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Babylon
(4000 BC - 500 BC)
Egypt
(5000 BC - 500 BC)
The geometry of Babylon (in 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
(see:
Pythagoras's theorem in Babylonian mathematics). But
there is no evidence that they logically deduced geometric facts from
basic principles. Nevertheless, they established the framework that
inspired Greek geometry. A detailed analysis of Egyptian
mathematics is given in the book:
Mathematics in the Time of the Pharaohs.
One of the few surviving documents was written by:
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.
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India
(1500 BC - 200 BC)
Everything that we know about ancient Indian (Vedic) mathematics is
contained in:
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 main Sulbasutras, named after their
authors, are:
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.
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 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. |
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 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. |
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 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. |
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 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
. |
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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. |
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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. |
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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. |
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 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 Elements
by Thomas Heath. |
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 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. |
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 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. |
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 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. |
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 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. |
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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). |
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 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.
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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. |
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 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.
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Medieval Geometry
 Chinese
(100 BC - 1400 AD)
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Modern Geometry
(1600 - 2000 AD)
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 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. |
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 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. |
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 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. |
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 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). |
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 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. |
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 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. |
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 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. |
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 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. |
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 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. |
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 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. |
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 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. |
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 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. |
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 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. |
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 Donald
Coxeter (1907-2003) [Obit]
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). |
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History
