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Geometry in Ancient and Medieval India c6: Chapter I. Introduction

Geometry in Ancient and Medieval India c6

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English

Contents

Chapter I. Introduction

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The mathematical genius of the ancient Indians was mainly computational and led to spectacular achievements in arithmetic and algebra. But the basis and inspiration for the whole of Indian mathematics is geometry. The beginnings of algebra are to be traced to the constructional geometry of the Vedic priests preserved in the *Śulbasūtras*. In later periods also algebra must have leaned on geometry. For, down the ages, Indian mathematicians have shown a predilection for demonstrating algebraical truths geometrically (see Ch. IX). So too the arithmetical operations, multiplication and division were demonstrated geometrically. Trigonometry is, as its name proclaims, the geometry of the triangle. Indian trigonometry which employed sine and cosine chords instead of the ratios has always retained its geometrical character. The study of the Theory of Numbers was really the study of the right triangle and rational rectilinear figures.

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The branch of mathematics which received the earliest attention was also geometry. The *Śulbasūtras* (5th to 8th century B.C.) is a manual of geometrical constructions. A geometrical knowledge of this level could not have come into existence overnight. The absence of earlier records has led many scholars to posit large-scale indebtedness to foreign countries, chiefly Babylonia.¹ This is not warranted, though exchange of ideas could have and must have occurred. Even the scant remnants of the Indus Valley Civilisation excavated at Harappa and Mohenjo-daro reveal some acquaintance with geometry. In the words of E. J. H. Mackay, "It is surprising to find that an instrument was actually used for this purpose (i.e. for drawing circles) in the Indus Valley as early as 2500 B.C." Hence it seems more probable that Indian geometry developed gradually into the *Sulbasūtra* stage with the intervening links buried in the sands of time. (India's sands were never so kind to her records as Babylonia's sands have been to her clay tablets.) Even the *Sulbasūtras* may have preserved only a part of the mathematical knowledge of those days, the part that was necessary for constructing the sacrificial altars and for computing the calendar to regulate the performance of sacrifices.

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¹ M. Cantor, *Vorlesungen Uber Geschichte der Mathematik*, 4th Edition, Vol. I. p. 645.

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² E.J.H. Mackay, *Further Excavations at Mohenjo-daro*, 1938, p. 222.

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With the sacrificial cult of the later Vedic period waning in influence and the rising ascendency of the cosmography of the Jainas and the astronomy of the Hindus, geometry becomes circle-oriented, so that Indian geometry after the dawn of the Christian era can pertinently be termed chord-geometry. It was studied for the sake of astronomy and along with the rest of mathematics, forms part of astronomical treatises. But this does not mean there was a break in continuity. The *Sulbasūtra* theorem of the square of the diagonal continued as the chief tool in the hands of the astronomer-geometer. The circle-centred nature of Indian geometry was not fully appreciated by some of the Indian mathematicians themselves, the most notable amongst such being Āryabhaṭa II (10th cent.) and Bhāskara II (12th cent.) who were at a loss to understand Brahmagupta's (7th cent.) expressions for the area and diagonals of a cyclic quadrilateral. The amount of geometrical knowledge (as also the knowledge of complicated mathematical series) which went into the derivation of the easy-to-handle rules of computation found in astronomical manuals is also hardly appreciated by the practising astronomer and astrologer. The continuity of the tradition of geometrical knowledge is testified to by the achievements of the Āryabhaṭa school, who not only understand the significance and limitations of Brahmagupta's theorems about quadrilaterals but also give complete proofs of these.

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A more serious charge against Indian mathematics in general and geometry in particular is the absence of proofs and demonstrations, so beloved of the Greeks. Yoked as geometry was to practical and astronomical needs, it has remained largely mensurational. In the Indian mathematical texts there is no attempt at any proof or derivation and the earlier commentaries too do not improve the situation much. But luckily there are some commentaries and an independent work meant to be an aid to the understanding of an astronomical text which preserve many elaborate proofs and derivations of complicated mathematical series etc. This shows Indian mathematicians too were not satisfied unless they could prove or derive the results they used. It is also significant that these proofs and derivations are found in commentaries, not in the standard mathematical or astronomical treatises, which satisfy themselves with the enunciation of the results. As in *Āyurveda*, the great teachers were content to provide manuals for easy reference in writing while the explanation and the rationale were left to oral instruction. In fact the break in the oral tradition brought about by foreign invasions may have resulted in significant links in our scientific knowledge being irretrievably lost.

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But one has to concede that there was an important difference between the Indian proofs and their Greek counterparts. The Indian's aim was not to build up an edifice of geometry on a few self-evident axioms, but to convince the intelligent student of the validity of the theorem, so that visual demonstration was quite an accepted form of proof.

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This leads us to another characteristic of Indian mathematics which makes it differ profoundly from Greek mathematics. Knowledge for its own sake did not appeal to the Indian mind. Every discipline (*Śāstra*) must have a purpose. And since self-realisation and the resulting deliverance from birth and death was the most legitimate purpose of life, those sciences which were supposed to further this purpose directly or indirectly were most assiduously pursued. Thus astronomy was cultivated as an adjunct to the study of the Vedas and the performance of sacrifices with the rest of mathematics as an appendage to it. Hence the paucity of treatises devoted wholly to mathematics. The *Bakhshali* manuscript, the *Pāṭīgaṇita* and *Triśatikā* of Śrīdhara and the *Gaṇitasāra-saṃgraha* of Mahāvīra are the only early works dealing exclusively with mathematics.

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Other important developments in Indian mathematics are also the outcome of religious needs. The early geometry of the *Śulbasūtras* grew out of the requirements of altar-construction, mathematical series out of Jaina cosmography and later out of the need for improved methods of astronomical culculations, trigonometry and the geometry of cyclic figures out of the calculations in the celestial circle. Indeterminate analysis, for which the Indians are justly famous, is again intimately connected with astronomical calculations.

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Outside this religio-astronomical sphere only the problems of day-to-day life interested the Indian mathematicians. Such were the problems of mensuration, paying for digging and sawing (*khāta* and *krakaca*), barter (*bhāṇḍapratibhāṇḍa*), Rule of Three (*trairāśika*) etc.

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The study of permutations and combinations was inspired by literary criticism (the prosody part of it), a necessary accomplishment of a man of refined tastes. Mixing of ingredients in medicines, idol-making and Jaina religious speculations also contributed to the study.

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This leaning towards utilitarianism has had unfortunate results. The nonchalance with which the splendid achievements of Greek geometry were ignored, while the pseudo-science of Greek and Babylonian astrology was received with open hands, is perhaps the worst of these. It was only in the 18th century, nearly 2000 years after the active contact of the Indians with the Greeks, that Euclid's *Elements* were translated into Sanskrit¹ and even then perhaps the example of the Arabs provided the inspiration.

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¹ The translator was Jagannātha, attached to the court of the famous astronomer king, Jayasiṁha.

General Survey of the history of Geometry and Mathematics in India

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In the history of Indian geometry three distinct periods can be discerned:

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(1) the pre-Āryan period, the remains of whose civilization have been dug up in Harappa, Mohenjo-daro and other places in the Indus Valley.

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(2) The Vedic or *Śulbasūtra* period, and

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(3) The post-Christian period.

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The first of these, the period of the Indus Valley civilization, is the oldest period in which we can find traces of a civilization in India though eminent scholars like B. G. Tilak and Jacobi would assign a greater antiquity to the civilization of the Vedic Āryans. The lower limit of this civilization cannot be later than 2500 B.C. The most noteworthy feature of this civilization is well-planned towns for which some knowledge of geometry is indispensable. We also have other evidence of the geometrical sense of this ancient people.¹ A favourite pattern on pottery excavated at Mohenjo-daro is a series of intersecting circles apparently made by drawing a series of vertical lines to divide the surface of the jar into a number of nearly equal panels and then scratching circles with a pair of dividers. Other patterns are the square, the circle, two triangles joined at their apexes, two or four hemispheres with their curved edges towards the middle of the pattern, a series of linked triangles with hemispheres filling up the space and a rectangle with the four sides incurved. "The last of these is a frequently found motif which I have already compared to stretched hide." The motif of the inverted triangles is also fairly common.

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¹ *Further Excavations at Mohenjo-daro*, pp. 221-25.

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We do not yet know what the exact relationship between this city-civilization and the Vedic civilization was. We have to attach some significance to the fact that the commonest motif on the pottery of this period, namely that of the rectangle with incurved sides resembling a stretched hide, is preserved in the shape of the sacrificial altars of the next Vedic period. The most important altar (*vedi*) of Vedic sacrifices, the *Mahāvedi* is of this shape, while its corners are called *śroṇīs* (hips) and *aṃsas* (shoulders). This points to a close connection between the geometrical knowledge of the two periods, since the geometry of the Vedic period, as far as we know, existed for the sake of sacrificial altars.

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When the poetic vision of the Vedic seers was externalised in symbols, the ritual sacrifice with all its elaborate details came to occupy a pivotal place in their religion. Exact measurements and different geometrical shapes were prescribed for the altars (*vedi*) and fire-places (*agni*) to be used in the sacrifices performed for different purposes. To meet these demands the science of geometry grew up and was codified in the *Śulbasūtras*. But since the sacrificial lore is as old as the Vedas or even older (according to Oldenberg) the geometrical knowledge necessary for the construction of the altars and the fire-places must be equally old. The Vedas themselves, as prayers addressed to particular deities, are unlikely to show any traces of geometrical knowledge. Yet we find in them words like *triraśri* and *caturaśri* (*Ṛg Veda* 1, 21, 152, 2), *daśabhuji* (*Ibid.* 1, 10, 52, 11) and *tribhuja* (*Atharva Veda* VIII, 5, 9, 2,). Though these are of doubtful geometrical significance their presence in purely literary works may be an indication of the popularity of geometry in those days.

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The *Taittirīya Saṃhitā* of the *Yajur Veda*, the Veda of the performing priest, which gives the measurements of the *Mahāvedi* as 36 as altitude and 30, 24 as parallel sides so that half of 30 and 36 make the sides about the right angle of the Pythagorean triangle 36, 15, 39, must, according to A. Bürk,¹ have been acquainted with the theorem connecting the squares of the sides of a right triangle. The Brāhmaṇas of the *Yajur Veda* contain sporadic accounts of the construction of the *vedis* and *agnis* — the sacrificial altars and fire altars (*Śatapatha Brāhmaṇa* III. 5. 1 & X. 2. 1. 1-3). Finally, along with the codification of the various disciplines for the correct understanding of the Vedas and the performance of the sacrifices, the geometrical knowledge required for the construction of the altars was put together in the *Śulbasūtras*, which form supplements to the *Kalpasūtras* or are incorporated in them. The *Śulbasūtras* cannot be later than the 5th century B.C. and may be as old as the 8th or 9th century B.C. The enunciation in general terms of the theorem of the square of the diagonal and its application to various problems is the most important achievement of this period.

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¹ *Z.D.M.G.*, 1901, pp. 553-55.

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More or less overlapping this period of the sūtras and Brāhmaṇas we find a voluminous Jaina literature which contains a considerable amount of mathematical information. If the correct orientation and measurements of the sacrificial *vedis* were important to the adherents of the Vedic religion, correct calculations in their cosmography were equally important to the Jainas. To them *gaṇita* (calculations or more broadly mathematics) was one of the four *anuyogas*, adjuncts to religious instruction, just as *jyotiṣa* or astronomy was one of the accessories to Vedic study. Hence we find mensuration formulae connected with circles, segments, trapezia and trapezoidal solids and rules connected with mathematical series, permutations and combinations scattered through Jaina canonical works, while works like the *Sūryaprajñapti* and *Candraprajñapti* deal exclusively with astronomy as understood by the Jainas. Series mathematics especially seems to have got a great impetus from their cosmographical enquiries. So also the mensurational bent of Indian geometry may be a heritage from them.

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The early Jaina works with scattered mathematical material in them are the *Tattvārthādhigamasūtra* more especially its commentary by Umāsvāmin or Umāsvāti, the *Sthānāṅgasūtra*, the *Jambudvīpaprajñapti*, *Anuyogadvārasūtra*; the *Kṣetrasamāsas* etc. As a matter of fact almost all Jaina works, canonical or otherwise, tend to have something of mathematics in them. For the Jainas loved to calculate everything elaborately and precisely, however fantastic their original premises were. But trying to assign dates to these various works is almost a hopeless task. The *Sūryaprajñapti*, which as Thibaut shows,¹ has close affinities with the *Vedāṅga Jyotiṣa*, could not have been much removed from them in time. Moreover Bhadrabāhu, who is said to have written a commentary (*niryukti*) on the *Sūryaprajñapti*, lived 162 years after Mahāvīra i.e. in the 4th century B.C.² And the *Sūryaprajñapti* is but an *Upāṅga* (a minor *aṅga*). The *aṅgas* like the *Ācāraṅgasūtras* are likely to be much earlier. Umāsvāmin, commentator of the *Tattvārthādhigamasūtra* and author of the *Jambudvīpasamāsa*, lived in the second century B.C. according to the *Śvetāmbara* tradition and in the 2nd or 3rd century A.D. according to the *Digambara* tradition. Hence we can safely assume that most of these works belong to the centuries before Christ and some may be quite old, since, according to Jaina tradition, Mahāvīra (5th century B.C.) is the last of the Tīrthaṃkaras and Bhadrabāhu the last of the *Śrutakevalins* (those who knew all the 12 *aṅgas*). The *Karaṇagāthās* from which later authors and commentators quote anonymously also appear to be very old.

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¹ *J.A.S.B.* 1880, pp. 107-127 and 181-206.

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² Jaina School of Mathematics. *Bull. Cal. Math. Soc.* XXI-1929. See also Weber, Sacred Literature of the Jains, *Ind. Ant.* XXI p. 14 onwards,..for the antiquity of the *Sūryaprajñapti*.

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Few extant works of any mathematical interest belong to the opening centuries of the Christian era. The *Sūryasiddhānta* and the *siddhāntas* summarised in Varāhamihira's *Pañcasiddhāntikā* must have been composed during this period. And these astronomical treatises, unlike the *Vedāṅga Joytiṣa*, presuppose a considerable amount of mathematical knowledge. It is not improbable that treatises exclusively devoted to mathematics were also composed during this period. The *Bakhshali* manuscript (2nd or 3rd century A.D.)¹ may be a representative remnant of such works. Even this manuscript is not of much help in reconstructing the history of Indian geometry, since the geometrical portions it must certainly have contained, have been entirely destroyed.

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¹ 1The *Bakhshali Mathematics* by B.B. Datta, *Bull. Cal. Math. Soc.* XXI, 1929 and The *Bakhshali Manuscript* by A.F. Rudolf Hoernle, *Ind. Ant.* XVII, 1888 p. 33ff. for a discussion of the date. This early date is still a disputed point.

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With the fifth century after Christ we reach surer ground. Āryabhaṭa I (born 475 A.D.) dominated the Indian mathematical world as Plato did the Greek philosophical world, though the extant text of the *Āryabhaṭīya* does reveal some inaccuracies. But the terse language in which it clothes its rules and formulae bespeaks a long tradition of mathematical studies, making it difficult to ascertain what exactly Āryabhaṭa's actual contribution to mathematics is. It is equally difficult to gauge his influence on later mathematicians. Bhāskara I (*c.* 522 A.D.) and Lalla² were his followers. Brahmagupta seems to attack him mercilessly in his *Brahmasphuṭasiddhānta* but to be compelled to bow to his superior popularity in his *Khaṇḍakhādyakaraṇa*. The whole host of astronomers which Kerala has produced recognises him as their *ācārya* (master)³.

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² 2Lalla's date is uncertain. Pt. Sudhākara Dvivedi assigns him to 499 A.D. and S.B. Dixit to *c.*738 A.D. on rather insufficient grounds. K. Balagangadharan in his "A consolidated list of Hindu mathematical works" published in the *Mathematics Student*, XV, 1947, pp. 55-69, assigns *c.* 748 to him.

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³ 3The author of the *Kriyākramakarī*, a commentary on the *Līlāvatī*, refers to Āryabhaṭa not to Bhāskara as 'Ācārya'.

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Many of these like Mādhava of Saṃgamagrāma, Parameśvara of *Dṛggaṇita* fame and Nīlakaṇṭha are astronomers and mathematicians of no mean ability, though to them pointing out mistakes in the master is obviously a sacrilege.

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The Āryabhaṭa School of mathematics seems to have had some connection and sympathy with Jaina mathematics. Bhāskara I quotes three Prākṛt verses in his commentary on the *Āryabhaṭīya*.¹ A second piece of evidence is the large number of the manuscripts of the Jaina Mahāvīrācārya's *Gaṇita-sāra-saṃgraha* found in Kerala. If this hypothesis is correct, the Āryabhaṭa School may be said to have maintained a continuity of mathematical tradition with the Vedic period and the bloom of mathematics in India with Āryabhaṭa ceases to be sudden.

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¹ *Āryabhaṭīyabhāṣya*. 12806. B. Triv. Uni. Mss. Lib. pp. 5, 10. Also quoted by B.B. Datta in his "A lost Jaina treatise on Arithmetic", *Jaina Siddhānta Bhāskara*, Bhāga 3, Kiraṇa 2.

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Brahmagupta (628 A.D.) though well-acquainted with Āryabhaṭa's mathematics and astronomy, seems, as already pointed out, to have parted company `with the school. His geometry contains some remarkable new theorems about the cyclic quadrilateral. But it is significant that the elucidation and proof of these theorems were undertaken by the Āryabhaṭa School, while Bhāskara II, who closely followed Brahmagupta failed to do so.²

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² Vide the chapter on quadrilaterals.

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Bhāskara I has to be dated between 550 and 628 A.D., round about 574 A.D.,³ since Pṛthūdakasvāmin makes Brahmagupta later than Bhāskara. In the Āryabhaṭa School Bhāskara's astronomical works (which are really expositions of the *Āryabhaṭīya*)—the *Mahābhāskarīya* and the *Laghubhāskarīya*—are very popular, as is shown by the large number of commentaries on them. But his commentary on the *Āryabhaṭīya* is more interesting to the student of Indian mathematics; for this is the first work in which we come across the geometrical treatment of algebraical formulae which the mathematicians of the Āryabhaṭa School often resorted to. Even series were subjected to this treatment with remarkable success. The only other mathematicians to treat series diagrammatically were Śrīdhara and Nārāyaṇa Paṇḍita, whose method and object differed from those of the Āryabhaṭa School. The diagrammatical treatment of one particular type of algebraic equation, viz. ax+by+c=xy (known as *bhāvitā* in Sanskrit) is met with in Bhāskara II also.¹

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³ Vide Kuppanna Sastrigal's introduction to his edition of the *Mahābhāskarīya* (Govt., Oriental Mss. Lib. publication. Madras 1957).

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¹ *Bhāskarīya Bījagaṇitam*, Ānandāśrama Granthāvali, 99, pp. 195-202.

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Between Brahmagupta and Bhāskara II we get four treatises devoted to mathematics — the *Pātīgaṇita* and *Triśatikā* of Śrīdhara, the *Gaṇitasārasaṃgraha* of Mahāvīra and the *Gaṇitatilaka* of Śrīpati. The Jaina Mahāvīra, whose work is characteristically elaborate, lived in the reign of the Rāṣṭrakūṭa king Amoghavarṣa Nṛpatuṅga who ruled between 814 or 815 and 877 or 878 a.d.² Śrīdhara is probably earlier than Mahāvīra though K. S. Shukla places him between 850 and 950 a.d. Śrīpati, who also wrote the *Siddhāntaśekhara* and other astronomical and astrological treatises, flourished in the 11th century. His *Gaṇitatilaka* omits geometry and algebra altogether. To the 10th century belongs Āryabhaṭa II whose astronomical treatise is the *Mahāsiddhānta*. More or less complete lack of understanding of Brahmagupta's theorems about the cyclic quadrilateral, more and more correct approximate formulae for the volume of a sphere, and various approximate formulae for the arc and area of a segment mark these works.³

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² *Gaṇitasārasaṃgraha* ed. by Rangacharya, p. IX.

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³ The *Vaṭeśvarasiddhānta*, whose first volume has been published from Delhi, ought to contain a section on mathematics. But this section has not appeared in print.

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A few Jaina semi-religious works, the *Tiloyapaṇṇatti* (*Trilokaprajñapti*) of Yativṛṣabhācārya and the *Trilokasāra* and *Gommaṭasāra* of Nemicandra (c. 980 a.d.) belong to this period. Of these the *Tiloyapaṇṇatti* is perhaps a later redaction of a much earlier work.

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Bhāskara II (b. 1114 a.d.), wrongly considered the last great mathematician that India has produced, has many original contributions to make, especially in the field of algebra and trigonometry. The accurate formulas for the volume and surface area of a sphere and a close approximation for the length of an arc in terms of its chord and vice versa are some of his achievements in geometry.

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The last of these is a modification of the formula: $R sin θ = (Rθ (180-θ))/(5/440500-θ (180-θ))$ given by Bhāskara I in his *Mahābhāskarīya*.

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Nārāyaṇa Paṇḍita, son of Nṛsiṁha and author of the *Gaṇita-kaumudī* (1356 A.D.) is sometimes confused with Nārāyaṇa, commentator of the *Līlāvatī*. The *Gaṇitakaumudī* gives the impression of being a full and detailed treatment of the mathematical knowledge of Nārāyaṇa's age. A few of his theorems—three and only three diagonals are possible for a cyclic quadrilateral with given sides, the area of a cyclic quadrilateral is the product of the 3 diagonals divided by twice the diameter—are not met with in earlier works.¹ In his treatment of cyclic quadrilaterals, Nārāyaṇa has some affinity with the Āryabhaṭa School. But his *śreḍhīkṣetras* (diagrams of series) do not bear the stamp of that school, nor does he seem to be familiar with the very close approximations to the value of π which the school had arrived at probably by about the same time.

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¹ Caturveda Pṛthūdakasvāmin, the commentator of Brahmagupta (who lived in the 10th century A.D.), is said to have proved Ptolemy's theorem.

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Saṁgamagrāma Mādhava, Parameśvara, Nīlakaṇṭha, Putumana Somayājin and the author of the *Kriyākramakarī*, an elaborate commentary on Bhāskara's *Līlāvatī* and the author of the *Yuktibhāṣā* which purports to be an exposition in Malayāḷam of Nīlakaṇṭha's *Tantrasaṁgraha* — all these belong to a period of intensive astronomical studies in Kerala from the 14th to 17th centuries A.D.

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A method of integration based on subtle geometrical analysis and ingenious methods of summing up series, used to get infinite series for π, sin θ, cos θ and the arc, is the most important achievement of this period. This method was also successfully employed for finding the volume and surface area of a sphere. Strictly geometrical proofs for Brahmagupta's theorems, trigonometrical identities, etc. are also found in the works of this period.

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From the references scattered through the works of others, one gathers that the remarkable developments in the geometry of the circle found in this period are the achievements of Mādhava, though his extant works, the *Veṇvāroha* and the *Candra-vākyāni*, do not mention these developments. Mādhava lived in the latter half of the 14th century.¹ Parameśvara who is said to have spent 55 years observing the sky and then corrected the astronomical elements to tally with his observations in 1430 A.D., has a number of astronomical works to his credit. But unlike the *Kriyākramakarī*, his commentaries on the *Āryabhaṭīya* and the *Līlāvatī* do not introduce much new mathematical material. Nīlakaṇṭha's (1443-1545 A.D.) commentary on the *Āryabhaṭīya* is much more elaborate and mathematically interesting. His *Tantrasaṃgraha* also, though mainly astronomical, contains all the new developments in mathematics.²

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¹ K.V. Sharma's introduction in the *Veṇvāroha* ed. by K. Achuta Poduval, Sri Ravi Varma Sanskrit Series. No. 7, pp. 7-8.

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² The text of the *Tantrasaṃgraha* published from the Triv. Mss. Lib. does not contain these mathematical portions, which are found in another manuscript of the *Tantrasaṃgraha* with a Malayālam commentary, a transcript of which Sri Rama Varma Maru Thampuran kindly lent to me and a *Tantrasaṃgrahavyākhyā* in Sanskrit in the Trippūṇithura Skt. College library No. 275 (572).

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Putumana Somayājin's *Karaṇapaddhati* of uncertain date (but probably of the early 18th century) seems to be of the nature of a compendium. On the other hand the *Yuktibhāṣā*, as the only work which gives the rationale and poof or derivation of all the theorems and formulae then in use among the astronomers of Kerala, is a unique work. The author of the work is, according to K.V. Sharma,³ Jyeṣṭhadeva, who lived between A.D. 1475 and 1575. And from the manner of his writing he is no innovator in the line of demonstration, derivation and proof. In fact, his remarkable expositions recur in the *Kriyākramakarī*⁴ (in part) and in a Sanskrit commentary on the *Tantrasaṃgraha* preserved in the Sanskrit College Library at Trippunithura.¹ The author of the former is probably Śaṅkara Vāriyar (1556 a.d.), a pupil and commentator of Nīlakaṇṭha. Hence it is not unreasonable to believe that these proofs were known in the Āryabhaṭa School at least from the time of Mādhava, if not earlier.

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³ Jyeṣṭhadeva and his identification as the author of the *Yuktibhāṣā* by K.V. Sharma, *Adyar Library Bulletin* Vol. XXII, pp. 35-40.

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⁴ In the *Kerala Sāhitya Caritra* the *Kriyākramakarī* is given as an alternative name for Nārāyaṇa's *Karmapradīpikā*, another commentary on the *Līlāvatī*. But the two commentaries are not the same. The *Kriyākramakarī* is full and elaborate, the *Karmapradīpikā* seems to be indebted to Parameśvara's commentary on the *Līlāvatī*.

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¹ Transcript No. 275 (572). This is of unknown authorship. But at the end of every chapter there occurs the verse:

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The *Sadratnamālā*, a later work composed by Śaṅkaravarman, contains all these results without the proofs. No new ground was explored after this, chiefly because the needs of astrological astronomy were already satisfied. Secondly, the contact with the West, which, with the brute force of fire-arms overpowered the superior skill and artistic perfection of the military techniques of the native warriors, had a powerful degrading effect on all indigenous arts and studies.

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इत्येष परक्रोडावासद्विजवरसमीरितो योऽर्थः ।
स तु तन्त्रसंग्रहस्य ⋯ऽध्याये मया कथितः ॥

c6.s1.47

(with the chapter number before the word *adhyāya*). The commentator has derived his knowledge from a Brahmin of Parakroḍa, who may be Jyeṣṭhadeva of Parañṅot (Sanskrit—Parakroḍa) family, who is the author of the *Yuktibhāṣā*.

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