Search Results for Euclidean geometry

Biographies

1. 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.
     
   • A Klein bottle cannot be constructed in Euclidean space.
     
   • It is possible to construct a Klein bottle in non-Euclidean space.
     
   • History Topics: Non-Euclidean geometry .
     

2. 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 .
     

3. 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.
     
   • This is described in [Ideas of space : Euclidean, non-Euclidean, and relativistic (New York, 1979).',3)">3].
     
   • 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.
     
   • He had developed a non-euclidean trigonometry which he applied to a number of elementary problems.
     
   • 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.
     
   • He also realised that there were an infinite number of non-euclidean geometries and this, Taurinus claimed, was highly significant.
     
   • 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.
     

4. 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 .
     

5. 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.
     
   • is that, having 'in intellectu' a more general notion of space (in fact a notion of non-Euclidean space), we learn by experience that space (the physical space of our experience) is, if not exactly, at least to the highest degree of approximation, Euclidean space.
     
   • 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 .
     

6. 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.
     
   • Euclidean and hyperbolic groups of three-dimensional space (1927) both in the Annals of Mathematics.
     
   • 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.
     

7. 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.
     
   • Today we call these three geometries Euclidean, hyperbolic, and absolute.
     
   • 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 .
     

8. 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.
     
   • It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied.
     
   • 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.
     
   • In particular book seven is a self-contained introduction to number theory and contains the Euclidean algorithm for finding the greatest common divisor of two numbers.
     
   • Books eleven to thirteen deal with three-dimensional geometry.
     
   • Our earliest glimpse of Euclidean material will be the most remarkable for a thousand years, six fragmentary ostraca containing text and a figure ..
     
   • 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 .
     

9. 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.
     
   • He discovered a tensor, now called Weyl's conformal curvature tensor, whose vanishing is a necessary condition that the space be conformally flat, that is to say, that the space can be mapped conformally on the Euclidean space.
     
   • 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.
     

10. 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.
      
    • This, however, has not proved to be correct and experimental evidence now shows clearly that physical space is not euclidean.
      
    • 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 .
      

11. 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.
      
    • the generalisation, in higher space, of polyhedra), it is typical that this includes polytopes in non-euclidean space ..
      
    • 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 .
      

12. 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.
      

13. 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 .
      

14. 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.
      
    • No proof is required, Poncelet says, for one simply uses the Euclidean theorem and invokes his principle of continuity.
      
    • 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.
      

15. 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]:- .
      
    • Lobachevskian, and Euclidean geometries on the concept of the distance between two points.
      
    • geometry was the mathematical physics of distances.
      

16. 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.
      
    • In 1908 Titeica showed that for a surface in Euclidean 3-space the ratio of the Gaussian curvature to the fourth power of the distance from a fixed point to the tangent plane is invariant under an affine transformation fixing O.
      
    • Titeica was led to these studies starting from deep research concerning deformation theory of surfaces in three dimensional Euclidean space.
      
    • 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.
      

17. Wu Wen-Tsun biography
    
    • He went on to use the classes he had introduced to prove a beautiful result on embedding manifolds in Euclidean space.
      
    • I introduced the notion of imbedding classes, and established a theory of imbedding, immersion, and isotopy of polyhedra in Euclidean spaces which was published in book form later in 1965.
      
    • 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.
      

18. 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.
      
    • Following the publication of his thesis, Kuiper published papers such as On differentiable line systems of one dual variable (1948), On conformally-flat spaces in the large (1949), A closure theorem (1949), On compact conformally Euclidean spaces of dimension > 2 (1950), On linear families of involutions (1950), Compact spaces with a local structure determined by the group of similarity transformations in En (1950), Einstein spaces and connections (1950), and Distribution modulo 1 of some continuous functions (1950).
      
    • 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.
      

19. Helmholtz biography
    
    • Helmholtz had begun to investigate the properties of non-Euclidean space around the time his interests were turning towards physics in 1867.
      
    • 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.
      

20. 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.
      

21. Moser William biography
    
    • For instance, it deals with the crystallographic groups, with the fundamental groups of surfaces, with the symmetric and related groups, with the projective linear groups over finite fields, and with groups on real Euclidean space generated by reflections.
      
    • Let there be a finite number n (> 1) of points in the real Euclidean or the real projective plane, not all on one line, and let every pair of distinct points be joined by a line.
      
    • 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.
      

22. Robertson biography
    
    • Robertson was awarded his doctorate from the California Institute of Technology in 1925 after submitting his dissertation On the Dynamical Space-Time which Contains a Conformal Euclidean 3-Space.
      
    • 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.
      

23. 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.
      

24. 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.
      
    • In 1934 Coxeter classified all spherical and euclidean Coxeter groups.
      
    • 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).
      

25. 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.
      

26. 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.
      
    • all failed because he always relied, in the last analysis, on propositions that were "evident" from the Euclidean point of view.
      
    • 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 .
      

27. 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.
      

28. 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.
      

29. 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.
      
    • In fact, at first approximation in a geodesic coordinate system such a metric is flat Euclidean, in the same way that a curved surface up to higher-order terms looks like its tangent plane.
      
    • 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 .
      

30. 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.
      

31. Lambert biography
    
    • By assuming that the parallel postulate was false, he managed to deduce a large number of non-euclidean results.
      
    • 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 .
      

32. 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.
      

33. 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.
      

34. 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.
      

35. 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 .
      

36. 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.
      
    • Kotelnikov obtained his doctorate in 1899 for a thesis The Projective Theory of Vectors which generalised the vector calculus to the non-euclidean spaces of Lobachevsky and Riemann.
      
    • He also applied this to mechanics in non-euclidean spaces.
      
    • 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.
      

37. 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.
      

38. 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 ..
      
    • He also studied birational contact transformations and non-euclidean and non-archimedean geometries.
      
    • 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).
      

39. 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 .
      

40. 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 .
      

41. 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.
      

42. 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.
      

43. 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.
      

44. 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.
      

45. 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".
      
    • By 1907 Minkowski realised that the work of Lorentz and Einstein could be best understood in a non-euclidean space.
      
    • 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.
      

46. 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.
      

47. 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).',1)">1]:- .
      
    • 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.
      

48. Bianchi biography
    
    • He was promoted a number of times, to extraordinary professor in differential geometry, then extraordinary professor in projective geometry, then of analytic geometry.
      
    • He became a full professor of analytic geometry in 1890.
      
    • Bianchi made important contributions to differential geometry.
      
    • His work on non-euclidean geometries was used by Einstein in his general theory of relativity.
      
    • 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.
      
    • In particular he wrote Lectures on differential geometry (1894), Lectures on the theory of groups of substitutions (1900), Lectures on the theory of continuous groups (1918), Lectures on the theory of functions of a complex variable (1901) and Lectures on the theory of algebraic numbers (1923).
      

49. 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]:- .
      

50. 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.
      
    • Among his most significant papers written later in his career we mention (with U Abresch) On complete manifolds with nonnegative Ricci curvature (1990), (with M Dajczer) The Weierstrass representation for complete minimal real Kahler submanifolds of codimension 2 (1995), and (with G Walschap) The metric fibrations of euclidean spaces (2001).
      
    • In this last mentioned paper, the authors completed the classification of metric fibrations in Euclidean space which they had begun in a paper in 1997.
      

51. 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.
      
    • The principal problem (due to P Funk) solved in this paper is that of characterising in an invariant manner all two-dimensional Finsler spaces which can be mapped geodesically on a Euclidean plane ("Finsler spaces with rectilinear extremals").
      

52. 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.
      
    • Together with Henry Whitehead and Christopher Zeeman he published Imbedding of manifolds in euclidean space in 1961.
      
    • 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.
      

53. 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.
      
    • His publications from this period are Normal congruences and quadruply infinite systems of curves in space (1924), and A criterion for the conformal equivalence of a Riemann space to a Euclidean space (1925).
      
    • 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.
      

54. 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.
      
    • In non-euclidean spaces he extended results due to Appell and Mittag-Leffler.
      
    • 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.
      

55. 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.
      

56. Motzkin biography
    
    • One of the first papers which he published after arriving in the United States was on the Euclidean algorithm in principal ideal domains.
      
    • He proved that there are principal ideal domains which are not Euclidean domains.
      
    • The problem here is not in showing that this is not Euclidean with respect to the standard norm, which is an undergraduate exercise, but rather that it is not Euclidean in any norm.
      
    • The proof is very typical of Motzkin in that the Euclidean algorithm is given a new formulation, which at first seems to be leading away from the problem at hand, but is suddenly seen to be the decisive key to its solution.
      
    • 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.
      

57. 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'.
      

58. Donaldson biography
    
    • 4-dimensional differentiable manifolds which are topologically but not differentiably equivalent to the standard Euclidean 4-space R4.
      
    • 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.
      

59. 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.
      
    • Between 1936 and 1942 he looked at the problem of determining which real quadratic number fields Q(√d) have a ring of integers which is a Euclidean ring.
      
    • In these he found several of the 21 cases, also showing that many others do not have a Euclidean ring of integers.
      
    • He did, however, get one of these wrong for he 'proved' in the last of the three papers we mentioned that Q(√97) has a ring of integers which is a Euclidean ring.
      
    • In 1965 he wrote Begrundung der euklidischen und nichteuklidischen Geometrien nach F Klein which was translated into English and published in 1968 as Foundation of Euclidean and non-Euclidean geometries according to F Klein.
      

60. 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 1934 T Bonnesen and W Fenchel proved that in Euclidean n-space an entire subset is a convex body of constant width, and conversely.
      
    • 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.
      
    • The third text we mention is The non-Euclidean, hyperbolic plane : Its structure and consistency which Kelly wrote jointly with Gordon Matthews and published in 1981.
      

61. 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.
      
    • He recently heard of the unsolved problem about imbedding a Riemannian manifold isometrically in Euclidean space, felt that this was his sort of thing, provided the problem were sufficiently worthwhile to justify his efforts; so he proceeded to write to everyone in the math society to check on that, was told that it probably was, and proceeded to announce that he had solved it, modulo details, and told Mackey he would like to talk about it at the Harvard colloquium.
      
    • 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.
      
    • contains some surprising results on the C1-isometric imbedding into an Euclidean space of a Riemannian manifold with a positive definite C0-metric.
      

62. 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.
      

63. 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.
      
    • In Traite d'algebre Rolle used the Euclidean algorithm to find the greatest common divisor of two polynomials.
      
    • 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.
      

64. 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:- .
      

65. 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.
      

66. D'Ovidio biography
    
    • In 1869 D'Ovidio published a geometry text for schools and then, in 1872, Beltrami persuaded him to enter the competition for the Chair of Algebra and Analytic Geometry at the University of Turin.
      
    • Euclidean and noneuclidean geometry were the areas of special interest to D'Ovidio.
      
    • D'Ovidio and Corrado Segre built an important school of geometry at Turin.
      

67. 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 .
      

68. 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 .
      

69. 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.
      

70. 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 .
      

71. Yamabe biography
    
    • Yamabe began to produce partial results in papers such as On some properties of locally compact groups with no small subgroup (1951) written with Morikuni Goto, which shows that in a locally Euclidean group with no small subgroup every point sufficiently near the identity is on a unique one-parameter subgroup.
      
    • This was a period when his mathematical interests began to move away from Lie groups to differential equations and differential geometry.
      
    • This paper presents a new, elegant method for constructing Green's function G for the heat equation over any domain D in Euclidean space.
      
    • 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.
      

72. 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: .
      

73. 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.
      

74. 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.
      

75. Autolycus biography
    
    • That a remark of this kind should be genuine in any Greek mathematical treatise, Euclidean or not, seems to me utterly implausible; I would assume the obvious, i.e.
      
    • 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.
      
    • It is thought that Theodosius's Sphaerics and Autolycus's work On the Moving Sphere are based on the same pre-Euclidean textbook which is now lost.
      

76. 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.
      
    • However Rawlins [Isis 73 (1982), 259-265.',15)">15] believes that a continued fraction method was used to calculate the value 11/83 while Fowler [Isis 74 (274) (1983), 556-562.',9)">9] proposes that the anthyphairesis (or Euclidean algorithm) method was used (see also [The mathematics of Plato\'s academy : a new reconstruction (Oxford, 1987).',3)">3]).
      

77. 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.
      

78. 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'.
      
    • Much space is occupied by the treatment of systems of linear equations (the Gaussian algorithm), the theory of determinants and the theory of eigenvalues of linear mappings in 'Euclidean' vector spaces (transformation of principal axes).
      

79. Bombieri biography
    
    • For high-dimensional Euclidean space they were investigating the minimal varieties of the family of submanifolds.
      
    • In 1914 Sergei Bernstein had proved that a minimal surface in 3-dimensional Euclidean space of the form f: R2 → R, is a plane.
      
    • In 1965 this result had been extended by de Giorgi and others to n-dimensional Euclidean spaces with n ≤ 8.
      
    • 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.
      

80. 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.
      

81. 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:- .
      
    • Al-Khwarizmi has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind.
      
    • 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.
      

82. 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.
      

83. 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.
      
    • A method to obtain a good approximation to the trisector of an angle by Euclidean construction is also given.
      
    • 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.
      

84. 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.
      

85. Mohr biography
    
    • The book contains a proof that all Euclidean constructions can be carried out with compasses alone.
      
    • Mascheroni, who is credited with proving that all Euclidean constructions can be carried out with compasses alone, did not prove this until 125 years after Mohr's book was published.
      
    • 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.
      
    • In "Euclides Curiosus", Georg Mohr in 1673 gave an exposition of the fact that it is possible to do all the Euclidean constructions of the first five books with a straight ruler and a compass with one single opening.
      

86. 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.
      

87. 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.
      

88. 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.
      
    • It is thought that Sphaerics is based on some pre-Euclidean textbook which is now lost.
      
    • 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.
      

89. Study biography
    
    • Study became a leader in the geometry of complex numbers.
      
    • He reformulated, independently of Severi, the fundamental principles of enumerative geometry due to Schubert.
      
    • With Corrado Segre, Study was one of the leading pioneers in the geometry of complex numbers.
      
    • In 1903 he published Geometrie der Dynamen which considered euclidean kinematics and the mechanics of rigid bodies.
      

90. 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.
      

91. 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.
      

92. Clifford biography
    
    • Influenced by the work of Riemann and Lobachevsky, Clifford studied non-euclidean geometry.
      
    • Clifford generalised the quaternions (introduced by Hamilton two years before Clifford's birth) to what he called the biquaternions and he used them to study motion in non-euclidean spaces and on certain surfaces.
      

93. Kendall biography
    
    • He has written on stochastic geometry and its applications, and the statistical theory of shape.
      
    • His recent work includes two articles How to look at objects in a five-dimensional shape space (1994-95) and The Riemannian structure of Euclidean shape spaces: a novel environment for statistics (1993).
      
    • 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).
      

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

95. 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.
      
    • In this work Mascheroni proved that all Euclidean constructions can be made with compasses alone, so a straight edge in not needed.
      
    • He was moved initially by a desire to make a contribution to elementary geometry.
      

96. 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.
      

97. Brill biography
    
    • He contributed to the study of algebraic geometry, trying to bring the rigour of algebra into the study of curves.
      
    • his papers on three-dimensional algebraic curves (1907) and on pseudospherical three-dimensional space (1885), where the impossibility of putting such a space into Euclidean four-dimensional space and the possibility of its being placed in a Euclidean five-dimensional space are proved.
      

98. Iwasawa biography
    
    • Iwasawa's results are related to Hibert's fifth problem which asks whether any locally Euclidean topological groups is necessarily a Lie group.
      
    • 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.
      

99. 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.
      

100. 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.
       

101. 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.
       
     • Another famous result is the 'Poncelet-Steiner theorem' which shows that only one given circle and a straight edge are required for Euclidean constructions.
       
     • Steiner disliked algebra and analysis and believed that calculation replaces thinking while geometry stimulates thinking.
       

102. 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.
       
     • In all, Spanier published moer than forty papers in algebraic topology, contributing to most to most of the major research areas in the field, including cohomology operations, obstruction theory, homotopy theory, imbeddability of polyhedra in Euclidean spaces, and topology of function spaces.
       

103. 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.
       
     • The set functions studied are defined either in a perfectly abstract set, or else in metric spaces (and, in particular, Euclidean spaces); the intermediate cases of locally compact spaces, normal spaces and topological groups are not discussed.
       

104. Jordan biography
     
     • Jordan's use of the group concept in geometry in 1869 was motivated by studies of crystal structure.
       
     • He considered the classification of groups of Euclidean motions.
       
     • Jordan's interest in groups of Euclidean transformations in three dimensional space influenced Lie and Klein in their own theories of continuous and discontinuous groups.
       

105. 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.
       

106. 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.
       
     • In trying to prove the parallels postulate he accidentally proved properties of figures in non-euclidean geometries.
       

107. Stott biography
     
     • Alicia Boole experimented with the cubes and soon developed an amazing feel for 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.
       

108. 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.
       

109. 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.
       
     • (2) 3 (1995), 125-206.',6)">6]) provides a vivid picture of the reactions of both the French and the Italian mathematical communities against the non-Euclidean geometries.
       
     • In particular they debated the use of Euclid's Elements as a textbook for teaching elementary geometry in schools.
       

110. 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.
       
     • In particular he proved theorems about the embedding of an n-dimensional differentiable manifold in Euclidean space and he discovered characteristic classes at the same time as Stiefel.
       
     • for his fundamental work in algebraic topology, differential geometry and differential topology.
       

111. 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.
       
     • The following year he wrote on number theory, making a contribution to the theory of the Euclidean algorithm.
       

112. 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.
       

113. Dickson biography
     
     • Dickson studied widely within mathematics but specialised in Halsted's own subjects of euclidean and non-euclidean geometry.
       

114. 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.
       

115. 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.
       

116. 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.
       

117. Cosserat biography
     
     • In the first part of his career he made observations of double stars, observed planets and comets and did research in geometry.
       
     • Cosserat also worked on mechanics based on euclidean laws and built into an original and coherent theory.
       

118. Schwarzschild biography
     
     • At a meeting of the German Astronomical Society in Heidelberg in 1900 Schwarzschild discussed the possibility that space was non-Euclidean.
       
     • 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.
       

119. 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.
       
     • What militates against its being read in its original form is the great extent of the exposition (it contains 387 separate propositions), due partly to the Greek habit of proving particular cases of a general proposition separately from the proposition itself, but more to the cumbersomeness of the enunciations of complicated propositions in general terms (without the help of letters to denote particular points) and to the elaborateness of the Euclidean form, to which Apollonius adheres throughout.
       

120. Beckenbach biography
     
     • His research at Rice was supervised by Lester Ford, and Beckenbach was awarded his doctorate in 1931 for his dissertation Minimal Surfaces in Euclidean N-Space .
       
     • 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).
       

121. 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.
       

122. Al-Karaji biography
     
     • The solutions of quadratics are based explicitly on the Euclidean theorems ..
       
     • So what he achieved here was defining the product of these terms without any reference to geometry.
       

123. Wilder biography
     
     • He continued to undertake research with this aim and in 1930, in A converse of the Jordan-Brouwer separation theorem in three dimensions, Wilder showed that a subset of Euclidean 3-space whose complementary domains satisfied certain homology conditions was a 2-sphere.
       
     • 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).
       

124. 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.
       
     • In fact Lipschitz rediscovered Clifford algebras and was the first to apply them to represent rotations of Euclidean spaces, thus introducing the spin groups Spin(n).
       

125. Engel biography
     
     • Engel collaborated with Stackel in studying the history of non-euclidean geometry.
       

126. Konig Julius biography
     
     • Konig worked on a wide range of topics in algebra, number theory, geometry, set theory, and analysis.
       
     • One of his early ideas was a paper of 1872 which looked at intuitive ways to prove the consistency of non-Euclidean geometries.
       

127. Finck biography
     
     • His texts include books on algebra, mechanics, geometry and analysis.
       
     • In [Historia Mathematica 21 (1994), 401-419.',1)">1] Finck's work on algorithms is discussed, in particular his work on analysing the Euclidean algorithm.
       

128. Ostrowski biography
     
     • By 1973 the third edition of this monograph appeared, this time with a new title: Solution of equations in Euclidean and Banach spaces.
       
     • 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.
       

129. Straus biography
     
     • geometry, convexity, combinatorics, group theory and linear algebra.
       
     • Their work was again basic in developing a new theory, this time it was Euclidean Ramsey theory.
       

130. Friedrichs biography
     
     • Knowledge of elementary Euclidean geometry is presupposed, and some familiarity with the basic notions of physics will be helpful.
       

131. 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.
       

132. 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).
       

133. Padoa biography
     
     • Padoa spoke on A new system of definitions for Euclidean geometry but began with a summary of his lecture at the Philosophy Congress.
       

134. Fermi biography
     
     • This paper gave an important result about the Euclidean nature of space near a world line in the geometry of general relativity.
       

135. 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.
       
     • He felt just as much at home in the ordinary three dimensional Euclidean space as in the four dimensional space-time of relativity.
       

136. 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.
       
     • Fibonacci numbers and the Euclidean algorithm .
       

137. 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.
       
     • In lighter moments he may turn to the theory of the game of Nim, while on more austere occasions he may study the question of Euclidean algorithms in algebraic fields, or the Rogers-Ramanujan identities in the theory of partitions.
       

138. Bieberbach biography
     
     • It was an important work on groups of euclidean motions, and it was a major step towards proving Hilbert's eighteenth problem.
       
     • Bieberbach was appointed professor of mathematics in Basel in Switzerland, Frankfurt am Main in Germany, and the University of Berlin where he held the chair of geometry.
       

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

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

141. Klugel biography
     
     • His work is cited by almost all later contributors to non-euclidean geometry.
       

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.
     Go directly to this paragraph
     
   • Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • Elementary geometry was by this time engulfed in the problems of the parallel postulate.
     Go directly to this paragraph
     
   • D'Alembert, in 1767, called it the scandal of elementary geometry.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy.
     Go directly to this paragraph
     
   • However in some sense Bolyai only assumed that the new geometry was possible.
     Go directly to this paragraph
     
   • However the real breakthrough was the belief that the new geometry was possible.
     Go directly to this paragraph
     
   • Nor is Bolyai's work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • Riemann briefly discussed a 'spherical' geometry in which every line through a point P not on a line AB meets the line AB.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • The first person to put the Bolyai - Lobachevsky non-Euclidean geometry on the same footing as Euclidean geometry was Eugenio Beltrami (1835-1900).
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.
     Go directly to this paragraph
     
   • Geometry and topology index .
     
   • http://www-history.mcs.st-andrews.ac.uk/HistTopics/Non-Euclidean_geometry.html .
     

2. 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.
     
   • F J Duarte, On the non-Euclidean geometries : Historical and bibliographical notes (Spanish), Revista Acad.
     
   • 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, Ideas of Space : Euclidean, non-Euclidean and Relativistic (Oxford, 1989).
     
   • 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.
     
   • H Karzel, Development of non-Euclidean geometries since Gauss, Proceedings of the 2nd Gauss Symposium (Berlin, 1995).
     
   • 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.
     
   • R J Trudeau, The non-Euclidean revolution (Boston, Mass., 1987).
     
   • 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] .
     

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.
     
   • F J Duarte, On the non-Euclidean geometries : Historical and bibliographical notes (Spanish), Revista Acad.
     
   • 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, Ideas of Space : Euclidean, non-Euclidean and Relativistic (Oxford, 1989).
     
   • 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.
     
   • H Karzel, Development of non-Euclidean geometries since Gauss, Proceedings of the 2nd Gauss Symposium (Berlin, 1995).
     
   • 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.
     
   • R J Trudeau, The non-Euclidean revolution (Boston, Mass., 1987).
     
   • 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. 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 .
     

5. 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.
     

6. 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.
     Go directly to this paragraph
     
   • Geometry had began to lose its 'metric' character with projective and non-euclidean geometries being studied.
     Go directly to this paragraph
     
   • Also the movement to study geometry in n dimensions led to an abstraction in geometry itself.
     Go directly to this paragraph
     
   • The difference between metric and incidence geometry comes from the work of Monge, his student Carnot and perhaps most importantly the work of Poncelet.
     Go directly to this paragraph
     
   • Non-euclidean geometry was studied by Lambert, Gauss, Lobachevsky and Janos Bolyai among others.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • Steiner in 1832 studied notions of synthetic geometry which were to eventually become part of the study of transformation groups.
     Go directly to this paragraph
     
   • Klein proposed the Erlangen Program in 1872 which was the group theoretic classification of geometry.
     Go directly to this paragraph
     

7. History overview
   
   • Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • In geometry Plucker produced fundamental work on analytic geometry and Steiner in synthetic geometry.
     Go directly to this paragraph
     
   • Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann.
     Go directly to this paragraph
     
   • His work in differential geometry was to revolutionise the topic.
     Go directly to this paragraph
     
   • Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann.
     Go directly to this paragraph
     
   • Think about it and realise how difficult it was to invent non-euclidean geometries, groups, general relativity, set theory, ..
     

8. Kepler's Laws
   
   • Section 5 Essential 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.
     
   • So the resulting radius vector AP that finally satisfied Kepler (in Ch.58) was quantified geometrically from the constructed rectangle AKQR, by applying nothing more than a Euclidean - straightedge-and-compasses - construction, as shown in Figure (3): .
     
   • 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.
     

9. 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.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • 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.
     Go directly to this paragraph
     
   • Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy and also used formulas involving sin and tan.
     Go directly to this paragraph
     

10. 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.
      

11. 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.
      Go directly to this paragraph
      
    • The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.
      Go directly to this paragraph
      
    • Listing had examined connectivity in three dimensional Euclidean space but Betti extended his ideas to n dimensions.
      Go directly to this paragraph
      
    • However Frechet was able to extend the concept of convergence from Euclidean space by defining metric spaces.
      Go directly to this paragraph
      
    • Geometry and topology index .
      

12. 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.
      Go directly to this paragraph
      
    • Before that however he had written a paper in October 1914 nearly half of which is a treatise on tensor analysis and differential geometry.
      

13. 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.
      

14. 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.
      
    • The question of whether physical space is euclidean is not a meaningful one to ask.
      

15. Bourbaki 2
    
    • Chapters VI and VII are concerned with elementary properties of Euclidean spaces, projective spaces, complex numbers, complex projective spaces, quaternions, etc.
      
    • 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.
      

16. Nine chapters
    
    • The Euclidean algorithm method for finding the greatest common divisor of two numbers is given.
      
    • 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.
      

Famous Curves

1. Tractrix
   
   • This is a surface of constant negative curvature and was used by Beltrami in 1868 in his concrete realisation of non-euclidean geometry.
     

Societies etc

1. European Mathematical Society Prizes
   
   • 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.
     
   • One of the earlier contributions is his surprising solution of the symplectic packing problem, completing work of Gromov, McDuff and Polterovich, showing that compact symplectic manifolds can be packed by symplectic images of equally sized Euclidean balls without wasting volume if the number of balls is not too small.
     
   • 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.
     

2. International Congress Speakers
   
   • 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.
     

3. 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.
     
   • In these papers he showed for the first time how to use the powerful tools of probability theory to attack the hard analytic questions of constructive quantum field theory, controlling renormalizations with Lp estimates in the first paper, and in the second turning Euclidean quantum field theory into a subset of the theory of stochastic processes.
     
   • 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).
     

4. 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 .
     

5. 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.
     

6. BMC 2007
   
   • Connes, A Recent developments on non-commutative geometry .
     
   • Leader, I B Euclidean Ramsey theory .
     
   • Singer, M Special metrics in Kahler geometry .
     
   • Wendland, K From dualities to geometry .
     
   • Series, CIndra's pearls: geometry and symmetry .
     

7. 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.
     

8. BMC 1981
   
   • Gibbons, G WThe Euclidean approach to quantum gravity .
     
   • Jones, J D SThe Kervaire invariant in geometry and homotopy theory .
     

9. BMC 1960
   
   • Eggleston, H GHausdorff's measure in Euclidean space .
     

References

1. References for Lobachevsky
   
   • 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.
     
   • N A Chernikov, Introduction of Lobachevskii geometry into the theory of gravitation, Soviet J.
     
   • N Daniels, Lobachevsky : some anticipations of later views on the relation between geometry and physics, Isis 66 (231) (1975), 75-85.
     
   • L E Evtushik and A K Rybnikov, The influence of Lobachevskii's ideas on the development of differential geometry, Moscow Univ.
     
   • K Fili, Lobachevskii's 'imaginary' geometry and the Russian avant-garde (Russian), Voprosy Istor.
     
   • G E Izotov, On the history of the publishing by N I Lobachevskii of his works on 'imaginary' geometry (Russian), Voprosy Istor.
     
   • S B Kadomtsev, E G Poznyak, and A G Popov, Lobachevskii's geometry: the discovery and a path to the present (Russian), Priroda (7) (1993), 19-27.
     
   • B Mayorga, Lobachevskii and non-Euclidean geometry (Spanish), Lect.
     
   • J M Montesinos Amilibia, Non-Euclidean geometries : Gauss, Lobachevskii and Bolyai (Spanish), in History of mathematics in the XIXth century (Part 1) (Madrid, 1992), 65-114.
     
   • A Vucinich, Nicolai Ivanovich Lobachevskii: The Man Behind the First Non-Euclidean Geometry, Isis 53 (1962), 465-481.
     

2. References for Taurinus
   
   • J Gray, Ideas of space : Euclidean, non-Euclidean, and relativistic (New York, 1979).
     
   • A Dou, Origins of non-Euclidean geometry : Saccheri, Lambert and Taurinus (Spanish), in History of mathematics in the XIXth century Madrid, 1991 1 (Madrid, 1992), 43-63.
     
   • S Xambo, Non-Euclidean geometry : from Euclid to Gauss (Catalan), Butl.
     

3. References for Archimedes
   
   • W R Knorr, Archimedes and the pseudo-Euclidean 'Catoptrics' : early stages in the ancient geometric theory of mirrors, Arch.
     
   • D C Gazis and R Herman, Square roots geometry and Archimedes, Scripta Math.
     
   • W R Knorr, Archimedes and the pre-Euclidean proportion theory, Arch.
     
   • 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.
     

4. 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.
     

5. References for Gauss
   
   • B Szenassy, Remarks on Gauss's work on non-Euclidean geometry (Hungarian), Mat.
     
   • K Zormbala, Gauss and the definition of the plane concept in Euclidean elementary geometry, Historia Math.
     

6. References for Lambert
   
   • A Dou, Origins of non-Euclidean geometry: Saccheri , Lambert and Taurinus (Spanish), in History of mathematics in the XIXth century, Part 1, Madrid, 1991 (Madrid, 1992), 43-63.
     
   • J Folta, Lambert's 'Architectonics' and the foundations of geometry, Acta Historiae Rerum Naturalium necnon Technicarum, 1973 (Prague, 1974), 145-163.
     

7. References for Beltrami
   
   • T R Chandrasekhar, Non-Euclidean geometry from early times to Beltrami, Indian J.
     

8. References for Killing
   
   • T Hawkins, Non-euclidean geometry and Weierstrassian mathematics : The background to Killing's work on Lie algebras, Historia Mathematica 7 (1980), 289-342.
     

9. References for Osipovsky
   
   • B A Rozenfeld, A History of Non-Euclidean Geometry : Evolution of the Concept of a Geometric Space (Springer, 1988).
     

Additional material

1. Poincaré on non-Euclidean geometry
   
   • Poincare on non-Euclidean geometry .
     
   • It contains a number of articles written by Poincare over quite a number of years and we present below a version of one of these articles, namely the one on Non-Euclidean geometries .
     
   • Non-Euclidean geometries .
     
   • 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.
     
   • Interpretation of Non-Euclidean Geometries.
     
   • 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.
     
   • http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_non-Euclidean.html .
     

2. 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.
     
   • The question whether the structure of this continuum is Euclidean, or in accordance with Riemann's general scheme, or otherwise, is, according to the view which is here being advocated, properly speaking a physical question which must be answered by experience, and not a question of a mere convention to be selected on practical grounds.
     
   • 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.
     
   • Further, the spherical surface is a non-Euclidean continuum of two dimensions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane.
     
   • 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.
     
   • In this way creatures which cannot leave the spherical surface, and cannot even peep out from the spherical surface into three-dimensional space, might discover, merely by experimenting with discs, that their two-dimensional "space" is not Euclidean, but spherical space.
     
   • 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.
     
   • We must carefully bear in mind that our statement as to the growth of the disc-shadows, as they move away from S towards infinity, has in itself no objective meaning, as long as we are unable to employ Euclidean rigid bodies which can be moved about on the plane E for the purpose of comparing the size of the disc-shadows.
     
   • 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 .
     

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.
     
   • But about that time mathematicians were already beginning to see their subject in a new light, as a branch of study not directly dependent on experience, and this change of outlook was encouraged by the discovery, early in the nineteenth century, of the non-euclidean geometries, systems consistent within themselves but incompatible with Euclid's system.
     
   • 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'.
     
   • In the language now in use, we would say that the geometrical (or, more accurately, the euclidean) properties of a body are invariant with respect to the operation of displacement in space; and invariance with respect to a certain kind of operation at once suggests the existence of an underlying group of operations.
     
   • 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.
     
   • Since the euclidean group, consisting of all displacements of the plane, may be shown to be a proper subgroup of the projective group, it follows at once that every projectively invariant property is also a euclidean invariant, whereas not every euclidean property is projective.
     
   • 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 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.
     
   • the generalization, in higher space, of polyhedra): it is typical that this includes polytopes in Non-Euclidean Space.
     
   • 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.
     
   • On Links and Knots in Euclidean Space of n Dimensions.
     
   • 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.
     
   • Space-filling Tetrahedra in Euclidean Space, 41, pp.
     
   • Peaucellier's Cell and other Linkages in Non-Euclidean Geometry, 44, pp.
     

5. 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.
     

6. 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 .
     

7. 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.
     
   • This opened the way to new geometries - they were called non-Euclidean geometries - based on axioms differing from Euclid's.
     
   • 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.
     
   • But shortly after, on purely mathematical grounds, the knowledge of consistent non-Euclidean geometries had been conceived, i.e.
     
   • 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.
     
   • And, what is of even greater importance, the coincidence between the Kepler orbit and the orbit of a planet moving according to the theory of relativity in a quasi non-Euclidean space, is not complete.
     
   • 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.
     

8. Born Inaugural
   
   • The idea that a science can be logically reduced to a small number of postulates or axioms is due to the great Greek mathematicians, who first tried to formulate the axioms of geometry and to derive the complete system of theorems from them.
     
   • The discovery of non-Euclidean geometry by Lobachevsky and Bolyai shook the a priori standpoint.
     
   • Gauss has frankly expressed his opinion that the axioms of geometry have no superior position as compared with the laws of physics, both being formulations of experience, the former stating the general rules of the mobility of rigid bodies and giving the conditions for measurements in space.
     
   • This standpoint denies the existence of a priori principles in the shape of laws of pure reason and pure intuition; and it declares that the validity of every statement of science (including geometry as applied to nature) is based on experience.
     
   • For it is of course not meant that every fundamental statement-as, for instance, the Euclidean axioms of geometry-is directly based on special observations.
     
   • It has not only doubted the a priori validity of Euclidean geometry as the great mathematicians did a hundred years ago, but has really replaced it by new forms of geometry; it has even made geometry depend on physical forces, gravitation, and it has revolutionised in the same way nearly all categories a priori, concerning time, substance, and causality.
     
   • Minkowski has shown that it is possible to get a description of the connection of all events which is independent of the observer, or invariant, as the mathematicians say, by considering them as points in a four-dimensional continuum with a quasi-Euclidean geometry.
     
   • The generalisation which was conceived by Einstein in 1915 combining the geometry of the space-time world with gravitation rested, and still rests, on a rather slender empirical basis.
     
   • In the case of space and time these are the laws of the four-dimensional geometry of Minkowski.
     
   • It is a mathematical expression first used in analytical geometry to handle quantitatively spatial Gestalten, which are simple shapes of bodies or configurations of such.
     
   • The methods of mathematical physics are just the same as those of geometry, starting with generalised co-ordinates and eliminating the accidental things.
     

9. 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.
     

10. Von Neumann: 'The Mathematician
    
    • One of its main branches, geometry, actually started as a natural, empirical science.
      
    • The first example is, as it should be, geometry.
      
    • Geometry was the major part of ancient mathematics.
      
    • Apart from all other evidence, the very name "geometry" indicates this.
      
    • But one might well argue that a similar interpretation of Euclid is possible, especially from the viewpoint of antiquity, before geometry had acquired its present bimillennial stability and authority - an authority which the modern edifice of theoretical physics is clearly lacking.
      
    • Furthermore, while the de-empirization of geometry has gradually progressed since Euclid, it never became quite complete, not even in modern times.
      
    • The discussion of non-Euclidean geometry offers a good illustration of this.
      
    • Since most of the discussion took place on a highly abstract plane, it dealt with the purely logical problem whether the "fifth postulate" of Euclid was a consequence of the others or not; and the formal conflict was terminated by F Klein's purely mathematical example, which showed how a piece of a Euclidean plane could be made non-Euclidean by formally redefining certain basic concepts.
      
    • The discovery of general relativity forced a revision of our views on the relationship of geometry in an entirely new setting and with a quite new distribution of the purely mathematical emphases, too.
      
    • And these two seemingly conflicting attitudes are perfectly compatible in one mathematical mind; thus Hilbert made important contributions to both axiomatic geometry and to general relativity.
      
    • This is geometry, but post-Euclidean, and, at the epoch in question, non-axiomatic, empirical geometry.
      
    • Second, that the empirical origin of mathematics is strongly supported by instances like our two earlier examples (geometry and calculus), irrespective of what the best interpretation of the controversy about the "foundations" may be.
      

11. A D Aleksandrov's view of Mathematics
    
    • Algebra, geometry, analysis and topology are well represented and there are chapters on number theory, numerical analysis and computing machines.
      
    • Similarly in geometry we consider, for example, straight lines and not stretched threads, the concept of a geometric line being obtained by abstraction from all other properties, excepting only extension in one direction.
      
    • The demand for a proof of a theorem is well known in high school geometry, but it pervades the whole of mathematics.
      
    • Mathematics demands that this result be deduced from the fundamental concepts of geometry, which at the present time, in view of the fact that geometry is nowadays developed on a rigorous basis, are precisely formulated in the axioms.
      
    • our expenses or geometry to calculate the floor area of an apartment.
      
    • Another example, equally impressive, is provided by non-Euclidean geometry, which arose from the efforts, extending for 2000 years from the time of Euclid, to prove the parallel axiom, a problem of purely mathematical interest.
      
    • N I Lobachevsky himself, the founder of the new geometry, was careful to label his geometry "imaginary," since he could not see any meaning for it in the actual world, although he was confident that such a meaning would eventually be found.
      
    • The results of his geometry appeared to the majority of mathematicians to be not only "imaginary" but even unimaginable and absurd.
      
    • Nevertheless, his ideas laid the foundation for a new development of geometry, namely the creation of theories of various non-Euclidean spaces; and these ideas subsequently became the basis of the general theory of relativity, in which the mathematical apparatus consists of a form of non-Euclidean geometry of four-dimensional space.
      
    • For a preliminary clarification of these questions, it is sufficient to examine the foundations of arithmetic and elementary geometry, to which we now turn.
      

12. EMS obituary
    
    • Originally it was a technique rather than a separate branch of mathematics, providing as it did a way of writing theorems of differential geometry and the calculus in a form at once concise and general, and it was not until after the development of relativity, followed shortly afterwards by Levi-Civita's definition of parallelism in Riemannian geometry, that it assumed the full place it now holds as one of the main branches of modern mathematics.
      
    • Reference has already been made to his definition of parallelism in Riemannian geometry.
      
    • This was published in 1917 in a paper "Nozione di parallelismo in una varieta qualunque e conseguente specificazione della curvatura Riemanniana." The basic idea was simple, but its influence on the development of differential geometry was profound.
      
    • Consider an ordinary surface S in three-dimensional Euclidean space.
      
    • This discovery by Levi-Civita, together with the contemporary development of general relativity and the search for a unified theory of gravitation and electromagnetism by Weyl, Eddington, Einstein and others, quickly led to generalisations of Riemannian geometry.
      
    • On the purely mathematical side, perhaps the most interesting consequence of these generalisations was the rapprochement, if not the complete reconciliation, which they brought about between differential geometry and the geometries of the Erlanger Programm.
      
    • In 1870 Felix Klein had defined a geometry to be the invariant theory of a transformation group, a definition which included such geometries as Euclidean, affine and projective, but did not include Riemannian, In the light of Levi-Civita's definition of parallelism it was seen that the spaces of differential geometry could, so to speak, be regarded as an assemblage of isomorphic, Klein spaces, each associated with a point of an "underlying space," and in this way there grew an extensive literature directly inspired by the work of Levi-Civita.
      
    • So extensive, indeed, did the literature become, that differential geometry seemed in the late 1920's and early 1930's to be almost in danger of self-suffocation: a Pelion of detail, painstakingly worked out by research students the world over, was piled upon an Ossa of greater and greater generalisations.
      
    • Nevertheless, his interest in geometry as such remained with him until the end, some of his last papers, published between 1934 and 1938, being concerned with such topics as the trigonometry of curvilinear triangles on surfaces, and families of isoparametric surfaces in ordinary Euclidean space.
      
    • By his early work with Ricci on tensor analysis and by his later discovery of infinitesimal parallelism, Levi-Civita laid the foundations both for relativity and for the establishment of differential geometry as one of the great branches of modern mathematics.
      

13. Tullio Levi-Civita
    

14. Levi-Civita.html
    
    • Originally it was a technique rather than a separate branch of mathematics, providing as it did a way of writing theorems of differential geometry and the calculus in a form at once concise and general, and it was not until after the development of relativity, followed shortly afterwards by Levi-Civita's definition of parallelism in Riemannian geometry, that it assumed the full place it now holds as one of the main branches of modern mathematics.
      
    • Reference has already been made to his definition of parallelism in Riemannian geometry.
      
    • This was published in 1917 in a paper "Nozione di parallelismo in una varieta qualunque e conseguente specificazione della curvatura Riemanniana." The basic idea was simple, but its influence on the development of differential geometry was profound.
      
    • Consider an ordinary surface S in three-dimensional Euclidean space.
      
    • This discovery by Levi-Civita, together with the contemporary development of general relativity and the search for a unified theory of gravitation and electromagnetism by Weyl, Eddington, Einstein and others, quickly led to generalisations of Riemannian geometry.
      
    • On the purely mathematical side, perhaps the most interesting consequence of these generalisations was the rapprochement, if not the complete reconciliation, which they brought about between differential geometry and the geometries of the Erlanger Programm.
      
    • In 1870 Felix Klein had defined a geometry to be the invariant theory of a transformation group, a definition which included such geometries as Euclidean, affine and projective, but did not include Riemannian, In the light of Levi-Civita's definition of parallelism it was seen that the spaces of differential geometry could, so to speak, be regarded as an assemblage of isomorphic, Klein spaces, each associated with a point of an "underlying space," and in this way there grew an extensive literature directly inspired by the work of Levi-Civita.
      
    • So extensive, indeed, did the literature become, that differential geometry seemed in the late 1920's and early 1930's to be almost in danger of self-suffocation: a Pelion of detail, painstakingly worked out by research students the world over, was piled upon an Ossa of greater and greater generalisations.
      
    • Nevertheless, his interest in geometry as such remained with him until the end, some of his last papers, published between 1934 and 1938, being concerned with such topics as the trigonometry of curvilinear triangles on surfaces, and families of isoparametric surfaces in ordinary Euclidean space.
      
    • By his early work with Ricci on tensor analysis and by his later discovery of infinitesimal parallelism, Levi-Civita laid the foundations both for relativity and for the establishment of differential geometry as one of the great branches of modern mathematics.
      

15. Hans Hahn: 'The crisis in intuition
    
    • : [Kant believed that] geometry, as it has been taught since ancient times, deals with the properties of the space that is fully and exactly presented to us by pure intuition ..
      
    • [These quotes] narrow the subject to geometry and intuition, and attempt to show how it came about that, even in the branch of mathematics which would seem to be its original domain, intuition gradually fell into disrepute and at last was completely banished ..
      
    • [To avoid such advanced] branches of mathematics, I propose to examine an occurrence of failure of intuition at the very threshold of geometry.
      
    • But what are we to say to the often heard objection that only conventional geometry is usable, for it is the only one that satisfies intuition? My first comment on this score ..
      
    • is that every geometry ..
      
    • Traditional physics is responsible for the fact that until recently the logical construction of three-dimensional Euclidean, Archimedean space has been used exclusively for the ordering of our experience.
      
    • This habituation to the use of ordinary geometry for the ordering of our experience explains why we regard this geometry as intuitive, and every departure from it unintuitive, contrary to intuition, and intuitively impossible.
      
    • But as we have seen, such intuitional impossibilities, also occur in ordinary geometry.
      
    • If the use of [new] geometries for the ordering of our experience continues to prove itself so that we become more and more accustomed to dealing with these logical constructs; if they penetrate into the curriculum of the schools, if we, so to speak, learn them at our mother's knee, as we now learn three-dimensional Euclidean geometry, then nobody will think of saying that these geometries are contrary to intuition.
      
    • They will be considered as deserving of intuitive status as three-dimensional Euclidean geometry is today.
      

16. Bolzano's publications
    
    • 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.
      

17. EMS 1913 Colloquium 3.html.html
    
    • 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.
      

18. The Works of Sir John Leslie
    
    • We look on the reform of Euclidean geometry as modern, but even in 1805 Leslie was a keen reformer.
      
    • Though so devoted an admirer of the ancient Greeks, his constructive bent makes his geometry practical.
      
    • His University course, as displayed in his University text books, by neglecting, or subordinating computation, algebra, coordinate geometry, differential and integral calculus, is enough to startle a mathematician.
      
    • But his setting out of the Euclidean herbaceous border will be viewed with interest by the teacher, who may find incidentally many valuable rooted cuttings, suitable for transporting to the forcing frame of the examination paper.
      
    • He has a great enthusiasm for geometrical analysis, but this is not analytical (co-ordinate) geometry.
      
    • The second volume consists of three treatises, Geometrical Analysis, Geometry of Lines of the Second Order, and Geometry of the Higher Curves.
      
    • In his Geometry of Higher Curves he has collected into one text "all the remarkable curves above lines of the second order," which had been till then scattered through the pages of continental writers in "volumes difficult of access".
      
    • The third volume was to contain (i) Descriptive Geometry; (ii) Theory of Solids (including Perspective); (iii) Projection of the Sphere; (iv) Spherical Trigonometry.
      
    • I into "Rudiments of Geometry," which he intended to act as a course of what is now called Practical Mathematics.
      

19. 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.
      
    • So the resulting radius vector AP that finally satisfied Kepler (in Ch.58) was quantified geometrically from the constructed rectangle AKQR, by applying nothing more than a Euclidean - straightedge-and-compasses - construction, as shown in Figure (3): .
      
    • 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.
      

20. EMS 1913 Colloquium
    
    • 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.
      
    • At 11.30 Professor Whittaker explained practical Harmonic Analysis and Periodogram Analysis; and at two o'clock Dr Sommerville of St Andrews expounded the mysteries of non-Euclidean Geometry and the Foundations of Geometry.
      
    • Newtonian dynamics, we found, was only a first approximation to the dynamics of our visible universe; while the Euclidean space in which this universe was vulgarly believed to move and have its being was a crude assumption from an axiom of ignorance.
      
    • The tendency of modern Physical Theory was in the direction of still further atomising the atom; yet it was necessary in geometry to have an assumption of continuity, so that all possible numbers might be brought into correspondence with an infinitude of points on a finite line.
      
    • The dictum of the logician that we cannot define by means of a negation seemed to have no terror to the modern geometer with his glib talk of non-Euclidean, non-Pascalian, non-Desarguesian, and even non-Archimedean.
      

21. Gibson History 7 - Robert Simson
    
    • When he entered Glasgow University Trail in his life of Simson states that "at this time, from temporary circumstances, it happened that no Mathematical Lectures were given in the College; but young Simson's inquisitive mind, from some fortunate incident having been directed to Geometry, he soon perceived the study of that science to be congenial to his taste and capacity.
      
    • In the higher geometry he prelected from his own Conics, and he gave a small specimen of the linear problems of the ancients by explaining the properties sometimes of the conchoid, sometimes of the cissoid, with their application to the solution of such problems.
      
    • Simson is known almost solely as an exponent of the Greek geometry, but I think it is worth noticing that when he was still a young man he showed a full command of one branch of analysis.
      
    • But analysis of this kind did not really interest Simson, and it is for his devotion to the ancient geometry that he is specially memorable.
      
    • If we are to form a fair judgment we should consider not merely what has been done since Simson's time for the study of elementary geometry but, quite as much, what was the state of the geometrical textbook before the issue of Simson's Euclid.
      
    • He was simply steeped in the ancient geometry and one should be very sure of one's ground before questioning any deliberate judgment of Simson's on the facts of any Greek textbook.
      
    • The book was designed to stem the tide that had begun to set in in favour of Analytical or Algebraic Geometry (on the lines of de I'Hopital's well-known work); though it was based on Apollonius the cone was not used in defining the conic.
      
    • Of course all the proofs are strictly Euclidean and very little is taken for granted; no important property is shoved into a Corollary.
      
    • To the study of Greek geometry Simson may almost be said to have dedicated his life, and he found ample scope for his ingenuity in his effort to recover some of the more important of the treatises of Euclid and Apollonius that had been lost but whose contents had been to a certain extent described by Pappus.
      
    • Trail was neither unprejudiced nor very critical, but he had a competent knowledge of Greek geometry and was thoroughly familiar with Simson's work, and I quote his estimate of the edition:- "Such is the elegance of method and the ingenious contrivance of demonstration in this work that he has truly exhibited a copy, or at least very nearly a copy, of the work of Apollonius, that little regret need be had for the loss of the original." We need not indorse this eulogium in its entirety but it is not altogether wide of the mark.
      
    • In recent times the geometry developed in the Determinate Section has often been represented as in many respects an equivalent of the modern theory of Involution, though it is not at all from that standpoint that Simson considered it.
      
    • There are however many propositions that can be readily adapted to the geometry of involution.
      
    • Heiberg is less enthusiastic, though I do not attach quite the same weight to Heiberg's views in this connection as in other fields in which he has rendered such great service to Greek geometry.
      
    • He had a competent knowledge of fluxions but he never really set himself to master algebraical analysis, and he held views that were completely antiquated on the nature and possibilities of algebra and of algebraic geometry.
      

22. Kepler's Planetary Laws
    
    • Section 5 Essential 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.
      
    • So the resulting radius vector AP that finally satisfied Kepler (in Ch.58) was quantified geometrically from the constructed rectangle AKQR, by applying nothing more than a Euclidean - straightedge-and-compasses - construction, as shown in Figure (3): .
      
    • 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.
      

23. Kuratowski: 'Introduction to Topology
    
    • Their generality is sufficient for the majority of important applications; in particular, subsets of n-dimensional Euclidean space, sequence, spaces (of Hilbert.
      
    • In the further chapters (XIII-XVIII) we gradually confine ourselves to more specific spaces: we give the important properties of separable spaces (still embracing the majority of spaces arising in applications), complete spaces (with the Baire theorem and its consequences), compact spaces (which form the generalization of closed bounded subsets of Euclidean space), connected spaces (connectedness is the precise statement of the concept of the continuity of a set) and locally connected spaces (as it turns out, curves, surfaces, multi-dimensional varieties or manifolds, with which we have to deal in differential geometry are as a rule locally connected continua).
      
    • We shall concern ourselves in more detail with the properties of the n-dimensional simplex, which is the fundamental concept of classical multi-dimensional geometry, in Chapter XX.
      
    • The latter has various applications in differential and algebraic geometry, the calculus of variations, and in other branches of analysis.
      
    • Finally, the last chapter, XXII, conceptually closely related to geometry, concerns theorems on the separation of the plane.
      
    • Set-theoretical topology, formerly called the theory of point sets, and concerning arbitrary subsets of Euclidean space, was begun by G Cantor, the creator of the theory of sets (circa 1880).
      
    • This period was preceded by the transition from the investigation of subsets of Euclidean space in set-theoretic topology to the investigation of arbitrary topological spaces.
      

24. EMS 1913 Colloquium 1.html.html
    
    • 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.
      

25. David Hilbert: 'Mathematical Problems
    
    • So, for example, the problem of the shortest line plays a chief and historically important part in the foundations of geometry, in the theory of curved lines and surfaces, in mechanics and in the calculus of variations.
      
    • And how convincingly has F Klein, in his work on the icosahedron, pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry, in group theory, in the theory of equations and in that of linear differential equations.
      
    • The same is true of the first problems of geometry, the problems bequeathed us by antiquity, such as the duplication of the cube, the squaring of the circle; also the oldest problems in the theory of the solution of numerical equations, in the theory of curves and the differential and integral calculus, in the calculus of variations, the theory of Fourier series and the theory of potential - to say nothing of the further abundance of problems properly belonging to mechanics, astronomy and physics.
      
    • Such a one-sided interpretation of the requirement of rigour would soon lead to the ignoring of all concepts arising from geometry, mechanics and physics, to a stoppage of the flow of new material from the outside world, and finally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of the irrational number.
      
    • But what an important nerve, vital to mathematical science, would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever, from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, mathematical ideas come up, the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms, that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.
      
    • Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigour a difficult theorem on the continuity of functions or the existence of points of condensation? Who could dispense with the figure of the triangle, the circle with its centre, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences? .
      
    • On the contrary we apply, especially in first attacking a problem, a rapid, unconscious, not absolutely sure combination, trusting to a certain arithmetical feeling for the behaviour of the arithmetical symbols, which we could dispense with as little in arithmetic as with the geometrical imagination in geometry.
      
    • Let us look at the principles of analysis and geometry.
      
    • The most suggestive and notable achievements of the last century in this field are, as it seems to me, the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor, and the discovery of non-euclidean geometry by Gauss, Bolyai, and Lobachevsky.
      

26. Whittaker EMS Obituary.html
    
    • Consequently, lectures were given by D M Y Somerville on Non-euclidean geometry and the foundations of geometry and by A W Conway [Arthur Conway] on The theory of relativity and the new physical ideas of space and time.
      
    • His interest in Relativity manifested itself also at the undergraduate level, for the Honours course entitled Higher Algebra and Geometry contained neither Algebra nor Geometry in the ordinary sense of these terms but comprised Tensor Calculus with Riemannian Geometry and its generalisations.
      
    • (2) Explain the difference between projective and elliptic geometry.
      
    • One of the few subjects on which I have never heard him discourse is Geometry in the Bakerian sense; Geometry for him meant Riemannian geometry which is used in Relativity and contributes to our understanding of the material universe.
      

27. G H Hardy addresses the British Association in 1922, Part 1
    
    • The most obvious example is to be found in the science of geometry.
      
    • Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions.
      
    • The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real.
      
    • It may be the seven-point geometry that fits the facts the best, for anything that mathematicians have to say.
      

28. G H Hardy addresses the British Association in 1922
    
    • The most obvious example is to be found in the science of geometry.
      
    • Mathematicians have constructed a very large number of different systems of geometry, Euclidean or non-Euclidean, of one, two, three, or any number of dimensions.
      
    • The old-fashioned geometry of Euclid, the entertaining seven-point geometry of Veblen, the space-times of Minkowski and Einstein, are all absolutely and equally real.
      
    • It may be the seven-point geometry that fits the facts the best, for anything that mathematicians have to say.
      

29. Heath: Everyman's Library 'Euclid' Introduction
    
    • book in the world could be more suitable for inclusion in the Library than this, the greatest textbook of elementary mathematics that there has ever been or is likely to be, a book which, ever since it was written twenty-two centuries ago., has been read and appealed to as authoritative by mathematicians great and small, from Archimedes and Apollonius of Perga onwards? No textbook, presumably, can ever be without flaw (especially in a subject like geometry, where some first principles, postulates or axioms, have to be assumed without proof, and any number of alternative systems are possible), and flaws there are in Euclid; but it is safe to say that no alternative to the Elements has yet been produced which is open to fewer or less serious objections.
      
    • The only general criticism of it which is deserving of consideration is that it is unsuitable as a textbook for very young boys and girls who are just beginning to learn the first things about geometry.
      
    • The simple truth is that it was not written for schoolboys or schoolgirls, but for the grown man who would have the necessary knowledge and judgment to appreciate the highly contentious matters which have to be grappled with in any attempt to set out the essentials of Euclidean geometry as a strictly logical system, and, in particular, the difficulty of making the best selection of unproved postulates or axioms to form the foundation of the subject.
      
    • My advice would, therefore, be: if you must spoon-feed the very young, do so; but when they have shown a taste for the subject and attained the standard necessary for, passing honours examinations, let them then be introduced to Euclid in his original form as an antidote to the more or less feeble echoes of him that are to be found in the ordinary school textbooks of "geometry." I should be surprised if such qualified readers, making the acquaintance of Euclid for the first time, did not find it fascinating, a book to be read in bed or on a holiday, a book as difficult as any detective story to lay down when once begun.
      
    • Nor does the reading of it require the "higher mathematics." Any intelligent person with a fair recollection of school work in elementary geometry would find it (progressing as it does by gradual and nicely contrived steps) easy reading, and should feel a real thrill in following its development, always assuming that enjoyment of the book is not marred by any prospect of having to pass an examination in it! This is why I applaud the addition of this great classic to Everyman's Library; for everybody ought to read it who can, that is, all educated persons except the very few who are constitutionally incapable of mathematics.
      
    • Steeped in the subject, he even made important attempts, with the aid of indications of content, etc., given in the Collection of Pappus of Alexandria, to restore three lost works, the Porisms of Euclid (a difficult treatise in higher geometry) and two minor works of Apollonius.
      
    • The merits of Simson, both as interpreter and as critic of Euclid, are very great; and it was mainly due to the excellence of his edition that the words "Euclid" and "geometry" became almost synonymous terms in this country.
      

30. EMS 1913 Colloquium 6.html.html
    
    • At 11.30 Professor Whittaker explained practical Harmonic Analysis and Periodogram Analysis; and at two o'clock Dr Sommerville of St Andrews expounded the mysteries of non-Euclidean Geometry and the Foundations of Geometry.
      
    • Newtonian dynamics, we found, was only a first approximation to the dynamics of our visible universe; while the Euclidean space in which this universe was vulgarly believed to move and have its being was a crude assumption from an axiom of ignorance.
      
    • The tendency of modern Physical Theory was in the direction of still further atomising the atom; yet it was necessary in geometry to have an assumption of continuity, so that all possible numbers might be brought into correspondence with an infinitude of points on a finite line.
      
    • The dictum of the logician that we cannot define by means of a negation seemed to have no terror to the modern geometer with his glib talk of non-Euclidean, non-Pascalian, non-Desarguesian, and even non-Archimedean.
      

31. EMS 1934 Colloquium
    
    • Courses of lectures are being delivered on "World-Structure by the Kinetic Methods of the Special Theory of Relativity," by Professor E A Milne, Oxford; on "Ramanujan's Note-Books and their Place in Modern Mathematics," by Professor B M Wilson, Dundee; on "Pictorial Geometry," by Professor H W Turnbull, St Andrews; and on "Some Expansions Relating to the Problem of Lattice Points," by Mr W L Ferrar, Fellow of Hertford College, Oxford.
      
    • In addition, the assumption that space-time is Euclidean appears to be made implicitly.
      
    • It was also not made quite clear in Prof Milne's theory how much of the results depended on the initial assumption that space-time was Euclidean and how much was really independent of the geometry.
      
    • In the field of pure mathematics, Professor H W Turnbull (St Andrews) spoke on Pictorial Geometry, dealing with such questions as the generalised construction for the ellipse and hyperbola, the densest and loosest packing of spheres of equal radius within a given volume and the problem of configurations.
      
    • A like interest was displayed in the discussion on Geometry led by Professor J G Semple (Belfast), Dr Timms (St Andrews) and Mr W L Edge (Edinburgh), who, taking a theorem in the theory of three associated quartic curves, each gave a proof of it from a different angle.
      

32. Edmund Whittaker: 'Physics and Philosophy
    
    • In order to build up a theory of geometry Whittaker assumes as an axiom based on experience that a material body can change its situation without being changed in any other way.
      
    • In three-dimensional Euclidean geometry there is a fixed relation between the ten mutual distances of any five points, and such a relation may be verified by measurement for the ten mutual distances of five material particles.
      
    • We assert that the relation between the mutual distances of actual particles are those characteristic of Euclidean geometry.
      
    • The motion of predictability involves the notion of laws of nature, e.g., gravitation, electromagnetics, theoretical geometry; the laws of nature form a rational structure underlying nature; we are finally led to conclude that reality is at every point in intimate relation with this structure and that a knowledge of the structure may be used to determine fresh constituents of reality.
      

33. Rota's lecture on 'Mathematical Snapshots
    
    • Mathematicians triumphantly point to mechanics as the example of a theory that began as an empirical science, and that eventually made its way into mathematics as a generalized geometry, geometry with time added.
      
    • In other words, if a set A in three-dimensional Euclidean space can be rigidly moved onto a set B, then A and B have the same volume.
      
    • In technical language, volume is invariant under the group of Euclidean motions.
      
    • A similar characterization of volume holds in n-dimensional Euclidean space for any finite dimension n.
      
    • Take two convex sets A and B in three dimensional Euclidean space, and suppose that A is contained in B.
      

34. Mathematics at Aberdeen 3
    
    • He entered Glasgow University at the age of eleven where he developed an interest in geometry and graduated at fifteen with a publicly defended thesis On the power of gravity.
      
    • At the same time he furthered his already considerable research in geometry.
      
    • Geography, Chronology and Natural and Civil History in the semi (second) year were accompanied by a compulsory Mathematics course consisting of Arithmetic, Euclidean Geometry, Plane Trigonometry, Practical Geometry and Elementary Algebra, in preparation for Natural Philosophy in the tertian (third) year.
      
    • 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.
      

35. Halmos: creative art
    
    • He used mathematics to find out facts about the universe, and that he successfully used certain parts of differential geometry for that purpose adds a certain piquancy to the appeal of differential geometry.
      
    • Withal, relativity theory and differential geometry are not the same thing.
      
    • The problem-solvers make geometric constructions, the theory-creators discuss the foundations of Euclidean geometry; the problem-solvers find out what makes switching diagrams tick, the theory-creators prove representation theorems for Boolean algebras.
      
    • Perhaps the ideal is to have a spice of reality always present, but not to crowd it the way descriptive geometry, say, does in mathematics, and medical illustration, say, does in painting.
      

36. EMS 1913 Colloquium 2.html.html
    
    • 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.
      

37. EMS 1934 Colloquium 2.html
    
    • In addition, the assumption that space-time is Euclidean appears to be made implicitly.
      
    • It was also not made quite clear in Prof Milne's theory how much of the results depended on the initial assumption that space-time was Euclidean and how much was really independent of the geometry.
      
    • In the field of pure mathematics, Professor H W Turnbull (St Andrews) spoke on Pictorial Geometry, dealing with such questions as the generalised construction for the ellipse and hyperbola, the densest and loosest packing of spheres of equal radius within a given volume and the problem of configurations.
      
    • A like interest was displayed in the discussion on Geometry led by Professor J G Semple (Belfast), Dr Timms (St Andrews) and Mr W L Edge (Edinburgh), who, taking a theorem in the theory of three associated quartic curves, each gave a proof of it from a different angle.
      

38. EMS obituary
    
    • This primal was encountered by Coble, in 1906, who saw that it has 45 nodes, and gave some details of their configuration, Now H F Baker, on a visit to Gottingen to study under Klein, had there met Burkhardt who gave him off-prints of his papers; these Baker studied and copiously annotated and when, nearly 50 years on, retirement from his Cambridge chair had brought comparative leisure he set out to describe, without any dependence on theta-functions, the geometry of the 45-nodal primal, calling it Burkhardt's primal.
      
    • Todd, who had found in 1936 a representation of this primal on [3], read the proof-sheets of the tract and noticed that the geometry afforded a means of partitioning the 25920 projectivities into 15 classes such that operations conjugate in G necessarily belonged to the same class.
      
    • But they are known in other contexts; not only are they groups of symmetries of regular polytopes in Euclidean space, but they are also the groups of automorphisms of .
      
    • Her results for G7 were not anticipated and those for G5 and G6, which were, are mere corollaries once the geometry in [7] has been set out in detail.
      
    • The geometry of the 27 lines of a cubic surface is classical, so is that of the 28 bitangents of a quartic curve.
      

39. EMS 1913 Colloquium 4.html.html
    
    • 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.
      

40. Einstein: 'Ether and Relativity
    
    • Nor do we know whether it is only in the proximity of ponderable masses that its structure differs essentially from that of the Lorentzian ether; whether the geometry of spaces of cosmic extent is approximately Euclidean.
      
    • But we can assert by reason of the relativistic equations of gravitation that there must be a departure from Euclidean relations, with spaces of cosmic order of magnitude, if there exists a positive mean density, no matter how small, of the matter in the universe.
      
    • The contrast between ether and matter would fade away, and, through the general theory of relativity, the whole of physics would become a complete system of thought, like geometry, kinematics, and the theory of gravitation.
      

41. Mathematics at Aberdeen 4
    
    • When it was first awarded in 1795, for solutions of questions in geometry, the dies were made larger than intended, but, according to William Knight (a later Professor of Natural Philosophy), this was 'better, as there is less temptation in future time to give away a large than a small medal'.
      
    • Although later renowned as a philosopher at Glasgow, Reid, who was a friend of Professor John Stewart of Marischal College where they had both graduated in 1726, was himself a capable mathematician with an interest in geometry, which led him to foresee the possibility of non-euclidean systems.
      
    • The syllabus for this senior class included Spherical Trigonometry, Conic Sections, Analytical Geometry and Differential and Integral Calculus.
      

42. The Edinburgh Mathematical Society: the first hundred years (1883-1983) Part 2
    
    • A W Conway, who was Professor of Mathematical Physics at University College, Dublin, spoke on 'The Theory of Relativity and the new Physical Ideas of Space and Time', Dr D M Y Sommerville, who was then Lecturer in Mathematics at St Andrews, spoke on 'Non-Euclidean Geometry and the Foundations of Geometry', while Whittaker himself lectured on 'Practical harmonic analysis; an illustration of Mathematical Laboratory practice.' The colloquium was a striking success, being attended by 77 participants from all over Great Britain.
      

43. EMS 1913 Colloquium
    
    • 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." .
      

44. Dickson: 'Theory of Equations
    
    • In particular every teacher of algebra should read the proof of the fundamental theorem of algebra and the work on graphing; while every teacher of geometry, should read the proofs given in Chap.
      
    • These puzzle students and often teachers, partly because the problem is not clearly understood, and partly because there is so obviously a solution; and yet their impossibility may readily be made plausible to a student familiar with coordinate geometry and is here rigorously proved in an elementary way.
      
    • We are also shown why there can be no construction, in the Euclidean sense, of regular polygons of 7 and 9 sides.
      

45. Eddington: 'Mathematical Theory of Relativity' Introduction
    
    • The pure mathematician proceeds differently; he defines distance as an attribute of the two points which obeys certain laws - the axioms of the geometry which he happens to have chosen - and he is not concerned with the question how this "distance" would exhibit itself in practical observation.
      
    • For example, until recently the practical man was never confronted with problems of non-Euclidean space, and it might be suggested that he would be uncertain how to construct a straight line when so confronted; but as a matter of fact he showed no hesitation, and the eclipse observers measured without ambiguity the bending of light from the "straight line." The appropriate practical definition was so obvious that there was never any danger of different people meaning different loci by this term.
      

46. The Edinburgh Mathematical Society: the first hundred years
    
    • The subjects discussed in these early papers cover a wide range, but with a preponderance of Euclidean geometry.
      

47. Ferrar: 'Textbook of Convergence
    
    • Though, of course, the order of proof is as important here as it is in the development of Euclidean geometry: we must not use A to prove B, and then use B to prove A.
      

48. Aitken: 'Statistical Mathematics
    
    • These laws are expressed, if possible, in the form of logical or numerical axioms, resembling those of Euclidean geometry.
      

49. Napier Tercentenary
    
    • A paper by Dr Somerville, of St Andrews, on Napier's rules and trigonometrically equivalent polygons with extension to non-Euclidean space touched on the other side of Napier's mathematical activity.
      
    • Mr R Cunningham, M.A., Fellow of St John's College Cambridge, will deliver four lectures on "Critical Studies of Modern Electric Theories of the Constitution of Matter, Gravitation, Spectroscope, etc." Mr H W Richmond, M.A., F.R.S., Fellow of King's College, Cambridge, will give four lectures on "Infinity in Geometry," and Professor Whittaker will supplement the lectures of Professor d'Ocagne by demonstrating the arithmetical methods of solving certain classes of equations in the mathematical laboratory.
      

50. EMS obituary
    
    • Thereafter he studied mathematics in Cambridge, Italy (where he learned his non-Euclidean geometry), and Germany, and in 1896, when George A Gibson became professor at the Technical College, he was appointed assistant in the Mathematics Department at the University.
      

51. American Mathematical Society Colloquium
    
    • Projective Differential Geometry.
      
    • Forms of Non-Euclidean Space.
      

Quotations

1. Quotations by Poincare
   
   • At that 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 transformations I had used to define the Fuchsian functions were identical with non-Euclidean geometry.
     
   • It has adopted the geometry most advantageous to the species or, in other words, the most convenient.
     
   • Geometry is not true, it is advantageous.
     
   • 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.
     
   • Quoted in M J Greenberg, Euclidean and non-Euclidean geometries: Development and history (San Fransisco, 1980).
     

2. Quotations by Dieudonne
   
   • Analytical geometry has never existed.
     
   • There are only people who do linear geometry badly, by taking coordinates, and they call this analytical geometry.
     
   • For example, it is well known that Euclidean geometry is a special case of the theory of Hermitian operators in Hilbert spaces.
     

3. Quotations by Dirac
   
   • Non-euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world.
     

4. Quotations by Bolyai
   
   • [A reference to the creation of a non-euclidean geometry.] .
     

Chronology

1. Mathematical Chronology
   
   • It gives details of Egyptian geometry.
     
   • He uses geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore.
     
   • Pythagoras of Samos moves to Croton in Italy and teaches mathematics, geometry, music, and reincarnation.
     
   • Hippocrates of Chios writes the Elements which is the first compilation of the elements of geometry.
     
   • Autolycus of Pitane writes On the Moving Sphere which studies the geometry of the sphere.
     
   • Eudemus of Rhodes writes the History of Geometry.
     
   • Euclid gives a systematic development of geometry in his Stoicheion (The Elements).
     
   • He writes works on two- and three-dimensional geometry, studying circles, spheres and spirals.
     
   • Diocles writes On burning mirrors, a collection of sixteen propositions in geometry mostly proving results on conics.
     
   • Nicomachus of Gerasa writes Arithmetike eisagoge (Introduction to Arithmetic) which is the first work to treat arithmetic as a separate topic from geometry.
     
   • Pappus of Alexandria writes Synagoge (Collections) which is a guide to Greek geometry.
     
   • Boethius writes geometry and arithmetic texts which are widely used for a long time.
     
   • Alcuin of York writes elementary texts on arithmetic, geometry and astronomy.
     
   • Thabit ibn Qurra makes important mathematical discoveries such as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry.
     
   • Ibn al-Haytham (often called Alhazen) writes works on optics, including a theory of light and a theory of vision, astronomy, and mathematics, including geometry and number theory.
     
   • His work on mathematics covers 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.
     
   • His important mathematical work Kitab al-Shifa' (The Book of Healing) divides mathematics into four major topics, geometry, astronomy, arithmetic, and music.
     
   • Bhaskara II (sometimes known as Bhaskaracharya) writes Lilavati (The Beautiful) on arithmetic and geometry, and Bijaganita (Seed Arithmetic), on algebra.
     
   • Nicholas of Cusa studies geometry and logic.
     
   • Pacioli publishes Summa de arithmetica, geometria, proportioni et proportionalita which is a review of the whole of mathematics covering arithmetic, trigonometry, algebra, tables of moneys, weights and measures, games of chance, double-entry book-keeping and a summary of Euclid's geometry.
     
   • Kepler publishes Nova stereometria doliorum vinarorum (Solid Geometry of a Wine Barrel), an investigation of the capacity of casks, surface areas, and conic sections.
     
   • Mydorge works on optics and geometry.
     
   • Descartes publishes La Geometrie which describes his application of algebra to geometry.
     
   • Desargues begins the study of projective geometry, which considers what happens to shapes when they are projected on to a non-parallel plane.
     
   • De Beaune writes Notes brieves which contains the many results on "Cartesian geometry", in particular giving the now familiar equations for hyperbolas, parabolas and ellipses.
     
   • It is the first systematic development of the analytic geometry of the straight line and conic.
     
   • This work establishes analytic geometry as a major mathematical topic.
     
   • James Gregory publishes Vera circuli et hyperbolae quadratura which lays down exact foundations for the infinitesimal geometry.
     
   • Mohr publishes Euclides danicus in which he shows that all Euclidean constructions can be carried out with compasses alone.
     
   • In Euclides ab Omni Naevo Vindicatus Saccheri does important early work on non-euclidean geometry, although he considers it an attempt to prove the parallel postulate of Euclid.
     
   • Maclaurin publishes Treatise on Fluxions which aims to provide a rigorous foundation for the calculus by appealing to the methods of Greek geometry.
     
   • Monge begins the study of descriptive geometry.
     
   • By assuming that the parallel postulate is false, he manages to deduce a large number of results about non-euclidean geometry.
     
   • D'Alembert calls the problems to elementary geometry caused by failure to prove the parallel postulate "the scandal of elementary geometry".
     
   • Legendre publishes Elements de geometrie, an account of geometry which would be a leading text for 100 years.
     
   • It becomes the prototype of later geometry texts.
     
   • Mascheroni proves in Geometria del compasso that all Euclidean constructions can be made with compasses alone and so a ruler in not required.
     
   • Lazare Carnot publishes Geometrie de position in which sensed magnitudes are first used systematically in geometry.
     
   • Poncelet develops the principles of projective geometry in Traite des proprietes projectives des figures (Treatise on the Projective Properties of Figures).
     
   • This work contains fundamental ideas of projective geometry such as the cross-ratio, perspective, involution and the circular points at infinity.
     
   • Janos Bolyai completes preparation of a treatise on a complete system of non-Euclidean geometry.
     
   • Steiner develops synthetic geometry.
     
   • Mobius publishes Der barycentrische Calkul on analytical geometry.
     
   • It becomes a classic and includes many of his results on projective and affine geometry.
     
   • Gauss introduces differential geometry and publishes Disquisitiones generales circa superficies.
     
   • Lobachevsky develops non-euclidean geometry, in particular hyperbolic geometry, and his first account of the subject is published in the Kazan Messenger.
     
   • (Systematic Development of the Dependency of Geometrical Forms on One Another) which gives a treatment of projective geometry based on metric considerations.
     
   • Janos Bolyai's work on non-Euclidean geometry is published as an appendix to an essay by Farkas Bolyai, his father.
     
   • Cayley is the first person to investigate "geometry of n dimensions" which occurs in the title of his paper of that year.
     
   • It is the first work to completely free projective geometry from any metrical basis.
     
   • In his 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 he defines an n-dimensional space and gives a definition of what today is called a "Riemannian space".
     
   • Plucker makes further advances in geometry when he defines a four dimensional space in which straight lines rather than points are the basic elements.
     
   • Beltrami publishes Essay on an Interpretation of Non-Euclidean Geometry which gives a concrete model for the non-euclidean geometry of Lobachevsky and Bolyai.
     
   • He defines geometry as the study of the properties of a space that are invariant under a given group of transformations.
     
   • Cesaro publishes Lezione di geometria intrinseca in which he formulates intrinsic geometry.
     
   • Hilbert publishes Grundlagen der Geometrie (Foundations of Geometry) putting geometry in a formal axiomatic setting.
     
   • Castelnuovo publishes Geometria analitica e proiettiva his most important work in algebraic geometry.
     
   • The third and final volume will appear three years later, while a fourth volume on geometry was planned but never completed.
     
   • Weyl publishes Die Idee der Riemannschen Flache which brings together analysis, geometry and topology.
     
   • Borsuk publishes his theory of retracts in metric differential geometry.
     
   • Baer introduces the concept of an injective module, then begins studying group actions in geometry.
     
   • Weil publishes Foundations of Algebraic Geometry.
     
   • Grothendieck receives a Fields Medal for his work on geometry, number theory, topology and complex analysis.
     
   • Thurston is awarded the Oswald Veblen Geometry Prize of the American Mathematical Society for his work on foliations.
     
   • Mandelbrot publishes The fractal geometry of nature which develops his theory of fractal geometry more fully than his work of 1975.
     
   • Shing-Tung Yau is awarded a Fields Medal 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.
     
   • Donaldson publishes Self-dual connections and the topology of smooth 4-manifolds which leads to totally new ideas concerning the geometry of 4-manifolds.
     
   • Witten publishes Supersymmetry and Morse theory containing ideas that have become of central importance in the study of differential geometry.
     
   • Connes publishes a major text on noncommutative geometry.
     
   • Borcherds is awarded a Fields Medal for his work in automorphic forms and mathematical physics; Gowers receives one for his work in functional analysis and combinatorics; Kontsevich receives one for his work in algebraic geometry, algebraic topology, and mathematical physics; and McMullen receives one for his work on holomorphic dynamics and geometry of 3-dimensional manifolds.
     

2. Chronology for 1820 to 1830
   
   • Poncelet develops the principles of projective geometry in Traite des proprietes projectives des figures (Treatise on the Projective Properties of Figures).
     
   • This work contains fundamental ideas of projective geometry such as the cross-ratio, perspective, involution and the circular points at infinity.
     
   • Janos Bolyai completes preparation of a treatise on a complete system of non-Euclidean geometry.
     
   • Steiner develops synthetic geometry.
     
   • Mobius publishes Der barycentrische Calkul on analytical geometry.
     
   • It becomes a classic and includes many of his results on projective and affine geometry.
     
   • Gauss introduces differential geometry and publishes Disquisitiones generales circa superficies.
     
   • Lobachevsky develops non-euclidean geometry, in particular hyperbolic geometry, and his first account of the subject is published in the Kazan Messenger.
     

3. Chronology for 1860 to 1870
   
   • Plucker makes further advances in geometry when he defines a four dimensional space in which straight lines rather than points are the basic elements.
     
   • Beltrami publishes Essay on an Interpretation of Non-Euclidean Geometry which gives a concrete model for the non-euclidean geometry of Lobachevsky and Bolyai.
     

4. Chronology for 1760 to 1780
   
   • Monge begins the study of descriptive geometry.
     
   • By assuming that the parallel postulate is false, he manages to deduce a large number of results about non-euclidean geometry.
     
   • D'Alembert calls the problems to elementary geometry caused by failure to prove the parallel postulate "the scandal of elementary geometry".
     

5. Chronology for 500 to 900
   
   • Boethius writes geometry and arithmetic texts which are widely used for a long time.
     
   • Alcuin of York writes elementary texts on arithmetic, geometry and astronomy.
     
   • Thabit ibn Qurra makes important mathematical discoveries such as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-euclidean geometry.
     

6. Chronology for 1650 to 1675
   
   • It is the first systematic development of the analytic geometry of the straight line and conic.
     
   • This work establishes analytic geometry as a major mathematical topic.
     
   • James Gregory publishes Vera circuli et hyperbolae quadratura which lays down exact foundations for the infinitesimal geometry.
     
   • Mohr publishes Euclides danicus in which he shows that all Euclidean constructions can be carried out with compasses alone.
     

7. Chronology for 1830 to 1840
   
   • (Systematic Development of the Dependency of Geometrical Forms on One Another) which gives a treatment of projective geometry based on metric considerations.
     
   • Janos Bolyai's work on non-Euclidean geometry is published as an appendix to an essay by Farkas Bolyai, his father.
     

8. Chronology for 1780 to 1800
   
   • Legendre publishes Elements de geometrie, an account of geometry which would be a leading text for 100 years.
     
   • It becomes the prototype of later geometry texts.
     
   • Mascheroni proves in Geometria del compasso that all Euclidean constructions can be made with compasses alone and so a ruler in not required.
     

9. Chronology for 1720 to 1740
   
   • In Euclides ab Omni Naevo Vindicatus Saccheri does important early work on non-euclidean geometry, although he considers it an attempt to prove the parallel postulate of Euclid.