Ancient World from 4000 BC to 1 BC

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