^{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.