Series" and to be ascribed to the tenth century or even later) is impossible, as a comparison of the two texts would have shown.
I feel justified in assuming that the Āryahhaṭīya on the whole is genuine. It is, of course, possible that at a later period some few stanzas may have been changed in wording or even supplanted by other stanzas. Noteworthy is I, 4, of which the true reading bhūḥ, as preserved in a quotation of Brahmagupta, has been changed by Parameśvara or by some preceding commentator to bham in order to eliminate Āryabhaṭa's theory of the rotation of the Earth.
Brahmagupta criticizes some astronomical matters in which Āryabhaṭa is wrong or in regard to which Āryabhaṭa's method differs from his own, but his bitterest and most frequent criticisms are directed against points in which Āryabhaṭa was an innovator and differed from smṛti or tradition. Such criticism would not arise in regard to mathematical matters which had nothing to do with theological tradition. The silence of Brahmagupta here may merely indicate that he found nothing to criticize or thought criticism unnecessary. Noteworthy is the fact that Brahmagupta does not give rules for the volume of a pyramid and for the volume of a sphere, which are both given incorrectly by Āryabhaṭa (II, 6-7) . This is as likely to prove ignorance of the true values on Brahmagupta's part as lateness of the rules of Āryabhaṭa. What other rules of theGaṇitapāda could be open to adverse criticism? On the positive side may be pointed out the very close correspondence in termi-
