Modern Geometry



edieval Geometry

Chinese Geometry (100 BC - 1300 AD)

Hindu Geometry (500 - 1100 AD)

Arabic Geometry (800 - 1500 AD)

Modern Geometry (1600 - 2000 AD)

The major Modern Geometers are listed in this chronological timeline. Click on a name or picture for an expanded biography.

Rene Descartes (1596-1650)
in an appendix "La Geometrie" of his 1637 manuscript "Discours de la method ...", he applied algebra to geometry and created analytic geometry. A complete modern English translation of this appendix is available in the book “The Geometry of Rene Descartes“. Also, the recent book “Descartes's Mathematical Thought” reconstructs his intellectual career, both mathematical and philosophical.

Girard Desargues (1591-1661)
invented perspective geometry in his most important work titled "Rough draft for an essay on the results of taking plane sections of a cone" (1639). In 1648, he published his famous “Desargues’ Theorem” for two triangles in perspective, which later evolved into projective geometry.

Pierre de Fermat (1601-1665)
is also recognized as an independent co-creator of analytic geometry which he first published in his 1636 paper "Ad Locos Planos et Solidos Isagoge". He also developed a method for determining maxima, minima and tangents to curved lines foreshadowing calculus. Descartes first attacked this method, but later admitted it was correct. The story of his life and work is described in the book “The Mathematical Career of Pierre de Fermat”.

Blaise Pascal (1623-1662)
was the co-inventor of modern projective geometry, published in his "Essay on Conic Sections" (1640). He later wrote "The Generation of Conic Sections" (1648-1654). He proved many projective geometry theorems, the earliest including "Pascal's mystic hexagon" (1639).

Giovanni Saccheri (1667-1733)
was an Italian Jesuit who did important early work on non-euclidean geometry. In 1733, the same year he died, Saccheri published his important early work on non-euclidean geometry, “Euclides ab Omni Naevo Vindicatus”. Although he saw it as an attempt to prove the 5th parallel axiom of Euclid. His attempt tried to find a contradiction to a consequence of the 5th axiom, which he failed to do, but instead developed many theorems of non-Euclidean geometry. It was 170 years later that the significance of the work realised. However, the discovery of non-Euclidean geometry by Nikolai Lobachevsky and Janos Bolyai was not due to this masterpiece by Saccheri, since neither ever heard of him.

Leonhard Euler (1707-1783)
was extremely prolific in a vast range of subjects, and is the greatest modern mathematician. He founded mathematical analysis, and invented mathematical functions, differential equations, and the calculus of variations. He used them to transform analytic into differential geometry investigating surfaces, curvature, and geodesics. Euler, Monge, and Gauss are considered the three fathers of differential geometry. In classical geometry, he discovered the “Euler line” of a triangle; and in analytic geometry, the “Euler angles” of a vector. He also discovered that the "Euler characteristic" (V-E+F) of a surface triangulation depends only on it’s genus, which was the genesis of topology. Euler made other breakthrough contributions to many branches of math. Famous formulas he discovered include “Euler’s formula” (eix = cos x + i sin x), “Euler’s identity” (eiπ + 1 = 0), and many formulas with infinite series. The list of his discoveries goes on and on. A representative selection of his work (in 8 different fields) is given in the popular book “Euler: The Master of Us All”. In 1766, Euler became almost totally blind, after which he produced nearly half of all his work, dictating his papers to assistants. He published over 800 papers and books, and his collected works fill 25,000 pages in 79 volumes. A large repository of his work is now available online at The Euler Archive.

Gaspard Monge (1746-1818)
is considered the father of both descriptive geometry in "Geometrie descriptive" (1799); and differential geometry in "Application de l'Analyse a la Geometrie" (1800) where he introduced the concept of lines of curvature on a surface in 3-space.

Adrien-Marie Legendre (1752-1833)

Legendremade important contributions to many fields of math: differential equations, ballistics, celestial mechanics, elliptic functions, number theory, and (of course) geometry. In 1794 Legendre published “Elements de Geometrie” which was the leading elementary text on the topic for around 100 years. In his "Elements" Legendre greatly rearranged and simplified many of the propositions from Euclid's "Elements" to create a more effective textbook. His 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, including those being used today. Although he was born into a wealthy family, in the 1793 French Revolution he lost his capital, and became dependent on his academic salary. Then in 1824, Legendre refused to vote for the government's candidate for the French Institut National; and as a result, his academic pension was stopped. In 1833 he died in poverty.

Carl Friedrich Gauss (1777-1855)
invented non-Euclidean geometry prior to the independent work of Janos Bolyai (1833) and Nikolai Lobachevsky (1829), although Gauss' work on this topic was unpublished until after he died. With Euler and Monge, he is considered a founder of differential geometry. He published "Disquisitiones generales circa superficies curva" (1828) which contained "Gaussian curvature" and his famous "Theorema Egregrium" that Gaussian curvature is an intrinsic isometric invariant of a surface embedded in 3-space. The story of his life and work is given in the popular book “The Prince of Mathematics: Carl Friedrich Gauss”.

Nikolai Lobachevsky (1792-1856)
Lobachevskypublished the first account of non-Euclidean geometry to appear in print. Instead of trying to prove Euclid’s 5th axiom (about a unique line through a point that is parallel to another line), he studied the concept of a geometry in which that axiom may not be true. He completed his major work Geometriya in 1823, but it was not published until 1909. In 1829, he published a paper on hyperbolic geometry, the first paper to appear in print on non-Euclidean geometry, in a Kazan University journal. But his papers were rejected by the more prestigious journals. Finally in 1840, a paper of his was published in Berlin; and it greatly impressed Gauss. There has been some speculation that Gauss influenced Lobachevsky’s work, but those claims have been refuted. In any case, his great mathematical achievements were not recognised in his lifetime, and he died without a notion of  the importance that his work would achieve

Janos Bolyai (1802-1860)
Bolyaiwas a pioneer of non-Euclidean geometry. His father, Farkas, taught mathematics, and raised his son to be a mathematician. His father knew Gauss, whom he asked to take Janos as a student; but Gauss rejected the idea. Around 1820, Janos began to  follow his father’s path to replace Euclid's parallel axiom, but he gave up this approach within a year, since he was starting to develop the basic ideas of  hyperbolic and absolute geometry. In 1825, he explained his discoveries to his father, who was clearly disappointed. But by 1831, his father’s opinion had changed, and he encouraged Janos to publish his work as the Appendix of another work. This Appendix came to the attention of Gauss, who both praised it, and also claimed that it coincided with his own thoughts for over 30 years. Janos took this as a severe blow, became irritable and difficult with others, and his health deteriorated. After this he did little serious mathematics. Later, in 1848, Janos discovered Lobachevsky’s 1829 work, which greatly upset him. He accused Gauss of spiteful machinations through the fictitious Lobachevsky. He then gave up any further work on math. He had never published more than the few pages of the Appendix, but he left more than 20000 pages of mathematical manuscripts, which are now in a Hungarian library.

Jean-Victor Poncelet (1788-1867)
was one of the founders of modern projective geometry. He had studied under Monge and Carnot, but after school, he joined Napoleon’s army. In 1812, he was left for dead after a battle with the Russians, who then imprisoned him for several years. During this time, he tried to remember his math classes as a distraction from the hardship, and started to develop the projective properties of conics, including the pole, polar lines, the principle of duality, and circular points at infinity. After being freed (1814), he got a teaching job, and finally  published his ideas in “Traite des proprietes projectives des figures (1822), from which the term “projective geometry” was coined. He was then in a priority dispute about the duality principle that lasted until 1829. This pushed Poncelet away from projective geometry and towards mechanics, which then became his career. Fifty years later, he incorporated his innovative geometric ideas into his 2-volume treatise on analytic geometry “Applications d'analyse et de geometrie (1862, 1864). He had other unpublished manuscripts, which survived until World War I, when they vanished.

Hermann Grassmann (1809-1877)
was the creator of vector analysis and the vector interior (dot) and exterior (cross) products in his books "Theorie der Ebbe and Flut" studying tides (1840, but 1st published in 1911), and "Ausdehnungslehre" (1844, revised 1862). In them, he invented what is now called the n-dimensional exterior algebra in differential geometry, but it was not recognized or adopted in his lifetime. Professional mathematicians regarded him as an obscure amateur (who had never attended a university math lecture), and mostly ignored his work. He gained some notoriety when Cauchy purportedly plagiarized his work in 1853 (see the web page Abstract linear spaces for a short account). A more extensive description of Grassmann's life and work is given in the interesting book “A History of Vector Analysis”.

Arthur Cayley (1821-1895)
was an amateur mathematician (he was a lawyer by profession) who unified Euclidean, non-Euclidean, projective, and metrical geometry. He introduced algebraic invariance, and the abstract groups of matrices and quaternions which form the foundation for quantum mechanics.

Bernhard Riemann (1826-1866)
was the next great developer of differential geometry, and investigated the geometry of "Riemann surfaces" in his PhD thesis (1851) supervised by Gauss. In later work he also developed geodesic coordinate systems and curvature tensors in n-dimensions. An engaging and readable account of Riemann’s life and work is given in the book “Bernhard Riemann 1826-1866: Turning Points in the Conception of Mathematics”.

Felix Klein (1849-1925)
is best known for his work on the connections between geometry and group theory. He is best known for his "Erlanger Programm" (1872) that synthesized geometry as the study of invariants under groups of transformations, which is now the standard accepted view. He is also famous for inventing the well-known "Klein bottle" as an example of a one-sided closed surface.

David Hilbert (1862-1943)
first worked on invariant theory and proved his famous "Basis Theorem" (1888). He later did the most influential work in geometry since Euclid, publishing "Grundlagen der Geometrie" (1899) which put geometry in a formal axiomatic setting based on 21 axioms. In his famous Paris speech (1900), he gave a list of 23 open problems, some in geometry, which provided an agenda for 20th century mathematics. The story of his life and mathematics are now in the acclaimed biography “Hilbert”.

Oswald Veblen (1880-1960)
developed "A System of Axioms for Geometry" (1903) as his doctoral thesis. Continuing work in the foundations of geometry led to axiom systems of projective geometry, and with John Young he published the definitive "Projective geometry" in 2 volumes (1910-18). He then worked in topology and differential geometry, and published "The Foundations of Differential Geometry" (1933) with his student Henry Whitehead, in which they give the first definition of a differentiable manifold.

Donald Coxeter (1907-2003)
is regarded as the major synthetic geometer of the 20th century, and made important contributions to the theory of polytopes, non-Euclidean geometry, group theory and combinatorics. Coxeter is noted for the completion of Euclid's work by giving the complete classification of regular polytopes in n-dimensions using his "Coxeter groups". He published many important books, including Regular Polytopes (1947, 1963, 1973) and Introduction to Geometry (1961, 1989). He was a Professor of Math at Univ. of Toronto from 1936 until his death at the age of 96. When asked about how he achieved a long life, he replied: "I am never bored". Recently, a biography of his remarkable life has been published in the interesting book “King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry”.



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