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Here's a list of mathematicians and their contribution to geometry.
Ancient Geometry (30000 BC - 500 BC)
Babylon (4000 BC - 500 BC)
Egypt (5000 BC - 500 BC)
Ahmes (1680-1620 BC)
wrote the Rhind Papyrus (aka the Ahmes Papyrus).
In it, he claims to not be the author, but merely a scribe of material from an earlier work of about 2000 BC. It contains rules for division, and has 87 problems including the solution of equations, progressions, volumes of granaries, etc.
India (1500 BC - 200 BC)
The Sulbasutras
which are appendices to the Vedas giving rules for constructing sacrificial altars. To please the gods, an altar's measurements had to conform to very precise formula, and mathematical accuracy was very important. It is not historically clear whether this mathematics was developed by the Indian Vedic culture, or whether it was borrowed from the Babylonians. Like the Babylonians, results in the Sulbasutras are stated in terms of ropes; and "sutra" eventually came to mean a rope for measuring an altar. Ultimately, the Sulbasutras are simply construction manuals for some basic geometric shapes. It is noteworthy, though, that all the Sulbasutras contain a method to square the circle (one of the infamous Greek problems) as well as the converse problem of finding a circle equal in area to a given square.:
The Baudhayana (800 BC)
Baudhayana was the author of the earliest known Sulbasutra. Although he was a priest interested in constructing altars, and not a mathematician, his Sulbasutra contains geometric constructions for solving linear and quadratic equations, plus approximations of p (to construct circles) and Ö2 = 577 / 408 (which is accurate to 5 decimal places). It also gives the special case of the Pythagorian theorem for the diagonal of a square.
The Manava (750 BC)
contains approximate constructions of circles from rectangles, and squares from circles, which give approximations of p.
The Apastamba (600 BC)
considers the problems of squaring the circle, and of dividing a segment into 7 equal parts. It also gives an accurate approximation of Ö2 .
The Katyayana (200 BC)
gives the general case of the Pythagorian theorem for the diagonal of any rectangle.
Greek Geometry (600 BC - 400 AD)
Thales of Miletus (624-547 BC)
was one of the Seven pre-Socratic Sages, and brought the science of geometry from Egypt to Greece. He is credited with the experimental discovery of five facts of elementary geometry (including that an angle in a semicircle is a right angle), but some historians dispute this and give the credit to Pythagoras.
Pythagoras of Samos (569-475 BC)
is regarded as the first pure mathematician to logically deduce geometric facts from basic principles. He is credited with proving many theorems such as the angles of a triangle summing to 180 deg, and the infamous "Pythagorian Theorem" for a right-angled triangle (which had been known experimentally in Egypt for over 1000 years). The Pythagorian school is considered as the (first documented) source of logic and deductive thought, and may be regarded as the birthplace of reason itself. As philosophers, they speculated about the structure and nature of the universe: matter, music, numbers, and geometry. Their legacy is described in Pythagoras and the Pythagoreans : A Brief History.
Hippocrates of Chios (470-410 BC)
wrote the first "Elements of Geometry" which Euclid may have used as a model for his own Books I and II more than a hundred years later. In this first "Elements", Hippocrates included geometric solutions to quadratic equations and early methods of integration. He studied the classic problem of squaring the circle showing how to square a "lune". He worked on duplicating the cube which he showed equivalent to constructing two mean proportionals between a number and its double. Hippocrates was also the first to show that the ratio of the areas of two circles was equal to the ratio of the squares of their radii.
Plato (427-347 BC)
founded "The Academy" in 387 BC which flourished until 529 AD. He developed a theory of Forms, in his book "Phaedo", which considers mathematical objects as perfect forms (such as a line having length but no breadth). He emphasized the idea of 'proof' and insisted on accurate definitions and clear hypotheses, paving the way to Euclid, but he made no major mathematical discoveries himself. The state of mathematical knowledge in Plato's time is reconstructed in the scholarly book: The Mathematics of Plato's Academy .
Theaetetus of Athens (417-369 BC)
was a student of Plato's, and the creator of solid geometry. He was the first to study the octahedron and the icosahedron, and thus construct all five regular solids. This work of his formed Book XIII of Euclid's Elements. His work about rational and irrational quantities also formed Book X of Euclid.
Eudoxus of Cnidus (408-355 BC)
foreshadowed algebra by developing a theory of proportion which is presented in Book V of Euclid's Elements in which Definitions 4 and 5 establish Eudoxus' landmark concept of proportion. In 1872, Dedekind stated that his work on "cuts" for the real number system was inspired by the ideas of Eudoxus. Eudoxus also did early work on integration using his method of exhaustion by which he determined the area of circles and the volumes of pyramids and cones. This was the first seed from which the calculus grew two thousand years later.
Menaechmus (380-320 BC)
was a pupil of Eudoxus, and discovered the conic sections. He was the first to show that ellipses, parabolas, and hyperbolas are obtained by cutting a cone in a plane not parallel to the base.
Euclid of Alexandria (325-265 BC)
is best known for his 13 Book treatise "The Elements" (~300 BC), collecting the theorems of Pythagoras, Hippocrates, Theaetetus, Eudoxus and other predecessors into a logically connected whole. A good modern translation of this historic work is The Thirteen Books of Euclid's Elementsby Thomas Heath.
Archimedes of Syracuse (287-212 BC)
is regarded as the greatest of Greek mathematicians, and was also an inventor of many mechanical devices (including the screw, the pulley, and the lever). He perfected integration using Eudoxus' method of exhaustion, and found the areas and volumes of many objects. A famous result of his is that the volume of a sphere is two-thirds the volume of its circumscribed cylinder, a picture of which was inscribed on his tomb. He gave accurate approximations to p and square roots. In his treatise "On Plane Equilibriums", he set out the fundamental principles of mechanics, using the methods of geometry, and proved many fundamental theorems concerning the center of gravity of plane figures. In "On Spirals", he defined and gave fundamental properties of a spiral connecting radius lengths with angles as well as results about tangents and the area of portions of the curve. He also investigated surfaces of revolution, and discovered the 13 semi-regular (or "Archimedian") polyhedra whose faces are all regular polygons. Translations of his surviving manuscripts are now available as The Works of Archimedes. A good biography of his life and discoveries is also available in the book Archimedes: What Did He Do Beside Cry Eureka?. He was killed by a Roman soldier 212 BC.
Apollonius of Perga (262-190 BC)
was called 'The Great Geometer'. His famous work was "Conics" consisting of 8 Books In Books 5 to 7, he studied normals to conics, and determined the center of curvature and the evolute of the ellipse, parabola, and hyperbola. In another work "Tangencies", he showed how to construct the circle which is tangent to three objects (points, lines or circles). He also computed an approximation for p better than the one of Archimedes. English translations of his Conics: Books I - III and Conics Books V to VII are now available.
Hipparchus of Rhodes (190-120 BC)
is the first to systematically use and document the foundations of trigonometry, and may have invented it. He published several books of trigonometric tables and the methods for calculating them. He based his tables on dividing a circle into 360 degrees with each degree divided into 60 minutes. This is the first recorded use of this subdivision. In other work, he applied trigonometry to astronomy making it a practical predictive science.
Heron of Alexandria (10-75 AD)
wrote "Metrica" (3 Books) which gives methods for computing areas and volumes. Book I considers areas of plane figures and surfaces of 3D objects, and contains his now-famous formula for the area of a triangle = sqrt[s(s-a)(s-b)(s-c)] where s=(a+b+c)/2 [but some historians attribute this result to Archimedes]. Book II considers volumes of 3D solids. Book III deals with dividing areas and volumes according to a given ratio, and gives a method to find the cube root of a number. He wrote in a practical manner, and has other books, notably in Mechanics.
Menelaus of Alexandria (70-130 AD)
developed spherical geometry in his only surviving work "Sphaerica" (3 Books). In Book I, he defines spherical triangles using arcs of great circles which marks a turning point in the development of spherical trigonometry. Book 2 applies spherical geometry to astronomy; and Book 3 deals with spherical trigonometry including "Menelaus's theorem" about how a straight line cuts the three sides of a triangle in proportions whose product is (-1).
Claudius Ptolemy (85-165 AD)
wrote "Almagest" (13 Books) giving the mathematics for the geocentric theory of planetary motion. Considered a masterpiece with few peers, Almagest remained the major work in astronomy for 1400 years until it was superceded by the heliocentric theory of Copernicus. Nevertheless, in Books 1 and 2, Ptolemy refined the foundations of trigonometry based on the chords of a circle established by Hipparchus. One infamous result that he used, known as "Ptolemy's Theorem", states that for a quadrilateral inscribed in a circle, the product of its diagonals is equal to the sum of the products of its opposite sides. From this, he derived the (chord) formulas for sin(a+b), sin(a-b), and sin(a/2), and used these to compute detailed trigonometric tables.
Pappus of Alexandria (290-350 AD)
was the last of the great Greek geometers. His major work in geometry is "Synagoge" or the "Collection" (in 8 Books), a handbook on a wide variety of topics: arithmetic, mean proportionals, geometrical paradoxes, regular polyhedra, the spiral and quadratrix, trisection, honeycombs, semiregular solids, minimal surfaces, astronomy, and mechanics. In Book VII, he proved "Pappus' Theorem" which forms the basis of modern projective geometry; and also proved "Guldin's Theorem" (rediscovered in 1640 by Guldin) to compute a volume of revolution.
Hypatia of Alexandria (370-415 AD)
was the first woman to make a substantial contribution to the development of mathematics. She learned mathematics and philosophy from her father Theon of Alexandria, and assisted him in writing an eleven part commentary on Ptolemy's Almagest, and a new version of Euclid's Elements. Hypatia also wrote commentaries on Diophantus's Arithmetica, Apollonius's Conics and Ptolemy's astronomical works. About 400 AD, Hypatia became head of the Platonist school at Alexandria, and lectured there on mathematics and philosophy. Although she had many prominent Christians as students, she ended up being brutally murdered by a fanatical Christian sect that regarded science and mathematics to be pagan. Nevertheless, she is the first woman in history recognized as a professional geometer and mathematician.
Medieval Geometry
Arabic / Islamic (600 - 1500 AD)
Chinese (100 BC - 1400 AD)
Modern Geometry (1600 - 2000 AD)
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.
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.
Girard Desargues (1591-1661)
invented modern projective geometry in his most important work titled "Rough draft for an essay on the results of taking plane sections of a cone" (1639). His famous 'perspective theorem' for two triangles was published in 1648.
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).
Leonhard Euler (1707-1783)
was extremely prolific in a vast range of subjects, and founded mathematical analysis. He invented the idea of functions and used them to transform analytic into differential geometry investigating surfaces, curvature, and geodesics. He discovered (1752) that the well-known "Euler characteristic" (V-E+F) of a polyhedron depends only on the surface topology. Euler, Monge, and Gauss are considered the three fathers of differential geometry. He also made breakthroughs contributions to many other branches of math. A representative selection of his discoveries is given in Euler: The Master of Us All.
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.
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 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.
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. The professionals regarded him as an obscure amateur mathematician (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 (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.
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.
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" (1910-18). He then worked in topology and differential geometry, and published with his student Henry Whitehead "The Foundations of Differential Geometry" (1933) which gives the first definition of a differentiable manifold.
Donald Coxeter (1907-2003)
is regarded as the major synthetic geometer of the 20th century, and has made important contributions to the theory of polytopes, non-Euclidean geometry, group theory and combinatorics. His "Coxeter groups" give the complete classification of regular polytopes in n-dimensions. He has published many important books, including Regular Polytopes (1947, 1963, 1973) and Introduction to Geometry (1961, 1989).
EDUCATION
