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Greek Geometry

 

 


Greek Geometry (600 BC - 400 AD)


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


Thales of Miletus (624-547 BC)
Thales
was one of the Seven pre-Socratic Sages, and brought the science of geometry from Egypt to Greece. He is credited with the discovery of five facts of elementary geometry, including that an angle in a semicircle is a right angle (referred to as “Thales Theorem”). But some historians dispute this and give the credit to Pythagorus. There is no evidence that Thales used logical deduction to prove geometric facts.


Pythagorus 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 "Pythagorean Theorem" for a right-angled triangle (which had been known experimentally in Babylon and Egypt for over 1000 years). The Pythagorean 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 Pythagorus 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 to be 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 to construct all five regular solids. His work 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 Pythagorus, 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 the inventor of many mechanical devices (including the screw, pulley, and 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 pi 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 in 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 pi better than the one of Archimedes. English translations of his Conics Books I - III, Conics Book IV, 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 = Eqn_heron-triangle where s=(a+b+c)/2 [note: 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 the "Almagest" (13 Books) giving the mathematics for the geocentric theory of planetary motion. Considered a masterpiece with few peers, the 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.

 

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