The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. This article discounts mere speculation as discovery, although imperfect reasoned arguments, arguments based on elegance/simplicity, and numerically/experimentally verified conjectures qualify (as otherwise no scientific discovery before the late 19th century would count). The timeline begins at the Bronze Age, as it is difficult to give even estimates for the timing of events prior to this, such as of the discovery of counting, natural numbers and arithmetic.
To avoid overlap with timeline of historic inventions, the timeline does not list examples of documentation for manufactured substances and devices unless they reveal a more fundamental leap in the theoretical ideas in a field.
Bronze Age Many early innovations of the Bronze Age were prompted by the increase in trade, and this also applies to the scientific advances of this period. For context, the major civilizations of this period are Egypt, Mesopotamia, and the Indus Valley, with Greece rising in importance towards the end of the third millennium BC. The Indus Valley script remains undeciphered and there are very little surviving fragments of its writing, thus any inference about scientific discoveries in that region must be made based only on archaeological digs. The following dates are approximations.
3000 BC: Units of measurement are developed in the Americas as well as the major Bronze Age civilizations: Egypt, Mesopotamia, Elam and the Indus Valley. 3000 BC: The first deciphered numeral system is that of the Egyptian numerals, a sign-value system (as opposed to a place-value system). 2650 BC: The oldest extant record of a unit of length, the cubit-rod ruler, is from Nippur. 2600 BC: The oldest attested evidence for the existence of units of weight, and weighing scales date to the Fourth Dynasty of Egypt, with Deben (unit) balance weights, excavated from the reign of Sneferu, though earlier usage has been proposed. 2100 BC: The concept of area is first recognized in Babylonian clay tablets, and 3-dimensional volume is discussed in an Egyptian papyrus. This begins the study of geometry. 2100 BC: Quadratic equations, in the form of problems relating the areas and sides of rectangles, are solved by Babylonians. 2000 BC: Pythagorean triples are first discussed in Babylon and Egypt, and appear on later manuscripts such as the Berlin Papyrus 6619.Richard J. Gillings, Mathematics in the Time of the Pharaohs, Dover, New York, 1982, 161. 2000 BC: Multiplication tables in a base-60, rather than base-10 (decimal), system from Babylon. 2000 BC: Primitive positional notation for numerals is seen in the Babylonian cuneiform numerals. However, the lack of clarity around the notion of zero made their system highly ambiguous (e.g. would be written the same as ). Early 2nd millennium BC: Similar triangles and side-ratios are studied in Egypt for the construction of pyramids, paving the way for the field of trigonometry. Early 2nd millennium BC: Ancient Egyptians study anatomy, as recorded in the Edwin Smith Papyrus. They identified the heart and its vessels, liver, spleen, kidneys, hypothalamus, uterus, and bladder, and correctly identified that blood vessels emanated from the heart (however, they also believed that tears, urine, and semen, but not saliva and sweat, originated in the heart, see Cardiocentric hypothesis). 1800 BC: The Middle Kingdom of Egypt develops Egyptian fraction notation. 1800 BC – 1600 BC: A numerical approximation for the square root of two, accurate to 6 decimal places, is recorded on YBC 7289, a Babylonian clay tablet believed to belong to a student. 1800 BC – 1600 BC: A Babylonian tablet uses = 3.125 as an approximation for , which has an error of 0.5%. 1550 BC: The Rhind Mathematical Papyrus (a copy of an older Middle Kingdom text) contains the first documented instance of inscribing a polygon (in this case, an octagon) into a circle to estimate the value of .
Iron Age The following dates are approximations. 700 BC: Pythagoras's theorem is discovered by Baudhayana in the Hindu Shulba Sutras in Upanishadic India. However, Indian mathematics, especially North Indian mathematics, generally did not have a tradition of communicating proofs, and it is not fully certain that Baudhayana or Apastamba knew of a proof. 700 BC: Pell's equations are first studied by Baudhayana in India, the first diophantine equations known to be studied. 700 BC: Grammar is first studied in India (note that Sanskrit Vyākaraṇa predates Pāṇini). 600 BC: Thales of Miletus is credited with proving Thales's theorem. 600 BC – 200 BC: The Sushruta Samhita shows an understanding of musculoskeletal structure (including joints, ligaments and muscles and their functions) (3.V). Alt URL It refers to the cardiovascular system as a closed circuit. In (3.IX) it identifies the existence of nerves.
500 BC – 1 BC The following dates are approximations. 500 BC: Hippasus, a Pythagorean, discovers irrational numbers.. 500 BC: Anaxagoras identifies moonlight as reflected sunlight. 5th century BC: The Greeks start experimenting with straightedge-and-compass constructions.Bold, Benjamin. Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982 (orig. 1969). 5th century BC: The earliest documented mention of a spherical Earth comes from the Greeks in the 5th century BC. 460 BC: Empedocles describes thermal expansion. Late 5th century BC: Antiphon discovers the method of exhaustion, foreshadowing the concept of a limit. 4th century BC: Greek philosophers study the properties of logical negation. 4th century BC: The first true formal system is constructed by Pāṇini in his Sanskrit grammar.Bhate, S. and Kak, S. (1993) Panini and Computer Science. Annals of the Bhandarkar Oriental Research Institute, vol. 72, pp. 79-94. 4th century BC: Eudoxus of Cnidus states the Archimedean property. 4th century BC: Thaetetus shows that square roots are either integer or irrational. 4th century BC: Thaetetus enumerates the Platonic solids, an early work in graph theory. 4th century BC: Menaechmus discovers conic sections.. "It was consequently a signal achievement on the part of Menaechmus when he disclosed that curves having the desired property were near at hand. In fact, there was a family of appropriate curves obtained from a single source – the cutting of a right circular cone by a plane perpendicular to an element of the cone. That is, Menaechmus is reputed to have discovered the curves that were later known as the ellipse, the parabola, and the hyperbola. [...] Yet the first discovery of the ellipse seems to have been made by Menaechmus as a mere by-product in a search in which it was the parabola and hyperbola that proffered the properties needed in the solution of the Delian problem." 4th century BC: Menaechmus develops co-ordinate geometry.. "Menaechmus apparently derived these properties of the conic sections and others as well. Since this material has a strong resemblance to the use of coordinates, as illustrated above, it has sometimes been maintained that Menaechmus had analytic geometry. Such a judgment is warranted only in part, for certainly Menaechmus was unaware that any equation in two unknown quantities determines a curve. In fact, the general concept of an equation in unknown quantities was alien to Greek thought. It was shortcomings in algebraic notations that, more than anything else, operated against the Greek achievement of a full-fledged coordinate geometry." 4th century BC: Mozi in China gives a description of the camera obscura phenomenon. 4th century BC: Around the time of Aristotle, a more empirically founded system of anatomy is established, based on animal dissection. In particular, Praxagoras makes the distinction between arteries and veins. 4th century BC: Aristotle differentiates between near-sighted and far-sightedness. Graeco-Roman physician Galen would later use the term "myopia" for near-sightedness. 4th century BC: Pāṇini develops a full-fledged formal grammar (for Sanskrit). Late 4th century BC: Chanakya (also known as Kautilya) establishes the field of economics with the Arthashastra (literally "Science of wealth"), a prescriptive treatise on economics and statecraft for Mauryan India. 4th – 3rd century BC: In Mauryan India, The Jain mathematical text Surya Prajnapati draws a distinction between countable and uncountable infinities. 350 BC – 50 BC: Clay tablets from (possibly Hellenistic-era) Babylon describe the mean speed theorem. 300 BC: Finite geometric progressions are studied by Euclid in Ptolemaic Egypt. 300 BC: Euclid proves the infinitude of primes. 300 BC: Euclid proves the Fundamental Theorem of Arithmetic. 300 BC: Euclid discovers the Euclidean algorithm. 300 BC: Euclid publishes the Elements, a compendium on classical Euclidean geometry, including: elementary theorems on circles, definitions of the centers of a triangle, the tangent-secant theorem, the law of sines and the law of cosines.. "Trigonometry, like other branches of mathematics, was not the work of any one man, or nation. Theorems on ratios of the sides of similar triangles had been known to, and used by, the ancient Egyptians and Babylonians. In view of the pre-Hellenic lack of the concept of angle measure, such a study might better be called "trilaterometry", or the measure of three sided polygons (trilaterals), than "trigonometry", the measure of parts of a triangle. With the Greeks we first find a systematic study of relationships between angles (or arcs) in a circle and the lengths of chords subtending these. Properties of chords, as measures of central and inscribed angles in circles, were familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had used ratios and angle measures in determining the size of the earth and the relative distances of the sun and the moon. In the works of Euclid there is no trigonometry in the strict sense of the word, but there are theorems equivalent to specific trigonometric laws or formulas. Propositions II.12 and 13 of the Elements, for example, are the laws of cosines for obtuse and acute angles respectively, stated in geometric rather than trigonometric language and proved by a method similar to that used by Euclid in connection with the Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the broken chord can readily be translated into trigonometric language analogous to formulas for sines of sums and differences of angles." 300 BC: Euclid's Optics introduces the field of geometric optics, making basic considerations on the sizes of images. 3rd century BC: Archimedes relates problems in geometric series to those in arithmetic series, foreshadowing the logarithm.Ian Bruce (2000) "Napier’s Logarithms", American Journal of Physics 68(2):148 3rd century BC: Pingala in Mauryan India studies binary numbers, making him the first to study the radix (numerical base) in history. 3rd century BC: Pingala in Mauryan India describes the Fibonacci sequence. 3rd century BC: Pingala in Mauryan India discovers the binomial coefficients in a combinatorial context and the additive formula for generating them ,A. W. F. Edwards. Pascal's arithmetical triangle: the story of a mathematical idea. JHU Press, 2002. Pages 30–31. i.e. a prose description of Pascal's triangle, and derived formulae relating to the sums and alternating sums of binomial coefficients. It has been suggested that he may have also discovered the binomial theorem in this context. 3rd century BC: Eratosthenes discovers the Sieve of Eratosthenes. 3rd century BC: Archimedes derives a formula for the volume of a sphere in The Method of Mechanical Theorems.