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## Ancient World (from 4000 BC to 1 BC)

Different civilizations and forms of government form on the banks of big rivers in Afrika (Egypt), Asia (including Mesopotamia, India, China), later also in the Mediterranean Area (Greece, Roman Empire). Although different in regions and time of origin, all these state forms have still a lot in common: a hierarchical system of social classes, with aristocracy on the top and many specialized classes below, including priests, scribes, soldiers, craftsmen, farmers, and slaves. Most of these state forms lasted for millennia and centuries. This allowed the melting of the religion with the state forms.

The mathematics in these different state forms was developed to serve practical means, for instance, administration or collection of taxes. But over the centuries, this practical respect of mathematics became more and more independent from the practical application. Scribes and priests began to bring more abstraction into the practical calculations, to think about the concepts behind the pure practical meaning of arithmetics and geometry.

Unfortunately, the knowledge of mathematics of this time is very limited, because only a little evidence is left. It did not survive, either because the material used by scribes to write what they knew was not so resistant (e.g. bamboo, papyrus, tree bark, clay), or because of wars, fire, or intentional destruction of generations which came after.

Year Mathematician Achievements Mathematics Category
about 2050 BC unknown Place-value sexagesimal system, numerical notation Sumerian
about 2000 BC Megalithic Stone and Wood Settings Western
ca. 1850 BC unknown solving linear, quadratic and some cubic equations, square roots, interpolation of logarithms, Pythagoras’ theorem; all theorems without proofs Babylonian Mathematics
ca. 1400 BC unknown Decimal numeration carved on oracle bones Chinese
1680 BC Ahmes solving simple practical problems using additive mathematics, Pythagorean theorem (without proof), volume formula for a pyramid trunk Egyptian Mathematics
800 BC Baudhayana Co-author of Sulbasutras, written for concrete religious purposes (e.g. building altars or sacrificial offerings), Pythagoras’ theorem, approximation of $\sqrt{2}$, constructing squares with sides equal to the diameter of a given circle Indian
750 BC Manava Co-author of Sulbasutras Indian
624 BC Thales of Miletus First Theorems in Geometry Greek
611 BC Anaximander of Miletus First idea of the Universe: Sun, Moon, and planets revolving around the Earth, construction of a sundial Greek
600 BC Apastamba Co-author of Sulbasutras Indian
569 BC Pythagoras of Samos The Pythagoreans find interconnections between number theory, geometry, astronomy, and music. Greek
520 BC Panini Forerunner of the modern formal language theory, notation analogous to modern Backus-Naur Form used to specify the syntax of computer languages. Indian
499 BC Anaxagoras of Clazomenae Proposition that the Moon reflects light from the “red-hot stone” which was the Sun; first understanding of centrifugal force; first known trials of squaring the circle with ruler and compasses (which was proven impossible not before 1882. Ionian
492 BC Empedocles of Acragas Four element theory of the world: fire, air, water, and earth; beginnings of empiric science: experiment showing that air exists and is not just empty space by observing that water did not enter a vessel when placed under water. Greek
490 BC Oenopides of Chios First estimation of the period after which the motions of the sun and moon came to repeat themselves to 59 years. Development of a theory for the Nile floods. Greek
490 BC Zeno of Elea Book containing forty paradoxes concerning the continuum (“paradoxes of motion”), some of which had an influence on the later development of mathematics. Greek
480 BC Leucippus of Miletus Together with Democritus, joint founder of the atomic theory, i.e. the theory that matter and space are not infinitely divisible. Greek
480 BC Antiphon the Sophist First to propose a “method of exhaustion”, i.e. calculating an area by approximating it by the areas of a sequence of polygons. Greek
470 BC Hippocrates of Chios In his attempts to square the circle, Hippocrates was able to find the areas of “lunes”, i.e. crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii. Greek
465 BC Theodorus of Cyrene Contribution to the development of irrational numbers, Theodorus proved that $\sqrt 3, \sqrt 5, \ldots, \sqrt {17}$ were not commensurable in length with the unit length. Greek
460 BC Democritus of Abdera Together with Leucippus, joint founder of the atomic theory, i.e. the theory that matter and space are not infinitely divisible. Greek
460 BC Hippias of Elis Probably, the inventor of “quadratrix” which may have been used by him for trisecting an angle and squaring the circle. Greek
450 BC Bryson of Heraclea Bryson claimed that the circle was greater than all inscribed polygons and less than all circumscribed polygons. Greek
428 BC Archytas of Tarentum Finding two mean proportionals between two line segments; a solution to the problem of duplicating the cube; proof, that there can be no number which is a geometric mean between two numbers in the ratio $\frac{n+1}n.$ Greek
427 BC Plato Main contributions are in philosophy, mathematics, and science. Plato’s name is attached to the Platonic solids representing the “elements” i.e. cube (=earth), tetrahedron (=fire), octahedron (=air), icosahedron (=water). Plato associated the dodecahedron with the whole universe. Greek
417 BC Theaetetus of Athens Greek
408 BC Eudoxus of Cnidus theory of proportion, astronomy, exhaustion method Greek
400 BC Gan De
400 BC Thymaridas of Paros Greek
396 BC Xenocrates of Chalcedon Greek
390 BC Dinostratus Greek
387 BC Heraclides of Pontus Greek
384 BC Aristotle Beginnings of Propositional Logic Greek
380 BC Menaechmus Greek
370 BC Callippus of Cyzicus Greek
370 BC Aristaeus the Elder Greek
360 BC Autolycus of Pitane
350 BC Eudemus of Rhodes
325 BC Euclid of Alexandria
310 BC Aristarchus of Samos
287 BC Archimedes of Syracuse Greek
280 BC Chrysippus of Soli Greek
280 BC Conon of Samos
280 BC Nicomedes
280 BC Philon of Byzantium
276 BC Eratosthenes of Cyrene
262 BC Apollonius of Perga
250 BC Dionysodorus
240 BC Diocles of Carystus
200 BC Katyayana
200 BC Zenodorus
190 BC Hipparchus of Rhodes
190 BC Hypsicles of Alexandria
160 BC Theodosius of Bithynia
150 BC Zeno of Sidon
135 BC Posidonius of Rhodes
130 BC Luoxia Hong
85 BC Marcus Vitruvius Pollio
10 BC Geminus
0 BC Hippasus of Metapontum

| | | | created: 2016-08-22 23:25:49 | modified: 2019-07-04 08:35:11 | by: bookofproofs | references: [8244]

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### Bibliography (further reading)

[8244] Struik, D.J.: “Abriss der Geschichte der Mathematik”, Studienbücherei, 1976