Abstract
A renewed interest for contextualization in indological studies,1 is but slowly affecting publications on Indian mathematics. The isolation of history of mathematics within the general field of indology derives partly from a lively historiographical trend of technical and patriotic history of mathematics which remains oblivious to social science. Preservation plays a role as well: precious little information exists on the context in which mathematics and astronomy were practiced in India in the past.2 To overcome this problem some historians of science have turned to periods (XVIth–XIXth century) and places where institutions, libraries and many texts help us contextualize their mathematical and astronomical ideas.3
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
For the fundamental project headed by Sheldon Pollock, “Sanskrit Knowledge Systems at the Eve of Colonialism” (SKEC), see http://www.columbia.edu/itc/mealac/pollock/sks, (Pollock 2002). For the results of this approach in literature (Pollock 2006).
- 2.
(Plofker 2009) struggles to link a technical history to social questions, but both fields remain estranged from one another today.
- 3.
Among the publications on history of science, produced within Pollock’s SKEC, see the works of Christopher Minkowski and Dominik Wujastyk, listed at http://www.columbia.edu/itc/mealac/pollock/sks/papers/index.html. We may add to this the recently defended thesis of Toke Lindegaard Knudsen, (Knudsen 2008).
- 4.
A list of various categories of astronomical/astrological texts is given in (Pingree 1981) and reformulated in (Plofker 2009). Different kinds of texts, categorized by the subjects or styles they are likely to adopt, have long been identified by Indologists. Kim Plofker, 105–108, briefly contrasts the kind of trigonometrical astronomy stated in karaṇas with the astronomy given in siddhāntas. Mathematical difference and textual variations are noted, but not closely investigated. Plofker remarks that both kinds of texts received commentaries.
- 5.
K. Chemla has published extensively on this question. Her latest synthesis is (Chemla 2004).
- 6.
A similar set of questions about another kind of text, the colonial archives, can be found in (Stoler 2002).
- 7.
Stated more precisely, these commentaries were written between the seventh and twelfth century about texts which had been composed between the fifth and tenth century.
- 8.
Jyoti⋅a (lit. “the 〈sky’s〉 luminaries”), which is commonly translated as “astronomy,” more exactly means “astral science” because the field includes horoscopy and mathematics together with observational and computational astronomy. See (Pingree 1981, Introduction and Table of contents).
- 9.
(Pingree 1970–95). In 1955, D. Pingree started (see CESS I, preface.) a survey of manuscripts on jyoti⋅a that was remained unfinished when he died fifty years later in November, 2005. The CESS spans 5 volumes, and Pingree’s death interrupted the completion of Volume 6. For authors who have not yet been treated in the published volumes of CESS, one can refer to (Sen, Bag, and Sarma 1966) and to individual library catalogs. Despite its name, the CESS lists texts in many languages of the Indian subcontinent. Pingree cast a large net when undertaking his census and included texts that may refer to some part of jyoti⋅a only in passing. While some manuscripts doubtless existing in private collections have escaped classification by libraries or still others have been misclassified, most known manuscripts are probably included in his census.
- 10.
Because no electronic version of the text was available, the entries were counted manually: the evaluation may be subject to human error. Nonetheless, the investigation gives a general idea of the proportions involved. Since this investigation, the CESS has been partly digitalized (Volumes 1, 2 and 4) on http://books.google.com/. I have counted only manuscripts and have thus excluded references to authors for which there is no remaining text, as well as twentieth century publications by modern authors for which no manuscript remains.
- 11.
Indeed, of the 3,686 texts I have recorded in the first five volumes of the CESS, only 102 (2.7%) are clearly devoted at least partly to mathematics (gaṇita).
- 12.
This list does not include texts which were edited and translated. However, among the 102 texts listed, a significant number are in vernacular languages (particularly, oriya and tamil) and have seldom been edited or even translated. A majority of the parts of the Sanskrit texts concerning gaṇita have been edited and translated.
- 13.
Of the 3,686 texts and 2,972 authors devoted to jyoti⋅a, 816 (22%) are commentaries and 646 authors (21.7%) are commentators. Before starting this article, I believed that commentaries on mathematics had been largely neglected in the historiography of Indian science, and that they were an important part of the past tradition. Indeed, I mentioned this in the introduction to my book, (Keller 2006). This error was noted and rightly criticized by S. R. Sharma. (Sarma 2006, 144).
- 14.
- 15.
Commentaries on the treatises enumerated here have been written after our period but these commentaries are not listed here. We will return to this situation below.
- 16.
K. S. Shukla, who edited the text, believes that the commentary shares features with texts from the chosen time span. He especially draws similarities with the Bhakshālī Manuscript and the BSS. (Shukla 1959, xxviii–xxxiv.)
- 17.
See (Shukla and Sharma 1976, xxv–lviii). We have included Prabhākara in this account, although no extant commentary is known. Nonetheless, he is quoted by Bhāskara.
- 18.
CESS IV 255 b; V 239 b.
- 19.
See (Shukla and Sharma, 1976), (M. S. Dvivedin, 1902), (Hayashi, 1995), (K. S. Shukla, 1959), and (Rangacarya 1912). There have been numerous printed editions of L and BG, two texts which are noted in CESS 4 308 a and 311 b. See the translation given in (Brahmagupta; Bhāskarācārya; Colebrooke 1817).
- 20.
- 21.
The mathematical part is sometimes translated in (Colebrooke 1817)’s footnotes. Part of PBSS’s astronomical commentary has been studied, edited and translated by Setsuro Ikeyama (Ikeyama, 2003). Pṛthudak⋅vamin’s commentary on the twelfth chapter of the BSS, is found in a manuscript at the Indian Office and in what appears as a copy of this manuscript used by S. Dvivedi in Benares (CESS A. IV. 221 b), (Ikeyama, 2003, S5–7). This last commentary has not been edited entirely, probably because the only recension is at times quite difficult to understand.
- 22.
(Shukla 1976).
- 23.
CESS 1 51a–52b; 2 15b; 3 16a; 4 27b.
- 24.
(Shukla 1976), CESS IV 297b.
- 25.
(Sarma 1976, xvii–xxv).
- 26.
(Sen et al. 1966, 202), CESS I-II 51a.
- 27.
CESS IV 254 b-255b, CESS V 239 b-240 a.
- 28.
CESS IV 221 a, CESS V 224 a .
- 29.
(Ikeyama 2003, p. S7).
- 30.
- 31.
For an attempt to characterize “scholastic Sanskrit” of commentaries (in the case of grammatical, philosophical and logical texts), see (Tubb 2007).
- 32.
In the case of commentaries, many different technical names are recorded either in the titles or by D. Pingree in the CESS. Keeping in mind that these numbers and percentages should be used cautiously, the initial results may be reported. First, ⃛īkās compose 549 titles, (67.2% of all commentaries), but other technical names and titles appear. Vyākhyās represent 85 titles (or 10.4%); vivṛttis number 50 (or, 6.1%); and bhā⋅yas number 34 (or, 4.1%). Smaller numbers of avacūrṇis, vārtikkas, tippaṇaṇis, and vivāraṇas also occur. This diversity raises the general question of how the texts were titled. Unlike the compositions examined by C. Minkowski, the names of patrons seldom appear here. Moreover, were the titles composed by authors or by later scholars? Is it possible that the titles were modified by those who copied texts? These questions can be extended to all texts in the census. I count only 205 texts (or, 5.5%) with siddhānta in the title. Likewise, only 18 are described as śāstras and tantras. In 141 cases (or, 3.8%), the titles (saṅgraha, jñana) express the fact that the compositions bestow knowledge; in 118 cases (or, 3.2%), the titles (phala, sāra) convey the idea of providing an essence of something. Sometimes the titles indicate both of these concerns: (sārasaṅgraha is quite a common title compound). The metaphor of light (dīpika, prakāśa) occurs 289 times (or, 7.8%) and somewhat less frequently in the titles of 28 commentaries (or, 3.4%).
- 33.
(Pingree 1981). I have counted in the CESS 130 texts (or, 3.5%) bearing the name karaṇa or some associated title and 66 almanacs (or, 1.7%) bearing the name(pa \({\tilde c}\) āṇgas) or texts explaining how to make almanacs. The CESS further notes a number of non-standard texts, such as the Aparājitapṛcchā of Bhuvanadeva (fl. twelfth century), a text on architecture written as a dialogue (CESS V 264 a.).
- 34.
- 35.
(Shukla 1959, 1).
- 36.
See the maṅgalācaraṇam of BAB.2 : vyākhyānaṃ gurupādalabdham adhunā kiñcin mayā likhyate, (Shukla 1976, 43).
- 37.
Thus, at the end of the introduction which begins his commentary, Yajvan writes evam upodghātaṃ pradarśya śastraṃ vyākhyāyate. (Sarma 1976, 7) However, at the end of that chapter’s commentary, he refers to his own text as a prakāśa. (Sarma 1976, 32, 79 (note 11), 117, 185). Again, at the end of SYAB.2, Yajvan relies on the verbal root vyākh-.
- 38.
(Shukla 1959, 1).
- 39.
- 40.
See (Bronkhorst 1990).
- 41.
One should recall however that PG, given in Shukla’s edition, is known only from this single recension (Shukla 1959).
- 42.
According to CESS IV 221b, the text of PBSS corresponds to I.1–3, XXI.1–XXII.3, I.4–II.68, XV.1–9, III.1–XIV.55, XV.1à–XX.19, and XXII.4–XXIV.13, respectively, with chapters denoted by roman numerals and sections within chapter indicated by arabic numerals. It remains to be clarified what logic directed the composition of PBSS in this order.
- 43.
However, they also all quote other texts, albeit not completely. Thus, SYAB often quotes PG, and APG cites BSS. SYAB frequently paraphrases BAB. Therefore, commentaries may be described as composite texts, assembled from parts of previously composed texts which are sometimes rewritten or paraphrased, and intertwined with passages original to the commentary. The commentaries also frequently share versified examples and these can be considered a form of quotation as well.
- 44.
Perhaps this standard arose in modern (eighteenth-nineteenth century) scribal traditions, and may very well have its ultimate origin in practices introduced by European texts, or as suggested in (Plofker 2009, A.3, 305) by Islamic ones.
- 45.
Note also that a strong stylistic criteria which separates commentaries from treatises would help philologists who edit manuscripts to determine what portion belongs to the original text and what portion belongs to the commentary.
- 46.
See, for instance, (Shukla 1976, Introduction).
- 47.
(Hayashi 1995, 85).
- 48.
In his edition of PG and APG, Shukla seems hesitant: are the examples part of the treatise or part of the commentary? All editions of L consider examples part of the treatise.
- 49.
In several parts of her book, (Plofker 2009) offers several rough characteristics of mathematical commentaries: their uses of proofs and language games essentially.
- 50.
(Sarma 1976, Introduction, xvii–xx).
- 51.
(Sarma 1976, Introduction, xix)
- 52.
Oddly, SYAB states this clearly in its general introduction to Ab (Sarma 1976, xxv–xxvi, 2–4). Āryabha⃛a can scarcely be considered a ritualist, given the fame he garnered for taking puranic cosmology lightly.
- 53.
(Sarma 1976, xviii).
- 54.
The “Index of Manuscripts” of the library notes that “In 1940 it possessed 3000 manuscripts, 142 publications in Sanskrit, 63 in Malayalam. Travancore University (which became the University of Kerala) organized after its establishment (1938) a manuscript preservation and collection department. Both were amalgamated in 1940. In 1958 there was (sic) 28 000 Codices in Sanskrit; 5 000 in Malayalam.”
- 55.
- 56.
- 57.
(Brahmagupta; Bhāskarācārya; Colebrooke, 1817).
- 58.
Op.cit.; p. xvi.
- 59.
(Kejariwal 1988, 111–112).
- 60.
CESS V 453 a.
- 61.
Op.cit. Note A p. xxv and p.xxvii
- 62.
Op.cit. Introduction, sections G to I pp. xxxvii–xiv.
- 63.
Op.cit. p.v Colebrooke writes that he has a copy of “Srídhara’s compendium of arithmetic”, which is probably the Triśatika.
- 64.
(Brahmagupta; Bhāskarācārya; Colebrooke 1817, iii)
- 65.
- 66.
(Keller 2006b).
- 67.
(Kern 1874).
- 68.
The tradition of copying manuscripts can, of course, also be understood as the editorial tradition of classical India, but printed books are referred to here.
- 69.
(Rodet 1879)
- 70.
(Dikshit 1896).
- 71.
(Kaye 1908).
- 72.
(Sengupta 1927).
- 73.
(Clark 1930).
- 74.
(Datta 1935).
- 75.
Despite this tendency, they also embraced the dying tradition of mastering Sanskrit texts on jyoti⋅a. At the end of his life, B. Datta was addressed as pandit. The enduring quality of their translations reflect their mastery of the Indian intellectual tradition. Indeed, “Hindu Mathematics” effectively blends two traditions, the lore of jyoti⋅a and the modern history of mathematics. A detailed scrutiny of Datta & Singh’s works could probably yield much useful information about the tradition of jyoti⋅a.
- 76.
(Datta 1935, Volume I, 125).
- 77.
(Datta 1935, Volume I, 66–67; 196; 211. Volume II, 93–95).
- 78.
(Datta 1935, Volume I, 170). See also, SYAB as in op. cit.[Volume II, 91, footnote 4].
- 79.
(Datta 1935, Volume I, 80, 82, 87, 130, 204, 239; Volume II, 87, 238).
- 80.
Thus, the whole portion of Volume 2 devoted to the ku⃛⃛aka quotes all the different texts which appear in Bhāskara I, even though these passages sometimes present Bhāskara’s own algorithms, and at other times only explain algorithms in Ab.
- 81.
(Shukla 1959).
- 82.
- 83.
- 84.
How closely can these attitudes toward commentaries be linked to developments in the field of Indology generally? Indeed, Indology has developed a special emphasis on the study of treatises and the contents of important commentaries but has somewhat neglected any reflection on the commentary as a specific kind of text. In the last five-to-ten years, however, a renewed interest in this kind of text has surfaced, as illustrated by the recent conference titled “Forms and Uses of the Commentary in the Indian world”, held at Pondicherry in February 2005. See http://www.ifpindia.org/Forms-and-Uses-of-the-Commentary-in-the-Indian-World.html., or the previously cited publication (Tubb 2007).
- 85.
Before publishing his edition of BAB, Shukla published a number of analyses, which pinpointed the mathematical relevance of the text. (Shukla 1972)
- 86.
Srinivas (1990), Patte (2004), and Plofker (2009). Strangely enough, few reflections on comments connected to the definition of the field of gaṇita have been published. Such reflections might help explain why chapters on gaṇita contain algorithms with little astronomical application, although they are included in treatises on astronomy. See Pingree (1981), Keller (2007), and Plofker (2009).
- 87.
- 88.
bhāgaṃ hared avargān nityaṃ dviguṇena vargamūlena|
vargād varge śuddhe labdhaṃ sthānāntare mūlam||
For an explanation of the algorithm, see Shukla (1976, 36–37).
- 89.
Shukla (1976, 36, line 15)
atra gaṇite vi⋅ amam sthānam vargaḥ
- 90.
Sarma (1976, 36, line 15).
saṃkhyā vinyāsasthāne⋅u vi⋅amasthānāni vargasaṃjñāni
- 91.
See Shukla (1959, 18 for the Sanskrit, 9–10 of the part in English for an explanation of the procedure as described in APG)
vi amāt padas tyaktyvā vargaṃsthānacyutena mūlena|
dviguṇena bhajec che⋅aṃ labdhaṃviniveśayet paṇktau ||
tadvargaṃsaṃśodhya dviguṇaṃ kurvīt purvaval labdham|
utsārya tato vibhajec śe⋅aṃ dviguṇīkṛtaṃ dalayet ||:
- 92.
Shukla (1959, 18–19 of the Sanskrit, 9–10 of the English version. Taken together, they reveal Shukla’s interpretation of how numbers are initially disposed and how they change during the execution of an algorithm, according to APG).
- 93.
Shukla (1959, 18 line 15–16).
- 94.
(Keller 2006b).
- 95.
bhāgaṃ hared avargān nityaṃ dvigu \(n_{\hskip -4pt.}\) ena vargamūlena|
vargād varge śuddhe labdhaṃ sthānāntare mūlam||
For an explanation of the algorithm, see (K. S. Shukla: 1976, 36–37).:
- 96.
BAB introduces this assessment through a linguistic analysis of the term avarga, noting: (Shukla 1976, 52)
tasya eva nañā vi⋅amatve pratisiddhe avargaḥ iti samam sthānam yataḥ hi vi⋅amam samam ca sthānam
Since a non-square 〈takes place〉 when oddness is denied, by means of 〈the affix〉 nañ 〈the expression refers to〉 an even (sama) place, because, indeed, a place is either odd or even.
- 97.
See Shukla (1959, 18 for the Sanskrit, 9–10 of the part in English for an explanation of the procedure as described in APG:
vi⋅amāt padas tyaktyvā vargaṃ sthānacyutena mūlena|
dvigu \(n_{\hskip -4pt.}\) ena bhajec che⋅aṃ labdhaṃ viniveśayet paṇktau ||
tadvargaṃ saṃśodhya dviguṇaṃ kurvīt purvaval labdham|
utsārya tato vibhajec śe⋅aṃ dviguṇīkṛtaṃ dalayet ||:
- 98.
Shukla (1959 18, line 10–12)
vargarāśer vi⋅amāt padād ojākhyād ekatṛtīyapañcamasaptamāder ekaśatāyutaprayutādisthānebhyo `nyatamasthānād antyāt padāt sambhavinaṃṃtyajet:
- 99.
The adjective saṃbhavin conveys both the meaning “appropriate” and “conjectured”, the subtext is thus understood an “appropriately conjectured” square.
- 100.
Shukla (1959, 18, line 19–22)
ānulomyena ekasthānāc catu ⋅ kāt prabh ṛ ti vi ⋅ ama ṃ sama ṃ vi ⋅ ama ṃ samam iti saṃ jñākara ṇ am /
Table 6 Setting down: atra catuḥ⋅aḍa⋅⃛akāni ekaśatāyutasthānāni vi⋅amapadāni tebhyo ‘yutasthānastham a⋅⃛akam antyaṃ vi⋅amapadaṃ:
- 101.
eka however is used here and not prathāma.
References
Aklujkar, A. 2001. Paṇḍita and Pandits in history. In The Pandit, traditional scholarship in India, ed. A. Michaels, (P. 17–38). New Delhi: Manohar.
Bag, A.K. 1979. Mathematics in ancient and medieval India (No. 16). Chaukhambha oriental research studies. Varanasi, India: Chaukhambha Orientalia.
Bayly, C.A. 1996. Empire and information : intelligence gathering and social communication in India, 1780–870. Cambridge, NY: Cambridge University Press.
Bernard, A. 2003. Comment définir la nature des textes mathématiques de l’antiquité tardive? Proposition de réforme de la notion de “Textes Deutéronomique”. Revue d’Histoire des mathématiques 9: 131–173.
Billard, A. 1971. L’astronomie Indienne. EFEO.
Brahmagupta; Bhāskarācārya; Colebrooke, H.T. 1817. Algebra, with Arithmetic and mensuration. London J. Murray.
Bronkhorst, J. 1990. Vārttika. Wiener Zeitschrift fuer die Kunde Suedasiens 34.
Chemla, K. 1999. Commentaires, Editions et autres textes seconds: Quel enjeu pour l’histoire des mathématiques? Réflexions inspirées par annote de Reviel Netz. Revue d’Histoire des mathématiques 5: 127–148.
Chemla, K. 2004. History of Science, History of Text: An Introduction. In History of science, history of text, ed. K. Chemla, xxvii. Boston, MA: Springer.
Clark, E.C. 1930. The Āryabha⃛īya of Āryabha⃛a. Chicago: University of Chicago Press.
Datta, A.N., Bibhutibhushan and Singh. 1935. History of Hindu Mathematics Vol. 2. Lahore: Motilal Banarsidass.
Dikshit, S.B. 1896. Bhā⃛iya Jyotiḥśāstra (Reprinted Poona 1931. Hindi translation b S. Jhārkhaḍī in Lucknow 1957. ed.). Poona.
Dvivedin, M.S. 1902. Brāhmasphu⃛asiddhānta and Dhyānagrahopade⋅ādhyāya by Brahmagupta edited with his own commentary The paṇḍit XXIV: 454.
Dvivedin, S. 1910. The Mahāsiddhānta of Āryabha⃛a. Benares Sanskrit Studies 148, 149, 150.
Dvivedin, S. 1910. The Mahāsiddhānta of Āryabha⃛a. Benares Sanskrit Studies 148, 149, 150.
Dvivedin, S.e. 1899. Triśatikà by Śridharàchàrya. Jagannath Mehta.
Hayashi, T.é. 1995. The Bakhshālī Manuscript. An ancient Indian mathematical treatise. Egbert Forsten.
Ikeyama, S. 2003. Brāhmasphu⃛asiddhānta (Ch. 21) of Brahmagupta with Commentary of Pṛthudaka, critically ed. with Eng. tr. and Notes (Vol 38). New Delhi, India Indian Journal of History of Science.
Kāpadīā, H.R. 1937. Ganitatilaka, edited with introduction by H. R. Kapadia and commentary of Simhatilaka Sūri. Gaekwad Oriental Series (78).
Kaye, G.R. (March 1908). Notes on Indian Mathematics. no. 2. Āryabha⃛a. Journal of the Asiatic Society of Bengal 8.
Kejariwal, O.P. 1988. The Asiatic Society of Bengal and the discovery of India’s Past. Oxford: Oxford University Press.
Keller, A. 2006. Expounding the mathematical seed, Bhāskara and the mathematical chapter of the Āryabha⃛īya (Vol 2 volumes). Basel: Birkhäuser.
Keller, A. 2006b. Comment on a écrit les nombres dans le sous- continent indien, histoires et enjeux. In Scéance du 17 Novembre 2006, Hommage rendu à Jean Filliozat, ed. P. -S. Filliozat, Paris: Académie des Belles Lettres et Société Asiatique.
Keller, A. 2007. Qu’est ce que les mathématiques? Les réponses taxinomiques de Bhāskara un commentateur, mathématicien et astronome du VIIème siècle In Sciences et frontières M. Hert Phillipe; Paul-Cavalier ed. P. 29–61. Kimé
Kern, H. 1874. The Āryabha⃛īya, with the commentary Bhatadīpika of Parameśvara (R. from Eastern Book Linkers, Ed).
Knudsen, L.T. 2008. The Siddhāntasundara of Jñānarāja, A critical edition of select chapters with English translation and commentary. Unpublished doctoral dissertation, Brown University.
Netz, R. 1998. Deuteronomic Texts: Late Antiquity and the History of Mathematics. Revue d’Histoire des mathématiques 4: 261–288.
Patte, F. 2004. Le Siddhāntaśiromāni : l’oeuvre mathématique et astronomique de Bhāskarācarya = Siddhāntaśiromāniḥ : Śri-Bhāskarācarya-viracitaḥ. Droz.
Pingree, D. 1970–1995. Census of the Exact Sciences in Sanskrit (CESS) (Vol. 5 volumes). Philadelphia, PA: American Philosophical Society.
Pingree, D. 1981. Jyotiḥśāstra : astral and mathematical literature. Wiesbaden: Harrassowitz.
Plofker, K. 2009. Mathematics in India. Princeton, NJ: Princeton University Press.
Pollock, S. 2002. Introduction: Working papers on Sanskrit knowledge systems on the eve of colonialism. Journal of Indian Philosophy 30(5): 431–9.
Pollock, S. 2006. The Language of the Gods in the world of men: sanskrit, culture and power in premodern India. Berkeley, CA: University of California Press.
Raina, D. 1999. Nationalism, institutional Science and the politics of knowledge; Ancient indian astronomy and mathematics in the landscape of french enlightenment historiography. unpublished doctoral dissertation, Goteborgs University.
Raina, D. 2003. Images and contexts: the historiography of science and modernity in India (Vol. xi 234). New-Delhi: Oxford University Press.
Rangacarya, M. 1912. The Gaṇitasārasaṃgraha of Mahāvīra, edited with translation and notes. Madras, India Madras Goverment Publication.
Rodet, L. 1879. Leçons de calcul d’Āryabhā⃛a. Journal Asiatique 7: 393–434.
Sarma, K.V. 1976. Āryabha⃛īya of Aryabha⃛a with the commentary of Sūryadeva yajvan. New-Delhi, India INSA.
Sarma, S.R. 1966. Grahagaṇitādhyāya of the Mahāsiddhānta of Āryabha⃛a II Marburg.
Sarma, S.R. 2006. Review of expounding the mathematical seed. Aestimatio. 3: 142–146.
Sen, S.N., Bag, A.K. and Sarma, R.S. 1966. A bibliography of Sanskrit works on astronomy and mathematics. National Institute of Sciences of India.
Sengupta, P.C. 1927. The Aryabhatiyam, translation., XVI.
Shukla, K.S. 1959. Pā⃛īgaṇita of Śrīdharācarya. Lucknow: Lucknow University.
Shukla, K.S. 1960. MahābhāskarIya, edited and translated into English, with explanatory and critical notes, and comments, etc. Lucknow: University, Department of Mathematics Lucknow.
Shukla, K.S. 1963. Laghubhāskarīya, edited and translated into English, with explanatory and critical notes, and comments, etc. Lucknow: Department of Mathematics and Astronomy, Lucknow University.
Shukla, K.S. 1972. Hindu mathematics in the seventh century as found in Bhāskara I’s commentary of the Āryabha⃛īya. Gaṇita IV (2): 41–50.
Shukla, K.S. 1976. Āryabha⃛īya of Āryabha⃛a, with the commentary of Bhāskara I and Someśvara. New Dehli: Indian National Science Academy.
Shukla, K.V. and Sharma, K.S. 1976. Āryabha⃛īya of Āryabha⃛a, critically edited with translation. New Delhi: Indian National Science Academy.
Srinivas, M.D. 1990. The methodology of Indian mathematics and its contemporary relevance. In History of science and technology in India; Mathematics and astronomy (Vol. II, Chap.2). New Delhi: Sundeep Prakashan.
Stoler, A.L. 2002. Colonial archives and the arts of governance. Archival Science 2: 87–109.
Tubb, E.R. Gary A. and Boose. 2007. Scholastic Sanskrit, A manual for students. New York: American Institute of Buddhist Studies, Colombia University’s Center for Buddhist Studies, Tibet House U S.
Acknowledgment:
I would like to thank K. Chemla, F. Bretelle, C. Proust and M. Ross for their comments, suggestions, improvements and encouragement offered on several drafts of this article.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media B.V.
About this chapter
Cite this chapter
Keller, A. (2010). On Sanskrit Commentaries Dealing with Mathematics (Fifth–Twelfth Century). In: Bretelle-Establet, F. (eds) Looking at it from Asia: the Processes that Shaped the Sources of History of Science. Boston Studies in the Philosophy of Science, vol 265. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3676-6_7
Download citation
DOI: https://doi.org/10.1007/978-90-481-3676-6_7
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-3675-9
Online ISBN: 978-90-481-3676-6
eBook Packages: Humanities, Social Sciences and LawHistory (R0)