[INDOLOGY] Brahmagupta's Tantraparīkṣā in Brāhmasphuṭasiddhānta

[Dean Michael Anderson]

I ran across this mention of Brahmagupta criticizing Āryabhaṭa
Does anyone know where I might find an English translation or discussion of this? I fear I would soon be lost in a sea of technical mathematical and astronomical terms if I tried to read it in Sanskrit!
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. 

Best,
Dean

https://en.wikisource.org/wiki/The_Aryabhatiya_of_Aryabhata
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, Brahmagupta refers to Āryabhaṭa some sixty times. Most of these passages contain very general criticism of Āryabhaṭa as departing from smṛti or being ignorant of astronomy, but for some 30 stanzas it can be shown that the identical stanzas or stanzas of identical content were known to Brahmagupta and ascribed to Āryabhaṭa. In XI, 8 Brahmagupta names the Aryāṣṭaśata as the work of Āryabhaṭa, and XI, 43, jānāty ekam api yato nāryabhaṭo gaṇitakālagolānām, seems to refer to the three sections of our Aryāṣṭaśata. These three sections contain exactly 108 stanzas. No stanza from the section on mathematics has been quoted or criticized by Brahmagupta, but it is hazardous to deduce from that, as Kaye does,[4] that this section on mathematics is spurious and is a much later addition.[5] To satisfy the conditions demanded by Brahmagupta's name Aryāṣṭaśata there must have been in the work of Āryabhaṭa known to him exactly 33 other stanzas forming a more primitive and less developed mathematics, or these 33 other stanzas must have been astronomical in character, either forming a separate chapter or scattered through the present third and fourth sections. This seems to be most unlikely. I doubt the validity of Kaye's contention that the Gaṇitapāda was later than Brahmagupta. His suggestion that it is by the later Āryabhaṭa who was the author of the Mahāsiddhānta (published in the "Benares Sanskrit 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 terminology and expression between the fuller text of Brahmagupta, XVIII, 3-5 and the more enigmatical text of Āryabhaṭīya, II, 32-33, in their statements of the famous Indian method (kuttaka) of solving indeterminate equations of the first degree. It seems probable to me that Brahmagupta had before him these two stanzas in their present form. It must be left to the mathematicians to decide which of the two rules is earlier.

The only serious internal discrepancy which I have been able to discover in the Āryabhaṭīya is the following. Indian astronomy, in general, maintains that the Earth is stationary and that the heavenly bodies revolve about it, but there is evidence in the Āryabhaṭīya itself and in the accounts of Āryabhaṭa given by later writers to prove that Āryabhaṭa maintained that the Earth, which is situated in the center of space, revolves on its axis, and that the asterisms are stationary. Later writers attack him bitterly on this point. Even most of his own followers, notably Lalla, refused to follow him in this matter and reverted to the common Indian tradition. Stanza IV, 9, in spite of Parameśvara, must be interpreted as maintaining that the asterisms are stationary and that the Earth revolves. And yet the very next stanza (IV, 10) seems to describe a stationary Earth around which the asterisms revolve. Quotations by Bhaṭṭotpala, the Vāsanāvārttika, and the Marīci indicate that this stanza was known in its present form from the eleventh century on. Is it capable of some different interpretation? Is it intended merely as a statement of the popular view? Has its wording been changed as has been done with I, 4? I see at present no satisfactory solution of the problem.

Colebrooke[6] gives caturviṁṡaty aṁṡaiṡ cakram ubhayato gacchet as a quotation by Munīśvara from the Āryāṣṭaśata of Āryabhaṭa. This would indicate a knowledge of a libration of the equinoxes. No such statement is found in our Āryāṣṭaśata . The quotation should be verified in the unpublished text in order to determine whether Colebrooke was mistaken or whether we are faced by a real discrepancy. The words are not found in the part of the Marīci which has already been published in the Pandit.

The following problem also needs elucidation. Although Brahmagupta (XI, 43-44)

⁠jānāty ekam api yato nāryabhaṭo gaṇitakālagolānām |
⁠na mayā proktāni tataḥ pṛthak pṛthag dūṣaṇāny eṣām ||
⁠āryabhaṭadūṣaṇānām saṁkhyā vaktuṁ na śakyate yasmāt |
⁠tasmād ayam uddeśo buddhimatānyāni yojyāni ||

sums up his criticism of Aryabhata in the severest possible way, yet at the beginning of his Khaṇḍakhādyaka, a karaṇagrantha which has recently been edited by Babua Misra Jyotishacharyya (University of Calcutta, 1925), we find the statement vakṣyāmi Khaṇḍakhādyakam ācāryāryabhaṭatulyaphalam. It is curious that Brahmagupta in his Khaṇḍakhādyaka should use such respectful language and should follow the authority of an author who was damned so unmercifully by him in Tantraparīkṣā of his Brāhmasphuṭasiddhānta. Moreover, the elements of the Khaṇḍakhādyaka seem to differ much from those of the Āryabhaṭīya.[7] Is this to be taken as an indication that Brahmagupta here is following an older and a different Āryabhaṭa? If so the Brāhmasphuaṭasiddhānta gives no clear indication of the fact. Or is he following another work by the same Āryabhaṭa? According to Dīkṣit,[8]


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The Aryabhatiya of Aryabhata - Wikisource, the free online library


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 the Khaṇḍakhādyaka agrees in all essentials with the old form of the Sūryasiddhānta rather than with the Brāhmasphuaṭasiddhānta. Just as Brahmagupta composed two different works so Āryabhaṭa may have composed two works which represented two different points of view. The second work may have been cast in a traditional mold, may have been based on the old Sūryasiddhānta, or have formed a commentary upon it.




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