Abstract
In 1817, Henry Thomas Colebrooke (1765–1837) published English translations of different Sanskrit mathematical texts, including a translation of the twelfth chapter of Brahmagupta’s Brāhmasphuṭasiddhānta (‘Theoretical Treatise on the true Brahmā [school of astral science]’ 628 CE), with the help of Pṛthūdaka’s Vāsanābhāṣya (‘Commentary with explanations’ fl. 960). Almost ninety years later, in 1902, Sudhākara Dvivedin (1855–1910/11) published an edition of the Brāhmasphuṭasiddhānta with his own commentary, which refers also to Pṛthūdaka’s commentary. Since both texts were crucial to the access one could have to Brahmagupta’s mathematics, their point of view durably marked the historiography. Dvivedin’s and Colebrooke’s publications are striking for the different ways they deal with the structure of Brahmagupta’s mathematical chapter, their relation to Pṛthūdaka’s commentary, or numerical practices.
I will argue that these authors’ views on commentaries, structure and number went together with a specific idea of what a Sanskrit mathematical text was to be, informing their editorial choices, and shaping the views we have of the text today.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 269804. I would like to thank A. Graheli, K. Preisendanz, M. Schneider, X. Zhou, C. Montelle, K. Margolis, K. Chemla and D. Bayuk for their helpful comments on previous drafts of this chapter.
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Notes
- 1.
Colebrooke (1817), which also includes translations of Chap. 18 of the Brāhmasphuṭasiddhānta, and of the Līlavatī and Bījagaṇita by Bhāskara (b. 1114).
- 2.
Dvivedin (1902).
- 3.
- 4.
As evidenced by CESS (A4:256–257). The first published edition of the Khaṇḍakhādyaka was Babuā (1925). Dvivedin evokes this text and quotes from some of its commentaries, which shows that he had access to this work, probably through manuscript copies.
- 5.
- 6.
The oldest known commentary of both works, by Balabhadra, has not been preserved. See Pingree (1983).
- 7.
- 8.
See below for a discussion of Dvivedin (1902: 3–4). It seems that he mainly relied on a corrected copy of Colebrooke’s manuscript. As an editor, I noticed that Dvivedin’s copy often corrected some mistakes of Colebrooke’s manuscripts but added others.
- 9.
(Colebrooke 1817: iv–vi; xxviii–xxxiii).
- 10.
Their manuscript copies are now the basis for today’s editions of PBSS. Setsuro Ikeyama edited Pṛthūdaka’s commentary on Chap. 21 of the BSS (Ikeyama 2003). I have started an edition of Pṛthūdaka’s commentary on Chap. 12 of the BSS using the only three known manuscripts of this part of the commentary: a manuscript that belonged to Colebrooke and was annotated by him, a copy of this manuscript and a copy (Pingree’s actual copy) of the manuscript used by Sudhākara Dvivedin to edit Brahmagupta’s text with detailed annotated readings and his own handwritten corrections.
- 11.
See, for instance, the contemporary attempt by Kichenassamy (2010, 2012) to give a new interpretation of Brahmagupta’s rules on cyclic quadrilaterals, arguing that the Sanskrit tradition actually misunderstood them. By contrast, B. Chatterjee gave a very literal translation of the text of the KK, peppered with Sanskrit words, as interpreted from Utpalla’s point of view; see Chatterjee (1970). S. Pollock’s chapter in this volume touches on the wider implications of these questions for the Sanskrit philological tradition.
- 12.
See Ikeyama (2003:2–3); Colebrooke (1817: Appendix B, xxviii–xxix); Dvivedin (1902: 2). It contains in total 25 chapters, if you count the versified tables (dhyanagrahopadeṣādhyāya) that Dvivedin attached to his edition and that have progressively been considered as belonging to the treatise as well.
- 13.
See Ikeyama (2003: S7 and S9). Here, Pṛthūdhaka may also be writing a specific kind of commentary which could include both adding explanations and not following the order of the treatise. It is possible that he may be following Balabhadra in this respect. Pṛthūdaka does the same with the KK. Both Utpalla and Āmarāja would also rearrange the verses of the KK in their commentaries, which quote and are inspired by PBSS and PKK. Rearrangement for coherence and the genre of commentary probably go hand in hand here.
- 14.
parikarmaviṃśatiṃ saṅkalitādyāṃ pṛthag vijānāti| aṣṭau ca vyavahārān chāyāntaṃ bhavati gaṇakas saḥ||
- 15.
Dvivedin (1902:172).
- 16.
See Rocher and Rocher (2012) for a detailed biography.
- 17.
Colebrooke (1839: 16–17): ‘The original object for which he sought to acquire a knowledge of Sanscrit (sic) language had a reference to the treatises on Algebra, which he afterwards translated and gave to the world. However, when he had mastered the language, his attention was diverted to other subjects, and that design was not executed until 20 years afterwards.’Colebrooke (1839:50–51): ‘His earliest labour, after his arrival in England, was to prepare for publication a work on which he had been engaged during his homeward voyage. It consisted of a translation of the most celebrated treatises on Indian Algebra, accompanied by a dissertation on the state of the science as cultivated by the Hindus. The subject is interesting in the history of his writings, as being (as I have already mentioned) that which first led him to the study of the Sanscrit language, and which was laid aside for about 20 years’
- 18.
Along the way (sometime in the first half of the twentieth century), this topic seems to have ‘fallen off the Indological wagon’ and it became a separate technical part of the history of science in India, undertaken by trained mathematicians rather than by Indologists.
- 19.
The expression is used in Colebrooke (1817: ii).
- 20.
Colebrooke (1817: ii): (a quotation respecting his yet non standardised orthography of Sansrkit names) ‘The treatises in question, which occupy the present volume, are the Víja-gańita and Lílávatí of Bháscara áchárya and the Gańitád’haya and Cuťťacád’hyaya of Brahmegupta. The two first mentioned constitute the preliminary portion of Bháscara’s Course of Astronomy, entitled Sidd’hánta-śirómańi. The two last are the twelfth and eighteenth chapters of a similar course of astronomy, by Brahmegupta, entitled Brahma-sidd’hánta.’
- 21.
Colebrooke (1817: Appendix D, xxxi–xxxiii) confirmed the fact that he had read other parts of the BSS as well.
- 22.
Since this is well known and documented, we will not dwell on it. See Raina (2012:231–239; 242–246), Rocher and Rocher (2012:133–135) and Smadja (2016:19). Indeed, at the outset, Colebrooke (1817: i): writes ‘observations will be added (to the translations), tending to a comparison of the Indian, with the Arabian, the Grecian and the modern Algebra.’
- 23.
See, for instance, the consequence in Peacock’s understanding of Indian mathematics, as analyzed in Raina (2011: 243): ‘It is interesting that in tracing the linkages between Āryabhaṭa and Bhāskara, Peacock is gesturing towards not only a genealogy of the arithmetical tradition in India but also towards a continuity of the tradition spanning several centuries.’ Peacock, we know, read Colebrooke carefully.
- 24.
Colebrooke (1817: i).
- 25.
But we know he was intent on finding a way of having Devanagari printed, as evidenced by his correspondence with Schlegel and his proximity to William Carrey. Rocher (2013: 76–77).
- 26.
The British Library has a copy of Colebrooke’s manuscript of Pṛthūdaka’s commentary in book form and clear reading, partly corrected, which, as suggested to me by A. Graheli, might have been a preprint copy of the manuscript. However, I do not know when or by whom such a copy was made.
- 27.
Colebrooke (1817: i) (my emphasis): ‘The object of the present publication is to exhibit the science in the state in which the Hindus possessed it, by an exact version of the most approved treatise on it in the ancient language of India, with one of the earlier treatises (the only extant one) from which it was compiled...’ He annotated the manuscript he used. Further study might reveal something of how he worked in relation to the text.
- 28.
Colebrooke (1817: ii).
- 29.
Colebrooke (1817: v–vi).
- 30.
Thus at the beginning of the Dissertation the authenticity of the texts is discussed. The collation establishes ‘the original’ text. In the appendixes, however, commentaries help additionally to provide a correct interpretation.
- 31.
- 32.
Colebrooke (1817: 278).
- 33.
Note that Colebrooke hesitated concerning the authorship of the versified examples found in the manuscript of the BSS—after all, the Līlavatī and the Bījagaṇita included such examples in the base text: ‘It is not quite clear whether the examples are the author’s or the commentator’s. The metre of them is different from that of the rules; and they are not comprehended, either in this or in the chapter on Algebra, in the summed contents at the close of each. They are probably the commentator’s; and consigned therefore to notes.’ (Colebrooke 1817:278).
- 34.
See also Raina (2011: 240).
- 35.
Colebrooke (1817: v).
- 36.
Grafton (1997: 23).
- 37.
Grafton (1997: 26–29). In the book itself, Brahmagupta’s text appears after the translations of Bhāskara’s mathematical texts. For Bhāskara, several commentators are used, in relation to the main text. In the case of the commentary to Brahmagupta’s text, Colebrooke, at just one crucial point concerning cyclic quadrilaterals, in a part of the manuscript he considers to be corrupt, supplies not only a translation of Pṛthūdaka’s explanations but also a part of Gaṇeśa’s commentary on the Līlāvatī, that he actually understands as dealing with this part of Brahmagupta’s text. See also Smadja (2015).
- 38.
Smadja (2015).
- 39.
Colebrooke (1817:1) specifies that the ‘axioms’ found at the beginning of Bhāskara’s text are ‘consisting in definitions of technical terms’.
- 40.
Looking more closely at Colebrooke’s sub-sectionings further highlights such difficulties. Bhāskara’s texts are subdivided into chapters and sections, giving the feeling that each is an individual treatise, while Brahmagupta’s chapters are each subdivided in sections. For instance, in the Līlāvatī, Section III (p. 122) and Sect. V (p. 125) of the chapter ‘Miscellaneous rules’, which precisely includes unclassified procedures, have no title. The first sections of Chaps. 12 and 18 of the BSS have no title either. Some chapters or sections are long, and others very short; thus in Chap. 12, the section on plane figures has 16 pages (p. 295–311), while those covering the topics of stacks (p. 314) and saws (p. 315) have but one page; the sectioning thus makes sense here in relation to the definition of mathematics but not in relation to the chapter’s scope of investigation. In addition, the specifications of titles are sometimes given in headers, which appears to be another level of sub-sectioning in Colebrooke’s published text.
- 41.
Bhāskara’s algebraical text is subtitled ‘Elemental Arithmetic or Algebra’ (Colebrooke 1817: 129).
- 42.
(Hayashi 2009: 15, note 97)
- 43.
(Hayashi 2009: 3–4, 7–8, 15). As underlined by Hayashi, six types of quantities each have six vidhas, which amounts to thirty-sixty.
- 44.
Colebrooke (1817: 339).
- 45.
Colebrooke (1817: 5).
- 46.
Colebrooke (1817: 129).
- 47.
Colebrooke (1817: 277). The reader can compare the translation of verse 1 with the translation in Colebrooke’s Footnote 1. The header on page 279 is ‘Logistics: Fractions’.
- 48.
Colebrooke (1817: 339).
- 49.
Raina (2011: 244–245) underlines how Peacock paid particularly careful attention to the definitions of arithmetic in its relation to algebra in Colebrooke’s translation.
- 50.
Hutton (1795) seems to have been regularly re-edited and augmented. Sections on algebra and arithmetic in Vol.II of the 1815 edition are abbreviated from Hutton (1812). Colebrooke may have read both. Indeed, E.T. Colebrooke’s biography transcribes many letters from Colebrooke to his family; in one dating from 1798–1799, we see him asking, among many other books, for Charles Hutton’s Dictionary of Mathematics; Colebrooke (1839: 30). Hutton’s dictionary is quoted twice in Colebrooke’s dissertation; Colebrooke (1817: x, lxxiii). The section on algebra in Hutton’s mathematical dictionary contains a detailed historical exposition, including a synthesis of Davis and Stratchey’s publications in which he claims that algebra probably originated in India (Hutton 1812: 65, Column 2). Colebrooke may have borrowed Hutton’s definition of algebra as analysis Hutton (1795: Vol. I, 60); of logistics as related to practical arithmetic, sexagesimal astronomical computation and occasionally algebra (Hutton 1795: Vol. I, 300). He also defined algorithms in relation to algebra (Hutton 1795: Vol. I, 103). Further, Colebrooke’s table of contents seems to resonate in Hutton’s Practical Arithmetic.
- 51.
Datta and Singh (1935: Vol. I, 124).
- 52.
- 53.
Sriram 2013.
- 54.
This is examined in Keller and Morice-Singh (2022).
- 55.
Hutton (1775: 23–25), for example.
- 56.
Colebrooke (1817: 339–340).
- 57.
Datta & Singh (1935: Vol I, p. 241).
- 58.
- 59.
- 60.
Dodson (2007a: 162).
- 61.
Dvivedin (1867).
- 62.
Dvivedin (1899). See, for instance, the footnote on page 21.
- 63.
Thibaut and Dvivedin (1889).
- 64.
Thibaut & Dvivedin (1889: v).
- 65.
Thibaut & Dvivedin (1889: lxi).
- 66.
Thibaut & Dvivedin (1889: lxi) also note in the case of Agatsya’s theory of heliacal rising: ‘The considerable deviation from the text of the manuscripts which our emended text exhibits in that place appears to be absolutely called for by the general principles on which such calculations have to be performed according to all scientific Hindû treatises.’
- 67.
Although we shall return below to the similar principles on which Dvivedin edited Brahmagupta’s text, let us note here that Thibaut insisted in his preface and introduction how closely he worked with Dvivedin, whom he referred to several times as his ‘collaborator’. See especially Thibaut & Dvivedin (1889: vi).
- 68.
Dvivedin (1902:2–3).
- 69.
Dvivedin (1902: 3): etatkṛtasyāsya siddhāntagranthasyaikā pratiḥ kāśikarājakīyapāṭhālayato dvitīyā ḍā. thibo sāhiba-ahmahāśayatas tṛtīyā cāyodhyānareśapradhānajyotir vicchrīyajñadata śamaṇo mayā labdhā| (…) idaṃ pustakatrayam atīvāśuddhaṃ bahucc skhalitaṃ cāsti yatpustakānusāreṇa vyaktāvyaktādhyāyayor dvādaśāṣṭādaśasaṃkhyayor aṅgulabhāṣāyām anuvādaḥ kolabrūkasāhibena kṛtas tat pustakam etat trayato bhinnam ity asaṃśayaṃ vibhāti paṭhavibhedāt| tat pustakasya kuṭṭakādhyāyaḥ saṃprati iṇḍīyā-āphis-sarasvatībhavane vartate. (See the Catalogue of the Sanskrit Manuscripts in the Library of India Office, Part V, p. 995).
I have obtained manuscripts (grantha) of this siddhānta, a first from the Rājakīyapāṭha in Kāśikā (Benares), a second from the Mahāśaya by Mister (sāhib) G. Thibaut and a third from Vicchrī Yajñadata Śarman, the head astrologer of the king of Ayodhyā, <each manuscript> had been produced for them. (…) These three manuscripts (pustaka) are very faulty indeed and are incomplete. A translation in English was made by Mister Colebrooke with an authoritative manuscript having the chapters on arithmetic and algebra numbered 12 and 18 which is different from the three others since it is devoid of difficulties <or > dissent (reading vimati for vibhāti) which would originate from divergent readings. The Chapter of this manuscript on Pulverizers can be found at present in the Indian Office Library. (See Catalogue of the Sanskrit Manuscripts in the Library of India Office, Part V, p. 995).’
- 70.
At times, he seems to assume that there were two manuscripts: one with commentary, another without. But mostly, he evokes the manuscript with which Colebrooke made his translation, in the singular.
- 71.
Dvivedin (1902: 3, footnote) idaṃ pustakaṃ kasyacin maithilākṣair likhitasya pustakasya pratyantaram iti niḥsaṃśayaṃ pratibhāti| 11 patre 9 ity asya sthāne 7 iti lekhāt| evam anyatrāpi | visargāt parataḥ ‘ka’ sthāne ‘kva’ iti lekhāt| yathā 12 patreḥ 6 paṅkto | tathā visargasthāne ‘v’ iti lekhāt, yathā 12 patreḥ 9 paṅkte| vākyavirāme ‘pi sandhikaraṇāt | yathā golādhyāyasya 21 patre 2 paṅkte sarvam upapapannam uktam akhaṇḍena| kutracit ‘nta’ sthāne ‘nū’ iti lekhāt| ekasmin patre methilākṣare ‘oṃgaṇeśāya namaḥ’ iti lekhāc ca|
This manuscript, which belongs to a codex written in Maithili script, appears devoid of error. On Sheet 11, 9 is written in the place of 7. It is like this elsewhere as well. When a ‘ka’ stands behind a visarga it is written ‘kva’. For instance, for Sheet 12, Line 6 in place of a visarga, there is a v, it is likewise for Sheet 12, on line 9. At the end of a sentence also sandhi is performed. For instance in the Chapter on the Sphere, in Leaf 21, Line 2, the whole explanation (upapapanna) is stated without any caesura. Sometimes ‘nū’ is written instead of ‘nta’. On one page, ‘Om, homage to Gaṇeśa’ is written in Maithili script.
- 72.
Dvivedin (1902: 3–4): aho iṇḍiyāphis sarasvatībhavane asya pustakasya puṭaka-bandhanakāle ‘navadhāna-tayā pattrāṇy asaṇgatāni jātāni, bahuc khaṇḍitāni ca santi| tāni kadācid an-upayuktapattrāṇāṃ madhye syur iti teṣāṃ samyag-anveṣaṇaṃ samucitam. (See Catalogue of the Sanskrit Manuscripts in the Library of India Office, Part V, p. 993–995).
Now, in the Indian Office library, at the time of the bundling (bandhana) of this manuscript, leaves were doubled over as a pocket (puṭaka), independent (asaṅgata) sheets (pattra) were produced by inattention and many were cut off (khanḍita). Some of these unattached (anupayukta) leaves were in the middle. (See Catalogue of the Sanskrit Manuscripts in the Library of India Office, Part V, p. 993–995)’.
This statement of Dvivedin’s preface is followed by a list of loose and cut out sheets, their number and the parts of the BSS they belonged to.
- 73.
Dvivedin (1902: 4–5).
- 74.
The manuscript’s alternative readings (pāṭhāntaram) are given in the footnotes. See, for instance, Dvivedin (1902: 38).
- 75.
Dvivedin (1902: 4–6).
- 76.
Dvivedin (1902: 7).
- 77.
The published edition marks some variants, but given the fact that Dvivedin noted the existence of very corrupt manuscripts, he does not seem to have adopted ‘a maximally inclusive edition’ of the kind that could be found for Sanskrit literature, as described by Pollock in the present volume.
- 78.
Dvivedin (1902:6): asya siddhāntasya dvādaśādhyāye gaṇitādhyāye pāṭīgaṇitarūpe mayā pāṭhakānāṃ vinodāya nijaṭīkāyāṃ caturvedācāryapṛthūdakaṭīkāt udāharaṇāni saṃhṛtya likhitāni|
For this treatise, in the twelfth chapter —which is the mathematical chapter, consisting of Pāṭīgaṇita (arithmetic)—after collecting examples from Master Caturveda, that is, Pṛthūdaka’s commentary, I have written a commentary for students, for pleasure. (Dvivedin 1902:6)
- 79.
Dvivedin (1902: 3).
- 80.
This title is repeated in his preface. See Dvivedin (1902: 5).
- 81.
See, for instance, the tables of BSS.1.15–24 in Dvivedin (1902: 6–7 in the main text).
- 82.
See, for instance, BSS.2.49 in Dvivedin (1902: 42).
- 83.
Colebrooke’s numbering is also given as an alternative BSS.18.35–36. See Colebrooke (1817: 339–340).
- 84.
Dvivedin (1902: 310).
- 85.
Dodson (2007b: 53–54).
- 86.
Dodson (2007a: 174).
- 87.
Dodson (2007a: 176).
- 88.
Keller (2011). Milieus dealing with astral science in Varanasi seem to have been located in a middle ground between the practitioners of ayurveda encountered in K. Preisendanz in the present volume (Chap. 6), and A. Graheli’s (Chap. 5) conservative epistemologists based on the outskirts of Calcutta. Indeed, a specialist in jyotiṣa could equally be a scholar studying ancient texts and a practicing astrologer, or sometimes both at the same time.
- 89.
Thus, in the sentence of his preface quoted above, the examples of PBSS seem to have been included in his commentary for the enjoyment of students.
- 90.
Dodson (2007b).
- 91.
Raina (2012: 245–246).
- 92.
In the following the non-standard transliteration of Sanskrit words have been preserved when giving titles of nineteenth century works.
Abbreviations
- BSS:
-
Brāhmasphuṭasiddhānta of Brahmagupta
- PBSS:
-
Pṛthūdaka’s Vāsanabhāṣya, a commentary on the BSS
- KK:
-
Khaṇḍakhādyaka of Brahmagupta
- PKK:
-
Pṛthūdaka’s commentary on the Khaṇḍakhādyaka of Brahmagupta
References
In the following the non-standard transliteration of Sanskrit words have been preserved when giving titles of nineteenth century works.
Critical Editions and Some Other Publications by S. Dvivedin
Dvivedin, Sudhākara. 1867. The S’ishyadhi’vṛddhida, the treatise on astronomy by Lall’ach’arya edited by Pandit Sudh’akara Dvivedi’. Benares: Medical Hall.
———. 1885. A catalogue of Sanskrit manuscripts in the North-Western provinces, compiled by order of government. Part IX. Allahabad: Printed at the M. -W. F. and Oudh Goverment Press.
———. 1892 [1933]. Gaṇakataraṅgiṇī. Benares: Hemant Sanskrit Series Granthamala.
———. 1899. Triśatikà by Śridharàchàrya. Benares: Jagannath Mehta.
———. 1902 Brāhmasphuṭasiddhānta and Dhyanagrahopadeśādhyāya by Brahmagupta edited with his own commentary. Reprint from The Pandit, no. 12, Vol. 24. Benares: Printed at the Medical Hall Press.
———. 1908. Yājusha-Jyautisha, with the Bhāshyas of Somākara Śesha & Sudh’akara Dvivedin and Ārcha-Jyautisha with the Bhāshya of Sudh’akara Dvivedin and Professor Mural’idhar Jh’a’s explanatory. Benares.
———. 1910a. The Mahāsiddhānta of Āryabhaṭa; Mah’asiddh’anta, a treatise on astronomy by ‘Aryabhaṭ, edited with his own commentary by Mah’amahop’adhy’aya Sudh’akara Dvivedi, first professor, Govt. Sanskrit college, Benares, and fellow of the Allahabad University &c. Benares Sanskrit series 148, 149, 150. Benares: Printed at the Medical Hall Press.
Dvivedin, Sudhākra. 1910b. Gaṇita Kā Itihāsa, a history of mathematics, Phalā Bhāga, Pāṭīgaṇita, first part, Arithmetics. Benares: Prabhakar Printing Works.
Dvivedin, Sudhākara & Śrī Muralīdhara Jhā. 1927. Bījagaṇita (Elements of algebra) of S’rī Bhāskarāchārya with expository notes and illustrative examples by M. M. Pandit S’rī Sudhākara Dvivedi. Edited with further notes by Mahāmahopādhyāya Pandit S’rī Muralīdhara Jhā, first professor, Sanskrit college, Benares. Benares Sanskrit series 159. Benares: Krishna Das Gupta.
Thibaut, Georges F., and S. Dvivedin. 1889. Pañchasiddhântikâ. Benares: EJ Lazarus and Company.
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Colebrooke, H.T. 1817. Algebra, with arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bhàscara. London: J. Murray.
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Datta, Bibhutibhushan, and Narayan Avadesh Singh. 1935. History of Hindu mathematics. Lahore: Motilal Banarsidass.
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———. 1795 [1815] A philosophical and mathematical dictionary, containing an explanation of the terms, and an account of the several subjects, comprised under the heads mathematics, astronomy, and philosophy both natural and experimental … Also memoirs of the lives and writings of the most eminent authors, both ancient and modern. London: J. Johnson; and GG and J. Robinson.
———. 1812. Tracts on mathematical and philosophical subjects. Vol. II. London: Printed for F. C and J. Rivington; G. Wilkik and J. Robinson; J. Walker. Lackington, Allen and Co.; Cadell and Dayies & I. J. Cuthill; D. and R. Crosby and Co. & J. J Richardson; J. M. Richardson & J. H. Baldwin and A. Robinson.
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———. 2007b. Contesting translations: Orientalism and the interpretation of the Vedas. Modern Intellectual History 4 (01): 43–59.
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Keller, A. (2024). Shaping a Mathematical Text in Sanskrit: H. T. Colebrooke, Sudhākara Dvivedin, and Pṛthūdaka’s Commentary on the Twelfth Chapter of the Brāhmasphuṭasiddhānta. In: Keller, A., Chemla, K. (eds) Shaping the Sciences of the Ancient and Medieval World. Archimedes, vol 69. Springer, Cham. https://doi.org/10.1007/978-3-031-49617-2_12
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