Indian mathematics (English Wikipedia)

Datta, Bibhutibhusan; Singh, Awadhesh Narayan (2019). "Use of permutations and combinations in India". In Kolachana, Aditya; Mahesh, K.; Ramasubramanian, K. (eds.). Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla. Sources and Studies in the History of Mathematics and Physical Sciences. Springer Singapore. pp. 356–376. doi:10.1007/978-981-13-7326-8_18. S2CID 191141516.. Revised by K. S. Shukla from a paper in Indian Journal of History of Science 27 (3): 231–249, 1992, MRMR1189487. See p. 363.

amstat.org

"Statisticians in History: Calyampudi R. Rao". American Statistical Association. 30 November 2016. Archived from the original on 3 March 2016. Retrieved 4 October 2009.

archive.org

(Bourbaki 1998, p. 46): "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus." Bourbaki, Nicolas (1998), Elements of the History of Mathematics, Berlin, Heidelberg, and New York: Springer-Verlag, 301 pages, ISBN 978-3-540-64767-6.
(Bourbaki 1998, p. 49): Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. Leonardo of Pisa wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by Brahmagupta during 7th century AD. "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic." Bourbaki, Nicolas (1998), Elements of the History of Mathematics, Berlin, Heidelberg, and New York: Springer-Verlag, 301 pages, ISBN 978-3-540-64767-6.
(Bourbaki 1998, p. 126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (Aristarchus, Hipparchus, Ptolemy) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the chord of the arc cut out by an angle $\theta <\pi$ on a circle of radius r, in other words the number $2r\sin \left(\theta /2\right)$ ; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)." Bourbaki, Nicolas (1998), Elements of the History of Mathematics, Berlin, Heidelberg, and New York: Springer-Verlag, 301 pages, ISBN 978-3-540-64767-6.
(Joseph 2000, p. 229) 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.
Cooke, Roger (1997), "The Mathematics of the Hindus", The History of Mathematics: A Brief Course, Wiley-Interscience, p. 197, ISBN 978-0-471-18082-1, The word Siddhanta means that which is proved or established. The Sulva Sutras are of Hindu origin, but the Siddhantas contain so many words of foreign origin that they undoubtedly have roots in Mesopotamia and Greece.
Cooke, Roger (1997). The history of mathematics : a brief course. Internet Archive. New York : Wiley. ISBN 978-0-471-18082-1.
Satis Chandra Vidyabhusana (1920). A History of Indian Logic: Ancient, Mediaeval and Modern Schools. Delhi: Motilal Banarsidas. p. 405. ISBN 9788120805651.

books.google.com

Gupta, R. C. (2000), "History of Mathematics in India", in Hoiberg, Dale; Ramchandani, Indu (eds.), Students' Britannica India: Select essays, Popular Prakashan, p. 329
Vidyabhusana, Satis Chandra (1920). A History of Indian Logic: Ancient, Mediaeval and Modern Schools. Delhi: Motilal Banarsidass. pp. 405–6. ISBN 9788120805651.
Florian Cajori (2010). "A History of Elementary Mathematics – With Hints on Methods of Teaching". p.94. ISBN 1-4460-2221-8
Agarwal, Ravi P.; Sen, Syamal K. (11 November 2014). Creators of Mathematical and Computational Sciences. Springer. ISBN 978-3-319-10870-4.

britannica.com

(Kim Plofker 2007, p. 1) Kim Plofker (2007), "mathematics, South Asian", Encyclopaedia Britannica Online, pp. 1–12, retrieved 18 May 2007.
"algebra" 2007. Britannica Concise Encyclopedia Archived 29 September 2007 at the Wayback Machine. Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra."
(Kim Plofker 2007, p. 6) Kim Plofker (2007), "mathematics, South Asian", Encyclopaedia Britannica Online, pp. 1–12, retrieved 18 May 2007.

crraoaimscs.org

"C.R.Rao in News". C.R.Rao Advanced Institute of Mathematics, Statistics and Computer Science.

currentscience.ac.in

"The mathematician in Asutosh Mukhopadhyay" (PDF). Current Science. Retrieved 29 September 2017.
"Sir Ashutosh Mukherjee: educationist, leader and institution-builder" (PDF). Current Science. Retrieved 29 September 2017.

doi.org

(Pingree 2003, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century." Pingree, David (2003), "The logic of non-Western science: mathematical discoveries in medieval India", Daedalus, 132 (4): 45–54, doi:10.1162/001152603771338779, S2CID 57559157, archived from the original on 15 January 2011, retrieved 1 September 2017.
(Filliozat 2004, pp. 140–143) Filliozat, Pierre-Sylvain (2004), "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature", in Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et al. (eds.), History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht: Springer Netherlands, 254 pages, pp. 137–157, pp. 360–375, doi:10.1007/1-4020-2321-9_7, ISBN 978-1-4020-2320-0.
(Stillwell 2004, p. 173) Stillwell, John (2004), Mathematics and its History, Undergraduate Texts in Mathematics (2 ed.), Springer, Berlin and New York, 568 pages, doi:10.1007/978-1-4684-9281-1, ISBN 978-0-387-95336-6.
(Bressoud 2002, p. 12) Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use." Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal, 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972.
(Plofker 2001, p. 293) Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)" [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalised to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here" Plofker, Kim (2001), "The "Error" in the Indian "Taylor Series Approximation" to the Sine", Historia Mathematica, 28 (4): 283–295, doi:10.1006/hmat.2001.2331.
(Pingree 1992, p. 562) Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution." Pingree, David (1992), "Hellenophilia versus the History of Science", Isis, 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257, S2CID 68570164
(Katz 1995, pp. 173–174) Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today." Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine, 68 (3): 163–174, doi:10.2307/2691411, JSTOR 2691411.
Coppa, A.; et al. (6 April 2006), "Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population", Nature, 440 (7085): 755–6, Bibcode:2006Natur.440..755C, doi:10.1038/440755a, PMID 16598247, S2CID 6787162.
(Staal 1999) Staal, Frits (1999), "Greek and Vedic Geometry", Journal of Indian Philosophy, 27 (1–2): 105–127, doi:10.1023/A:1004364417713, S2CID 170894641.
Ingerman, Peter Zilahy (1 March 1967). ""Pānini-Backus Form" suggested". Communications of the ACM. 10 (3): 137. doi:10.1145/363162.363165. ISSN 0001-0782. S2CID 52817672.
(Fowler 1996, p. 11) Fowler, David (1996), "Binomial Coefficient Function", The American Mathematical Monthly, 103 (1): 1–17, doi:10.2307/2975209, JSTOR 2975209.
(Singh 1936, pp. 623–624) Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1 (1): 606–628, doi:10.1086/368443, JSTOR 301627, S2CID 144760421
Datta, Bibhutibhusan; Singh, Awadhesh Narayan (2019). "Use of permutations and combinations in India". In Kolachana, Aditya; Mahesh, K.; Ramasubramanian, K. (eds.). Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla. Sources and Studies in the History of Mathematics and Physical Sciences. Springer Singapore. pp. 356–376. doi:10.1007/978-981-13-7326-8_18. S2CID 191141516.. Revised by K. S. Shukla from a paper in Indian Journal of History of Science 27 (3): 231–249, 1992, MRMR1189487. See p. 363.
(Filliozat 2004, p. 137) Filliozat, Pierre-Sylvain (2004), "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature", in Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et al. (eds.), History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht: Springer Netherlands, 254 pages, pp. 137–157, pp. 360–375, doi:10.1007/1-4020-2321-9_7, ISBN 978-1-4020-2320-0.
(Pingree 1988, p. 637) Pingree, David (1988), "Reviewed Work(s): The Fidelity of Oral Tradition and the Origins of Science by Frits Staal", Journal of the American Oriental Society, 108 (4): 637–638, doi:10.2307/603154, JSTOR 603154.
(Filliozat 2004, p. 139) Filliozat, Pierre-Sylvain (2004), "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature", in Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et al. (eds.), History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht: Springer Netherlands, 254 pages, pp. 137–157, pp. 360–375, doi:10.1007/1-4020-2321-9_7, ISBN 978-1-4020-2320-0.
(Filliozat 2004, pp. 140–141) Filliozat, Pierre-Sylvain (2004), "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature", in Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et al. (eds.), History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht: Springer Netherlands, 254 pages, pp. 137–157, pp. 360–375, doi:10.1007/1-4020-2321-9_7, ISBN 978-1-4020-2320-0.
(Yano 2006, p. 146) Yano, Michio (2006), "Oral and Written Transmission of the Exact Sciences in Sanskrit", Journal of Indian Philosophy, 34 (1–2), Springer Netherlands: 143–160, doi:10.1007/s10781-005-8175-6, S2CID 170679879
(Filliozat 2004, pp. 143–144) Filliozat, Pierre-Sylvain (2004), "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature", in Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et al. (eds.), History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht: Springer Netherlands, 254 pages, pp. 137–157, pp. 360–375, doi:10.1007/1-4020-2321-9_7, ISBN 978-1-4020-2320-0.
(Filliozat 2004, p. 144) Filliozat, Pierre-Sylvain (2004), "Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature", in Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et al. (eds.), History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht: Springer Netherlands, 254 pages, pp. 137–157, pp. 360–375, doi:10.1007/1-4020-2321-9_7, ISBN 978-1-4020-2320-0.
(Pingree 1988, p. 638) Pingree, David (1988), "Reviewed Work(s): The Fidelity of Oral Tradition and the Origins of Science by Frits Staal", Journal of the American Oriental Society, 108 (4): 637–638, doi:10.2307/603154, JSTOR 603154.
(Katz 1995) Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine, 68 (3): 163–174, doi:10.2307/2691411, JSTOR 2691411.
(Stillwell 2004, p. 77) Stillwell, John (2004), Mathematics and its History, Undergraduate Texts in Mathematics (2 ed.), Springer, Berlin and New York, 568 pages, doi:10.1007/978-1-4684-9281-1, ISBN 978-0-387-95336-6.
(Stillwell 2004, p. 87) Stillwell, John (2004), Mathematics and its History, Undergraduate Texts in Mathematics (2 ed.), Springer, Berlin and New York, 568 pages, doi:10.1007/978-1-4684-9281-1, ISBN 978-0-387-95336-6.
(Stillwell 2004, pp. 72–73) Stillwell, John (2004), Mathematics and its History, Undergraduate Texts in Mathematics (2 ed.), Springer, Berlin and New York, 568 pages, doi:10.1007/978-1-4684-9281-1, ISBN 978-0-387-95336-6.
(Stillwell 2004, pp. 74–76) Stillwell, John (2004), Mathematics and its History, Undergraduate Texts in Mathematics (2 ed.), Springer, Berlin and New York, 568 pages, doi:10.1007/978-1-4684-9281-1, ISBN 978-0-387-95336-6.
(Roy 1990) Roy, Ranjan (1990), "Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha", Mathematics Magazine, 63 (5): 291–306, doi:10.2307/2690896, JSTOR 2690896.
(Bressoud 2002) Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal, 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972.
(Singh 1936) Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1 (1): 606–628, doi:10.1086/368443, JSTOR 301627, S2CID 144760421
(Whish 1835) Whish, Charles (1835), "On the Hindú Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the diameter exhibited in the four S'ástras, the Tantra Sangraham, Yucti Bháshá, Carana Padhati, and Sadratnamála", Transactions of the Royal Asiatic Society of Great Britain and Ireland, 3 (3): 509–523, doi:10.1017/S0950473700001221, JSTOR 25581775
Rajagopal, C.; Rangachari, M. S. (1977), "On an untapped source of medieval Keralese mathematics", Archive for History of Exact Sciences, 18 (2): 89–102, doi:10.1007/BF00348142, S2CID 51861422.
Rajagopal, C.; Rangachari, M. S. (1986), "On Medieval Kerala Mathematics", Archive for History of Exact Sciences, 35 (2): 91–99, doi:10.1007/BF00357622, S2CID 121678430.
Langlands, Robert P. (1985). "Harish-Chandra. 11 October 1923 – 16 October 1983". Biographical Memoirs of Fellows of the Royal Society. 31: 198–225. doi:10.1098/rsbm.1985.0008. JSTOR 769925.
Varadarajan, V. S. (1984). "Harish-Chandra (1923–1983)". The Mathematical Intelligencer. 6 (3): 9–13. doi:10.1007/BF03024122. S2CID 122014700.

dx.doi.org

Divakaran, P. P. (2018), "From 500 BCE to 500 CE", Sources and Studies in the History of Mathematics and Physical Sciences, Singapore: Springer Singapore, pp. 143–173, ISBN 978-981-13-1773-6, retrieved 18 June 2024

etymonline.com

- Harper, Douglas (2011). "Zero". Etymonline Etymology Dictionary. Archived from the original on 3 July 2017. figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", c. 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher", translation of Sanskrit sunya-m "empty place, desert, naught
frontlineonnet.com
- Ramachandran, R. (7–20 April 2007). "Science of chance". Frontline. India. Archived from the original on 11 December 2007.
harvard.edu
ui.adsabs.harvard.edu
- (Pingree 1992, p. 562) Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution." Pingree, David (1992), "Hellenophilia versus the History of Science", Isis, 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257, S2CID 68570164
- Coppa, A.; et al. (6 April 2006), "Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population", Nature, 440 (7085): 755–6, Bibcode:2006Natur.440..755C, doi:10.1038/440755a, PMID 16598247, S2CID 6787162.
heidelberg-laureate-forum.org
- "Subhash Khot - Heidelberg Laureate Forum". - Heidelberg Laureate Forum. Retrieved 3 July 2024.
hindustantimes.com
- "Indian wins Norway's Abel Prize for Mathematics". Hindustan Times. 23 March 2007.
ias.ac.in
- (Dani 2003) Dani, S. G. (25 July 2003), "On the Pythagorean triples in the Śulvasūtras" (PDF), Current Science, 85 (2): 219–224.
infinityfoundation.com
- "Panini-Backus". infinityfoundation.com. Retrieved 16 March 2018.
jainworld.com
- Singh, A. N., Mathematics of Dhavala, Lucknow University, archived from the original on 11 May 2011, retrieved 31 July 2010
jstor.org
- (Bressoud 2002, p. 12) Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use." Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal, 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972.
- (Pingree 1992, p. 562) Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution." Pingree, David (1992), "Hellenophilia versus the History of Science", Isis, 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257, S2CID 68570164
- (Katz 1995, pp. 173–174) Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today." Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine, 68 (3): 163–174, doi:10.2307/2691411, JSTOR 2691411.
- (Fowler 1996, p. 11) Fowler, David (1996), "Binomial Coefficient Function", The American Mathematical Monthly, 103 (1): 1–17, doi:10.2307/2975209, JSTOR 2975209.
- (Singh 1936, pp. 623–624) Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1 (1): 606–628, doi:10.1086/368443, JSTOR 301627, S2CID 144760421
- (Pingree 1988, p. 637) Pingree, David (1988), "Reviewed Work(s): The Fidelity of Oral Tradition and the Origins of Science by Frits Staal", Journal of the American Oriental Society, 108 (4): 637–638, doi:10.2307/603154, JSTOR 603154.
- (Pingree 1988, p. 638) Pingree, David (1988), "Reviewed Work(s): The Fidelity of Oral Tradition and the Origins of Science by Frits Staal", Journal of the American Oriental Society, 108 (4): 637–638, doi:10.2307/603154, JSTOR 603154.
- (Katz 1995) Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine, 68 (3): 163–174, doi:10.2307/2691411, JSTOR 2691411.
- (Roy 1990) Roy, Ranjan (1990), "Discovery of the Series Formula for $\pi$ by Leibniz, Gregory, and Nilakantha", Mathematics Magazine, 63 (5): 291–306, doi:10.2307/2690896, JSTOR 2690896.
- (Bressoud 2002) Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal, 33 (1): 2–13, doi:10.2307/1558972, JSTOR 1558972.
- (Singh 1936) Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1 (1): 606–628, doi:10.1086/368443, JSTOR 301627, S2CID 144760421
- (Whish 1835) Whish, Charles (1835), "On the Hindú Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the diameter exhibited in the four S'ástras, the Tantra Sangraham, Yucti Bháshá, Carana Padhati, and Sadratnamála", Transactions of the Royal Asiatic Society of Great Britain and Ireland, 3 (3): 509–523, doi:10.1017/S0950473700001221, JSTOR 25581775
- Langlands, Robert P. (1985). "Harish-Chandra. 11 October 1923 – 16 October 1983". Biographical Memoirs of Fellows of the Royal Society. 31: 198–225. doi:10.1098/rsbm.1985.0008. JSTOR 769925.
macfound.org
- "Subhash Khot - MacArthur Foundation".
mathgenealogy.org
- Indian mathematics at the Mathematics Genealogy Project
mathunion.org
- "Fields Medal 2014" (Press release). International Mathematical Union. Retrieved 10 February 2021.
- "Press Release - Manjul Bhargava" (PDF). International Mathematical Unioin. 10 February 2021.
- "Fields Medallists 2014 awardees with brief citations | International Mathematical Union (IMU)". www.mathunion.org. Retrieved 9 February 2021.
nih.gov
pubmed.ncbi.nlm.nih.gov
- Coppa, A.; et al. (6 April 2006), "Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population", Nature, 440 (7085): 755–6, Bibcode:2006Natur.440..755C, doi:10.1038/440755a, PMID 16598247, S2CID 6787162.
nio.org
drs.nio.org
- Rao, S. R. (July 1992). "A Navigational Instrument of the Harappan Sailors" (PDF). Marine Archaeology. 3: 61–62. Archived from the original (PDF) on 8 August 2017.
ox.ac.uk
arch.ox.ac.uk
- Mason, Robyn (14 September 2017). "Oxford Radiocarbon Accelerator Unit dates the world's oldest recorded origin of the zero symbol". School of Archaeology, University of Oxford. Archived from the original on 14 September 2017. Retrieved 14 September 2017.
bodleian.ox.ac.uk
- "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 14 September 2017. Retrieved 14 September 2017.
questia.com
- (Pingree 2003, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century." Pingree, David (2003), "The logic of non-Western science: mathematical discoveries in medieval India", Daedalus, 132 (4): 45–54, doi:10.1162/001152603771338779, S2CID 57559157, archived from the original on 15 January 2011, retrieved 1 September 2017.
royalsociety.org
- "Professor Subhash Khot". Retrieved 27 July 2024.
sciencemuseum.org.uk
blog.sciencemuseum.org.uk
- "Illuminating India: Starring the oldest recorded origins of 'zero', the Bakhshali manuscript".
semanticscholar.org
api.semanticscholar.org
- (Pingree 2003, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century." Pingree, David (2003), "The logic of non-Western science: mathematical discoveries in medieval India", Daedalus, 132 (4): 45–54, doi:10.1162/001152603771338779, S2CID 57559157, archived from the original on 15 January 2011, retrieved 1 September 2017.
- (Pingree 1992, p. 562) Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution." Pingree, David (1992), "Hellenophilia versus the History of Science", Isis, 83 (4): 554–563, Bibcode:1992Isis...83..554P, doi:10.1086/356288, JSTOR 234257, S2CID 68570164
- Coppa, A.; et al. (6 April 2006), "Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population", Nature, 440 (7085): 755–6, Bibcode:2006Natur.440..755C, doi:10.1038/440755a, PMID 16598247, S2CID 6787162.
- (Staal 1999) Staal, Frits (1999), "Greek and Vedic Geometry", Journal of Indian Philosophy, 27 (1–2): 105–127, doi:10.1023/A:1004364417713, S2CID 170894641.
- Ingerman, Peter Zilahy (1 March 1967). ""Pānini-Backus Form" suggested". Communications of the ACM. 10 (3): 137. doi:10.1145/363162.363165. ISSN 0001-0782. S2CID 52817672.
- (Singh 1936, pp. 623–624) Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1 (1): 606–628, doi:10.1086/368443, JSTOR 301627, S2CID 144760421
- Datta, Bibhutibhusan; Singh, Awadhesh Narayan (2019). "Use of permutations and combinations in India". In Kolachana, Aditya; Mahesh, K.; Ramasubramanian, K. (eds.). Studies in Indian Mathematics and Astronomy: Selected Articles of Kripa Shankar Shukla. Sources and Studies in the History of Mathematics and Physical Sciences. Springer Singapore. pp. 356–376. doi:10.1007/978-981-13-7326-8_18. S2CID 191141516.. Revised by K. S. Shukla from a paper in Indian Journal of History of Science 27 (3): 231–249, 1992, MRMR1189487. See p. 363.
- (Yano 2006, p. 146) Yano, Michio (2006), "Oral and Written Transmission of the Exact Sciences in Sanskrit", Journal of Indian Philosophy, 34 (1–2), Springer Netherlands: 143–160, doi:10.1007/s10781-005-8175-6, S2CID 170679879
- (Singh 1936) Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris, 1 (1): 606–628, doi:10.1086/368443, JSTOR 301627, S2CID 144760421
- Rajagopal, C.; Rangachari, M. S. (1977), "On an untapped source of medieval Keralese mathematics", Archive for History of Exact Sciences, 18 (2): 89–102, doi:10.1007/BF00348142, S2CID 51861422.
- Rajagopal, C.; Rangachari, M. S. (1986), "On Medieval Kerala Mathematics", Archive for History of Exact Sciences, 35 (2): 91–99, doi:10.1007/BF00357622, S2CID 121678430.
- Varadarajan, V. S. (1984). "Harish-Chandra (1923–1983)". The Mathematical Intelligencer. 6 (3): 9–13. doi:10.1007/BF03024122. S2CID 122014700.
st-andrews.ac.uk
mathshistory.st-andrews.ac.uk
- O'Connor, John J.; Robertson, Edmund F., "Indian mathematics", MacTutor History of Mathematics Archive, University of St Andrews
stanford.edu
plato.stanford.edu
- Ganeri, Jonardon (2023), "Analytic Philosophy in Early Modern India", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 23 January 2024
theguardian.com
- Devlin, Hannah (13 September 2017). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Retrieved 14 September 2017.
theintellibrain.com
- "Vedic Maths Complete Detail". ALLEN IntelliBrain. Retrieved 22 October 2022.
timescrest.com
- "The Numberdars". Times Crest. 1 October 2001. Archived from the original on 23 March 2016. Retrieved 14 March 2016.
ubc.ca
math.ubc.ca
- Mathematics Department, University of British Columbia, The Babylonian tabled Plimpton 322 Archived 17 June 2020 at the Wayback Machine.
web.archive.org
- "algebra" 2007. Britannica Concise Encyclopedia Archived 29 September 2007 at the Wayback Machine. Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra."
- (Pingree 2003, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century." Pingree, David (2003), "The logic of non-Western science: mathematical discoveries in medieval India", Daedalus, 132 (4): 45–54, doi:10.1162/001152603771338779, S2CID 57559157, archived from the original on 15 January 2011, retrieved 1 September 2017.
- Rao, S. R. (July 1992). "A Navigational Instrument of the Harappan Sailors" (PDF). Marine Archaeology. 3: 61–62. Archived from the original (PDF) on 8 August 2017.
- Mathematics Department, University of British Columbia, The Babylonian tabled Plimpton 322 Archived 17 June 2020 at the Wayback Machine.
- - Harper, Douglas (2011). "Zero". Etymonline Etymology Dictionary. Archived from the original on 3 July 2017. figure which stands for naught in the Arabic notation," also "the absence of all quantity considered as quantity", c. 1600, from French zéro or directly from Italian zero, from Medieval Latin zephirum, from Arabic sifr "cipher", translation of Sanskrit sunya-m "empty place, desert, naught
  - Mason, Robyn (14 September 2017). "Oxford Radiocarbon Accelerator Unit dates the world's oldest recorded origin of the zero symbol". School of Archaeology, University of Oxford. Archived from the original on 14 September 2017. Retrieved 14 September 2017.
  - Singh, A. N., Mathematics of Dhavala, Lucknow University, archived from the original on 11 May 2011, retrieved 31 July 2010
  - "The Numberdars". Times Crest. 1 October 2001. Archived from the original on 23 March 2016. Retrieved 14 March 2016.
  - "Statisticians in History: Calyampudi R. Rao". American Statistical Association. 30 November 2016. Archived from the original on 3 March 2016. Retrieved 4 October 2009.
  - Ramachandran, R. (7–20 April 2007). "Science of chance". Frontline. India. Archived from the original on 11 December 2007.
  worldcat.org
  search.worldcat.org
  - Ingerman, Peter Zilahy (1 March 1967). ""Pānini-Backus Form" suggested". Communications of the ACM. 10 (3): 137. doi:10.1145/363162.363165. ISSN 0001-0782. S2CID 52817672.
  - Devlin, Hannah (13 September 2017). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Retrieved 14 September 2017.
  zenodo.org
  - (Whish 1835) Whish, Charles (1835), "On the Hindú Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the diameter exhibited in the four S'ástras, the Tantra Sangraham, Yucti Bháshá, Carana Padhati, and Sadratnamála", Transactions of the Royal Asiatic Society of Great Britain and Ireland, 3 (3): 509–523, doi:10.1017/S0950473700001221, JSTOR 25581775

Indian mathematics (English Wikipedia)

abelprize.no

ams.org

mathscinet.ams.org

amstat.org

archive.org

books.google.com

britannica.com

crraoaimscs.org

currentscience.ac.in

doi.org

doi.org

dx.doi.org

etymonline.com

frontlineonnet.com

harvard.edu

ui.adsabs.harvard.edu

heidelberg-laureate-forum.org

hindustantimes.com

ias.ac.in

infinityfoundation.com

jainworld.com

jstor.org

macfound.org

mathgenealogy.org

mathunion.org

nih.gov

pubmed.ncbi.nlm.nih.gov

nio.org

drs.nio.org

ox.ac.uk

arch.ox.ac.uk

bodleian.ox.ac.uk

questia.com

royalsociety.org

sciencemuseum.org.uk

blog.sciencemuseum.org.uk

semanticscholar.org

api.semanticscholar.org

st-andrews.ac.uk

mathshistory.st-andrews.ac.uk

stanford.edu

plato.stanford.edu

theguardian.com

theintellibrain.com

timescrest.com

ubc.ca

math.ubc.ca

web.archive.org

worldcat.org

search.worldcat.org

zenodo.org