The Babylonian sexagesimal number system is used to record and predict the positions of the Sun, Moon and planets. (See this History Topic.)
Panini's work on Sanskrit grammar is the forerunner of the modern formal language theory.
Hippasus writes of a "sphere of 12 pentagons", which must refer to a dodecahedron.
Hippocrates of Chios writes the Elements which is the first compilation of the elements of geometry.
Babylonians use a symbol to indicate an empty place in their numbers recorded in cuneiform writing. There is no indication that this was in any way thought of as a number. (See this History Topic.)
Plato founds his Academy in Athens
Autolycus of Pitane writes On the Moving Sphere which studies the geometry of the sphere. It is written as an astronomy text.
Euclid gives a systematic development of geometry in his Stoicheion (The Elements). He also gives the laws of reflection in Catoptrics.
Aristarchus of Samos uses a geometric method to calculate the distance of the Sun and the Moon from Earth. He also proposes that the Earth orbits the Sun.
In On the Sphere and the Cylinder, Archimedes gives the formulae for calculating the volume of a sphere and a cylinder. In Measurement of the Circle he gives an approximation of the value of π with a method which will allow improved approximations. In Floating Bodies he presents what is now called "Archimedes' principle" and begins the study of hydrostatics. He writes works on two- and three-dimensional geometry, studying circles, spheres and spirals. His ideas are far ahead of his contemporaries and include applications of an early form of integration.
Eratosthenes of Cyrene estimates the Earth's circumference with remarkable accuracy finding a value which is about 15% too big.
Diocles writes On burning mirrors, a collection of sixteen propositions in geometry mostly proving results on conics.
Possible earliest date for the classic Chinese work Jiuzhang suanshu or Nine Chapters on the Mathematical Art. (See this History Topic.)
Date of earliest Chinese document Suanshu shu (A Book on Arithmetic). (See this History Topic.)
Hypsicles writes On the Ascension of Stars. In this work he is the first to divide the Zodiac into 360 degrees.
Hipparchus discovers the precession of the equinoxes and calculates the length of the year to within 6.5 minutes of the correct value. His astronomical work uses an early form of trigonometry.