would like to eliminate it since it seems to furnish evidence for Āryabhata's knowledge of place-value. Nothing is gained by doing so since Lalla gives in numerical words the most important numerical elements of Āryabhata without change, and even without this paribhāsā stanza the rationale of the alphabetical notation in general could be worked out and just as satisfactory evidence of place-value furnished. Further, Brahmagupta (Brāhmasphutasiddhānta, XI, 8) names the Dasagītika as the work of Āryabhata, gives direct quotations (XI, 5; I, 12 and XI, 4; XI, 17) of stanzas 1, 3, and 4 of our Dasagītika, and XI, 15 (although corrupt) almost certainly contains a quotation of stanza 5 of our Dasagītika. Other stanzas are clearly referred to but without direct quotations. Most of the Dasagītika as we have it can be proved to be earlier than Brahmagupta (628 A.D.).
The second section in 33 stanzas deals with mathematics. The third section in 25 stanzas is called Kālakriyā, or "The Reckoning of Time." The fourth section in 50 stanzas is called Gola, or "The Sphere." Together they contain 108 stanzas.
The Brāhmasphutasiddhānta of Brahmagupta was composed in 628 A.D., just 129 years after the Āryabhatīya, if we accept 499 A.D., the date given in III, 10, as being actually the date of composition of that work. The eleventh chapter of the Brāhmasphutasiddhānta, which is called "Tantraparīksā," and is devoted to severe criticism of previous works on astronomy, is chiefly devoted to criticism of Āryabhata. In this chapter, and in other parts of his work,
