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Published online by Cambridge University Press: 15 March 2011
MANY writers have enlarged upon the subject of our indebtedness to India in matters intellectual, and in particular have drawn attention to ancient Hindu mathematics, which they consider exhibit in a marked degree the intellectual superiority of the Hindus in early times. They not only inform us that a system of mathematics was developed in India in early times, but imply that in this direction the Hindus were the benefactors of the rest of mankind. The latest authoritative statement of this kind is as follows: “In the mathematical sciences the achievements of the Indians have been very considerable. As the inventors of the numerical figures with which the whole world reckons, and of the decimal system connected with the use of these figures, they naturally became the greatest calculators of antiquity, just as the Greeks were the greatest geometricians . . . The later mathematicians made more progress in trigonometry, especially by the invention of the sine table. The greatness of the Indian mathematical writers who belong to the fifth century and later lies in their arithmetical and algebraical investigations . . . The raising of numbers to various powers and the extraction of the square or cube root were but elementary operations to these mathematicians. They also calculated arithmetical progressions, perhaps first suggested by the chess-board of sixty-four squares, which was known in India before the beginning of our era. They attained the greatest eminence in algebra, which they developed to a degree beyond anything ever achieved by the Greeks.”
page 750 note 1 Imperial Gazetteer of India, 1908, vol. ii, p. 265Google Scholar.
page 750 note 2 Elphinstone, , History of India, seventh edition, edited by Cowell, , p. 141 f.Google Scholar; Monier-Williams, , Modern India and the Indians, p. 286Google Scholar; Dutt, R. C., A History of Civilisation in Ancient India, vol. ii, p. 246Google Scholar; Mazelière, La, Essai sur l'évolution de la Civilisation indienne, vol. i, p. 81Google Scholar; etc., etc.
page 751 note 1 History of India, p. 142.
page 751 note 2 See the Metrica, viii, p. 18 f., and the Dioptra, xxx, p. 280 f., ed. Schöne.
page 751 note 3 By Vincent, A. H. in Notices et extraits des MSS. de la Biblioth. Impér., 1858, p. 157 fGoogle Scholar.
page 752 note 1 Bürk, A., “Das Āpastamba-Śulba-Sütra”: Zeitschrift der deutschen morgenländischen Gesellschaft, 1901, p. 543 f.Google Scholar; 1902, p. 327 f.
page 752 note 2 Ibid.
page 752 note 3 Rodet, L., Leçons de Calcul d'Āryabhaṭa, p. 22Google Scholar; Kaye, , JASB., 1908, p. 122Google Scholar.
page 752 note 4 Ibid., p. 123.
page 752 note 5 Ibid., p. 135.
page 752 note 6 Colebrooke, , Algebra with Arithmetic, and Mensuration from the Sanscrit, p. 363 fGoogle Scholar.
page 752 note 7 See H. Konen's Geschichte der Gleichuny, t2 – Du 2 = 1.
page 753 note 1 Colebrooke, p. 296.
page 753 note 2 See Bühler's introduction to Āpastamba in the Sacred Books of the East, vol. ii.
page 753 note 3 e.g. compare the sets of rational right-angled triangles given by Baudhāyana and Āpastamba.
page 753 note 4 See Vogt's, H. paper in the Bibliotheca Mathematica, 1906, p. 6 f.Google Scholar: “Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?” and Heath's, H. T.The Thirteen Books of Euclid's Elements, vol. i, p. 360Google Scholar.
page 753 note 5 Euclid's Elements, vol. i, p. 363.
page 754 note 1 A Short History of Greek Mathematics, p. 299.
page 754 note 2 Albiruni, , India, i, p. 168Google Scholar.
page 754 note 3 See the Pañchasiddhāntika, ed. Thibaut, cli. iv; Burgess, J., Indian Antiquary, 1891, p. 228Google Scholar; Kaye, , JASB., 1908, p. 125Google Scholar.
page 754 note 4 The Chinese and Japanese Repository, vol. i, p. 411 f.; Cantor, , Vorlesung über Geschichte der Mathematik, vol. i, p. 685 (3rd ed., 1907)Google Scholar.
page 755 note 1 Colebrooke, p. 366, § 72.
page 755 note 2 Ibid., § 70.
page 755 note 3 Ibid., §77.
page 755 note 4 Ibid., §73.
page 755 note 5 Ibid., § 64.
page 755 note 6 Vīja-Gaṇita, § 208.
page 756 note 1 Fleet, , Indian Antiquary, vol. xxx, p. 205Google Scholar.
page 756 note 2 For details see my previous paper in the JASB., 1907, p. 481 f.
page 756 note 3 Bīja Gannita: or the Algebra of the Hindus, p. 17.
page 756 note 4 Sūrya siddhānta, p. 335.
page 756 note 5 Lilāwati: or a Treatise on Arithmetic and Geometry by Bhāscara Achārya, p. 11.
page 756 note 6 Kaye, , “The Use of the Abacus in Ancient India”: JASB., 1908, p. 293 fGoogle Scholar.
page 757 note 1 Thibaut, , JASB., 1875, p. 261Google Scholar.
page 757 note 2 Possibly these did not come from Ptolemy, but indirectly from Hipparchus.
page 758 note 1 These problems may be compared with those in the Palatine Anthology.
page 758 note 2 Colebrooke, p. 295.
page 758 note 3 e.g. a square and an isosceles-parallel-trapezium.
page 758 note 4 Līlāvatī, § 172.
page 759 note 1 At least three embassies to the Roman Emperors and a large number to China are recorded. C. Mabel Duff, The Chronology of India.
