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Ancient Geometry
(3000 BC - 600 BC)
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| The geometry of Mesopotamia and Egypt was mostly experimentally derived rules used by the engineers of those civilizations. They knew how to compute areas, and even knew the "Pythagorian Theorem" 1000 years before the Greeks. But there is no evidence that they logically deduced these facts from basic principles. Nevertheless, they established the framework that inspired Greek geometry. |
| Time Line of Greek Mathematicians | |
| Major Greek Geometers (listed
cronologically) [click on a name or picture for an expanded biography]. |
 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 for 1000 years). |
 Hippocrates of Chios (470-410
BC)wrote the first "Elements of Geometry" which Euclid may have used as a model for Books I and II. In his "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. |
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| 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. |
| 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 of Geometry" (~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 Elements by 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. 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 studied semi-regular polyhedra. |
 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. |
 Hipparchus of Rhodes (190-120
BC)is the first to systematically use and document 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 [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". |
| Pappus
of Alexandria (290-350 AD) was the last of the great Greek geometers. One of his theorems forms the basis of projective geometry. 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" (the basis of projective geometry); and also "Guldin's Theorem" (rediscovered in 1640 by Guldin) to compute a volume of revolution. |
| Time Lines of Modern Mathematicians | |
| Major Modern Geometers (listed
cronologically) [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. |
 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. |
 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. |
 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- )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 a 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). |
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Plato