This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.
Rhetorical stage Before 1000 BC ca. 70,000 BC – South Africa, ochre rocks adorned with scratched geometric patterns (see Blombos Cave).Art Prehistory, Sean Henahan, January 10, 2002. ca. 35,000 BC to 20,000 BC – Africa and France, earliest known prehistoric attempts to quantify time (see Lebombo bone).How Menstruation Created Mathematics, Tacoma Community College, (archive link). c. 20,000 BC – Nile Valley, Ishango bone: possibly the earliest reference to prime numbers and Egyptian multiplication. c. 3400 BC – Mesopotamia, the Sumerians invent the first numeral system, and a system of weights and measures. c. 3100 BC – Egypt, the earliest known decimal system allows indefinite counting by way of introducing new symbols. c. 2800 BC – Indus Valley Civilization on the Indian subcontinent, earliest use of decimal ratios in a uniform system of ancient weights and measures; the smallest unit of measurement used is 1.704 millimetres, and the smallest unit of mass used is 28 grams. 2700 BC – Egypt, precision surveying. 2400 BC – Egypt, precise astronomical calendar, used even in the Middle Ages for its mathematical regularity. c. 2000 BC – Mesopotamia, the Babylonians use a base-60 positional numeral system and compute the first known approximate value of π at 3.125. c. 2000 BC – Scotland, carved stone balls exhibit a variety of symmetries, including all of the symmetries of Platonic solids, though it is not known if this was deliberate. c. 1800 BC – The Plimpton 322 Babylonian tablet records the oldest known examples of Pythagorean triples. and 1800 BC – Egypt, Moscow Mathematical Papyrus, finding the volume of a frustum. c. 1800 BC – Berlin Papyrus 6619 (Egypt, 19th dynasty) contains a quadratic equation and its solution. 1650 BC – Rhind Mathematical Papyrus, a copy of a lost scroll from around 1850 BC, the scribe Ahmes presents one of the first known approximate values of π at 3.16, the first attempt at squaring the circle, the earliest known use of a sort of cotangent, and knowledge of solving first-order linear equations. The earliest recorded use of combinatorial techniques comes from problem 79 of the Rhind papyrus, which dates to the 16th century BCE.
Syncopated stage
1st millennium BC c. 1000 BC – Simple fractions used by the Egyptians. However, only unit fractions are used (i.e., those with 1 as the numerator), and interpolation tables are used to approximate the values of the other fractions.Carl B. Boyer, A History of Mathematics, 2nd Ed. first half of 1st millennium BC – Vedic India – Yajnavalkya, in his Shatapatha Brahmana, describes the motions of the Sun and the Moon and advances a 95-year cycle to synchronize the motions of the Sun and the Moon. c. 800 BC – Baudhayana, author of the Baudhayana Shulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, calculates the square root of two correctly to five decimal places, and contains "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."* c. 8th century BC – the Yajurveda, one of the four Hindu Vedas, contains the earliest concept of infinity and states, "If you remove a part from infinity or add a part to infinity, still what remains is infinity." 1046 BC to 256 BC – China, Zhoubi Suanjing, arithmetic, geometric algorithms, and proofs. 624 BC – 546 BC – Greece, Thales of Miletus has various theorems attributed to him. c. 600 BC – India, the other Vedic "Shulba Sutras" ("rule of chords" in Sanskrit) use Pythagorean triples, contain a number of geometrical proofs, and approximate π at 3.16. second half of 1st millennium BC – The Luoshu Square, the unique normal magic square of order three, was discovered in China. 530 BC – Greece, Pythagoras studies propositional geometry and vibrating lyre strings; his group also discovers the irrationality of the square root of two. c. 510 BC – Greece, Anaxagoras c. 500 BC – Indian grammarian Pāṇini writes the Aṣṭādhyāyī, which contains the use of metarules, transformations, and recursions, originally for the purpose of systematizing the grammar of Sanskrit. c. 500 BC – Greece, Oenopides of Chios 470 BC – 410 BC – Greece, Hippocrates of Chios utilizes lunes in an attempt to square the circle. 490 BC – 430 BC – Greece, Zeno of Elea creates Zeno's paradoxes. 5th century BC – India, Apastamba, author of the Apastamba Shulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places. 5th c. BC – Greece, Theodorus of Cyrene 5th century – Greece, Antiphon the Sophist 460 BC – 370 BC – Greece, Democritus 460 BC – 399 BC – Greece, Hippias 5th century (late) – Greece, Bryson of Heraclea 428 BC – 347 BC – Greece, Archytas 423 BC – 347 BC – Greece, Plato 417 BC – 317 BC – Greece, Theaetetus c. 400 BC – India, write the Surya Prajnapti, a mathematical text classifying all numbers into three sets: enumerable, innumerable, and infinite. It also recognizes five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. 408 BC – 355 BC – Greece, Eudoxus of Cnidus 400 BC – 350 BC – Greece, Thymaridas 395 BC – 313 BC – Greece, Xenocrates 390 BC – 320 BC – Greece, Dinostratus 380–290 – Greece, Autolycus of Pitane 370 BC – Greece, Eudoxus states the method of exhaustion for area determination. 370 BC – 300 BC – Greece, Aristaeus the Elder 370 BC – 300 BC – Greece, Callippus 350 BC – Greece, Aristotle discusses logical reasoning in Organon. 4th century BC – China, Counting rods 330 BC – China, the earliest known work on Chinese geometry, the Mo Jing, is compiled. 310 BC – 230 BC – Greece, Aristarchus of Samos 390 BC – 310 BC – Greece, Heraclides Ponticus 380 BC – 320 BC – Greece, Menaechmus 300 BC – India, Bhagavati Sutra, which contains the earliest information on combinations. 300 BC – Greece, Euclid, in his Elements studies geometry as an axiomatic system, proves the infinitude of prime numbers, and presents the Euclidean algorithm; he states the law of reflection in Catoptrics, and he proves the fundamental theorem of arithmetic. c. 300 BC – India, Brahmi numerals (ancestor of the common modern base 10 numeral system) 370 BC – 300 BC – Greece, Eudemus of Rhodes works on histories of arithmetic, geometry, and astronomy now lost. 300 BC – Mesopotamia, the Babylonians invent the earliest calculator, the abacus. c. 300 BC – Indian mathematician Pingala writes the Chandaḥśāstra, which contains the first Indian use of zero as a digit (indicated by a dot) and also presents a description of a binary numeral system, along with the first use of Fibonacci numbers and Pascal's triangle. 3rd century BC – Indian texts use the Sanskrit word "Shunya" to refer to the concept of "void" (zero). 280 BC – 210 BC – Greece, Nicomedes (mathematician) 280 BC – 220BC – Greece, Philo of Byzantium 280 BC – 220 BC – Greece, Conon of Samos 279 BC – 206 BC – Greece, Chrysippus c. 3rd century BC – India, Kātyāyana 250 BC – 190 BC – Greece, Dionysodorus 262 -198 BC – Greece, Apollonius of Perga 260 BC – Greece, Archimedes proved that the value of π lies between 3 + 1/7 (approx. 3.1429) and 3 + 10/71 (approx. 3.1408), that the area of a circle was equal to π multiplied by the square of the radius of the circle, and that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He also gave a very accurate estimate of the value of the square root of 3. c. 250 BC – late Olmecs had already begun to use a true zero (a shell glyph) several centuries before Ptolemy in the New World. See 0 (number). 240 BC – Greece, Eratosthenes uses his sieve algorithm to quickly isolate prime numbers. 240 BC–190 BC– Greece, Diocles (mathematician) 225 BC – Greece, Apollonius of Perga writes On Conic Sections and names the ellipse, parabola, and hyperbola. 202 BC to 186 BC –China, Book on Numbers and Computation, a mathematical treatise, is written in the Han dynasty. 200 BC – 140 BC – Greece, Zenodorus (mathematician) 150 BC – India, Jain mathematicians in India write the Sthananga Sutra, which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. c. 150 BC – Greece, Perseus (geometer) 150 BC – China, A method of Gaussian elimination appears in the Chinese text The Nine Chapters on the Mathematical Art. 150 BC – China, Horner's method appears in the Chinese text The Nine Chapters on the Mathematical Art. 150 BC – China, negative numbers appear in the Chinese text The Nine Chapters on the Mathematical Art. 150 BC – 75 BC – Phoenician, Zeno of Sidon 190 BC – 120 BC – Greece, Hipparchus develops the bases of trigonometry. 190 BC – 120 BC – Greece, Hypsicles 160 BC – 100 BC – Greece, Theodosius of Bithynia 135 BC – 51 BC – Greece, Posidonius 78 BC – 37 BC – China, Jing Fang 50 BC – Indian numerals, a descendant of the Brahmi numerals (the first positional notation base-10 numeral system), begin development in India. mid-1st century Cleomedes (as late as 400 AD) final centuries BC – Indian astronomer Lagadha writes the Vedanga Jyotisha, a Vedic text on astronomy that describes rules for tracking the motions of the Sun and the Moon and uses geometry and trigonometry for astronomy. 1st C. BC – Greece, Geminus 50 BC – 23 AD – China, Liu Xin
1st millennium AD 1st century—Greece, Heron of Alexandria, Hero, the earliest, fleeting reference to square roots of negative numbers. c 100—Greece, Theon of Smyrna 60–120—Greece, Nicomachus 70–140—Greece, Menelaus of Alexandria, Spherical trigonometry 78–139—China, Zhang Heng c. 2nd century—In Greece, Ptolemy of Alexandria wrote the Almagest. 132–192—China, Cai Yong 240–300—Greece, Sporus of Nicaea 250—In Greece, Diophantus uses symbols for unknown numbers in terms of syncopated algebra and writes Arithmetica, one of the earliest treatises on algebra. 263—China, Liu Hui computes π using Liu Hui's π algorithm. 300 – the earliest known use of zero as a decimal digit is introduced by Indian mathematicians. 234–305—Greece, Porphyry (philosopher) 300–360—Greece, Serenus of Antinoöpolis 335–405—Greece, Theon of Alexandria c. 340—Greece, Pappus of Alexandria states his hexagon theorem and his centroid theorem. 350–415—Eastern Roman Empire, Hypatia c. 400—India, the Bakhshali manuscript, which describes a theory of the infinite containing different levels of infinity, shows an understanding of indices, as well as logarithms to base 2, and computes square roots of numbers as large as a million correct to at least 11 decimal places. 300–500—The Chinese remainder theorem is developed by Sun Tzu. 300–500—In China, a description of rod calculus is written by Sun Tzu. 412–485—Greece, Proclus 420–480—Greece, Domninus of Larissa b 440—Greece, Marinus of Neapolis "I wish everything was mathematics." 450—In China, Zu Chongzhi computes π to seven decimal places. This calculation remained the most accurate calculation for π for close to a thousand years. c. 474–558—Greece, Anthemius of Tralles