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Timeline of scientific discoveries

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See also: Timeline of historic inventions

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.

The Nippur cubit-rod, c. 2650 BCE, in the Archeological Museum of Istanbul, Turke
  • 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.[1][2]
  • 3000 BC: The first deciphered numeral system is that of the Egyptian numerals, a sign-value system (as opposed to a place-value system).[3]
  • 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.[4]
  • 2100 BC: The concept of area is first recognized in Babylonian clay tablets,[5] 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.[5]
  • 2000 BC: Pythagorean triples are first discussed in Babylon and Egypt, and appear on later manuscripts such as the Berlin Papyrus 6619.[6]
  • 2000 BC: Multiplication tables in Babylon.[7]
  • 2000 BC: Primitive positional notation for numerals is seen in the Babylonian cuneiform numerals.[8] However, the lack of clarity around the notion of zero made their system highly ambiguous (e.g. 13200 would be written the same as 132).[9]
  • 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.[10]
  • 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).[11]
  • 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.[12]
  • 1800 BC - 1600 BC: A Babylonian tablet uses 25⁄8 = 3.125 as an approximation for π, which has an error of 0.5%.[13][14][15]
  • 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 π.[16][17]

Iron Age

The following dates are approximations.

  • 600 BC - 200 BC: The Sushruta Samhita shows an understanding of musculoskeletal structure (including joints, ligaments and muscles and their functions) (3.V).[21] It refers to the cardiovascular system as a closed circuit.[22] In (3.IX) it identifies the existence of nerves.[21]

500 BC – 1 BC

The following dates are approximations.

  • 500 BC: Hippasus, a Pythagorean, discovers irrational numbers.[23][24]
  • 500 BC: Anaxagoras identifies moonlight as reflected sunlight.[25]
  • 5th century BC: The Greeks start experimenting with straightedge-and-compass constructions.[26]
  • 5th century BC: The earliest documented mention of a spherical Earth comes from the Greeks in the 5th century BC.[27] It is known that the Indians modeled the Earth as spherical by 300 BC[28]
  • 460 BC: Empedocles describes thermal expansion.[29]
  • 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.[30][31]
  • 4th century BC: Eudoxus of Cnidus states the Archimedean property.[32]
  • 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.[33]
  • 4th century BC: Menaechmus develops co-ordinate geometry.[34]
  • 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.[35] Graeco-Roman physician Galen would later use the term "myopia" for near-sightedness.
    Pāṇini's Aṣṭādhyāyī, an early Indian grammatical treatise that constructs a formal system for the purpose of describing Sanskrit grammar.
  • 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.[36]
  • 4th - 3rd century BC: In Mauryan India, The Jain mathematical text Surya Prajnapati draws a distinction between countable and uncountable infinities.[37]
  • 350 BC - 50 BC: Clay tablets from (possibly Hellenistic-era) Babylon describe the mean speed theorem.[38]
  • 300 BC: Greek mathematician Euclid in the Elements describes a primitive form of formal proof and axiomatic systems. However, modern mathematicians generally believe that his axioms were highly incomplete, and that his definitions were not really used in his proofs.
  • 300 BC: Finite geometric progressions are studied by Euclid in Ptolemaic Egypt.[39]
  • 300 BC: Euclid proves the infinitude of primes.[40]
  • 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.[41]
  • 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.[42]
  • 3rd century BC: Pingala in Mauryan India studies binary numbers, making him the first to study the radix (numerical base) in history.[43]
  • 3rd century BC: Pingala in Mauryan India describes the Fibonacci sequence.[44][45]
  • 3rd century BC: Pingala in Mauryan India discovers the binomial coefficients in a combinatorial context and the additive formula for generating them ,[46][47] 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.[48]
  • 3rd century BC: Eratosthenes discovers the Sieve of Eratosthenes.[49]
  • 3rd century BC: Archimedes derives a formula for the volume of a sphere in The Method of Mechanical Theorems.[50]
  • 3rd century BC: Archimedes calculates areas and volumes relating to conic sections, such as the area bounded between a parabola and a chord, and various volumes of revolution.[51]
  • 3rd century BC: Archimedes discovers the sum/difference identity for trigonometric functions in the form of the "Theorem of Broken Chords".[41]
  • 3rd century BC: Archimedes makes use of infinitesimals.[52]
  • 3rd century BC: Archimedes further develops the method of exhaustion into an early description of integration.[53][54]
  • 3rd century BC: Archimedes calculates tangents to non-trigonometric curves.[55]
  • 3rd century BC: Archimedes uses the method of exhaustion to construct a strict inequality bounding the value of π within an interval of 0.002.
  • 3rd century BC: Archimedes develops the field of statics, introducing notions such as the center of gravity, mechanical equilibrium, the study of levers, and hydrostatics.
  • 3rd century BC: Eratosthenes measures the circumference of the Earth.[56]
  • 260 BC: Aristarchus of Samos proposes a basic heliocentric model of the universe.[57]
  • 200 BC: Apollonius of Perga discovers Apollonius's theorem.
  • 200 BC: Apollonius of Perga assigns equations to curves.
  • 200 BC: Apollonius of Perga develops epicycles. While an incorrect model, it was a precursor to the development of Fourier series.
  • 2nd century BC: Hipparchos discovers the apsidal precession of the Moon's orbit.[58]
  • 2nd century BC: Hipparchos discovers Axial precession.
  • 2nd century BC: Hipparchos measures the sizes of and distances to the Moon and Sun.[59]
  • 190 BC: Magic squares appear in China. The theory of magic squares can be considered the first example of a vector space.
  • 165 BC - 142 BC: Zhang Cang in Northern China is credited with the development of Gaussian elimination.[60]

1 AD – 500 AD

Mathematics and astronomy flourish during the Golden Age of India (4th to 6th centuries AD) under the Gupta Empire. Meanwhile, Greece and its colonies have entered the Roman period in the last few decades of the preceding millennium, and Greek science is negatively impacted by the Fall of the Western Roman Empire and the economic decline that follows.

  • 1st to 4th century: A precursor to long division, known as "galley division" is developed at some point. Its discovery is generally believed to have originated in India around the 4th century AD,[61] although Singaporean mathematician Lam Lay Yong claims that the method is found in the Chinese text The Nine Chapters on the Mathematical Art, from the 1st century AD.[62]
  • 60 AD: Heron's formula is discovered by Hero of Alexandria.[63]
  • 2nd century: Ptolemy formalises the epicycles of Apollonius.
  • 2nd century: Ptolemy publishes his Optics, discussing colour, reflection, and refraction of light, and including the first known table of refractive angles.
  • 2nd century: Galen studies the anatomy of pigs.[64]
  • 100: Menelaus of Alexandria describes spherical triangles, a precursor to non-Euclidean geometry.[65]
  • 150: The Almagest of Ptolemy contains evidence of the Hellenistic zero. Unlike the earlier Babylonian zero, the Hellenistic zero could be used alone, or at the end of a number. However, it was usually used in the fractional part of a numeral, and was not regarded as a true arithmetical number itself.
  • 150: Ptolemy's Almagest contains practical formulae to calculate latitudes and day lengths.
    Diophantus' Arithmetica (pictured: a Latin translation from 1621) contained the first known use of symbolic mathematical notation. Despite the relative decline in the importance of the sciences during the Roman era, several Greek mathematicians continued to flourish in Alexandria.
  • 3rd century: Diophantus discusses linear diophantine equations.
  • 3rd century: Diophantus uses a primitive form of algebraic symbolism, which is quickly forgotten.[66]
  • 210: Negative numbers are accepted as numeric by the late Han-era Chinese text The Nine Chapters on the Mathematical Art.[67] Later, Liu Hui of Cao Wei (during the Three Kingdoms period) writes down laws regarding the arithmetic of negative numbers.[68]
  • By the 4th century: A square root finding algorithm with quartic convergence, known as the Bakhshali method (after the Bakhshali manuscript which records it), is discovered in India.[69]
  • By the 4th century: The present Hindu–Arabic numeral system with place-value numerals develops in Gupta-era India, and is attested in the Bakhshali Manuscript of Gandhara.[70] The superiority of the system over existing place-value and sign-value systems arises from its treatment of zero as an ordinary numeral.
  • 4th to 5th centuries: The modern fundamental trigonometric functions, sine and cosine, are described in the Siddhantas of India.[71] This formulation of trigonometry is an improvement over the earlier Greek functions, in that it lends itself more seamlessly to polar co-ordinates and the later complex interpretation of the trigonometric functions.
  • By the 5th century: The decimal separator is developed in India,[72] as recorded in al-Uqlidisi's later commentary on Indian mathematics.[73]
  • By the 5th century: The elliptical orbits of planets are discovered in India by at least the time of Aryabhata, and are used for the calculations of orbital periods and eclipse timings.[74]
  • By 499: Aryabhata's work shows the use of the modern fraction notation, known as bhinnarasi.[75]
    Fragment of papyrus with clear Greek script, lower-right corner suggests a tiny zero with a double-headed arrow shape above it
    Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
  • 499: Aryabhata gives a new symbol for zero and uses it for the decimal system.
  • 499: Aryabhata discovers the formula for the square-pyramidal numbers (the sums of consecutive square numbers).[76]
  • 499: Aryabhata discovers the formula for the simplicial numbers (the sums of consecutive cube numbers).[76]
  • 499: Aryabhata discovers Bezout's identity, a foundational result to the theory of principal ideal domains.[77]
  • 499: Aryabhata develops Kuṭṭaka, an algorithm very similar to the Extended Euclidean algorithm.[77]
  • 499: Aryabhata describes a numerical algorithm for finding cube roots.[78][79]
  • 499: Aryabhata develops an algorithm to solve the Chinese remainder theorem.[80]
  • 499: Historians speculate that Aryabhata may have used an underlying heliocentric model for his astronomical calculations, which would make it the first computational heliocentric model in history (as opposed to Aristarchus's model in form).[81][82][83] This claim is based on his description of the planetary period about the Sun (śīghrocca), but has been met with criticism.[84]
  • 499: Aryabhata creates a particularly accurate eclipse chart. As an example of its accuracy, 18th century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations (based on Aryabhata's computational paradigm) of the duration of the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[85]

500 AD – 1000 AD

The age of Imperial Karnataka was a period of significant advancement in Indian mathematics.

The Golden Age of Indian mathematics and astronomy continues after the end of the Gupta empire, especially in Southern India during the era of the Rashtrakuta, Western Chalukya and Vijayanagara empires of Karnataka, which variously patronised Hindu and Jain mathematicians. In addition, the Middle East enters the Islamic Golden Age through contact with other civilisations, and China enters a golden period during the Tang and Song dynasties.

1000 AD – 1500 AD

  • 11th century: Alhazen discovers the formula for the simplicial numbers defined as the sums of consecutive quartic powers.[citation needed]
  • 11th century: Alhazen systematically studies optics and refraction, which would later be important in making the connection between geometric (ray) optics and wave theory.
  • 11th century: Shen Kuo discovers atmospheric refraction and provides the correct explanation of rainbow phenomenon[citation needed]
  • 11th century: Shen Kuo discovers the concepts of true north and magnetic declination.
  • 11th century: Shen Kuo develops the field of geomorphology and natural climate change.
  • 1000: Al-Karaji uses mathematical induction.[100]
  • 1058: al-Zarqālī in Islamic Spain discovers the apsidal precession of the Sun.
  • 12th century: Bhāskara II develops the Chakravala method, solving Pell's equation.[101]
  • 12th century: Al-Tusi develops a numerical algorithm to solve cubic equations.
  • 12th century: Jewish polymath Baruch ben Malka in Iraq formulates a qualitative form of Newton's second law for constant forces.[102][103]
  • 1220s: Robert Grosseteste writes on optics, and the production of lenses, while asserting models should be developed from observations, and predictions of those models verified through observation, in a precursor to the scientific method.[104]
  • 1267: Roger Bacon publishes his Opus Majus, compiling translated Classical Greek, and Arabic works on mathematics, optics, and alchemy into a volume, and details his methods for evaluating the theories, particularly those of Ptolemy's 2nd century Optics, and his findings on the production of lenses, asserting “theories supplied by reason should be verified by sensory data, aided by instruments, and corroborated by trustworthy witnesses", in a precursor to the peer reviewed scientific method.
  • 1290: Eyeglasses are invented in Northern Italy,[105] possibly Pisa, demonstrating knowledge of human biology and optics, to offer bespoke works that compensate for an individual human disability.
  • 1295: Scottish priest Duns Scotus writes about the mutual beneficence of trade.[106]
  • 14th century: French priest Jean Buridan provides a basic explanation of the price system.
  • 1380: Madhava of Sangamagrama develops the Taylor series, and derives the Taylor series representation for the sine, cosine and arctangent functions, and uses it to produce the Leibniz series for π.[107]
  • 1380: Madhava of Sangamagrama discusses error terms in infinite series in the context of his infinite series for π.[108]
  • 1380: Madhava of Sangamagrama discovers continued fractions and uses them to solve transcendental equations.[109]
  • 1380: The Kerala school develops convergence tests for infinite series.[107]
  • 1380: Madhava of Sangamagrama solves transcendental equations by iteration.[109]
  • 1380: Madhava of Sangamagrama discovers the most precise estimate of π in the medieval world through his infinite series, a strict inequality with uncertainty 3e-13.
  • 15th century: Parameshvara discovers a formula for the circumradius of a quadrilateral.[110]
  • 1480: Madhava of Sangamagrama found pi and that it was infinite.
  • 1500: Nilakantha Somayaji discovers an infinite series for π.[111][112]
  • 1500: Nilakantha Somayaji develops a model similar to the Tychonic system. His model has been described as mathematically more efficient than the Tychonic system due to correctly considering the equation of the centre and latitudinal motion of Mercury and Venus.[93][113]

16th century

The Scientific Revolution occurs in Europe around this period, greatly accelerating the progress of science and contributing to the rationalization of the natural sciences.

17th century

18th century

1800–1849

1850–1899

1900–1949

1950–1999

21st century

See also: List of years in science § 2000s, and Breakthrough of the Year § Breakthrough of the Year
Further information: 2023 in science and 2024 in science

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  45. ^ Knuth, Donald (1968), The Art of Computer Programming, vol. 1, Addison Wesley, p. 100, ISBN 978-81-7758-754-8, Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed)...
  46. ^ A. W. F. Edwards. Pascal's arithmetical triangle: the story of a mathematical idea. JHU Press, 2002. Pages 30–31.
  47. ^ a b c Edwards, A. W. F. (2013), "The arithmetical triangle", in Wilson, Robin; Watkins, John J. (eds.), Combinatorics: Ancient and Modern, Oxford University Press, pp. 166–180
  48. ^ Amulya Kumar Bag (6 January 1966). "Binomial theorem in Ancient India" (PDF). Indian J. Hist. Sci.: 68–74.
  49. ^ Hoche, Richard, ed. (1866), Nicomachi Geraseni Pythagorei Introductionis arithmeticae libri II, Leipzig: B.G. Teubner, p. 31
  50. ^ Archimedes (1912), The method of Archimedes recently discovered by Heiberg; a supplement to the Works of Archimedes, Cambridge University Press
  51. ^ Eves, Howard (1963), A Survey of Geometry (Volume One), Boston: Allyn and Bacon
  52. ^ Archimedes, The Method of Mechanical Theorems; see Archimedes Palimpsest
  53. ^ O'Connor, J.J. & Robertson, E.F. (February 1996). "A history of calculus". University of St Andrews. Retrieved 7 August 2007.
  54. ^ K., Bidwell, James (30 November 1993). "Archimedes and Pi-Revisited". School Science and Mathematics. 94 (3).{{cite journal}}: CS1 maint: multiple names: authors list (link)
  55. ^ Boyer 1991, "Archimedes of Syracuse" p. 127. "Greek mathematics sometimes has been described as essentially static, with little regard for the notion of variability; but Archimedes, in his study of the spiral, seems to have found the tangent to a curve through kinematic considerations akin to differential calculus. Thinking of a point on the spiral 1=r = aθ as subjected to a double motion — a uniform radial motion away from the origin of coordinates and a circular motion about the origin — he seems to have found (through the parallelogram of velocities) the direction of motion (hence of the tangent to the curve) by noting the resultant of the two component motions. This appears to be the first instance in which a tangent was found to a curve other than a circle.
    Archimedes' study of the spiral, a curve that he ascribed to his friend Conon of Alexandria, was part of the Greek search for the solution of the three famous problems."
  56. ^ D. Rawlins: "Methods for Measuring the Earth's Size by Determining the Curvature of the Sea" and "Racking the Stade for Eratosthenes", appendices to "The Eratosthenes–Strabo Nile Map. Is It the Earliest Surviving Instance of Spherical Cartography? Did It Supply the 5000 Stades Arc for Eratosthenes' Experiment?", Archive for History of Exact Sciences, v.26, 211–219, 1982
  57. ^ Draper, John William (2007) [1874]. "History of the Conflict Between Religion and Science". In Joshi, S. T. (ed.). The Agnostic Reader. Prometheus. pp. 172–173. ISBN 978-1-59102-533-7.
  58. ^ Jones, A., Alexander (September 1991). "The Adaptation of Babylonian Methods in Greek Numerical Astronomy" (PDF). Isis. 82 (3): 440–453. Bibcode:1991Isis...82..441J. doi:10.1086/355836. S2CID 92988054. Archived from the original (PDF) on 4 March 2016. Retrieved 5 March 2020.
  59. ^ Bowen A.C., Goldstein B.R. (1991). "Hipparchus' Treatment of Early Greek Astronomy: The Case of Eudoxus and the Length of Daytime Author(s)". Proceedings of the American Philosophical Society 135(2): 233–254.
  60. ^ Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth (Vol. 3), p 24. Taipei: Caves Books, Ltd.
  61. ^ Cajori, Florian (1928). A History of Elementary Mathematics. Vol. 5. The Open Court Company, Publishers. pp. 516–7. doi:10.1126/science.5.117.516. ISBN 978-1-60206-991-6. PMID 17758371. S2CID 36235120. It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure. {{cite book}}: |journal= ignored (help)
  62. ^ Lay-Yong, Lam (1966). "On the Chinese Origin of the Galley Method of Arithmetical Division". The British Journal for the History of Science. 3: 66–69. doi:10.1017/S0007087400000200. S2CID 145407605.
  63. ^ Heath, Thomas L. (1921). A History of Greek Mathematics (Vol II). Oxford University Press. pp. 321–323.
  64. ^ Pasipoularides, Ares (1 March 2014). "Galen, father of systematic medicine. An essay on the evolution of modern medicine and cardiology". International Journal of Cardiology. 172 (1): 47–58. doi:10.1016/j.ijcard.2013.12.166. PMID 24461486.
  65. ^ Boyer 1991, "Greek Trigonometry and Mensuration" p. 163. "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue – that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem A + B + C > 180° is established. The second book of the Sphaerica describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest. Book III, the last, contains the well known "theorem of Menelaus" as part of what is essentially spherical trigonometry in the typical Greek form – a geometry or trigonometry of chords in a circle. In the circle in Fig. 10.4 we should write that chord AB is twice the sine of half the central angle AOB (multiplied by the radius of the circle). Menelaus and his Greek successors instead referred to AB simply as the chord corresponding to the arc AB. If BOB' is a diameter of the circle, then chord A' is twice the cosine of half the angle AOB (multiplied by the radius of the circle)."
  66. ^ Kurt Vogel, "Diophantus of Alexandria." in Complete Dictionary of Scientific Biography, Encyclopedia.com, 2008. Quote: The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.
  67. ^ Struik, pp. 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
  68. ^ Luke Hodgkin (2005). A History of Mathematics: From Mesopotamia to Modernity. Oxford University Press. p. 88. ISBN 978-0-19-152383-0. Liu is explicit on this; at the point where the Nine Chapters give a detailed and helpful 'Sign Rule'
  69. ^ Bailey, David; Borwein, Jonathan (2012). "Ancient Indian Square Roots: An Exercise in Forensic Paleo-Mathematics" (PDF). American Mathematical Monthly. Vol. 119, no. 8. pp. 646–657. Retrieved 14 September 2017.
  70. ^ Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved 24 July 2007.
  71. ^ Boyer 1991, p. [page needed].
  72. ^ Reimer, L., and Reimer, W. Mathematicians Are People, Too: Stories from the Lives of Great Mathematicians, Vol. 2. 1995. pp. 22-22. Parsippany, NJ: Pearson ducation, Inc. as Dale Seymor Publications. ISBN 0-86651-823-1.
  73. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". In Katz, Victor J. (ed.). The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 530. ISBN 978-0-691-11485-9.
  74. ^ Hayashi (2008), Aryabhata I.[full citation needed]
  75. ^ Miller, Jeff (22 December 2014). "Earliest Uses of Various Mathematical Symbols". Archived from the original on 20 February 2016. Retrieved 15 February 2016.
  76. ^ a b Boyer 1991, "The Mathematics of the Hindus" p. 207. "He gave more elegant rules for the sum of the squares and cubes of an initial segment of the positive integers. The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes."
  77. ^ a b Bibhutibhushan Datta and Avadhesh Narayan Singh (1962). History of Hindu Mathematics A source Book Part II. Asia Publishing House. p. 92.
  78. ^ Aryabhata at the Encyclopædia Britannica
  79. ^ Parakh, Abhishek (2006). "Aryabhata's Root Extraction Methods". arXiv:math/0608793.
  80. ^ Kak 1986.
  81. ^ The concept of Indian heliocentrism has been advocated by B. L. van der Waerden, Das heliozentrische System in der griechischen, persischen und indischen Astronomie. Naturforschenden Gesellschaft in Zürich. Zürich:Kommissionsverlag Leeman AG, 1970.
  82. ^ B.L. van der Waerden, "The Heliocentric System in Greek, Persian and Hindu Astronomy", in David A. King and George Saliba, ed., From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E. S. Kennedy, Annals of the New York Academy of Science, 500 (1987), pp. 529–534.
  83. ^ Hugh Thurston (1996). Early Astronomy. Springer. p. 188. ISBN 0-387-94822-8.
  84. ^ Noel Swerdlow, "Review: A Lost Monument of Indian Astronomy," Isis, 64 (1973): 239–243.
  85. ^ Ansari, S.M.R. (March 1977). "Aryabhata I, His Life and His Contributions". Bulletin of the Astronomical Society of India. 5 (1): 10–18. Bibcode:1977BASI....5...10A. hdl:2248/502.
  86. ^ a b Kelley, David H. & Milone, Eugene F. (2011). Exploring Ancient Skies: A Survey of Ancient and Cultural Astronomy (2nd ed.). Springer Science+Business Media. p. 293. Bibcode:2011eas..book.....K. doi:10.1007/978-1-4419-7624-6. ISBN 978-1-4419-7624-6. OCLC 710113366.
  87. ^ Morris R. Cohen and I. E. Drabkin (eds. 1958), A Source Book in Greek Science (p. 220), with several changes. Cambridge, MA: Harvard University Press, as referenced by David C. Lindberg (1992), The Beginnings of Western Science: The European Scientific Tradition in Philosophical, Religious, and Institutional Context, 600 B.C. to A.D. 1450, University of Chicago Press, p. 305, ISBN 0-226-48231-6
  88. ^ Henry Thomas Colebrooke. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, London 1817, p. 339 (online)
  89. ^ Plofker 2007, pp. 428–434.
  90. ^ Tabak, John (2009), Algebra: Sets, Symbols, and the Language of Thought, Infobase Publishing, p. 42, ISBN 978-0-8160-6875-3
  91. ^ Kusuba 2004, pp. 497–516.
  92. ^ Gupta, R. C. (2000), "History of Mathematics in India", in Hoiberg, Dale; Ramchandani, Indu (eds.), Students' Britannica India: Select essays, Popular Prakashan, p. 329
  93. ^ a b Joseph, G. G. (2000), The Crest of the Peacock: The Non-European Roots of Mathematics, Princeton, NJ: Princeton University Press, 416 pages, ISBN 978-0-691-00659-8
  94. ^ Bina Chatterjee (introduction by), The Khandakhadyaka of Brahmagupta, Motilal Banarsidass (1970), p. 13
  95. ^ Lallanji Gopal, History of Agriculture in India, Up to C. 1200 A.D., Concept Publishing Company (2008), p. 603
  96. ^ Kosla Vepa, Astronomical Dating of Events & Select Vignettes from Indian History, Indic Studies Foundation (2008), p. 372
  97. ^ Dwijendra Narayan Jha (edited by), The feudal order: state, society, and ideology in early medieval India, Manohar Publishers & Distributors (2000), p. 276
  98. ^ http://spie.org/etop/2007/etop07fundamentalsII.pdf," R. Rashed credited Ibn Sahl with discovering the law of refraction [23], usually called Snell’s law and also Snell and Descartes’ law."
  99. ^ Smith, A. Mark (2015). From Sight to Light: The Passage from Ancient to Modern Optics. University of Chicago Press. p. 178. ISBN 9780226174761.
  100. ^ Katz 1998, p. 255.
  101. ^ Florian Cajori (1918), Origin of the Name "Mathematical Induction", The American Mathematical Monthly 25 (5), p. 197-201.
  102. ^ Crombie, Alistair Cameron, Augustine to Galileo 2, p. 67.
  103. ^ Pines, Shlomo (1970). "Abu'l-Barakāt al-Baghdādī, Hibat Allah". Dictionary of Scientific Biography. Vol. 1. New York: Charles Scribner's Sons. pp. 26–28. ISBN 0-684-10114-9.
    (cf. Abel B. Franco (October 2003). "Avempace, Projectile Motion, and Impetus Theory", Journal of the History of Ideas 64 (4), p. 521-546 [528].)
  104. ^ "Robert Grosseteste". Stanford Encyclopaedia of Philosophy. Stanford.edu. Retrieved 6 May 2020.
  105. ^ "The invention of spectacles". The College of Optometrists. Retrieved 9 May 2020.
  106. ^ Mochrie, Robert (2005). Justice in Exchange: The Economic Philosophy of John Duns Scotus
  107. ^ a b Katz 1995.
  108. ^ J J O'Connor and E F Robertson (2000). "Madhava of Sangamagramma". MacTutor History of Mathematics archive. School of Mathematics and Statistics, University of St Andrews, Scotland. Archived from the original on 14 May 2006. Retrieved 8 September 2007.
  109. ^ a b Ian G. Pearce (2002). Madhava of Sangamagramma. MacTutor History of Mathematics archive. University of St Andrews.
  110. ^ Radha Charan Gupta (1977) "Parameshvara's rule for the circumradius of a cyclic quadrilateral", Historia Mathematica 4: 67–74
  111. ^ Roy 1990, pp. 101–102.
  112. ^ Brink, David (2015). "Nilakantha's accelerated series for π". Acta Arithmetica. 171 (4): 293–308. doi:10.4064/aa171-4-1.
  113. ^ Ramasubramanian, K.; Srinivas, M. D.; Sriram, M. S. (1994). "Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion". Current Science. 66: 784–790.
  114. ^ Beckmann, Petr (1971). A history of π (2nd ed.). Boulder, CO: The Golem Press. pp. 94–95. ISBN 978-0-88029-418-8. MR 0449960.
  115. ^ Burton, David. The History of Mathematics: An Introduction (7th (2010) ed.). New York: McGraw-Hill.
  116. ^ Bruno, Leonard C (2003) [1999]. Math and mathematicians: the history of math discoveries around the world. Baker, Lawrence W. Detroit, Mich.: U X L. p. 60. ISBN 0787638137. OCLC 41497065.
  117. ^ Volckart, Oliver (1997). "Early beginnings of the quantity theory of money and their context in Polish and Prussian monetary policies, c. 1520–1550". The Economic History Review. 50 (3). Wiley-Blackwell: 430–49. doi:10.1111/1468-0289.00063. ISSN 0013-0117. JSTOR 2599810.
  118. ^ Kline, Morris. A history of mathematical thought, volume 1. p. 253.
  119. ^ Jourdain, Philip E. B. (1913). The Nature of Mathematics.
  120. ^ Robert Recorde, The Whetstone of Witte (London, England: John Kyngstone, 1557), p. 236 (although the pages of this book are not numbered). From the chapter titled "The rule of equation, commonly called Algebers Rule" (p. 236): "Howbeit, for easie alteration of equations. I will propounde a fewe examples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in worke use, a paire of paralleles, or Gemowe [twin, from gemew, from the French gemeau (twin / twins), from the Latin gemellus (little twin)] lines of one lengthe, thus: = , bicause noe .2. thynges, can be moare equalle." (However, for easy manipulation of equations, I will present a few examples in order that the extraction of roots may be more readily done. And to avoid the tedious repetition of these words "is equal to", I will substitute, as I often do when working, a pair of parallels or twin lines of the same length, thus: = , because no two things can be more equal.)
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  122. ^ Katz, Victor J. (2004), "9.1.4", A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2
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