
    • The book contains six chapters treating in order the following topics: the general problem of knowledge and related matters; facts and theories and their interactions; the general problems of logic; the philosophical and psychological questions which are naturally raised in connection with the science of geometry; mechanics, its objective significance and the psychological development of its principles; the extension of mechanics into physics and the relation of the mechanical hypothesis to the phenomena of life.
    • It may be divided into five parts; the first is a general introduction and explanation of the author's position (which he calls Critical Positivism), the second deals with Logic and its applicability to the real world, the third deals with geometry, the fourth with the classical mechanics, and the last with electro-dynamics and the alterations which it has entailed in the mechanics of Newton.
    • It is the outcome of a suggestion made by certain leaders in the in the training of teachers of mathematics and it proceeds upon the principle that it is only by a historic survey of the classical textbooks of geometry that a sound basis can be laid for a critical study of the subject.
    • It is significant that the teachers of Italy, which now ranks among the half dozen leading nations in mathematical activity, should feel the need of such an edition of Euclid at a time when some of our American educators are proclaiming the uselessness and indeed the happy death of the science of geometry.
    • "And certainly in no field can this programme be better carried out than in that of elementary geometry, since the development of this subject through the centuries is completely dominated by the great work of Euclid, insomuch that its history is bound up with that of Euclidean criticism." .
    • Professor Enriques has brought his wide scholarship, his penetrating intelligence, and his customary charm of manner, to what is not quite, in the usual sense, a history of mathematics; history is there, in plenty, but to a great extent in outline only; and the historical development is broken off at the eighteenth century to make room for a longish digression, about a third of the book, on the relation of mathematics to other sciences, before returning to the modern development of various branches of analysis and geometry.

  19. Pappus on analysis and synthesis in geometry
    • Pappus on analysis and synthesis in geometry .
    • In Book VII of this work he discusses analysis and synthesis in geometry.

  20. Kepler's Planetary Laws
    • 5.nnEssential orthogonality of Euclid's geometry .
      Go directly to this paragraph
    • In Kepler's day modern algebraic notation and techniques were just being developed, but for his approach to astronomy Kepler depended exclusively on the traditional geometry of Euclid in which he had been trained at the University of Tubingen, as part of the standard preparation for the ministry.
    • Thus, the distinguishing feature of the geometry of Elements was that it relied on straight lines and circles alone.
    • 129-141 AD), Kepler made use of precisely three propositions from the work of Archimedes; one of these was vital in supplying the geometrical backing for Section 6 (the other two - one cited in Section 7, one in Section 11 - were concerned with an innovative approach to 'infinitesimal' considerations which went well beyond traditional geometry).
    • Meanwhile we reiterate Kepler's belief that Euclid's Elements encapsulated the only geometry that could properly be applied to the heavens, which after all was the realm of God.
    • So he finally rejected the idea that each planet moved in a single circle, and set out to find the actual curve that was the planet's path - naturally, this had to be constructed from a combination of (arcs of) circles by the geometry of Euclid, since Kepler recognized nothing else as appropriate for the heavens.
    • Unless its focus coincides with the fixed Sun (the origin), the investigation would have been too complicated to manage by geometry.
    • Kepler was able to formulate a complete account of planetary motion using only elementary geometry, and accordingly we will highlight the two overriding reasons for his achievement, putting them in a historical context.
    • A E L Davis: 'Some plane geometry from a cone: the focal distance of an ellipse at a glance', Mathematical Gazette, forthcoming July 2007.

  21. Ernest Hobson addresses the British Association in 1910, Part 2
    • That great modern development, Projective Geometry, has been so formulated as to be independent of all metric considerations.
    • More recently, questions relating to the foundations of geometry and rational mechanics have much occupied the attention of mathematicians.
    • The Mengenlehre, or theory of aggregates, had its origin in the critical study of the foundations of analysis, but has already become a great constructive scheme, is indispensable as a method in the investigations of analysis, provides the language requisite for the statement in precise form of analytical theorems of a general character, and, moreover, has already found important applications in geometry.
    • Both in geometry and in analysis our standard of what constitutes a rigorous demonstration has in the course of the nineteenth century undergone an almost revolutionary change.
    • That oldest text-book of science in the world, 'Euclid's Elements of Geometry,' has been popularly held for centuries to be the very model of deductive logical demonstration.
    • The method of superimposition, employed by Euclid with obvious reluctance, but forming an essential part of his treatment of geometry, is, when regarded from his point of view, open to most serious objections as regards its logical coherence.
    • In analysis, as in geometry, the older methods of treatment consisted of processes of deduction eked out by the more or less surreptitious introduction, at numerous points in the subject, of assumptions only justifiable by spatial intuition.
    • The, result of this deviation from the purely deductive method was more disastrous in the case of analysis than in geometry, because it led to much actual error in