Emergence of a New Era in the History of Indian Mathematics

It is now well established that the school of astronomy and mathematics, founded by Mādhava in Kerala in the 14th century, came up with two of the major discoveries, namely calculus and non-geocentric planetary models, a couple of centuries before they were to become the hallmarks of the modern European scientific revolution. The seminal discovery of the Kerala school, of the infinite series for \pi was brought to the notice of modern scholarship for the first time in the book Kālasa\dot{n}kalita of John Warren, published from Madras in 1825. This was soon followed by the detailed account presented in the pioneering articles of Charles Whish, which were published during 1827–1834. However, the work of the Kerala school was totally ignored by modern scholarship for over a hundred years. This was because Whish’s paper and the Kerala work on infinite series were denigrated and suppressed by the British academic establishment so much so that even reputed Indologists such as Albrecht Weber, and George Thibaut, and renowned historians of mathematics such as David Smith chose not to say anything about the work of the Kerala school even when they referred to the articles of Whish in their scholarly writings.

The work of the Kerala school was resurrected by the pioneering article on the “Hindu values of \pi” by Bibhutibhushan Datta in 1926. This was followed by the article “On the use of series in Hindu Mathematics” by Datta’s colleague, Avadhesh Narayan Singh, which was published in the first issue of Osiris in 1936. During 1940–1952, C.T. Rajagopal and his collaborators wrote a series of important articles explaining, in modern mathematical terms, all the important demonstrations for the infinite series for \pi, and the sine and the cosine functions, as given in the Malayalam text Yuktibhāṣā. Equally significant was the publication in 1948 of the mathematics part of this classic text, which was edited along with detailed notes in Malayalam, by Rama Varma Tampuran and Akhileswaraiyer. When the contents of these important publications were taken note of in the 1950s and 1960s, by some of the well-known historians of mathematics such as J.E. Hoffmann (1953), C.N. Srinivasiengar (1958, 1967), A.P. Yushkevich (1961, 1964), D.T. Whiteside (1961, 1968), H.L.L. Busard (1962), T.A. Sarasvati Amma (1963, 1979), A.K. Bag (1966, 1979) and M.E. Baron (1969), one could clearly see the emergence of a new era in the historiography of Indian mathematics.

Starting from the publication of Bhaṭadīpikā commentary of Parameśvara on Āryabhaṭīya by H. Kern in 1874, around fifteen source-texts of the Kerala school were published in the next eighty years. Still, there remained several gaps in our understanding of the history and achievements of the Kerala school. It is the monumental work of K.V. Sarma, whose prolific writings have contributed substantially to our present understanding of the history and contributions of the Kerala school. Starting from 1953, Sarma edited and published about twenty very important works of the leading savants of the Kerala school. Sarma also brought out a comprehensive history and bibliography of Kerala works on Astronomy and Mathematics in 1972, where he listed around 400 works. It was Sarma’s relentless efforts to bring out works which present yuktis, or demonstrations, which led to the publication of the Sanskrit commentaries, Kriyākramakarī and Yuktidīpikā of Śa\dot{n}kara, the compilation Gaṇitayuktayaḥ (containing 27 small tracts of yuktis), and the critical edition of Yuktibhāṣā of Jyeṣṭhadeva, along with English translation and explanations. Sarma’s publication of the seminal works of Nīlkaṇṭha such as Golasāra, Siddhāntadarpaṇa and Grahasphuṭānayane Vikṣepavāsanā, led to a clearer understanding of the revised planetary model that was proposed by Nīlkaṇṭha in his Tantrasa\dot{n}graha and Āryabhaṭīyabhāṣya. One of Sarma’s landmark publications has been the Jyotirmīmāṃsā of Nīlkaṇṭha, which discusses the crucial role of continuous parīkṣā, or examination of the theories by observation, and the consequent process of revision and preparation of newer and more accurate karaṇas or computational manuals.

Currently, there are still many important source-works of the Kerala school–-including works composed by legendary figures such as Mādhava, Parameśvara, Putumana Somayājī, Śa\dot{n}kara and Acyuta–-which are yet to be edited and published. Further, among the fifty major source-works (and about thirty small tracts) which have been published, only about a dozen have been translated; and even among them, there are only perhaps half a dozen works which have been studied in depth to explicate their full technical content. It will be befitting to the legacy of K.V. Sarma, and his illustrious predecessors such as Datta and Singh, Rama Varma Tampuran, C.T. Rajagopal and others, that our young scholars imbibe their great dedication and respect for our tradition, and carry forward their mission to ensure that the entire corpus of works of the Kerala school is rigorously and comprehensively studied and brought to light.

Denigration and Suppression of the Kerala work on Calculus, 1825–1925

The first notice of Indians’ knowledge of the infinite series for \pi (1825)

Lt. Col. John Warren (1769–1830), a Frenchman who was in charge of the observatory at Madras during 1805–10, seems to have been the first European observer to report in the 1820s that some of the Indian astronomers that he had met seemed to know the infinite series for \pi [1], [2]. In his book Kālasa\dot{n}kalita, which was published in Madras in 1825, Warren states that [3][pp.92–93]:

In the footnote marked (^*) to the above passage, Warren included a note from George Hyne (1800–1826), an officer in the East India Company’s medical service.¹ The note referred to a communication made to the Madras Literary Society by Charles Whish, who was in the East India Company’s service since 1812 and was posted at Malabar (northern Kerala). Hyne mentions that Whish, in his communication, had listed a number of infinite series for \pi that were employed by the Indian astronomers. However, Hyne was very emphatic that [3][p.93]:

After reproducing Hyne’s note, Warren states that [3][p.93]:

Later, in an Appendix to the Kālasa\dot{n}kalita, titled “On certain infinite series collected in different parts of India, by various Gentlemen, from various Native Astronomers…”, Warren reproduces a subsequent letter (dated August 17, 1825) from Hyne. Warren prefaces the communication of Hyne with the following comments [3][p.330]:

The Appendix then reproduces the letter of Hyne, which presents seven infinite series for \pi [3][pp.330–331]. It may be seen that these series are all part of the famous paper of Whish which was read at the Royal Asiatic Society, London, in December 1832, and published in its Transactions in 1834.

From all these passages in the Kālasa\dot{n}kalita, we can clearly conclude the following:

1. In the 1820s, both Warren (who was mainly in Madras and Pondicherry) and Whish (who was stationed in Malabar or northern Kerala) had met several Indian astronomers who were conversant with some of the infinite series for \pi.
2. By 1825, the Madras Literary Society had received a communication from Whish wherein he had referred to several Indian texts (śāstras) which mentioned these series.
3. Whish was however persuaded by Hyne who was a part of the elite establishment in Madras that, since the Indian astronomers as well as their śāstras seemed to be unaware of the demonstrations of these series, they must have been communicated to the Indians by Europeans in the modern times.

The career of Charles Whish

Charles Mathew Whish (1794–1833) came to India in 1812 after studying at the East India College in London. His career in India in the service of the East India Company is summarised below [4][p.153]:

• 1813: Writer
• 1815: Register of the Zillah of South Malabar
• 1823: Register of the Zillah of Malabar
• 1827: Assistant Judge and Joint Criminal Judge of Malabar
• 1830: Assistant Judge and Joint Criminal Judge of Cuddapah
• 1831: Out of employment
• 1832: Acting Judge and Joint Criminal Judge of Cuddapah
• 1833: Died on 14th April 1833, at Cuddapah.

Whish spent over 15 years in Malabar and seems to have written a Malayalam grammar and dictionary. He also collected around 300 Malayalam and Sanskrit manuscripts of important literary, philosophical and scientific works. After his death, these manuscripts were presented to the Royal Asiatic Society in 1836 by J.C. Whish, the brother of Charles Whish.

The notebook of Whish (c.1820)

We have already referred to the communication from Whish sent around 1825 to the Madras Literary Society regarding the Kerala work on infinite series. Some idea of the background of this communication may be had from an uncatalogued and unnumbered paper manuscript in the form of a notebook preserved in the Whish Collection at the Royal Asiatic Society, London. This is a manuscript of the Malayalam Commentary Kriyākalāpa by Śa\dot{n}kara Vāriyar on the Tantrasa\dot{n}graha of Nīlakaṇṭha. It also contains some notes by Whish in English, which seem to have been written from 1820 onwards. The details of this notebook have been presented in a recent study of Kriyākalāpa [5]:

In Kriyākalāpa, while commenting on the verses 3–6 of the second chapter (Sphuṭaprakaraṇa) of Tantrasa\dot{n}graha, Śa\dot{n}kara makes a long excursus where he cites several Sanskrit verses–-which describe the Mādhava series for \frac{\pi}{4} and its many transformed versions–-along with his Malayalam commentary. The Whish notebook mentioned above contains notes by Whish giving the mathematical form of all these series, and it may be seen that they appear in the same order in his famous article of 1834. In that article, Whish wrongly attributed the Sanskrit verses on infinite series, which are mentioned in the commentary Kriyākalāpa, to be a part of the Tantrasa\dot{n}graha itself–-an error that has been repeated in several later studies even after reliable editions of the Tantrasa\dot{n}graha were published by S.K. Pillai in 1958 and K.V. Sarma in 1977.

Initial publications of Whish (1827)

While his first communication on the Kerala work on infinite series was summarily rejected by Hyne, Whish did manage to get two of his papers published in 1827 in the very first issue of the Transactions of the Madras Literary Society. The first article on the “Alphabetic Notation of the Hindus” [6] deals with different systems of numeration employed in the Indian texts, such as the system used in the Āryabhaṭīya and the Kaṭapayādi system used in south India. As in illustration of the Kaṭapayādi system, Whish presents the verse from the Sadratnamāla of Śa\dot{n}karavarman which gives the value of \pi correct to 17 decimal places. Whish also cites the verses of Mādhava giving the table of 24 sine values which are accurate to the thirds. Again, Whish attributes the latter verses to the Tantrasa\dot{n}graha, though they are found only in the commentaries. In the natural course of events, just these two illustrations of the remarkable achievements of the Kerala school should have been sufficient to evoke considerable interest to locate and study the works of this school, even at this stage when the more detailed 1834 paper of Whish had not yet seen the light of the day.

The other paper of Whish’s is on a very different note and is titled, “On the Origin and Antiquity of the Hindu Zodiack”. At the beginning of this paper, Whish declares [7][pp.63–64]:

At this point, there is a footnote inserted in the paper, attributed to “G.H.”, which states [7][p.64]:

The unsigned editorial “Notice”, at the beginning of this volume of the Transactions of the Madras Literary Society mentions that G.H. refers to George Hyne. In fact, this article of Whish has several footnotes by G.H. inserted at different places. It seems that, by this article, which purported to show that the twelve-part Zodiac was communicated to the Indians by the Greeks, Whish had managed to some extent to win the approval of George Hyne. In any case, Hyne seems to have passed away sometime in 1826, as the same editorial note at the beginning of the Transactions also records its tribute to George Hyne, “by whose lamented death science has been deprived of an ardent and profound investigator”.

Whish’s landmark paper (1832)

On the title page of Whish’s landmark paper “On the Hindu Quadrature of the Circle”, it is mentioned that it was communicated by the “Madras Literary Society and Auxiliary Royal Asiatic Society” and was read at the Royal Asiatic Society, London, on the 15th of December 1832 [9][p.509]. It was published in the third volume of the Transactions of Royal Asiatic Society in 1834.³

Whish begins his paper with the famous verse from Āryabhaṭīya which gives an approximate value for \pi, and then goes on to cite some of the very accurate approximations given in the Kerala works, Tantrasa\dot{n}graha, Karaṇapaddhati, and Sadratnamāla. Whish then states [9][p.512]:

Whish then presents the verses which describe the Mādhava series for \pi with the three successive end-correction terms, corresponding to increasing levels of accuracy. As we mentioned earlier, Whish again wrongly attributes these verses, and the next fourteen verses that he cites (which give many transformed series for \pi), to Tantrasa\dot{n}graha without realising that they are being cited by Śa\dot{n}kara Vāriyar as a part of his commentary Kriyākalāpa.

Whish then goes on to cite a few verses from Karaṇapaddhati and Sadratnamāla which give some more series for \pi, and the series (the so-called Gregory Series) for the arc-tan. Before going on to a discussion of the authorship and the period of these works, Whish declares rather emphatically [9][p.521]:

It is not clear on what basis, and under what compulsions, Whish is making this unusually emphatic claim, which has often been repeated in the later studies. For, as may be seen from the passage that was cited from the Kālasa\dot{n}kalita (with which Whish would have been very familiar), Warren states that he had met an Indian astronomer (either in Madras or Pondicherry) who knew some of these series. Warren also mentions that these series were collected “from various parts of India by Gentlemen”. As regards a historically earlier period, it is well known that the sixteenth-century astronomer Sundararāja of Kanchipuram has mentioned, in his commentary on Vākyakaraṇa, that he was in correspondence with the great savant, Nīlakaṇṭha, who was kind enough to compose a tract Sundararājapraśnottara in answer to his queries. It may also be noted that there are Telugu and Grantha manuscripts of Karaṇapaddhati, and a manuscript of a Sanskrit version of the Malayalam text Yuktibhāṣā is deposited in GOML,⁴ Madras, which has been edited and published by K.V. Sarma in 2004.

Proceeding to a discussion of the date and authorship of the texts, Whish first notes the following regarding Sadratnamāla and its author [9][p.521]:

Whish then mentions 1733 as the time of Putumana Somayājī, the author of Karaṇapaddhati. As regards Tantrasa\dot{n}graha, Whish mentions [9][p.522]:

In the above passage, Whish refers to a college which “still exists, which is a place of reputable sanctity, and certainly of reputable learning”. In this context, it may be noted that the report dated August 5, 1823, sent by J. Vaughan, the Collector of Malabar, as a part of a survey of indigenous education ordered by Thomas Munro the then Governor of Madras Presidency, reproduces a note by the Samudrin Rāja of Kozhikode on the College being maintained by him in Tirunāvāya which had 75 scholars studying “Theology & Law”. This report of the Collector also mentions that, in addition, there were 808 students pursuing higher studies in “Astronomy” and 194 students studying “Medical Science” under reputed individual scholars [10][pp.199–203].

As regards Yuktibhāṣā, Whish mentions that it is a commentary on the Tantrasa\dot{n}graha and that its author, Cellalura Nambutiri, wrote another work Dṛkkaraṇa in 1608 CE. Though Whish promised at the beginning of the article that he will discuss the series for sine and cosine, his paper seems to end rather abruptly with a promise that they will be taken up in a further publication, along with the demonstrations of all the series as given in the Yuktibhāṣā [9][p.523]:

Unfortunately, Whish passed away in 1833, without fulfilling the above promise of presenting the demonstrations of the various series as given in Yuktibhāṣā. Around this period, there seems to have been some personal or other problem due to which Whish was “out of service” in 1831, after being transferred to Cuddapah in 1830. He rejoined as Acting Judge in Cuddapah in 1832, but passed away soon on April 14, 1833, without seeing his article come out in print.

It appears that Whish was clear from the various responses he received from around 1825, that his European peers and superiors would not accept any number of quotations from the Indian śāstras describing different infinite series as satisfactory evidence that the Indians had discovered these series, unless he could also show that some of the Indian texts also presented valid demonstrations for these series. And he was well aware of the proofs contained in the Malayalam text, Yuktibhāṣā, of which he had two readable palm-leaf manuscripts which are preserved in the Whish collection at the Royal Asiatic Society, London–-Manuscript Nos.124 and 138. In order to highlight the importance of this work, Yuktibhāṣā, and perhaps with the premonition that he might not be around to explain its significance, Whish inscribed a note on the verso of the first blank palm-leaf of Manuscript No. 124, which is reproduced below:

This important note of Whish seems to have been missed (or overlooked) by all the scholars who made any list or catalogue of the Whish Collection of manuscripts at the Royal Asiatic Society in London in the 19th as well as the 20th century. We shall return to this issue later in this article.

David Pingree’s explanation as to why Whish’s article was ignored

In a lecture delivered at the Department of History of Science of the Harvard University in 1990, which was later published in Isis in 1992, the renowned American historian of science, David Pingree, argued that Mādhava’s achievements in demonstrating various infinite series, though very notable, should not be confused with any “discovery of the calculus” as they were based on “geometrical and algebraic arguments”. He also offered an explanation as to why Whish’s article was totally ignored by modern Western scholarship for over a century, “presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish’s article was published.”[11][p.562]:

As we shall discuss below, contrary to the claims made by Pingree, Whish’s paper was, in fact, widely noted and referred to during the decade (1832–1842) after the paper was read in 1832. It was such wide notice, and the fact that the Transactions of the Royal Asiatic Society was regarded as a highly reputed journal on such matters, which seems to have alarmed the leading lights of the British academic establishment such as Augustus De Morgan, and scholar administrators such as Charles P. Brown, so much that they thought it necessary to publicly denounce the Kerala work on infinite series as a “clever hoax”. This seems to have led to a veritable consensus among the European Indologists and historians of mathematics to completely ignore the work of the Kerala school for the next hundred years or so. The important manuscripts of Malayalam works (such as the Yuktibhāṣā), collected by Whish and deposited in the library of the Royal Asiatic Society since 1836, were not even listed or catalogued; and reputed Indologists such as Albrecht Weber, George Thibaut and renowned historians of mathematics such as David Smith chose not to discuss the work of the Kerala school even while referring to the articles of Whish in their scholarly writings.

Initial response to Whish’s papers (1832–1842)

As we noted earlier, Whish’s paper “On the Hindu Quadrature of the Circle”, was read at the meeting of the Royal Asiatic Society held in London on December 15, 1832. The proceedings of the meeting are described as follows in the January 1833 issue of the Asiatic Journal and Monthly Register [12]:

Similar summaries of the paper of Whish were carried also in various other journals in 1832/33 [13]. It may be noted that all this reporting was done even prior to the publication of the article in 1834. Incidentally, Whish’s 1827 article “On the Alphabetical Notation of the Hindus”, was translated into French and published in the Journal Asiatique in 1835 [14].

One of the earliest references to the 1834 article of Whish was made by the leading German Indologist, Christian Lassen, in a paper on the systems of numerations used in Indian mathematics in 1837 [15]. An important bibliography of Sanskrit literature compiled by the German scholar Friedrich Adelung and published in 1837 [16], listed Warren’s Kālasa\dot{n}kalita, Whish’s article of 1834 and the four major Kerala works (śāstras) mentioned therein. More importantly, Whish’s article was cited in a volume published in 1841 on the development of calculus in Europe in the 17th century. Stephen Peter Rigaud and Stephen Jordan Rigaud in their Correspondence of Scientific men of Seventeenth Century [17] referred to the fact that a certain approximation to \pi mentioned in a letter of John Wallis to John Collins, dated September 1, 1767, was the same as the one attributed to Tantrasa\dot{n}graha in Whish’s article [17][pp.590–591]:

Augustus De Morgan dismisses Kerala work as a “clever hoax” (1843)

The kind of response that the paper of Whish–-and the Kerala work on infinite series that was discussed therein–-seems to have received, both in the Indological and mathematical circles, does seem to have seriously disturbed the high priests of British academia. Indeed their response was not much different from what was first pronounced (though in private correspondence) by George Hyne in 1825, namely that these results must have been recently received by the Indians from certain European sources. Now, the same pronouncement is made publicly by one of Britain’s leading mathematicians, Augustus De Morgan (1806–71),¹⁰ in a review article on Indian Astronomy and Mathematics published in the widely read Penny Cyclopaedia (brought out by The Society for the Diffusion of Useful Knowledge) in 1843 [18]. In this review (which carries the strange title “Viga Ganita”, purportedly referring to Indian algebra, Bījagaṇita), De Morgan declares at the outset that [18][pp.318–319]:

De Morgan devotes nearly five pages of his eight-page article to the issues surrounding the antiquity and development of Indian astronomy by reviewing the works of Guillaume Le Gentil, Jean-Sylvain Bailly, John Playfair, and the debate between Henry Colebrooke and John Bentley, and concludes that, “The astronomy of Hindus would have had little interest, but for their arithmetic and algebra” [18][p.323]. De Morgan then gives a cursory outline of the Līlāvatī and Bījagaṇita of Bhāskara and makes a brief mention of Indian trigonometry following Delambre. In his concluding section, De Morgan states [18][p.325]:

De Morgan is clear that but for the fact that the paper of Whish has appeared in the “Memoirs of the Society which is best able to judge of its contents”, he would have right away dismissed the paper as a “clever hoax”. De Morgan goes on to provide a brief introduction to the three texts Tantrasa\dot{n}graha (which he wrongly dates to 1608), Karaṇapaddhati and Sadratnamāla, and then presents the various series for \pi as given in the paper of Whish. He, however, makes no mention of the Yuktibhāṣā, or the claim of Whish that the demonstrations of the series for \pi, sine and cosine are presented in that text. Concluding his discussion on Whish’s paper, De Morgan declares that it is “a staggering assertion that works written in 1608 and 1733 give series which not only were not known in Europe at those periods, but must have been discovered by something answering to the integral calculus”. The fact that, according to Whish, these series are not known outside Malabar “even to this day”, makes it very likely that “they were introduced from Europe at a comparatively recent period”. He, therefore, suggests that several of these series must have been communicated to the Indians by the French astronomer Le Gentil, who was in “the southern parts of the Deccan in 1769”, and that the Catholic missionaries “may have given the rest” [18][pp. 325–326]:

There does not seem to be any further debate on the paper of Whish, or on the Kerala work on infinite series, after this devastating review in the Penny Cyclopaedia. Perhaps that was what was intended by De Morgan in the first place, since, without any ado, he removed all references to the paper of Whish when his article on “Viga Ganita” was reprinted fifteen years later in the English Cyclopaedia in 1858 [19]. Incidentally, the comprehensive bibliography of secondary sources on Indian astronomy and mathematics in the volumes of Census of Exact Sciences in Sanskrit compiled by David Pingree does not include any reference to either of these articles on Viga Ganita. De Morgan is better known to the historians of Indian mathematics through his patronage of the work of Ramchundra which he got republished in London in 1859 along with a glowing preface [20][pp. iii–xviii].

Charles Brown castigates Whish’s paper as a “Fraudulent Document” (1863)

The British scholar administrator in South India, Charles Phillip Brown (1798–1884),¹¹ wrote a monograph on Carnatic Chronology in 1863, which has a section “On Some Fraudulent Documents”. Brown refers to the article on Viga Ganita in the Penny Encyclopaedia, which “surmised” that the verses cited by Whish seemed to be “forgeries”. Brown, in fact, claims to have identified the Brahmin who had composed these verses attributed to Tantrasa\dot{n}graha in the article of Whish. The following is an extract from Brown’s diatribe against the paper of Whish, and even the text Tantrasa\dot{n}graha itself [21]:

The Malayalam manuscripts in the Whish Collection at the Royal Asiatic Society

As we noted earlier, J.C. Whish, the brother of Charles Whish donated his entire collection comprising nearly 300 valuable manuscripts in Sanskrit and Malayalam to the Royal Asiatic Society, London, at its meeting on July 2, 1836. The proceedings of the meeting include the following account of this donation [22]:

As we can see, the proceedings do not mention anything about manuscripts relating to astronomy or mathematics, which were perhaps included among the miscellaneous works. In any case, since scholars like De Morgan had declared that “we must suspend our opinion [on Whish’s paper] till these works are published”, it was to be hoped that some of the high scholars of Indology or history of mathematics in England would take the initiative of getting these manuscripts catalogued and studied. Nothing of that sort happened, and in 1863 we see Charles Brown demanding that these manuscripts may be sent to Madras, since “being in the Tamil, Malayalam and Telugu character, they will probably be useless in England”.

In the meeting of the Royal Asiatic Society held on December 16, 1867, Reinhold Rost, the then Secretary of the Society, gave an account of the Whish collection of Sanskrit manuscripts. We present the following extract from a report of the meeting [23]:

In the above summary, Rost refers to only the Sanskrit manuscripts collected by Whish, whom he refers to as the “author of several valuable articles on Hindu astronomy”. As regards the works of the Kerala school, only two commentaries of Parameśvara and the Tantrasa\dot{n}graha are mentioned, with no further details regarding the commentaries etc., of the latter.

In 1890, the Royal Asiatic Society published a list of titles of the Sanskrit manuscripts in the Whish collection [24]. Finally, in 1902, the Society published a descriptive catalogue of the Sanskrit Manuscripts of the Whish Collection prepared by the famous Austrian Indologist, Moriz Winternitz [25]. In his preface to the catalogue, Winternitz refers to the rare and valuable manuscripts in this collection pertaining to the Vedas, Mahabharata etc., but makes no mention of astronomy or mathematics. The catalogue later lists 400 works distributed among 215 manuscripts–-of which about 25 works are listed under astronomy and astrology. Apart from the two commentaries of Parameśvara, the only work of the Kerala school listed is “Kriyākalāpa of Tantrasa\dot{n}graha” (Manuscript No. 134, which actually contains the Sanskrit commentary Laghuvivṛti of Śa\dot{n}kara on Tantrasa\dot{n}graha)–- which is listed both under “Astronomy and Astrology” as well as “Tantra”.

Indeed, there is no reference to Yuktibhāṣā in the entire catalogue of Winternitz. Since many of the Sanskrit manuscripts were also in the Malayalam script, it would have taken little additional effort to catalogue the Malayalam works in the Whish collection. From the current digitized archives of the Royal Asiatic Society today, we see that there are only 13 manuscripts of Malayalam works in the entire Whish collection [26]. Had these also been included in the descriptive catalogue, then the world of modern scholarship would have come to know the details of the two palm-leaf manuscripts of Yuktibhāṣā, referred to earlier, including the important note inscribed by Whish in the Manuscript No.124, that it is a manuscript of “Yukti-Bhāshā containing the demonstration of the … infinite series in the Tantra Sangrahah”. We can see no reason for omitting the Malayalam works from the descriptive catalogue of 1902, except for the extreme reluctance that prevailed among the European Indologists to rake up the issues raised in Whish’s paper on the Kerala work on infinite series and their demonstrations which were supposed to be contained in the Malayalam work Yuktibhāṣā.

During 1904–1920, Winternitz published what was reputed to be the most comprehensive history of Indian literature, in three volumes. The second part of the third volume devotes about 200 pages to “The Scientific Literature” (Die Wissenschaftliche Litteratur), of which about 23 pages are devoted to astronomy, astrology and mathematics [27][pp.555–577]. However, Winternitz makes no reference to the paper of Whish or to any of the Kerala works on mathematics or astronomy–-even those that were included in his catalogue–-in his survey of the literature on mathematics and astronomy.

Suppression of all reference to Kerala work on infinite series (1850–1925)

It appears that by the 1850s there had emerged a strong consensus among the European Indologists and historians of mathematics and astronomy, not to make any reference to the Kerala work on the infinite series for \pi and the sine and cosine functions, even while they made a reference to the papers of Whish for one reason or the other. The articles of Whish were indeed well known to the prominent European Indologists, as may be seen from the fact that in 1863, Albrecht Weber cited the 1827 article of Whish (on the Indian systems of numeration) in order to explain the Kaṭapayādi system of numeration in his review of a new edition of the Rāmāyaṇa published from Bombay [28]. Both the 1827 and 1834 articles of Whish were again brought to the notice of the Indological community in 1865 by the well known Indian scholar, Bhau Daji, in his article on the date and works of Āryabhaṭa and others, which was published in the Journal of the Royal Asiatic Society [29].

During 1860–1899, the other 1827 article of Whish, on the Greek origin of the Hindu Zodiac, was frequently cited in several studies of Indian astronomy, such as the popular translation of the Sūryasiddhānta by Ebenezer Burgess and William Whitney [30], the review article by John Burgess of the European studies of Indian astronomy in the 18th and 19th centuries [31][pp.746–750], and the scholarly monograph of George Thibaut (in German) on Indian Astronomy, Astrology and Mathematics [32]. None of these works makes any mention of the other two articles of Whish which discuss the Kerala work on the infinite series for \pi–-notwithstanding the fact that these issues were germane to the subject of their discussions.

Even more intriguing is the fact that the renowned historian of mathematics, David Eugene Smith cites the 1834 article of Whish, without giving any account of its contents, in a famous article of his in 1913. Smith again refers to the article of Whish in his seminal two-volume history of mathematics completed in 1925. The 1913 article is on the geometry of the Hindus, and it is published in the very first issue of the Isis [33]. This article contains some discussion of the work of Āryabhaṭa, Brahmagupta, Mahāvīra and Bhāskara, who is referred to as the “last of the well-known geometers of India” [33][p.198]. Among the eleven references given at the end of the paper, we see the article of Whish “On the Hindu Quadrature of the Circle”. However, there is no citation made to the article in the entire text.

The story becomes even more curious when we come to the second volume of Smith’s History of Mathematics published in 1925 [34]. In the chapter on geometry, there is a small section on the squaring of the circle, which includes a discussion of the “Hindu values of \pi”. Here, Smith mentions the approximate values given by Āryabhaṭa, Brahmagupta and Bhāskara. Again, there is a reference to Whish’s article in a footnote, which is somewhat more explicit than the citation made in the 1913 paper, and states [34][p.309]:

However, there is no discussion of the contents of Whish’s paper or any mention of the Kerala work on infinite series for \pi in this comprehensive work on the history of mathematics by David Smith. This is all the more surprising because, as is well-known, Smith was the co-author with Louis Karpinski of the famous book on The Hindu-Arabic Numerals published in 1911. It seems that Smith is shying away from entering into any discussion on the issue of the work of the Kerala school. However, by prominently citing the 1834 article of Whish in his 1913 paper and the 1925 book, Smith was perhaps giving a lead to the Indian scholars to take up this issue for further investigation.

The resurrection of the work of the Kerala School (1926–1952)

Datta and Singh initiate the modern Indian scholarly study (1926–1936)

The reference to Whish’s article, made in Smith’s History of Mathematics, was promptly noted by Bibhutibhushan Datta (1888–1958), who had embarked on an intensive study of the source-works of Indian mathematics soon after completing his Ph.D. in applied mathematics from the Calcutta University in the early 1920s. In his 1926 paper “On the Hindu Values of \pi”, Datta refers to Smith’s book, and goes on to present a much more comprehensive account of the different approximations to \pi found in the Indian mathematical and astronomical works [35]. The final section of Datta’s paper was on “Values in Series”, where he states [35][p.41]:

Datta gives a summary account of all the series mentioned in Whish’s paper, for the first time, after a lapse of over eighty years from the time when they were last mentioned in the 1843 article of De Morgan.

As is well known, the monumental work of Datta and his collaborator Avadhesh Narayan Singh (1901–1954) on the History of Hindu Mathematics was published in two parts in 1935 and 1938. In the preface to the first volume, the authors mention that there would be a third part, which would contain “the history of geometry, trigonometry, calculus and various other topics such as magic squares, theory of series and permutations and combinations” [36]. The section on series of this proposed third volume was later edited and published by Kripa Shankar Shukla in 1993 [37].

While Datta had mentioned that he had no access to the manuscripts of the Kerala works in his 1926 article, several of these manuscripts seem to have been collected by him and Singh while working on their History. In 1936, Singh published an article on the use of series in India in the first issue of Osiris, where he declares at the very outset that [38][p.607]:

Singh discusses the arc-tangent series as given in the paper of Whish. He presents the reference to these verses as found in the manuscripts of Karaṇapaddhati and Sadratnamāla. He also gives, for the first time, a translation of the verses from Karaṇapaddhati which give the series for the sine and cosine functions, thereby fulfilling a part of the promise made by Whish over a hundred years ago in 1832. As we shall see below, the work Karaṇapaddhati was edited and published by K. Sambasiva Sastri in 1937.

The seminal work of C.T. Rajagopal and co-workers (1940–1952)

It was during the decade of 1940–1950 that the rest of the promises made in the article of Whish got fulfilled. This was mainly due to the seminal work of Cadambathur Thiruvenkatacharlu Rajagopal (1903–1978) (see [39] for an overview of his work) and his co-workers, who published a series of articles [40],[41],[42],[43],[44],[45],[46],[47],[48],[49] during 1940–1952¹² explaining, in modern mathematical terms, all the important demonstrations for the infinite series for \pi, the sine and the cosine functions, as given in the Malayalam text Yuktibhāṣā. Mukunda Marar and Rajagopal state right at the beginning of their publication of 19[44], titled “On the Hindu Quadrature of the Circle”, that [41][pp.65–66]:

As has been noted in the above article of Marar and Rajagopal, Prince Rama Varma Tampuran seems to have played a crucial role in supporting the work of Rajagopal and co-workers, perhaps by providing the manuscripts of the Kerala works and by helping in their translation etc. Soon, Rama Varma Tampuran, in collaboration with Akhileswaraiyer, came up with a pioneering edition, along with detailed notes in Malayalam, of the mathematics part of Yuktibhāṣā in 1948.

When the contents of all these important publications were taken note of in the 1950s and 1960s, by well-known historians of mathematics such as J.E. Hoffmann (1953) [50], C.N. Srinivasiengar (1958, 1967) [51],[52], A.P. Yushkevich (1961, 1964) [53], D.T. Whiteside (1961, 1968) [54],[55], H.L.L. Busard (1962) [56], T.A. Sarasvati Amma (1963, 1979) [57],[58], A.K. Bag (1966, 1979) [59],[60] and M.E. Baron (1969) [61], one could clearly see the emergence of a really new chapter in the historiography of Indian Mathematics.¹⁷

Publication of the source-works of the Kerala school (1874–1952)

The first source-work of the Kerala school to be published was the Bhaṭadīpikā commentary of Parameśvara on the Āryabhaṭīya in 1874. This also happened to be the first edition of the Āryabhaṭīya, and was carried out by Hendrik Kern, based on a manuscript in the Whish collection and another one from Arthur Burnell. In 1898, the Kavanodayam Press in Calicut published the Sadratnamālā (c.1819) of Śa\dot{n}karavarman (1774–1839) along with the author’s auto-commentary. However, this pioneering edition does not seem to have led to any study or discussion of the contents of this seminal work. The fourth chapter of this work, titled jyācāpādiprakaraṇam, presents a detailed discussion of the relation between the circumference and diameter of a circle, as well as the relation between arcs and their Rsines.

In all, fifteen source-works of the Kerala school were published during 1874–1952. These included the Gaṇitapāda and Kālakriyāpāda portions of the Āryabhaṭīyabhāṣya of Nīlakaṇṭha, and the Karaṇapaddhati of Putumana Somayājī, which were all edited by K. Sambasiva Sastri during 1930s. As we noted earlier, the most important work published during 1874–1952 was the pioneering edition of the Gaṇita section of the Yuktibhāṣā, which was published along with detailed notes in Malayalam by Rama Varma Tampuran and Akhileswaraiyer in 1948.

The following is the list of the fifteen Kerala source-works published during 1874–1952:

1. Āryabhaṭīya with Bhaṭadīpikā of Parameśvara, Ed. by H. Kern, E. J. Brill, Leiden, 1874.
2. Sadratnamāla of Śa\dot{n}karavarman with Auto-commentary, Kavanodayam Press, Kozhikode, 1898.
3. Goladīpikā of Parameśvara, Ed. by T. Ganapati Sastri, Trivandrum Sanskrit Series 49, Trivandrum, 1916.
4. Prakīrṇasa\dot{n}graha of Putumana Somayājī, included in Pañcabodhaṃ-Bhāṣā by Kanippayyoor Sankaran Nambutirippad, Kunnamkulam, 1927.
5. Pañcabodha of Putumana Somayājī, included in Jytotiṣaśāstrasubodhinī, Vol. II, by Punnasseri Nilakantha Sarma, Trissur, 1929.
6. Āryabhaṭīyabhāṣya of Nīlakaṇṭha Somayājī, Vol. I, Gaṇitapāda, Ed. by K. Sambasiva Sastri, Trivandrum Sanskrit Series 13, Trivandrum, 1930.
7. Āryabhaṭīyabhāṣya of Nīlakaṇṭha Somayājī, Vol. II, Kālakriyāpāda, Ed. by K. Sambasiva Sastri, Trivandrum Sanskrit Series 22, Trivandrum, 1931.
8. Karaṇapaddhati of Putumana Somayājī, Ed. by K. Sambasiva Sastri, Trivandrum Sanskrit Series 126, Trivandrum, 1937.
9. Mahābhāskarīya with Dīpikā of Parameśvara, Ed. by B.D. Apte, Anandasram Sanskrit Series 126, Pune, 1945.
10. Laghubhāskarīya with Vyākhyā of Parameśvara, Ed. by B.D. Apte, Anandasram Sanskrit Series 128, Pune, 1946.
11. Candravākya of Vararuci, Ed. by C. Kunhan Raja, Adyar Library, Chennai, 1948.
12. Haricarita of Parameśvara, Ed. by V. Krishnamacharya, Adyar Library, Chennai, 1948.
13. Yuktibhāṣā of Jyeṣṭhadeva, the Gaṇita Part Ed. with Detailed Notes in Malayalam by Rama Varma Tampuran and A.R. Akhileswaraiyer, Mangalodayam Press, Trissur, 1948.
14. Laghubhāskarīya with Vivaraṇa of Śa\dot{n}karanārāyaṇa, Ed. by P.K. Narayana Pillai, Trivandrum Sanskrit Series 162, Trivandrum, 1949.
15. Laghumānasa of Muñjāla with Vyākhyā of Parameśvara, Ed. by B.D. Apte, Anandasram Sanskrit Series 126, Pune, 1952.

In spite of the publication of these 15 major source-works during the period 1874–1952, and the landmark studies of Datta and Singh and Rajagopal and co-workers, there still remained huge gaps in our understanding of the history and achievements of the Kerala school. There was little clarity about the period and works of most of the illustrious astronomers of this school, starting from the legendary Mādhava himself. The authorship of even a seminal text such as the Yuktibhāṣā was not known. A large number of texts of Mādhava, Parameśvara, Nīlakaṇṭha, Śa\dot{n}kara and Acyuta were yet to be located and published. While there was some clarity on the contributions of the Kerala school in developing the infinite series and approximations for \pi and the sine and cosine functions, there was no awareness of the great contributions of this school in the field of astronomy.

The Monumental Contribution of K.V. Sarma (1943–2005)

Krishna Venkateswara Sarma (1919–2005) was born in Chengannur, presently in the Alappuzha District of Kerala. After completing a BSc, he obtained a Master’s degree in Sanskrit in 1942 from the Maharaja College of Arts, Thiruvananthapuram. He worked in the Manuscripts section of the Kerala University Oriental Research Institute during 1943–1951, and with the New Catalogus Catalogorum Project and the Department of Sanskrit in the University of Madras during 1951–1962. During 1962–1983, Sarma was associated with the Vishvesvaranand Institute of Sanskrit and Indological Studies at Hoshiarpur, affiliated with the Panjab University. Returning to Chennai, he founded the Sree Sarada Education Society and Research Centre, which has become the Prof. K.V Sarma Research Foundation after Sarma’s demise in 2005. During his very active and prolific research career spanning over sixty years, K.V. Sarma authored a large number of books and articles on various aspects of Indian Studies (see [64] for more details). Here, we shall focus on his monumental contribution in the field of astronomy and mathematics, especially the work of the Kerala school.

A comprehensive survey of the history of the Kerala school

As we noted earlier, notwithstanding the publication of about 15 major source-works during the period 1874–1952 (that we listed above), there were huge gaps in our understanding of the history and achievements of the Kerala school. This may be seen for instance from the titles of two landmark papers of Sarma which appeared in 1958. The first one is titled “Date of Mādhava: A Little-Known Indian Astronomer” [65]. It summarises some of the important verses and results in mathematics and astronomy attributed to Mādhava in the Āryabhaṭīyabhāṣya of Nīlakaṇṭha, and the then unpublished commentary Kriyākramakarī on Līlāvatī. The article determines, for the first time, the period of Mādhava as 1350–1410, by considering the epochs mentioned in his work Veṇvāroha, as also the epoch (1431) of the work Dṛggaṇita of his student Parameśvara.

The other article is titled “Jyeṣṭhadeva and his Identification as the Author of Yuktibhāṣā” [66]. There was still a lot of confusion about the name and identity of the author of Yuktibhāṣā, even 125 years after the work was first mentioned by Whish in 1832. Sarma identified Jyeṣṭhadeva as the author of Yuktibhāṣā by locating a statement found in Manuscript No. 755 of Kerala University Manuscripts Library, which contained a work called vikṣepacalanavāsanā–-which was later published by Sarma as a part of the compilation entitled Gaṇitayuktayaḥ in 1979. The statement, “jyeṣṭhadevo’pi bhāṣāyāṃ nādhikaṃ kiñciduktavān”, clearly showed that Jyeṣṭhadeva was the author of “bhāṣā” or Yuktibhāṣā. Further confirmation was obtained in a long passage found in Manuscript No. 9886 in the Oriental Institute Baroda. This is a Malayalam commentary on Sūryasiddhānta, which mentions that Jyeṣṭhadeva, like Nīlakaṇṭha, was the pupil of Dāmodara the son of Parameśvara, and that he was also the author of Yuktibhāṣā.

Sarma’s tireless efforts at compiling a detailed bibliography of the works of the Kerala school resulted in the publication in 1972 of the first ever comprehensive landmark survey of the work of Kerala school, A History of the Kerala School of Hindu Astronomy [67]. In this pioneering survey, K.V. Sarma listed around 400 works on mathematics and astronomy. Sarma also noted that of these, there were 211 Karaṇa texts, 34 Tantra texts and 6 Siddhānta texts. More than 150 works happened to be in Malayalam. Thirty years later, in 2002, Sarma published a survey of the Science Texts in Sanskrit in the Manuscripts Repositories of Kerala and Tamil Nadu, where he listed over 450 works in Sanskrit on mathematics and astronomy from Kerala [68].

Publication of editions and translations of the source-works (1953–2008)

With his expertise in Malayalam and Sanskrit, and his familiarity with the manuscript collections in Kerala and elsewhere, K.V. Sarma had an uncanny ability to identify important unpublished works. He embarked on the daunting task of editing and publishing them, right from the early 1950s. He brought out an edition and translation of Rāśigolasphuṭanīti of Acyuta Piṣāraṭi in 1953. Sarma was able to locate manuscripts of the famous seventh-century text on the Parahita system, the Grahacāranibandhana of Haridatta, which he brought out in 1954. In 1955, he edited and translated the Siddhāntadarpaṇa of Nīlakaṇṭha Somayājī. The next year 1956, saw the first-ever publication of a work of Mādhava, when Sarma published his celebrated work presenting an improved vākya method for computing the motion of the Moon, Veṇvāroha, along with the Malayalam commentary of Acyuta Piṣāraṭi. As an appendix to this edition, Sarma also included an edition of the accurate Candravākya of Mādhava.

During the decade 1957–66, Sarma published five works of Mādhava’s disciple, Parameśvara, including Goladīpikā with the author’s auto-commentary, and the famous karaṇa work of the Dṛk System, Dṛggaṇita, composed in c.1431. During 1970–1979, Sarma brought out the Sphuṭacandrāpti of Mādhava; the seminal works Golasāra, Candracchāyāgaṇita with auto-commentary, Siddhāntadarpaṇa-vyākhya and Jyotirmīmāṃsā of Nīlakaṇṭha; and Sphuṭanirṇayatantra of Acyuta. In 1979, he published a compilation Gaṇitayuktayaḥ, comprising twenty-seven short tracts, presenting yuktis (demonstrations) for various computational procedures in astronomy. Yuktis or demonstrations were also the central concern of two extensive commentaries, Kriyākramakarī and Yuktidīpikā of Śa\dot{n}kara Vāriyar, on Līlāvatī and Tantrasa\dot{n}graha respectively, which were published by Sarma in 1975 and 1977. These Sanskrit commentaries are in fact based on the seminal Malayalam work Yuktibhāṣā of Jyeṣṭhadeva. From the 1980s onwards, Sarma had been working on a critical edition and translation of the Yuktibhāṣā. He asked K. Ramasubramanian, M.D. Srinivas and M.S. Sriram to prepare detailed explanatory mathematical notes for this work in English. He had the satisfaction to oversee the finalisation of the press copy of the first volume just prior to his demise in 2005. The two-volume work was finally published in 2008. Sarma also published a Sanskrit version of Yuktibhāṣā, based on a manuscript in GOML Madras, in 2004.

In all, during the period 1953–2005, Sarma brought out pioneering editions of nineteen source-works of the Kerala school, apart from the compilation Gaṇitayuktayaḥ comprising twenty-seven short tracts. Following is the list of the works of the Kerala school published by Sarma:

1. Rāśigolasphuṭanīti of Acyuta Piṣāraṭi, Ed. and Tr. by K.V. Sarma, Adyar Library, Madras, 1953. Rev. Ed. with Appendices, Vishvesvaranand Institute, Hoshiarpur, 1977.
2. Grahacāranibandhana of Haridatta, Ed. by K.V. Sarma, K.S.R. Institute, Madras, 1954.
3. Siddhāntadarpaṇa of Nīlakaṇṭha Somayājī, Ed. and Tr. by K.V. Sarma, Adyar Library, Madras, 1955. Rev. Ed. with the Auto-commentary of Nīlakaṇṭha Somayājī, Vishvesvaranand Institute, Hoshiarpur, 1976.
4. Veṇvāroha of Mādhava with the Malayalam Commentary of Acyuta Piṣāraṭi, Ed. by K.V. Sarma, Sanskrit College, Tripunithura, 1956.
5. Goladīpikā of Parameśvara, Ed. with Auto-commentary and Tr. by K.V. Sarma, Adyar Library, Madras, 1957.
6. Grahaṇāṣṭaka of Parameśvara, Ed. by K.V. Sarma, K.S.R. Institute, Madras, 1959.
7. Dṛggaṇita of Parameśvara, Ed. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1963.
8. Grahaṇamaṇḍana of Parameśvara, Ed. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1965.
9. Grahaṇanyāyadīpikā of Parameśvara, Ed. and Tr. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1966.
10. Golasāra of Nīlakaṇṭha Somayājī, Ed. and Tr. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1970.
11. Sphuṭacandrāpti of Mādhava, Ed. & Tr. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1973.
12. Sphuṭanirṇayatantra of Acyuta Piṣāraṭi, Ed. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1974.
13. Līlāvatī with the Commentary Kriyākramakarī of Śa\dot{n}kara Vāriyar, Ed. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1975.
14. Candracchāyāgaṇita of Nīlakaṇṭha Somayājī, Ed. with Auto-commentary by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1976.
15. Tantrasa\dot{n}graha with the Commentary Yuktidīpikā of Śa\dot{n}kara Vāriyar, Ed. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1977.
16. Jyotirmīmāṃsā of Nīlakaṇṭha Somayājī, Ed. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1977.
17. Gaṇitayuktayaḥ (Collection of Twenty-seven Tracts), Ed. by K.V. Sarma, Vishvesvaranand Institute, Hoshiarpur, 1979.
18. Sadratnamāla of Śa\dot{n}karavarman, Ed. by K.V. Sarma, INSA, New Delhi, 2001.
19. Gaṇitayuktibhāṣā (Sanskrit Version), Ed. by K.V. Sarma, IIAS, Shimla, 2004.
20. Gaṇitayuktibhāṣā of Jyeṣṭhadeva, Ed. & Tr. by K.V. Sarma, with Notes by K. Ramasubramanian, M. S. Sriram and M. D. Srinivas, 2 Vols., Hindustan Book Agency, New Delhi, 2008. Rep. Springer, New York, 2009.

Apart from the above series of publications by Sarma, another twelve works of the Kerala school were published during 1953–2008. Eight of them were published during 1953–58 itself, and these included the very important editions, by Suranad Kunjan Pillai, of the Golapāda part of the Āryabhaṭīyabhāṣya of Nīlakaṇṭha and the Tantrasa\dot{n}graha of Nīlakaṇṭha with the commentary Laghuvivṛti of Śa\dot{n}kara; the editions of Karaṇapaddhati, firstly with detailed notes in Malayalam by P.K. Koru, and with two Malayalam commentaries by S.K. Nayar; the edition of the Siddhāntadīpikā super-commentary of Parameśvara on Mahābhāskarīya by T.S. Kuppanna Sastri; and, an edition of the entire text of Yuktibhāṣā brought out by the GOML, Madras.

The following is the list of the twelve source-works published during 1953–2008, in addition to the ones edited by Sarma during this period:

1. Gaṇitayuktibhāṣā of Jyeṣṭhadeva, Ed. by T. Chandrasekharan, GOML, Chennai, 1953.
2. Karaṇapaddhati of Putumana Somayājī, Ed. with Malayalam Notes by P.K. Koru, Cherp, 1953.
3. Uparāgaviṃśati of Acyuta Piṣāraṭi, Ed. by Rama Varma Tampuran, Trippunithura, 1954. Rev. Ed. with Malayalam Commentary by K. A. Poduval, Sanskrit College, Tripunithura, nd.
4. Karaṇapaddhati of Putumana Somayājī, Ed. with Two Malayalam Commentaries by S.K. Nayar, GOML, Madras, 1956.
5. Āryabhaṭīyabhāṣya of Nīlakaṇṭha Somayājī, Vol. III, Golapāda, Ed. by Suranad Kunjan Pillai, Trivandrum Sanskrit Series 185, Trivandrum, 1957.
6. Sūryasiddhānta with the Vivaraṇa of Parameśvara, Ed. with Notes by K. S. Shukla, Lucknow University, Lucknow, 1957.
7. Mahābhāskarīya with the Bhāṣya of Govindasvāmin and the super-commentary Siddhāntadīpikā of Parameśvara, Ed. with notes by T.S. Kuppanna Sastri, K.S.R. Institute, Chennai, 1957.
8. Tantrasa\dot{n}graha of Nīlakaṇṭha Somayājī, Ed. with the Laghuvivṛti of Śa\dot{n}kara Vāriyar, by Suranad Kunjan Pillai, Trivandrum Sanskrit Series 188, Trivandrum, 1958.
9. Karaṇottama of Acyuta Piṣāraṭi, Ed. with Auto-commentary by Raghavan Pillai, Trivandrum Sanskrit Series 213, Trivandrum, 1964.
10. Karaṇāmṛta of Citrabhānu, Ed. by V. Narayanan Namboodiri, Trivandrum Sanskrit Series 240, Trivandrum, 1975.
11. Mahājyānayanaprakāra, Ed. and Tr. by D. Gold and D. Pingree, Historia Scientiarum, 42, 1991, pp.49–65.
12. Pañcabodha, Ed. by Nataraja Pillai, Sreevidya Niketanam, Trivandrum, 1994.

Thus, with the lead given by K.V. Sarma, the period 1953–2008 indeed saw a spate of publications of thirty-one source-works of the Kerala school apart from a compilation including 27 small tracts.

Other publications of K.V. Sarma

It is important to note that, in addition to bringing out works of the Kerala school, Sarma also brought out landmark editions and translations of the Vedā\dot{n}gajyotiṣa, Āryabhaṭīya, Pañcasiddhāntikā and Bṛhatsaṃhitā, and also of Vākyakaraṇa, the basic text of the vākya system. Some of this work was done in collaboration with his reputed colleagues T.S. Kuppanna Sastri (1900–1982) and Kripa Shankara Shukla (1918–2007). These studies, especially of Āryabhaṭīya and Vākyakaraṇa are of crucial importance in understanding the later developments initiated by the Kerala school.

In collaboration with another reputed historian of Indian science, B.V. Subbarayappa (1925–2019), Sarma prepared a valuable Source Book which presented important passages from various source-works of Indian Astronomy along with translations, with all the passages being arranged theme-wise [68]. Sarma also gave a series of lectures on observational astronomy in ancient and medieval periods, which was published as a monograph by the Calicut University [69].

Sarma was a prolific writer of articles, both scholarly and popular, on all aspects of Indian Studies, especially Jyotiṣa, Gaṇita and manuscriptology. Some of his important articles have been compiled in two volumes, edited by Siniruddha Dash [70],[71]. Sarma also wrote meticulous summaries on the work of the great astronomers and mathematicians of India for various encyclopaedias. In particular, he contributed a large number of articles, especially on the savants of the Kerala school, for the first two editions of the Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, edited by Helaine Selin and published by Springer in 1997 and 2008. Some of these articles have been reproduced in the recent edition of the Encyclopaedia in 2016 [72].¹⁸

The Legacy of K.V. Sarma

In discussing the legacy of K.V. Sarma, we shall focus on mainly three issues that lie at the heart of much of his work on the Kerala school’s astronomy and mathematics. The first is his concern with bringing out editions and translations of works that present yuktis or demonstrations of the various results and procedures in astronomy and mathematics. While C.T. Rajagopal and co-workers and others (such as T.A. Sarasvati Amma, A.K. Bag and R.C. Gupta) did pioneering work in explaining the demonstrations discussed in Yuktibhāṣā, it was the painstaking effort of Sarma which resulted in the publication of the Sanskrit commentaries of Śa\dot{n}kara Vāriyar (Kriyākramakarī and Yuktidīpikā), and more crucially the preparation of the critical edition of Yuktibhāṣā of Jyeṣṭhadeva, along with English translation and notes, which ensured that all the demonstrations worked out in the Kerala tradition became widely accessible in their original form, along with translations and detailed mathematical explanations. This has made it possible for scholars to appreciate the rigour and sophistication of the Kerala school’s work on infinite series. It is now well established that the proofs found in this 16th century Kerala work are more like the modern proofs (with careful consideration of the orders of magnitudes and the limiting procedures) in comparison to the proofs given by the great European savants when they rediscovered these series in the 17th century. The volume Gaṇitayuktayaḥ edited by Sarma contains twenty-seven tracts presenting yuktis for various astronomical procedures. Most of these tracts are yet to be translated and studied. Amidst all his scholarly endeavours, Sarma was particularly keen to publish texts which give us first-hand knowledge of the way of thinking and working of the Indian astronomers and mathematicians.

The second concern of Sarma was to look for texts which introduced novel procedures or models in planetary theory, computation of eclipses, spherical astronomy and so on. The earliest publications of Sarma such as the Rāśigolasphuṭanīti of Acyuta Piṣāraṭi and Veṇvāroha of Mādhava were important texts which gave a better method for the `reduction to ecliptic’ and an improved vākya method for the computation of the lunar longitude. In collaboration with Kuppanna Sastri, Sarma brought out the 13th-century work, Vākyakaraṇa, which happens to be the basic text for understanding the vākya system of computations.

While working on the edition and translation of Yuktibhāṣā, Sarma noticed that, in the astronomy part of the text, Jyeṣṭhadeva was actually discussing a new planetary model as formulated by Nīlakaṇṭha. He presented this finding in an important paper on Yuktibhāṣā that he wrote with S. Hariharan in 1991 [73][p.193]:

This revised planetary model was presented in detail by Nīlakaṇṭha in his major works Tantrasa\dot{n}graha, Golasāra, Siddhāntadarpaṇa and its auto-commentary, Āryabhaṭīyabhāṣya and a small tract entitled Grahasphuṭānayane Vikṣepavāsanā (which is a part of the compilation Gaṇitayuktayaḥ edited by Sarma). The essential features of this model, as presented in the above works, were brought to light in a paper by K. Ramasubramanian, M.D. Srinivas and M.S. Sriram in 1994 [74]. Considerable work has now been done on this revised planetary model of Nīlakaṇṭha and its adoption by the later Kerala astronomers (for a recent overview see [75][pp.487-535]). Thus, it is now well established that the Kerala school came up with two of the major innovations, namely the discovery of calculus and of non-geocentric planetary models, more than a century before they were to become hallmarks of the European scientific revolution.

A third issue, that was repeatedly highlighted by K.V. Sarma, has been the importance given to the constant checking and updating of the theories and models in the Indian astronomical tradition using continued and accurate observations. While this was emphasised in several ancient texts, this was particularly highlighted in the works of Parameśvara, Nīlakaṇṭha and Jyeṣṭhadeva. Sarma had written about this in his editions of the works of Parameśvara, but he was indeed excited to chance upon a manuscript of the seminal text of Nīlakaṇṭha, the Jyotirmīmāṃsā, which discusses at length the crucial role of continuous parīkṣā, or examination of the theories by observation, and the consequent process of revision and bringing out newer and newer karaṇas, or computational manuals, that are accurate for the period under consideration. Sarma has discussed this issue extensively, along with elaborate quotations from the source-works, in his “Source Book”, the book on “Observational Astronomy”, and in several articles that he wrote on the subject [68],[69],[70].

Continuing the Legacy of the Great Savants (2005–2021)

Finally, we briefly touch upon the recent work on the contributions of the Kerala school. This is largely based on the substantial corpus of source-works of the Kerala school that have been edited and published, as also the scholarly studies that have been published in the last century. Thus, much of this work may be said to be in continuation with the seminal contributions of the great savants such as Śa\dot{n}karavarman, Charles Whish, B. Datta and A.N. Singh, K. Sambasiva Sastri, C.T. Rajagopal, Rama Varma Tampuran, Suranad Kunjan Pillai and K.V. Sarma and their illustrious colleagues.

Recent decades have seen the publication of a large number of scholarly articles and books, which deal with various aspects of the work of the Kerala school (see, for instance [72, 75-85], and the references cited therein). Even if we restrict ourselves to only the books published in English, we find that there are more than thirty-five books and monographs published in the last twenty-five years which include at least a section devoted to the discussion of the work of the Kerala school. The number of scholarly articles would indeed be much larger. From this vast literature, we shall present two recent assessments of the work of the Kerala school in order to illustrate how the discourse has changed from what we saw in the nineteenth century. The renowned mathematician David Mumford has characterised the work of the Kerala school as “the crown jewel of Indian mathematics”, in his 2010 review of Kim Plofker’s book on Mathematics in India [86][p.389]:

Roy Wagner, who has written extensively on the foundations of mathematics and also on the work of the Kerala school, refers to the Kerala mathematics of the 14–16 centuries as “the most advanced mathematical culture of its time throughout the world”, in a 2019 publication (which is the proceedings of a conference on the foundations of mathematics in 2016) [87][pp.390–391]:

While it is heartening to see that there is a growing literature on the work of the Kerala school, the situation is somewhat disappointing when we take stock of the number of source-works or their translations that have been published in the last decade and a half. We list below the few source-works that have been published since 2008:

1. Tantrasa\dot{n}graha of Nīlakaṇṭha Somayājī, Ed. & Tr. with Notes by K. Ramasubramanian and M.S. Sriram, Hindustan Book Agency, New Delhi, 2011. Rep. Springer, New York, 2011.
2. Sadratnamāla of Śa\dot{n}karavarman, Ed. & Tr. with Notes by S. Madhavan, K.S.R. Institute, Chennai, 2011.
3. Madhyamānayanaprakāra of Mādhava, Ed. & Tr. by U.K.V. Sarma, R. Venkateswara Pai and K. Ramasubramanian, Ind. Jour. Hist. Sc., 46(1), 2011, pp. T1-T29.
4. Karaṇapaddhati of Putumana Somayājī, Ed. & Tr. with Notes by R. Venkateswara Pai, K. Ramasubramanian, M.S. Sriram and M.D. Srinivas, Hindustan Book Agency, New Delhi, 2017. Rep. Springer, Singapore, 2017.

Currently, there are several important source-works of the Kerala school–-even some works composed by legendary figures such as Mādhava, Parameśvara, Nīlakaṇṭha, Jyeṣṭhadeva, Putumana Somayājī, Śa\dot{n}kara and Acyuta–-which are yet to be edited and published. Further, even among the fifty major works (and about thirty small tracts) which have been published in the last 150 years (1874–2021), only about a dozen have been translated; and even among them, there are only about half a dozen works which have been studied in depth to explicate their full technical content.

It will be befitting to the legacy of the great savants mentioned above, that our young scholars imbibe their great dedication, perseverance and love and respect for our tradition, and carry forward their mission to ensure that the entire corpus of works of the Kerala school is rigorously and comprehensively studied and brought to light. In this way, they would ensure that the great achievements of the Kerala school of Astronomy and Mathematics are not only accorded their due place in the history of science, but also serve as a source of ideas and inspiration for a resurgent Indian science in our times.

Acknowledgement: The author is deeply indebted to Prof. K. Ramasubramanian and Prof. M.S. Sriram for their valuable insights, advice and help which have contributed immensely to the preparation of this article. He is also grateful to Prof. Michel Danino and Prof. C.S. Aravinda for their kind encouragement and support. \blacksquare

References

1. U.K.V. Sarma, Vanishri Bhat, Venkateswara Pai and K. Ramasubramanian, “The discovery of Mādhava series by Whish: An episode in historiography of science”, Gaṇita Bhāratī, 32, 2010, pp.115–126.
2. K. Ramasubramanian, “The history and historiography of the discovery of calculus in India”, IIT Bombay preprint 2021 (To appear in the Proceedings of the International Congress of Mathematicians, St. Petersburg, 2022).
3. J. Warren, Kālasa\dot{n}kalita, A Collection of Memoirs on the Various Modes According to which the Nations of the Southern Part of India Divide Time}, College Press, Madras, 1825.
4. Charles C. Prinsep, Record of Services of the Honourable East India Company’s Civil Servants in the Madras Presidency from 1741 to 1858, Trubner & Co., London 1885.
5. R. Venkateswara Pai, K. Mahesh and K. Ramasubramanian, “A commentary of Tantrasa\dot{n}graha in Keralabhāṣā: Kriyākalāpa”, Ind. Jour. Hist. Sc., 45.2, 2010, pp.S1-S47.
6. C.M. Whish, “On the alphabetical notation of the Hindus”, Transactions of the Literary Society of Madras, Part I, 1827, pp.54–62.
7. C.M. Whish, “On the origin and antiquity of the Hindu zodiack”, Transactions of the Literary Society of Madras, Part I, 1827, pp.63–77.
8. William Jones, “On the Antiquity of the Indian Zodiac”, Asiatic Researches, Vol. II, 1790, pp. 289–306.
9. C.M. Whish, “On the Hindū quadrature of the circle, and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sāstras, the Tantra Sangraham, Yucti Bhāshā, Carana Paddhati and Sadratnamāla”, Transactions of the Royal Asiatic Society of Great Britain and Ireland, Vol. III, 1834, pp. 509–523.
10. Dharampal, The Beautiful Tree: Indigenous Indian Education in the Eighteenth Century, Rashtrotthana Sahitya, Bangalore, 2021. First published, Biblia Impex, Delhi, 1983.
11. D.E. Pingree, “Hellenophelia versus the history of science”, Isis, 83.4, 1992, pp.554–563.
12. The Asiatic Journal and Monthly Register for British and Foreign India China and Australia, Vol. X, January-April 1833, pp.81–82.
13. The London Literary Gazette and Journal of the Belles Letters, No. 831, December 22, 1832, p.810; The Gentleman’s Magazine and Historical Chronicle December 1832, Vol. CII, Part II, p.631; The New Monthly Magazine and Literary Journal, American Edition, Boston, Vol. I, January 1833, p.74.
14. E. Jacquet, “De la notation alphabetique des Indiens par M. C. M. Whish”, Jour. Asiatique, August 1835, pp.116–130.
15. C. Lassen, “Ueber den gebrauch der buchstaben zur bezeichnung der zahlen bei den Indischen mathematikern”, Zeitschrift fur die Kunde des Morgenlandes, Band II, Heft 3, 1837, pp.419–427.
16. F. Adelung, Bibliotheca Sanscrita. Litratur der Sanskrit-Sprache, St. Petersburg, 1837, pp.194, 197, 198, 202.
17. S.P. Rigaud and S.J. Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Letters of Barrow, Flamstead, Wallis and Newton from the Originals, Vol. II, Oxford, 1841.
18. Augustus De Morgan, Article on “Viga Ganita”, in The Penny Cyclopaedia, Vol. XXVI, Charles Knight & Co, London, 1843, pp.318–26.
19. Augustus De Morgan, Article on “Viga Ganita”, in The English Cyclopaedia: A New Dictionary of Universal Knowledge, Vol. VI, Bradbury and Evans, London, 1858, pp.371–81.
20. Ramchundra, Treatise on Problems of Maxima and Minima Solved by Algebra, W. H. Allen and Co, London, 1859.
21. C.P. Brown, Carnatic Chronology: The Hindu and Mahomedan Methods of Reckoning Time Explained, Bernard Quaritch, London 1863, pp.48–49.
22. Jour. Royal Asiatic Soc., Vol. 3.2, 1836, p.lxxiii.
23. Teubner’s American and Oriental Literary Record, No. 32, February 25, 1868, p.154.
24. Jour. Royal Asiatic Soc., 22.4, 1890, pp.805–813.
25. M. Winternitz, A Catalogue of South Indian Sanskrit Manuscripts: Especially those of the Whish Collection, Royal Asiatic Society, London, 1902.
26. https://royalasiaticcollections.org/south-asian-manuscripts/, accessed on December 20, 2021.
27. M. Winternitz, Geschichte der Indischen Litteratur, Bd. 3, Amelangs Verlag, Leipzig, 1920.
28. A. Weber, Indische Studien, Achter Band, Berlin 1863, p.160.
29. Bhau Daji, Brief notes on the age and authenticity of the works of Āryabhaṭa, Varāhamihira, Brahmagupta, Bhaṭṭotpala and Bhāskarācārya, Jour. Royal Asiatic Soc., Vol. I, 1865, pp.392–418.
30. E. Burgess, Sūryasiddhānta: A Text-Book of Hindu Astronomy; with Notes and an Appendix, The American Oriental Society, New Haven, 1860, p.174.
31. J. Burgess, Notes on Hindu astronomy and the history of our knowledge of it, Jour. Roy Asiatic Soc., 25, 717–761, 1893.
32. G. Thibaut, Astronomie, Astrologie und Mathematik, Verlag von Karl J. Trubner, Strassburg, 1899, p.2.
33. D.E. Smith, “The geometry of the Hindus”, Isis, Vol. 1, No. 2, 1913, pp.197–204.
34. D.E. Smith, History of Mathematics, Vol. II, Dover, New York, 1958 (Rep. of 1925 edition).
35. B. Datta, “Hindu values of \pi”, Jour. Asiatic Soc. Bengal, 22, 1926, pp.25–42.
36. B. Datta and A.N. Singh, History of Hindu Mathematics, A Source Book, Part I, Motilal Banarsi Das, Lahore 1935, p.ix.
37. B. Datta and A.N. Singh (Revised by K.S. Shukla), “Use of series in India”, Ind. Jour. Hist. Sc., 28(2), 1993, pp.103–129.
38. A.N. Singh, “On the use of series in Hindu mathematics”, Osiris, 1, 1936, pp.606–28.
39. M.S. Rangachari, “Prof. C.T. Rajagopal”, Indian J. Math., 22 (1), 1980, pp.i-xxix.
40. K. Mukunda Marar, “Derivation of the value of \pi–-How the ancients did it”, Teacher’s Magazine (Journal of the Cochin Teachers’ Association), 15, 1940, pp.28–34.
41. K. Mukunda Marar and C.T Rajagopal, “On the Hindu quadrature of the circle”, Jour. Bombay Branch Royal Asiatic Soc., 20, 1944, pp.65–82.
42. K. Mukunda Marar and C.T. Rajagopal, “Gregory’s series in the mathematical literature of Kerala”, Math. Student, 13, 1945, pp.92–98.
43. A. Venkataraman, “Some interesting proofs from Yuktibhāṣā”, Math. Student, 16, 1948, pp.1–7.
44. C.T. Rajagopal, “A neglected chapter of Hindu mathematics”, Scripta Math., 15, 1949, pp.201–209.
45. C.T. Rajagopal and A. Venkataraman, “The sine and cosine power series in Hindu mathematics”, Jour. Royal Asiatic Soc. Bengal, 15, 1949, pp.1–13.
46. C.T. Rajagopal and T. V. Vedamurthi Aiyar, “On the Hindu proof of Gregory’s series”, Scripta Math., 17, 1951, pp.65–74.
47. C.T. Rajagopal and T.V. Vedamurthi Aiyar, “A Hindu approximation to \pi”, Scripta Math., 18, 1952, pp.25–30.
48. C.T. Rajagopal and M.S. Rangachari, “On an untapped source of medieval Keralese mathematics”, Arch. History Exact Sci., 18, 1978, pp.84–102.
49. C.T. Rajagopal and M.S. Rangachari, “On medieval Kerala mathematics”, Arch. History Exact Sci., 35(2), 1986, pp.91–99.
50. J.E. Hoffmann, “Uber eine altindische berechnung von Pi und ihre allgemeine bedeutung”, in Mathematisch-Physikalische Semesterberichte, Bd. 3, H. 3/4, Hamburg, 1953, pp.193–216.
51. C.N. Srinivasiengar, Gaṇitaśāstrada Caritre (in Kannada), Mysore University, Mysore, 1958, pp.319–324.
52. C.N. Srinivasiengar, The History of Ancient Indian Mathematics, The World Press, Calcutta, 1967, pp.142–154.
53. A.P. Yushkevich, History of Medieval Mathematics (in Russian), Moscow 1961; German translation published as Geschichte der Mathematik im Mittelalter, Teubner, Leipzig, 1964 (see pp.168–174).
54. D.T. Whiteside, “Patterns of mathematical thought in the later seventeenth century”, Arch. Hist. Exact. Sci., 1, 1961, pp.179–388 (see pp.210, 255).
55. D.T. Whiteside, The Mathematical Papers of Isaac Newton, Vol. II, The Cambridge University Press, Cambridge, 1968, fn on p.237.
56. H.L.L. Busard, “Ueber unendliche reihen im mittelalter”, L’Enseignement Mathématique, 8, 1962, pp.281–290 (see p.289).
57. T.A. Saraswathi Amma, “The development of mathematical series in India after Bhāskara II”, Bull. Natl. Inst. Sc. in India, 21, 1963, pp.320–343.
58. T.A. Saraswathi Amma, Geometry in Ancient and Medieval India, Motilal Banarsidass, Delhi, 1979, pp.154–194.
59. A.K. Bag, “Trigonometrical series in Karaṇapaddhati”, Ind. Jour. Hist. Sc., 1(2), 1966, pp.98–106.
60. A.K. Bag, Mathematics in Ancient and Medieval India, Chowkhambha Orientalia, Varanasi, 1979, pp.263–298.
61. M.E. Baron, The Origins of the Infinitesimal Calculus, Pergamon, Oxford, 1969, pp.61–65.
62. C.H. Edwards, The Historical Development of the Calculus, Springer, New York, 1979.
63. G. Ferraro, The Rise and Development of the Theory of Series up to the Early 1820s, Springer, New York, 2008.
64. M.S. Sriram, “K.V. Sarma (1919–2005)”, Ind. Jour. Hist. Sc., 41, 2006, pp.231–246.
65. K.V. Sarma, “Date of Mādhava: A little-known Indian astronomer”, Qtr. Jour. Mythic Soc., 49(3), 1958, pp.183–186.
66. K.V. Sarma, “Jyeṣṭhadeva and his identification as the author of Yuktibhāṣā”, Adyar Lib. Bull., 22, 1958, pp.35–40.
67. K.V. Sarma, A History of the Kerala School of Hindu Astronomy, Vishvesvaranand Institute, Hoshiarpur, 1972.
68. B.V. Subbarayappa and K.V. Sarma, Indian Astronomy: A Source Book, Nehru Centre, Bombay, 1985.
69. K.V. Sarma, Observational Astronomy in India, University of Calicut, Kozhikode, 1990.
70. S. Dash (ed.), Facets of Indian Astronomy: A Collection of Articles of Prof. K.V. Sarma, Rashtriya Sanskrit University, Tirupati, 2009.
71. S. Dash (ed.), New Lights on Manuscriptology: A Collection of Articles of Prof. K.V. Sarma, Sree Sarada Education Society and Research Centre, Chennai, 2007.
72. H. Selin (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer, New York, 1997. Second edition, ibid., 2008. Third edition, ibid., 2016.
73. K.V. Sarma and S. Hariharan, “Yuktibhāṣā of Jyeṣṭhadeva, A book of rationales in Indian mathematics and astronomy: An analytical appraisal”, Ind. Jour. Hist. Sci., 26, 1991, pp.185–207.
74. K. Ramasubramanian, M.D. Srinivas and M.S. Sriram, “Modification of the ancient planetary model by the astronomers of the Kerala school and the implied heliocentric picture of the planetary motion”, Current Science, 66, 1994, pp.784–790.
75. Tantrasa\dot{n}graha of Nīlakaṇṭha Somayājī, Ed. & Tr. with Notes by K. Ramasubramanian and M.S. Sriram, Hindustan Book Agency, New Delhi, 2011.
76. S. Paramesvaran, The Golden age of Indian Mathematics, Kochi, 1998.
77. M.S. Sriram, K. Ramasubramanian and M.D. Srinivas (eds.), 500 Years of Tantrasaṅgraha: A Landmark in the History of Astronomy, IIAS, Shimla, 2002.
78. G.G. Emch, R. Sridharan and M.D. Srinivas, Eds., Contributions to the History of Indian Mathematics, Hindustan Book Agency, New Delhi, 2005.
79. C.K. Raju, Cultural Foundations of Mathematics: The Nature of Mathematical Proof and the Transmission of the Calculus from India to Europe in the 16thc.CE, Pearson Education, Delhi, 2007.
80. K. Plofker, Mathematics in India, Princeton University Press, Princeton, 2009.
81. C.S. Seshadri (ed.), Studies in the History of Indian Mathematics, Hindustan Book Agency, New Delhi, 2010.
82. R. Roy, Sources in the Development of Mathematics Series and Products from Fifteenth to Twenty-first Century, University Press, Cambridge, 2011.
83. G.G. Joseph, Indian Mathematics Engaging the World from Ancient to Modern Times, World Scientific, London, 2016.
84. K. Ramasubramanian, A. Sule and M. Vahia (eds.), History of Indian Astronomy A Handbook, IIT, Mumbai, 2016.
85. P.P. Divakaran, The Mathematics of India Concepts Methods Connections, Hindustan Book Agency, Delhi, 2018. Rep Springer, Singapore, 2018.
86. D. Mumford, “Review of Mathematics in India”, Notices of the AMS, 57 (3), 2010, pp.385–390.
87. R. Wagner, “Does mathematics need foundations?”, in S. Centrone, D. Kant and D. Sarikaya (eds.), Reflections on the Foundations of Mathematics, Springer, Cham, 2019, pp.381–396.

Footnotes

1. George Hyne arrived in India in 1817 and was appointed as an Assistant Surgeon in 1820. Hyne seems to have become an important member of the Madras Literary Society, which was founded in 1818. In 1826, Hyne was appointed by the Madras Government as the Secretary of the newly formed Committee of Public Instruction, though he passed away within a few months thereafter. ↩
2. The reference here is to the 1790 article of William Jones [8][p.302]. ↩
3. The Royal Asiatic Society of Great Britain and Ireland was the premier learned society founded to further the investigation of “science, literature and arts in relation to Asia”. It was started in 1823 at the initiative of the leading Indologist of England, Henry Thomas Colebrooke (1765–1837), who also served as its Founder Director during 1823–37. The King consented to be its patron and the Royal Charter was granted to the Society in 1824. The Asiatic Societies at Calcutta, Bombay etc., were made its associate Societies. The first volume of its Transactions was published in 1827. The third volume of the Transactions seems to have been issued in three parts in 1831, 1833 and 1834 and the consolidated volume bears the year 1835 on its title page. ↩
4. The Collections of Colonel Colin Mackenzie (1754–1821), Dr. Leyden and Mr. C. P. Brown constitute a large part of the vast collection of manuscripts preserved in Government Oriental Manuscripts Library (GOML), Madras, which consists of manuscripts of works in literature, history, philosophy and science, written in South Indian and Oriental languages, and of Kaifiyats and inscriptions, found in many places belonging to different periods. ↩
5. From this passage it appears that Whish might have had an occasion to meet Śa\dot{n}karavarman (1774–1839) who was the brother of the Rāja of Kaṭattanād and the author of Sadratnamāla, which was composed in Kali 4921 or 1819 CE.↩
6. The semantics of the word `vulgar’ has itself changed since the last two hundred years. Back when Whish used it, the word meant as something that referred to, or belonged to, the masses and the common people of the land. ↩
7. Despite the clear statement made by Whish that Tantrasa\dot{n}graha was composed in 4600 Kali or around 1500 CE, many later studies cite Whish as attributing this text to 1608.. ↩
8. The proof of the theorem of Pythagoras given by Whish, however, seems to differ significantly from the proof that is found in all the manuscripts of Yuktibhāṣā. ↩
9. This is the first instance of misquoting Whish on the date of the Tantrasa\dot{n}graha, which is more or less correctly given in the original paper as Kali 4,600 or around A.D. 1500.↩
10. De Morgan’s father and grandfather served as British officials in India and Augustus himself was born in Madurai. He was educated in Cambridge and later worked as a Professor of Mathematics in London University. ↩
11. Charles Brown was the son of an English missionary in Calcutta. He joined the East India Company’s service in 1817 and became well known for his proficiency in the Telugu language and literature. After returning to England in 1854, he worked in London University as a Professor of Telugu. ↩
12. The last two articles [48][49] were published much later in 1978 and 1986. ↩
13. Here is a footnote in the original giving the reference to the article of Whish. Regarding the four śāstras mentioned in the title of Whish’s article, it is noted that: “These works, with the exception of the third [Karaṇa-Paddhati], are available only in manuscript. Our study of the relevant portions of the works and the literature relating to the works has been helped not a little by the devoted labours of Prince Rāma Varma (the 12th prince of Cochin) and Mr. K. Balagangadharan, one of our colleagues.”. ↩
14. Here, there is a footnote in the original giving references to the 1937 edition of the Karaṇa-Paddhati and some of the available manuscripts of the other three works. ↩
15. The reference here is to G. R. Kaye, Indian Mathematics, Calcutta, 1915, p.24. ↩
16. This is a reference to the 1940 publication of K. Mukunda Marar [40]. ↩
17. This does not, however, mean that all the later historical accounts of mathematics (or even specialised works dealing with the development of infinite series or calculus) do take due note of the Kerala contributions wherever required. For instance, the scholarly account of The Historical Development of Calculus by Charles Edwards (1978) [62], or the more recent specialised treatise on The Rise and Development of the Theory of Infinite Series by Giovanni Ferraro (2008) [63], make no reference at all to the contributions of the Kerala school. ↩
18. K.V. Sarma contributed thirty-four important articles for the second edition of the Encyclopaedia which appeared in 2008, three years after his demise. In the third edition of the Encyclopaedia, published in 2016, sixteen of Sarma’s articles have been retained as such. However, we find that fifteen of the scholarly articles contributed by Sarma have been replaced by inaccurate and sloppy articles written by someone who has no scholarly publications to his credit either in astronomy or mathematics, or in the history of science. This is particularly distressing as the Encyclopaedia is otherwise a scholarly publication, and such casual attitude is not displayed in the choice of various contributions pertaining to China, West Asia or other cultures and civilisations. The editor and the publishers are best advised to replace these articles with the earlier articles of K.V. Sarma (the edition is also accessible on the internet), at least till such time that they organise new articles on these topics written by reputed and competent scholars. ↩