References analysis of the Wikipedia article "Brahmagupta" in the English version

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Sachau, Edward C. (1910), Alberuni's India, Volume I, London: Kegan Paul, Trench and Trubner, p. 153 – via archive.org, Brahma-siddhānta, so-called from Brahman, composed by Brahmagupta, the son of Jishnu, from the town of Bhillamāla between Multān and Anhilwāra, 16 yojana from the latter place (?)

Cai 2023, p. 114; Cooke 1997, p. 208 Cai, Tianxin (25 July 2023). A Brief History of Mathematics: A Promenade through the Civilizations of Our World. Springer Nature. ISBN 978-3-031-26841-0. Cooke, Roger (1997), The History of Mathematics: A Brief Course, Wiley-Interscience, ISBN 0-471-18082-3

Pingree, David E. (1970–1994). Pingree's Census of the Exact Sciences in Sanskrit. APS. pp. A4, 256 ff., A5, 239–240 et passim.

Boyer (1991, "China and India" p. 221) "he was the first one to give a general solution of the linear Diophantine equation ax + by = c, where a, b, and c are integers. [...] It is greatly to the credit of Brahmagupta that he gave all integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India – or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words." Boyer, Carl B. (1991), A History of Mathematics, John Wiley & Sons, Inc, ISBN 0-471-54397-7

Boyer (1991, p. 220): However, here again Brahmagupta spoiled matters somewhat by asserting that 0 ÷ 0 = 0, and on the touchy matter of a ÷ 0, he did not commit himself. Boyer, Carl B. (1991), A History of Mathematics, John Wiley & Sons, Inc, ISBN 0-471-54397-7

Joseph (2000, pp.285–86). Joseph, George G. (2000), The Crest of the Peacock, Princeton University Press, ISBN 0-691-00659-8

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Pickover, Clifford (2008). Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. p. 105. ISBN 978-0-19-979268-9.

Bose, Mainak Kumar (1988). Late classical India. A. Mukherjee & Co.^{[page needed]}

Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).

Ayyappappanikkar (1997). Medieval Indian Literature: Surveys and selections. Sahitya Akademi. p. 493. ISBN 978-81-260-0365-5.

Bhattacharyya 2011, p. 185: "Brahmagupta, one of the most celebrated mathematicians of the East, indeed of the world, was born in the year 598 CE, in the town of Bhillamala during the reign of King Vyaghramukh of the Chapa Dynasty." Bhattacharyya, R. K. (2011), "Brahmagupta: The Ancient Indian Mathematician", in B. S. Yadav; Man Mohan (eds.), Ancient Indian Leaps into Mathematics, Springer Science & Business Media, pp. 185–192, ISBN 978-0-8176-4695-0

Gupta 2008, p. 162. Gupta, Radha Charan (2008), "Brahmagupta", in Selin, Helaine (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, pp. 162–163, ISBN 978-1-4020-4559-2

Bhattacharyya 2011, pp. 185–186. Bhattacharyya, R. K. (2011), "Brahmagupta: The Ancient Indian Mathematician", in B. S. Yadav; Man Mohan (eds.), Ancient Indian Leaps into Mathematics, Springer Science & Business Media, pp. 185–192, ISBN 978-0-8176-4695-0

Gupta 2008, p. 163. Gupta, Radha Charan (2008), "Brahmagupta", in Selin, Helaine (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, pp. 162–163, ISBN 978-1-4020-4559-2

Bhattacharyya 2011, p. 185. Bhattacharyya, R. K. (2011), "Brahmagupta: The Ancient Indian Mathematician", in B. S. Yadav; Man Mohan (eds.), Ancient Indian Leaps into Mathematics, Springer Science & Business Media, pp. 185–192, ISBN 978-0-8176-4695-0

Young, M. J. L.; Latham, J. D.; Serjeant, R. B. (2 November 2006), Religion, Learning and Science in the 'Abbasid Period, Cambridge University Press, pp. 302–303, ISBN 978-0-521-02887-5

van Bladel, Kevin (28 November 2014), "Eighth Century Indian Astronomy in the Two Cities of Peace", in Asad Q. Ahmed; Benham Sadeghi; Robert G. Hoyland (eds.), Islamic Cultures, Islamic Contexts: Essays in Honor of Professor Patricia Crone, BRILL, pp. 257–294, ISBN 978-90-04-28171-4

Plofker (2007, pp. 428–434) Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Brahmagupta; Bhāskara II (1817). Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara. Translated by Henry Thomas Colebrooke. John Murray. p. 319.

Tiwari, Sarju (1992), Mathematics in History, Culture, Philosophy, and Science: From Ancient Times to Modern Age, Mittal Publications, pp. 91–, ISBN 978-81-7099-404-6

Plofker (2007, pp. 422) The reader is apparently expected to be familiar with basic arithmetic operations as far as the square-root; Brahmagupta merely notes some points about applying them to fractions. The procedures for finding the cube and cube-root of an integer, however, are described (compared the latter to Aryabhata's very similar formulation). They are followed by rules for five types of combinations: [...] Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Plofker (2007, pp. 421–427) Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Plofker (2007, p. 423) Here the sums of the squares and cubes of the first n integers are defined in terms of the sum of the n integers itself; Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Plofker (2007, p. 426) Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Stillwell (2004, pp. 44–46): In the seventh century CE the Indian mathematician Brahmagupta gave a recurrence relation for generating solutions of x² − Dy² = 1, as we shall see in Chapter 5. The Indians called the Euclidean algorithm the "pulverizer" because it breaks numbers down to smaller and smaller pieces. To obtain a recurrence one has to know that a rectangle proportional to the original eventually recurs, a fact that was rigorously proved only in 1768 by Lagrange. Stillwell, John (2004), Mathematics and its History (Second ed.), Springer Science + Business Media Inc., ISBN 0-387-95336-1

Stillwell (2004, pp. 72–74) Stillwell, John (2004), Mathematics and its History (Second ed.), Springer Science + Business Media Inc., ISBN 0-387-95336-1

Plofker (2007, p. 424) Brahmagupta does not explicitly state that he is discussing only figures inscribed in circles, but it is implied by these rules for computing their circumradius. Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Stillwell (2004, p. 77) Stillwell, John (2004), Mathematics and its History (Second ed.), Springer Science + Business Media Inc., ISBN 0-387-95336-1

Plofker (2007, p. 427) After the geometry of plane figures, Brahmagupta discusses the computation of volumes and surface areas of solids (or empty spaces dug out of solids). His straightforward rules for the volumes of a rectangular prism and pyramid are followed by a more ambiguous one, which may refer to finding the average depth of a sequence of puts with different depths. The next formula apparently deals with the volume of a frustum of a square pyramid, where the "pragmatic" volume is the depth times the square of the mean of the edges of the top and bottom faces, while the "superficial" volume is the depth times their mean area. Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Plofker (2007, p. 419) Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Plofker (2007, pp. 419–420) Brahmagupta's sine table, like much other numerical data in Sanskrit treatises, is encoded mostly in concrete-number notation that uses names of objects to represent the digits of place-value numerals, starting with the least significant. [...]
There are fourteen Progenitors ("Manu") in Indian cosmology; "twins" of course stands for 2; the seven stars of Ursa Major (the "Sages") for 7, the four Vedas, and the four sides of the traditional dice used in gambling, for 6, and so on. Thus Brahmagupta enumerates his first six sine-values as 214, 427, 638, 846, 1051, 1251. (His remaining eighteen sines are 1446, 1635, 1817, 1991, 2156, 2312, 2459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, 3270). The Paitamahasiddhanta, however, specifies an initial sine-value of 225 (although the rest of its sine-table is lost), implying a trigonometric radius of R = 3438 approx= C(')/2π: a tradition followed, as we have seen, by Aryabhata. Nobody knows why Brahmagupta chose instead to normalize these values to R = 3270. Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Plofker (2007, pp. 418–419) Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Plofker (2007, pp. 419–420) Brahmagupta discusses the illumination of the moon by the sun, rebutting an idea maintained in scriptures: namely, that the moon is farther from the earth than the sun is. In fact, as he explains, because the moon is closer the extent of the illuminated portion of the moon depends on the relative positions of the moon and the sun, and can be computed from the size of the angular separation α between them. Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

Plofker (2007, p. 420) Plofker, Kim (2007), "Mathematics in India", in Victor Katz (ed.), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press, ISBN 978-0-691-11485-9

worldcat.org

Kitāb al-Jawharatayn al-'atīqatayn al-mā'i'atayn min al-ṣafrā' wa-al-bayḍā': al-dhahab wa-al-fiḍḍah كتاب الجوهرتين العتيقتين المائعتين من الصفراء والبيضاء : الذهب والفضة. Cairo: Maṭba'at Dār al-Kutub wa-al-Wathā'iq al-Qawmīyah bi-al-Qāhirah. 2004. pp. 43–44, 87. OCLC 607846741.

Brahmagupta (English Wikipedia)

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