Re: [INDOLOGY] Brahmagupta's Tantraparīkṣā in Brāhmasphuṭasiddhānta

Agathe Keller kellera at
Tue Jan 5 20:11:20 UTC 2021

Dear Dean,

I fear that there is no English Translation of the 11th chapter of the BSS. Colebrooke translated chapter 12 and 18, Setsuro Ikeyama, edited and translated into English Chapter 21 with Pṛthūdaka’s commentary, and you can find also some parts of the BSS translated in Kim Plofker’s chapter called « mathematics in India » of a volume edited by Katz in 2007, but she skips over chapter 11.

I’m quite sure that S.N. Sen must have translated some parts of chapter 11 and discussed them, maybe in his History of Astronomy in India, a source book he co-published with K. S. Shukla in 1985 published by the INSA of which I do not have an accessible copy (I think it is out of print, and I admit i’d love to have a pdf of this text if anybody has one of it).

I was told that among his many projects, M. Yano wanted to publish a manuscript of D. Pingree which contained an English  translation of the BSS, translation that M. Yano wanted to correct and enhance, but I’m not sure whether this publication is on its way or not.

Chapter 21 contains some criticism of Āryabhaṭa (and of the Jains), by Brahmagupta expounded by Pṛthūdaka for which then there is thus Setsuro’s translation. Pingree wrote indirectly and not too technically about these criticism in his 1983 article in the JAOS .

As for secondary litterature, a bit of Brahmagupta’s criticism is discussed also in  Bongard-Levin, Grigorij M. « Āryabhaṭa and Lokāyatas ». Annals of the Bhandarkar Oriental Research Institute 58 (1977): 69-77.

But maybe some colleagues working on the history of astronomy can help pinch in some additional references on this question?



Ikeyama, Setsuro. Brāhmasphuṭasiddhānta (Ch. 21) of Brahmagupta with Commentary of Pṛthūdaka, Critically Ed. with Eng. Tr. and Notes. Reprinted from IJHS [Indian journal of history of science] 38. 1-4 (2003). New Delhi, India: Indian National Comission for the History of Science, Indian National Science Academy, 2003.

Pingree, David. « Brahmagupta, Balabhadra, Pṛthūdaka and Al-Bīrūnī » Journal of JAOS 103, no 2 (1983): 353‑60.

Plofker, Kim. « Mathematics in India ». In The Mathematics of Egypt, Mesopotamia, China, India and Islam. A Sourcebook., édité par Victor J. Katz, 385‑514. Princeton, N.J.: Princeton University Press, 2007.

Le 5 janv. 2021 à 16:11, Dean Michael Anderson via INDOLOGY <indology at<mailto:indology at>> a écrit :

Thanks! I would greatly appreciate the photos of just the English pages. If I need to go deeper, I can try to find the originals.

Yes, my first name is Dean and my middle name is Michael. Actually, you just solved a mystery for me: why I am occasionally called Michael. People must assume it's the academic title. Dean is a not uncommon name in the US ... regardless of educational qualifications. In fact, sometimes I wonder if there's not generally an inverse correlation. :-)



On Tuesday, January 5, 2021, 8:31:36 PM GMT+5:30, jmdelire via INDOLOGY <indology at<mailto:indology at>> wrote:

Dear Michael (or is Dean also your name ?),

I have at my disposal a discussion of the 11th chapter in an edition of
the BSS by "A Board of Editors headed by ACHARYAVARA RAM SWARUP SHARMA,
Chief Editor and Director of the Indian Institute of Astronomical and
Sanskrit Research" in 1966. The edition is accompanied by the Vasana,
Vijnana and Hindi Commentaries. Are you interested by a scan, or rather
photos because the (bulky) book is uneasy to scan. I also have the
edition (without translation nor discussion) by M.S.Dvivedin, Banaras,

Best regards,

Lecturer on Science and Civilization of India - Sanskrit Texts, IHEB
(University of Brussels, Belgium)

Le 04.01.2021 08:07, Dean Michael Anderson via INDOLOGY a écrit :
> 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
> [1]
> 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] [2] that this section on mathematics is spurious and is
> a much later addition.[5] [3] 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 the_Gaṇitapāda_
> 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] [4] 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
> Khaṇ_ḍakhādyaka_ seem to differ much from those of the
> _Āryabhaṭīya_.[7] [5] 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] [6]
>  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.
> Links:
> ------
> [1]
> [2]
> [3]
> [4]
> [5]

> [6]

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