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 commona 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.

- The Ishango Bone as a possibly the earliest known counting evidence

- Clay tokens used in the Middle East for accounting

- Sumerian mathematics
- Place-value sexagesimal system, numerical notation

- Megalithic stone and wood settings
- Found mainly on the British Isles; following astronomic and geometric principles.
- The most famous example is Stonehenge. The settings form circles, ellipses, egg-shaped rings or more complicated figures.

- Found mainly on the British Isles; following astronomic and geometric principles.

- Babylonian Mathematics
- Solving linear, quadratic and some cubic equations, square roots, interpolation of logarithms
* First occurrence of theorem, later known as
*Pythagoras' theorem** Babylonians did not formulate any proofs for the theorems they stated.

- Solving linear, quadratic and some cubic equations, square roots, interpolation of logarithms
* First occurrence of theorem, later known as

- Mathematical Rhind Papyrus written by Ahmes as an example of egyptian mathematics

- Beginnings of Ancient Chinese Mathematics
- Decimal numeration carved on oracle bones

- Thales Of Miletus formulates first theorems in geometry ("father of geometry")
- Probably first to predict an eclipse of the Sun in 585 BC.
- The first natural philosopher in the Milesian School.
- Probably the teacher of Anaximander

- The Pythagoreans formulate interconnections between numbers, geometry, astronomy, and music.
- Influenced by travels to Egypt, and southern Italy.

- Anaxagoras is first to introduce philosophy to the Athenians.
- In about 450 BC imprisoned for claiming that the Sun was not a god and that the Moon reflected the Sun's light.

- Empedocles Of Acragas formulates a four element theory of the worldfire, air, water, and earth. He also conducts on of the first
**empiric science experiments**showing that air exists and is not just empty space by observing that water did not enter a vessel when placed underwater. - First estimation of the period after which the motions of the sun and moon came to repeat themselves to 59 years by Oenopides Of Chios. Development of a theory for the Nile floods.
- Zeno Of Elea formulates paradoxes concerning the continuum ("paradoxes of motion"), some of which had an influence on the later development of mathematics.
- Influenced by the arguments of Parmenides and Plato he met in Athens.

- Beginnings of atomic theory by Leucippus Of Miletus, i.e. the theory that matter and space are not infinitely divisible.
- First proposal of the
**method of exhaustion**by Antiphon, i.e. calculating an area by approximating it by the areas of a sequence of polygons. - Finding the areas of "lunes" by Hippocrates Of Chios, 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.

- Contribution to the development of irrational numbers, Theodorus Of Cyrene proved that $\sqrt 3, \sqrt 5, \ldots, \sqrt {17}$ were not commensurable in length with the unit length.
- Further development of the atomic theory by Democritus Of Abdera i.e. the theory that matter and space are not infinitely divisible.
- Invention of "quadratrix" by Hippias Of Elis which may have been used by him for trisecting an angle and squaring the circle.

- First ideas towards the infinitesimal calculus: Bryson Of Heraclea claimed that the circle was greater than all inscribed polygons and less than all circumscribed polygons.

- Plato creates a new cosmology: he's name is attached to the Platonic solids representing the "elements of the universe" i.e. cube (=earth), tetrahedron (=fire), octahedron (=air), icosahedron (=water). Plato associated the dodecahedron with the whole universe.
- Plato founds his Academy in Athens, devoted to research and instruction in philosophy and the sciences
- Aristotle, Heraclides, and Xenocrates to join the Academy.

- Archytas Of Tarentum finds
- two mean proportionals between two line segments,
- a solution to the problem of duplicating the cube
- a proof that there can be no number which is a geometric mean between two numbers in the ratio $\frac{n+1}n.$.

- Further developments in the theory of proportion, astronomy, exhaustion method by Eudoxus Of Cnidus.
- Anaxagoras Of Clazomenae formulates a first understanding of centrifugal force. He also formulates first known trials of squaring the circle with ruler and compasses (which was proven impossible not before 1882).

- Beginnings of propositional logic by Aristotle
- Plato's nephew Speusippus becomes head of the Academy on Plato's death, but in 340 BC he sent for Xenocrates to return to Athens to prepare to become his successor.

- Aristotle founds his own school: the Lyceum in Athens.
- Aristarchus to join the Lyceum later

- Callippus
- Mades accurate determinations of the lengths of the seasons
- Constructs a 76 year cycle comprising 940 months to harmonise the solar and lunar years
- This calendar adopted in 330 BC and used by many later astronomers.

- Eudemus writes "History of Arithmetic" (two or more books), "History of Geometry" (two or more books), and "History of Astronomy" (two or more books).

- Philon writes his treatise Mechanics.
- Euclid Euclid of Alexandria.
- The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 years earlier.

- Eudoxus writes a book on geography called "Tour of the Earth".

- Eratosthenes to become the third librarian at Alexandria.

- Chrysippus to become the third head of the Stoa Poikile following the death of Cleanthes.

- Archimedes killed during the capture of Syracuse by the Romans in the Second Punic War after all his efforts to keep the Romans at bay with his machines of war had failed.

- Nicomedes criticises the method that Eratosthenes used to duplicate the cube.

- Hipparchus produces a star catalogue containing about 850 stars, probably not listed in a systematic coordinate system but using various different ways to designate the position of a star.

- Posidonius to become the head of the Stoic School in Rhodes.
- Using a distance of 5000 stadia between Rhodes and Alexandria, Posidonius estimates a value of 240000 stadia for the circumference of the Earth.

- Zeno of Sidon first to show about 250 years after Euclid that Euclid's propositions were not deduced from the postulates and axioms alone, and Euclid does make other subtle assumptions.

- Person: Ahmes
- Person: Baudhayana
- Person: Manava
- Person: Thales Of Miletus
- Person: Anaximander Of Miletus
- Person: Apastamba
- Person: Pythagoras Of Samos
- Person: Panini
- Person: Anaxagoras Of Clazomenae
- Person: Empedocles Of Acragas
- Person: Oenopides Of Chios
- Person: Zeno Of Elea
- Person: Antiphon
- Person: Leucippus Of Miletus
- Person: Hippocrates Of Chios
- Person: Theodorus Of Cyrene
- Person: Democritus Of Abdera
- Person: Hippias Of Elis
- Person: Bryson Of Heraclea
- Person: Archytas Of Tarentum
- Person: Plato
- Person: Theaetetus Of Athens
- Person: Eudoxus Of Cnidus
- Person: Gan De
- Person: Thymaridas Of Paros
- Person: Xenocrates Of Chalcedon
- Person: Dinostratus
- Person: Heraclides Of Pontus
- Person: Aristotle
- Person: Menaechmus
- Person: Aristaeus the Elder
- Person: Callippus Of Cyzicus
- Person: Autolycus Of Pitane
- Person: Eudemus Of Rhodes
- Person: Euclid Of Alexandria
- Person: Aristarchus Of Samos
- Person: Archimedes Of Syracuse
- Person: Chrysippus Of Soli
- Person: Conon Of Samos
- Person: Nicomedes
- Person: Philon Of Byzantium
- Person: Eratosthenes Of Cyrene
- Person: Apollonius Of Perga
- Person: Dionysodorus
- Person: Diocles Of Carystus
- Person: Katyayana
- Person: Zenodorus
- Person: Hipparchus Of Rhodes
- Person: Hypsicles Of Alexandria
- Person: Perseus
- Person: Theodosius Of Bithynia
- Person: Zeno Of Sidon
- Person: Posidonius Of Rhodes
- Person: Hong, Luoxia
- Person: Pollio, Marcus Vitruvius
- Person: Geminus
- Person: Hippasus of Metapontum
- Topic: Egyptian Mathematics
- Topic: Babylonian Mathematics
- Topic: Ancient Chinese Mathematics
- Topic: Indian Mathematics
- Topic: Sumerian Mathematics

**Struik, D.J.**: "Abriss der Geschichte der Mathematik", Studienbücherei, 1976

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive