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A History of Sanskrit Literature c35: XXVII. Astronomy, Astrology, and Mathematics

A History of Sanskrit Literature c35

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English

Contents

XXVII. Astronomy, Astrology, and Mathematics

1. The Pre-scientific Period

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THERE is a definite breach of continuity in Indian thought on astronomy, with which astrology and mathematics are ever closely connected.¹ In the Vedic period we find extremely little sign of astronomical study; the year is vaguely reckoned, and the twenty-seven or twenty-eight Nakṣatras, moon stations, are of dubious origin. At the close of the Vedic period we have more elaborate works on the calendar evinced in Sūtra notices and summed up in the *Jyotiṣa Vedāṅga*,² preserved in two versions, for the *Yajurveda* and the *Ṛgveda*; we find here a calendar arranged on the basis of a five-year Yuga, with a 366-day year, notices of the position of the sun and moon at the solstices, and at new and full moon with regard to the Nakṣatras. Some further development of a purely Indian type is found in the case of works like the *Gārgī Saṁhitā*, of which we have fragments, the astronomical hints of the *Vṛddha Gargasaṁhitā*, the fragment of Pauṣkarasādin preserved in the Weber MS., the Nakṣatra and other Pariçiṣṭas of the *Atharvaveda*, and the *Paitāmaha Siddhānta* recorded by Varāhamihira. The Jain texts, chiefly the *Sūryaprajñapti*,³ though they develop a fantastic view of their own, are essentially of this type. The epic, the Purāṇas, the Smṛtis, and old writers such as Parāçara known from fragments are of the same type.

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¹ See G. Thibaut, *Astronomie, Astrologie und Mathematik* (1899); Kaye, *Hindu Astronomy* (1924).

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² Ed. A. Weber, ABA. 1862; *Pandit*, N.S. xxix.

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³ See Thibaut, JASB. xlix. 108 ff.

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The characteristics of this period are a general ignorance of the mean motions of the sun and moon, resulting in faulty appreciation of the length of years and months; a total ignorance of the true motion as opposed to mean motion; the teaching of an equal daily increase or decrease of the length of the day; dividing the sphere into twenty-seven or twenty-eight Nakṣatras; entertaining fantastic ideas of the constitution of the earth and the universe, and a determination on false premisses to work out large numerical calculations. All save the Jain texts assume that the winter solstice fell at the beginning of the Nakṣatra Dhaniṣṭhā, but this datum is quite insufficient to enable us to fix in any way the date of the works. They contribute to the scientific period two ideas of great importance, if of no value: the conception of great Yugas, during which a complete change of the heavenly bodies is carried out, so that a new Yuga begins with all of them in the same places as the preceding Yuga; and the conception of the lunar day, Tithi, which is a thirtieth part of a synodical month, a strange and not convenient unit.

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In one sphere, however, distinctly interesting results were attained in geometry as a result of the care taken in the measurement of altars. These results are enshrined in the Çulbasūtras, works which are of the late Sūtra period, possibly of *c.* 200 B.C., though this is mere guesswork. They are concerned with the construction of squares and rectangles; the relation of the diagonal to the sides; the equivalence of rectangles and squares; and the construction of equivalent squares and circles. We find the Pythagorean problem stated generally, but there is nothing to show how far it was fully understood and what exactly was the Indian conception of the irrational. The question of influence on Pythagoras or influence of Greece or Egypt on India has been much discussed¹ without proving any dependence in either case. But in any event the theories of the Çulbasūtras for whatever reason had apparently no effect on the later progress of geometry.

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¹ Cf. Keith, JRAS. 1910, pp. 519–21; Kaye, JRAS. 1910, pp. 749–60; Thibaut, *op. cit.*, p. 78.

2. The Period of the Siddhāntas

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Varāhamihira, who is asserted to have died in A. D. 587, and who wrote perhaps *c.* 550, has preserved in his *Pañcasiddhāntikā*² information of the contents of five Siddhāntas of an earlier date. Of these the *Paitāmaha* belonged to the pre-scientific period, but the other four in various degrees showed a new spirit, which it is impossible not to ascribe to Greek influence, which displayed itself also indelibly in the case of astrology. It is significant in the extreme that two of these Siddhāntas bear non-Indian names, the *Romaka*, which must be connected with Rome, and the *Pauliça*, which reminds us of the name of Paulus Alexandrinus, of whom, however, we have preserved only an astrological treatise. The *Sūrya Siddhānta*, in the form in which we have it, asserts that it was revealed by Sūrya to Asura Maya in Romaka, which is significant. The *Romaka* adopts not the Indian Yuga system, but one of its own, namely the Metonic period of nineteen years multiplied by 150 which gives the smallest Yuga exactly divisible into integral numbers of lunar months and civil days. Further, it makes calculations for the meridian¹ of Yavanapura, city of the Greeks; and the *Pauliça*, which does not adopt a constant Yuga, but operates with specially constructed short periods of time, gives the difference in longitude between Yavanapura and Ujjain. The *Romaka* again alone of Indian works operates with the tropical revolutions of the sun and moon, while the *Sūrya Siddhānta* and probably even the *Pauliça* deal with sidereal revolutions. The *Sūrya*, it seems, shows us the process of adaptation of the new science to Indian ideas in its most pronounced state; thus it accepts the Kalpa system, while, on the other hand, it is more precise in doctrine than its rivals; it alone gives a general rule for the equation of the centre, and its full treatment of eclipses contrasts with the meagre rules of the *Romaka* and the rough formulae of the *Pauliça*. The mention of Romaka, of course, need not be interpreted as an allusion to Rome; it is due to the fame of the Roman Empire when the knowledge which came probably from Alexandria came to be associated with the name of the great metropolis.

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² Ed. G. Thibaut and Sudhākara Dvivedī, Benares, 1889. See also M. P. Kharegat JBRAS. xix. 109 ff.; V. B. Ketkar, POCP. 1919, ii. 457 f., who argues that the *Sūrya Siddhānta*'s fixation of the initial point of the ecliptic points to *c.* A. D. 290; cf. Bhandarkar, *Early History of India*, p. 69.

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¹ No doubt Alexandria. Kern, *Bṛhatsaṁhitā*, p. 54.

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The evidences of Greek derivation in the Siddhāntas, and still more plainly in later works, may be summed up as follows.² The division of the ecliptic into the Nakṣatras yields to that into the signs of the zodiac, with names borrowed from the Greek; the motions of the planets, hitherto neglected, come to be explained by the doctrine of epicycles; the notion of parallax and methods of calculating it were introduced; new methods of calculating eclipses appear; the heliacal rising and setting of heavenly bodies was studied, especially with astrological reference; correct measurements of day and night were achieved; the length of the year was revised; and the names of the planetary week-days were introduced. We find already in the *Pauliça*—perhaps also in the other Siddhāntas—an important contribution to Indian trigonometry, in the shape of a table of sines, which seems clearly to have been borrowed from Ptolemy's table of chords, the device being adopted of dividing the radius not into sixty parts with Ptolemy but into 120 parts, thus enabling the value given for the chords to be taken over bodily for the sines of half the angles. It is only in Āryabhaṭa that we find the radius as 3438' with the necessary change of sine values.

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² Kaye, *Hindu Astronomy*, pp. 39 ff.

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The exact mode and date of the introduction of these Greek elements has been disputed, and Whitney¹ suggested that it fell in a period before the *Syntaxis* of Ptolemy, a view supported by the constant difference in detail as in the figures of the epicycles of the planets. The question is rendered specially obscure by the fact that we do not know how Greek astronomy progressed between Hipparchos and Ptolemy. It is true that Hipparchos already settled the theory of the sun and the moon and had discovered the mean periods of the revolutions of the planets, and it is conceivable that the *Romaka Siddhānta* may have contented itself with treating of sun and moon only, in accordance with the necessities of the calendar and the practice of the earlier Indian period. But Ptolemy claims to have been the first to take into account the anomalies in planetary motions dependent on the distance of the planet from the sun and its distance from the apsis. The *Vāsiṣṭha* and *Pauliça Siddhāntas* seem to have taken some note of planetary anomalies, though exactly what is uncertain. But apart from the similarity as regards sine values noted above, which is strongly in favour of use of Ptolemy's results, the position may best be explained on the basis of Thibaut's suggestion as to the means by which Ptolemy's views reached India. It is indeed incredible that Indian astronomers should have deviated so largely and needlessly from his work, if they had real knowledge of it. But they probably learned their views from books of a very inferior type, the sort of manual used by astrologers and calendar makers, works which troubled not at all about the basis of their *résumés*, but simply gave results convenient for practical purposes. The Indian Siddhāntas, then, of the *Sūrya* type would represent not a mere borrowing nor an adaptation, but a combination and development on independent lines of elements borrowed in the shape of practical rules and vague hints of theory from mere manuals. The date of the borrowing cannot be determined with certainty. If, as is probable, the year 505 marks the date of Lāṭa, who commented on the *Romaka Siddhānta*, it is natural to place that text about A. D. 400 at latest, and, if we place the period of reception somewhere in the time between A. D. 300–500, we reach a plausible result, though not one admitting of strict proof. This accords with the period when the Gupta Empire was showing many signs of contact with the Roman Empire in other spheres of activity, and the Sassanian dynasty's rule may have promoted intercourse. But the old *Sūrya Siddhānta* shows us a specifically Indian reaction; it accepts where it thinks fit the new matter, but it fits it in as far as may be with the old; it revels in the theory of Kalpas, restores the pre-eminence of mount Meru at the north pole, finds room for the Nakṣatras, and so forth.

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¹ JAOS. vi. 470 ff. Cf. Thibaut, *Pañchasiddhântikâ*, pp. li ff.; *Astronomie*, pp. 47 ff.

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None of the Siddhāntas which Varāhamihira had before him has come down to us in its original form. We know that Bhaṭṭotpala had before him a *Pauliça* so changed as to render the retention of the same name anomalous. The *Paitāmaha Siddhānta* of Varāhamihira differed little from the pre-scientific period; it commenced, however, a Yuga in the third year of the Çaka epoch, which may give its date. It differed from the *Brahma Siddhānta* forming part of the *Viṣṇudharmottara Purāṇa* on which Brahmagupta's *Sphuṭa Brāhmasiddhānta* is on one view based, and from the *Brahma Siddhānta* or *Çākalya Siddhānta*,¹ all of which present the orthodox modern doctrines. The *Romaka Siddhānta* was perhaps touched up by Lāṭa *c.* A. D. 505, and certainly later was revised drastically in the modern sense by Çrīṣeṇa, who wrote after Lāṭa and before Brahmagupta. The *Vāsiṣṭha Siddhānta* appears before Brahmagupta's time to have been revised by Vijayanandin and then by Viṣṇucandra, but the *Laghu Vāsiṣṭha Siddhānta*¹ which we have is clearly not connected with the original or the revision, and the *Vṛddha Vāsiṣṭha Siddhānta*,² which exists in manuscripts, seems equally far removed. The *Sūrya Siddhānta*³ which we have in fourteen chapters of Çlokas, is clearly in many respects modernized from the original; possibly Lāṭa had a hand in this, as Albērūnī ascribes the work to him, and he commented on the *Romaka* and *Pauliça* texts.

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¹ Eggeling, IOC. i. 998 ff.

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¹ Ed. Benares, 1881.

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² Eggeling, IOC. i. 991.

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³ Ed. BI. 1854–8 and 1909 ff.; trans. W. D. Whitney, JAOS. vi. 141 ff.; cf. S. B. Dikshit, IA. xix. 45 ff.; for comm. IOC. i. 996 ff.; ii. 765 ff.

3. Āryabhaṭa and later Astronomers

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Before the discovery of the *Pañcasiddhāntikā* the credit of introducing the new ideas into Indian astronomy was usually given to Āryabhaṭa of Kusumapura, who was born in A.D. 476 and wrote in 499. We have of him only the *Āryabhaṭīya*,⁴ in the shape of 10 stanzas in Āryā verses; the *Daçagītikāsūtra*, in which he gives his numerical notation; and the *Āryāṣṭaçata*, 108 Āryās, divided into the *Gaṇita*, 33 stanzas on mathematics; *Kālakriyā*, 25 stanzas on measurement of time; and *Gola*, 50 stanzas on the sphere. His other works are lost; Albērūnī already could judge of him only by Brahmagupta's attacks. His fame in the light of our larger knowledge seems overdone, as he does not advance much beyond the old *Sūrya Siddhānta* and his views often agree with those of *Pauliça*, but he may have earned commendation by the brevity and elegance of his composition; moreover, his is the first work to show a distinct chapter on mathematics in relation to astronomy, and the division of astronomical topics may have seemed effective. It is, however, of very real interest that he held that the earth was a sphere and rotated on its axis; the idea was not approved by either Varāhamihira or Brahmagupta; if it were so, why can falcons return from the sky to their nests, and why are not flags always blown in one direction as a result of the motion? It is tempting to see here a borrowing by Āryabhaṭa from Greece, but obviously that is a mere guess. He did not believe, we learn, in the height of Meru, he equated the four Yugas despite traditional difference of length, and he ascribed eclipses not to the operation of Rāhu, but to the moon and the shadow of the earth, for which Brahmagupta severely censures him. From Āryabhaṭa a second writer of that name must be distinguished; he was known to Albērūnī and we have a work of considerable size, the *Ārya Siddhānta*,¹ which has been ascribed to *c.* 950, and which in its numerical notation differs entirely from Āryabhaṭa.

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⁴ Ed. H. Kern, Leiden, 1874. Cf. Kaye, JPASB. 1908, pp. 111 ff.

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¹ Ed. Benares, 1910. Cf. Fleet, JRAS. 1911, pp. 788 ff.; 1912, pp. 459 ff.

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In addition to Lāṭa and Āryabhaṭa, Varāhamihira mentions Siṅha, Pradyumna, and Vijayanandin. His own work lay mainly in the field of astrology, but his *Pañcasiddhāntikā* is of very high historical importance, despite its obscurity through the corruption of the text and lack of old commentaries. Albērūnī thought well of him, and he shows common sense, as when he declines to accept conjunctions of planets as explaining eclipses. Much more important was Brahmagupta, born A. D. 598, son of Jiṣṇu of Bhillamalla near Multan, who wrote his *Brāhma Siddhānta*² or *Sphuṭa Siddhānta* in 628; as has been said, this is believed traditionally to be based on a section of the *Viṣṇudharmottara*, but it may be rather that that version is borrowed from Brahmagupta. In 665 he wrote the *Khaṇḍakhādyaka*,³ a Karaṇa, that is, a practical treatise giving material in a convenient shape for astronomical calculations, but this was based on a lost work of Āryabhaṭa, who again agreed with the *Sūrya Siddhānta*. Brahmagupta is essentially on the same level as that text, but he is far more systematic and complete, and in chapter xi of the *Siddhānta* he attacks very severely Āryabhaṭa in a tone which called down upon him the just censure of Albērūnī. It is clear also that he was under the control of orthodoxy more than his predecessor, while, like him, he excelled in mathematics. One chapter of the *Siddhānta* he devotes to solving astronomical problems.

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² Ed. *Pandit*, N.S. xxiii and xxiv.

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³ Ed. Babuya Misra, Calcutta, 1925.

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Later than Brahmagupta probably must be put Lalla, author of the *Çiṣyadhīvṛddhitantra*,⁴ treatise to increase the pupil's intelligence, which was commented on by Bhāskara, though tradition makes him a pupil of Āryabhaṭa. To Bhōja is ascribed the *Rājamṛgāṅka*, a Karaṇa of 1042, and the *Bhāsvatī*,¹ a Karaṇa by Çatānanda, begins its reckoning from A.D. 1099. Far more important is the *Siddhāntaçiromaṇi*² of Bhāskarācārya, written in 1150. It falls into four parts, the *Līlāvatī* and *Bījagaṇita*, containing the mathematical part of his work, and the *Graha-gaṇita* and *Gola*, chapters giving astronomy proper. In the *Gola* there is a section on astronomical problems, a treatise on astronomical instruments, and a description of the seasons. His *Karaṇakutūhala*³ dates from 1178. His attitude is that of the *Sūrya Siddhānta* and Brahmagupta, but he is clear and precise, while his commentary on his Āryā stanzas has the merit of making his ambiguous phrases intelligible. After Bhāskara no progress can be recorded in Indian astronomy, though there were written popular works like the tables of Makaranda (1478), *Tithyādipattra*, or the *Grahalāghava* of Gaṇeça, son of Keçava, who wrote in 1520. The advent of Persian and Arabic influences has left Indian astronomy unchanged, nor has it ever been extinguished by western science.

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⁴ Cf. Kern, *Āryabhaṭīya*, p. vi.

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¹ Ed. Benares, 1883.

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² Ed. Benares, 1866; M. Jhā, *Pandit*, N.S. xxx-xxxiii.

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³ Ed. Benares, 1881.

4. Āryabhaṭa and later Mathematicians

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Āryabhaṭa, as we have seen, was the first to insert a definitely mathematical⁴ section in his astronomy. He deals in it with evolution and involution, area and volumes; then, after a semi-astronomical section dealing with the circle, shadow problems, &c., he proceeds to progressions and algebraic identities. The rest of the *Gaṇita* deals with examples, save at the close when indeterminate equations of the first degree are taken up (ax+by=c). We find also a remarkably accurate value⁵ of π, viz. 3·1416, and the rule known as the epanthem,⁶ and the type of definition not otherwise in use in India, ‘The product of three equal numbers is a cube and it also has twelve edges.’ On the other hand, we must set clear errors in the volume of a pyramid and a sphere. His notation¹ is unique; it uses the consonants *k* to *m* for 1 to 25, the rest, *y* to *h*, for 30 to 100, while the vowels denote multiplication by powers of 100, *a* being 100⁰ and *au* 100¹⁶.

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⁴ See Kaye, *Indian Mathematics* (1915); *Scientia*, xxv. 1 ff.

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⁵ The epic value is 3·5; Hopkins, *JAOS.* xxiii. 154 f.

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⁶ It is known to Thymaridas (A. D. 380) and Iamblichos (350).

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¹ Cf. Fleet, JRAS. 1911, pp. 109 ff.; IHQ. iii. 116.

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Brahmagupta's work covers very briefly the ordinary arithmetical operations, square and cube roots, rule of three, interest, progressions, geometry, including treatment of the rational right-angled triangle and the elements of the circle, elementary mensuration of solids, shadow problems, negative and positive quantities, cipher, surds, simple algebraic identities, indeterminate equations of the first and second degrees, in considerable detail, and simple equations of the first and second degrees which are briefly treated. Special attention is given to cyclic quadrilaterals. Later, in the ninth century under the Rāṣṭrakūṭa king Amoghavarṣa, we have the *Gaṇitasārasaṁgraha*² of Mahāvīrācārya, which insists on the importance of its subject for every kind of science from cooking to logic, and adapts in its exposition the elegance which is later still further affected by Bhāskara. The work is fuller but rather more elementary than that of Brahmagupta; it gives many examples of solutions of indeterminates, but not the ‘cyclic method’ of Brahmagupta; it introduces geometrical progressions and alone deals, inaccurately, with ellipses, but has no formal algebra. Çrīdhara, born 991, in his *Triçatī*³ is much on the same level as Mahāvīra, but is cited as having dealt with quadratic equations. On Çrīdhara's work and those of Brahmagupta and a certain Padmanābha was based the *Līlāvatī*⁴ of Bhāskara, in which a lovely maiden is addressed, and problems set to her; it includes combinations, while the *Bījagaṇita*, which agrees largely with Brahmagupta, is the fullest and most systematic account of Indian algebra. With Bhāskara ends the active period of Indian mathematics; a school to study his work was founded in 1205 by Caṅgadeva, his grandson,⁵ but its interest seems to have been given to astrology. Of dubious age is the mathematical manuscript known as the Bakhshālī manuscript,¹ from its place of discovery in Peshāwar. It is written in Sūtra style with examples in Çlokas, taken from daily life, and explanations in prose. The mixed Sanskrit in which it is written induced Hoernle to ascribe the work to the third or fourth century A.D., and the manuscript on palaeographic grounds was ascribed by him to the eighth or ninth century, but these conclusions are far from being certain, and the work may date much later.

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² Ed. and trans. M. Raṅgācārya, Madras, 1912.

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³ See N. Rāmānujācārya, *Bibl. Math.*, 1913, pp. 203 ff.

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⁴ See H. T. Colebrooke, *Algebra* (1817); his trans. is ed. H. Ch. Banerji, Calcutta, 1893. Cf. Brockhaus, BSGW. 1852, pp. 1-46.

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⁵ See EI. i. 338 ff.

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¹ Hoernle, OC. VII, i. 128 ff.; IA. xvii. 33 ff. Contrast Kaye, JPASB. 1907, pp. 498 ff.; 1912, pp. 349 ff.

5. Greece and Indian Mathematics

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The relation of India to Greek mathematics in this period is one of complexity and difficulty, and it cannot be disposed of by insisting on the indebtedness of India to Greece in respect either of astronomy or astrology, for in both cases the exact extent of that influence is obscure.² The question is obscured also by the fact that we have lost the works of Hypatia, murdered in 415 by the Alexandrian mob, and therefore cannot trace the progress of mathematics after Diophantos (*c.* 260). The visit of the philosophers expelled from the schools of Philosophy in Athens in 530 to the court of Chosrau of Persia in 532 was brief, and it is not much use speculating on its possibilities, though Damaskios and Simplikios, who had some repute in mathematics, were among those involved. The facts are that, as regards indeterminate equations, the Greeks by the fourth century had achieved rational solutions, not necessarily integral, of equations of the first and second degree and of some cases of the third degree. The Indian records go distinctly beyond this. Brahmagupta shows a complete grasp of the integral solution of ax ± by = c, and he indicates one method, called by Bhāskara the method by composition, of the solution of Du² + 1 = t². Bhāskara adds the cyclic method, as he calls it, and the combination of these two methods, which gives integral solutions, has been styled by Hankel the finest thing achieved in the theory of numbers before Lagrange. To find an ultimate Greek origin for these discoveries seems due rather to a *parti pris* than to justice.

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² Kaye (*Hindu Mathematics*) goes rather far in his claims for Greece. Contrast D. E. Smith in Raṅgācārya's *Gaṇitasārasaṁgraha*, pp. xxi ff. For older views see Hankel, *Gesch. der Math.* (1874), pp. 172 ff.; Cantor, *Gesch. der Math.*, i. 505 ff.; M. Simon, *Gesch. der Math.* (1909). See also J. L. Heiberg, *Mathematics and Physical Science in Classical Antiquity* (1922); D. E. Smith, *Hist. of Mathematics* (1925); Peet, *The Rhind Mathematical Papyrus* (1923); Heath, *Hist.* (1921).

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Another point on which special interest in India was centred was the question of obtaining integral solutions in the case of the rational right-angled triangle. The results achieved there are interesting and may be compared with similar work, not by any means identical, of Euclid and Diophantos, as well as solutions ascribed to Plato by Proclus. Brahmagupta, Mahāvīra, and Bhāskara all contribute to the topic, and the former first states certain historically interesting problems; the sum of the sides is 40 and area 60; the sum of the sides is 56 and the product 7 × 600; the area is numerically equal to the hypotenuse; or to the product of the sides. Brahmagupta further did important work on cyclic quadrilaterals, achieving as one of his results his theorem: x² = (ad + bc) (ac + bd) / (ab + cd), and y² = (ab + cd) (ac + bd) / (ad + bc), where x and y are the diagonals of the cyclic quadrilateral, a, b, c, d. Mahāvīra and Çrīdhara repeat some of his matter, but their commentators show ignorance of the principle, and Bhāskara severely condemns both him who puts such a question and him who answers it. It is interesting, but by no means a proof of borrowing, that a commentator on Brahmagupta constructs from triangles new triangles and actually uses the same examples as Diophantos. Nor can we draw any definite conclusion from the fact that Indian mathematics in regard to geometry shows an absence of definitions, does not deal with angles nor mention parallels, nor give a theory of proportion, while traditional inaccuracies are common and knowledge is in the later period steadily declining. The same facts are seen in Greek geometry from A.D. 300, and possibly we can best understand Indian facts as indicating borrowings from such a decadent school, but there is no cogency in the contention.

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The independence and originality of Indian mathematics have been defended on the score that the love of dealing with large numbers and making calculations is recorded early for India, where it is alleged the abacus¹ was invented, and that the numerals of the west are borrowed from India, where the place value system¹ was introduced. The abacus, on the other hand, is asserted to be of comparatively modern date, and it has been suggested that India borrowed it from Greece. The question of numbers is very dubious; the figures of the Brāhmī or Kharoṣṭhī notation have not place value, but their origin is uncertain. India knows, beside the unique system of Āryabhaṭa, the use of words for numbers, and place value is actually found in inscriptions from the ninth century onwards, but its earliest occurrence on an inscription of 595 is doubted, though the *Yogabhāṣya* clearly knows it, as did Āryabhaṭa and Varāhamihira.² But there is considerable antiquity for the tradition of the borrowing; the Indian figures were known in Syria in A.D. 662, and Mascūdi ascribes the origin to a congress of sages gathered together by king Brahma. The probability still remains that India did render a great service in this regard, and in any case excelled Greece. It is, of course, perfectly possible, and in view of the facts as regards astronomy and astrology not at all unlikely, that India borrowed its impulse to mathematics from Greece in the shape of those manuals whence she borrowed her astronomy, and this is certainly supported by the fact of Āryabhaṭa's evaluation of π, which is also ascribed to Puliça, and it was known to Apollonios and Ptolemy.

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¹ *Contra*, Kaye, JPASB. 1908, pp. 293 ff., but see Fleet, JRAS. 1911, pp. 121, 518 ff. *Cf.* IHQ. iii. 357 ff.

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¹ See Kaye, JPASB. 1907, pp. 475 ff.; Bubnow, *Arithmetische Selbständigkeit der europäischen Kultur* (1914); *contra*, D. E. Smith and L. C. Karpinski, *The Hindu Arabic Numerals* (1911); Nau, JA. sér. 10, xvi. 225-7; C. de Vaux, *Scientia*, 1917, pp. 273 f.; Sukumar Ranjan Das, IHQ. iii. 100 ff., 356 ff.

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² Woods, HOS. xvii. 216.

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Recently the claim of India to have inspired Arabic mathematics has also been attacked, on the score that Muḥammad ibn Mūsā (782) in his Algebra is not, as was long believed, really under Indian but under Greek influence, and a good case seems to have been made against any substantial importance of India in this regard; but there seems equally slight ground for the counterclaim that India borrowed after Brahmagupta from Arab mathematics at least in the period up to Bhāskara. But it must be remembered that Arabian science from A.D. 771 borrowed freely from Indian astronomy,³ translating and adapting both Āryabhaṭa and Brahmagupta, so that, if we hold that Arabia was independent in mathematics of India, we must recognize that borrowing of astronomy or astrology must not be adduced as conclusive for borrowing of mathematics. Coincidences with Chinese¹ mathematics are numerous and interesting, and it has, of course, long been urged that China invented the system of Nakṣatras found in early Indian astronomy,² but at present at least the case for dependence on China is not made out, and Indian influence on China is proved sufficiently by the history of Chinese Buddhism and the discoveries in Central Asia.

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³ Nallino, ERE. xii. 95.

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¹ See Yoshio Mikami, *Development of Mathematics in China and Japan* (1913); Kaye, *Indian Mathematics*, pp. 37–41; Smith, *Hist.*, i. 22 ff., 138 ff., 148 ff.

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² Cf. Oldenberg, GN. 1909, pp. 544 ff.

6. Varāhamihira and early Astrologers

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That the celestial bodies exercise influence on the fate of men, and that the future can be foretold from their aspect, is a very early belief in India, whether we believe it appeared there independently or was borrowed from Babylon. In the Brāhmaṇas and the Sūtras we find recognition of the idea of a lucky star, and the Dharmasūtras demand that the king shall have an astrologer just as he has a house chaplain, while the *Arthaçāstra* ranks court bards, the servants of the chaplain, and astrologers among the lower court functionaries. In war an astrologer is essential to foretell the result from the signs, and to encourage the army and terrify the foe. On the other side, we have the fact that, like a magician, an astrologer may be ritually impure, and the Buddhists denounce the occupation as they do many others.

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We need not doubt that text-books of astrology were numerous, and in fact Varāhamihira, whose great work caused all the older texts to disappear, mentions Asita Devala, Garga, Vṛddha Garga, Nārada, and Parāçara among authorities. We have possibly genuine fragments of these works, but the most considerable are those of the *Vṛddha Garga Saṁhitā*, or *Gārgī Saṁhitā*,³ which is well known for containing in pseudo-prophetic form some allusions to Greek rule in India. That it existed as early as the first century B.C. is a mere guess. It is important, however, for its definite assertion that the Greeks are barbarians,¹ yet among them the science of astrology was well established and those who knew it were honoured like seers; how much more so then a Brahman skilled in it?

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³ The relation of these texts is uncertain; Kern, *Bṛhatsaṁhitā*, pp. 33 ff. The astronomical data of the *Gārgī* are given by Weber, ABA. 1862, pp. 33 ff., 40 ff.; IS. ix. 460 ff. Garga is reputed the author of the Atharvan Pariçiṣṭas, li, lxii, lxiv. In the Weber MS. (JASB. lxii. 9) is a fragment of Pauṣkarasādin on astronomy.

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¹ ^1 *Cf.* *Bṛhatsaṁhitā*, ii. 15.

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Varāhamihira himself divides the science of Jyotis into three branches. The first, the astronomical and mathematical foundations, is called Tantra; the next, Horā, deals with horoscopes, and its name is obviously Greek; the third, Saṁhitā, covers the sphere of natural astrology. His own work on astronomy has been mentioned, but, valuable as it is, it is much less important than his *Bṛhatsaṁhitā*,² in which he shows himself a master of the learning of his day in wide fields of knowledge, and thoroughly skilled in language and metre, not at times without a real touch of poetic ability. The scope of the text is ample. After insisting on the importance of knowledge of astrology, he deals with the effects of the movements of the sun, of the changes of the moon, its conjunction with the planets and eclipses. Then he takes up the several constellations and describes their powers over the fate of man. Incidentally we have in chapter xiv an interesting sketch of Indian geography, and we learn what lands, people, and things stand under the aegis of each planet; the planetary movements also determine the wars of kings, and each year owes its fortune or mishap to the planet which presides over it. We are told also of the signs of weather, and how to foretell not merely the crop but the rise and fall of prices. The festival of the raising of Indra's banner (xliii) is poetically described and is followed by further religious matter. The importance of the astrologer in connexion with architecture, the digging of tanks, the laying out of gardens, and the making of images leads to valuable chapters on these topics (liii–lx). Then comes a description of the specific characteristics of oxen, dogs, cocks, tortoises, horses, elephants, man, woman, parasols, &c. (lxi–lxxiii). A praise of women, which is worthy of an anthology, occupies chapter lxxiv, and is followed by a section on the life of the harem, which shows affinity with the *Kāmasūtra* and *Arthaçāstra*. Couches and seats come next (lxxix), then jewels³ (lxxx–lxxxiii); short chapters deal with lamps and tooth sticks; then a long Çākuna, on augury, fills eleven chapters; of the rest, two chapters (c and ciii) deal with marriage, while cvi is a finale, followed by an index. The question of marriage is also taken up in the *Bṛhad Vivāhapaṭala* and the *Svalpa Vivāhapaṭala* of the author, while he deals with the wars of kings in his *Yogayātrā*,¹ marching out under favouring combinations, which exists also in two forms, in the first part of it he resumes the question of the relation of king and astrologer touched on in the *Bṛhatsaṁhitā*, insisting that the king has a part to play as well as the astrologer. In these works, as in the *Bṛhatsaṁhitā*, we have no reason to see anything but a development of the ideas prevalent in India itself.

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² ^2 Ed. H. Kern, BI. 1865; VizSS. 1895-7; trans. C. Iyer, Madura, 1884.

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³ ^3 Ed. and trans. L. Finot, *Les lapidaires indiens*, pp. 59 ff.

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¹ Ed. and trans. H. Kern, IS. x. 161 ff.; xiv. 312 ff. There are variant texts; IOC. i. 1057; *Nepal Catal.*, p. xxx.

7. Greece and Indian Astrology

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The case, however, is clearly different with the *Horā* section of astrology on which Varāhamihira has left us a *Bṛhaj*² and a *Laghu Jātaka*. The borrowing of the name and of other terms from Greece is flagrant and it is only interesting to note the efforts made to give the words an Indian tinge; the names of the Houses prove dependence beyond a peradventure: Horā, Paṇaphara, Āpoklima, Hibuka, Trikoṇa, Jāmitra, Meṣūraṇa; the signs of the zodiac include Kriya, Tāvuri, Jituma, Leya, Pāthona, Juka, Kaurpya, Taukṣika, Ānokero, Hṛdroga, and Itthya, as well as translations. Moreover, among his authorities, Maya, Satyācārya, Viṣṇugupta, Devasvāmin, Jīvaçarman, Piṇḍāyu, Pṛthu, Çaktipūrva, and Siddhasena, occur Maṇittha and Yavanācārya. The only difficulty is the period when the borrowing took place. Jacobi³ held that it fell not before the fourth century, as the stage reached seems to be that attained in the works of Firmicus Maternus (*c.* 350), but it is by no means certain that this view can any longer be accepted. A Nepalese manuscript⁴ of a *Yavana Jātaka* contains a very obscure and mutilated statement which appears to mean that a Yavaneçvara translated the work from his own speech in the year 91 of an unspecified era, while another person, a king Sphūrjidhvaja, brought out the work in the shape of 4,000 Indravajrā verses in 191. Now Bhaṭṭotpala, the commentator on Varāhamihira, tells us of one Yavaneçvara Sphujidhvaja who used the Çaka era, and who, therefore, may be a somewhat confused reminiscence of this set of two people—unless our fragmentary text has obscured the true facts. Kern's suggestion¹ that Yavaneçvara was younger than Varāhamihira ignores the fact that Varāhamihira cites a Yavanācārya who may well be meant for this writer, whose date then would be A.D. 169. We have later texts of the *Yavana Jātaka*, a *Vṛddha* of 8,000 verses, and another text ascribed to Mīnarāja Yavanācārya,² which need not be before Varāhamihira, but the evidence clearly renders it difficult to rely implicitly on Jacobi's dating. Maṇittha has been compared with Manetho, author of the *Apotelesmata*, and this view is strongly supported by the fact that he is given as agreeing with the ancient Greeks and disagreeing with Varāhamihira and Satya. The date of Jacobi is supported by Fleet, who lays stress on the order of the planets in Varāhamihira, beginning with the sun, as showing that India adopted a Jewish-Christian week, Jewish in respect of order and Christian in respect of the names. We know that, according to Dio Cassius, the calendrical use of the names of the planets was regular in his time, and in 321 Constantine gave the seven days' week its definite sanction by appointing Sunday as a day of rest. It is fair, however, to note that the use of planet names is to be traced much farther back than Dio Cassius, and that the argument is not altogether conclusive. But it is supported to some extent by the fact that the first case of the use of a name of this kind in an inscription is in A.D. 484, after which it is still rare down to A.D. 800.

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² Trans. N. Ch. Aiyar, Madras, 1905; SBH. 12, 1912. Cf. Haraprasād, *Nepal Catal.*, p. xxxi.

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³ *De astrologiae Indicae ‘Horâ’ appellatae originibus* (1872). Cf. Fleet, JRAS. 1912, pp. 1039 ff.

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⁴ Haraprasād, *Report I*, p. 8; *II*, p. 6; *Magadhan Literature*, p. 129; *Nepal Catal.*, p. xxx.

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¹ *Bṛhatsaṁhitā*, p. 51.

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² Eggeling, IOC. i. 1096. Minas as Minos is suggested by Brockhaus, BSGW. 1852, p. 18.

8. Varāhamihira's Poetry

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Varāhamihira is often effective and spirited in style and the existence of his work proves the long period of cultivation of the Kāvya prior to his time. Whether his own or Garga's, the plight of the king who has no guide is well phrased:

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*apradīpā yathā rātrir anādityaṁ yathā nabhaḥ*
*tathāsāṁvatsaro rājā bhramaty andha ivādhvani.*

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‘Like the night without a light, like the sky without the sun, even so the king without an astrologer wanders blindly on his way.’ Misfortunes are effectively portrayed:

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*vātoddhataç carati vahnir atipracaṇḍo: grāmān vanāni naga-*
*rāṇi ca saṁdidhakṣuḥ*
*hā heti dasyugaṇapātahatā raṭanti: niḥsvīkṛtā vipaçavo bhuvi*
*martyasaṅghāḥ.*

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‘A fire, fanned by the breeze, rages, seeking to devour villages, forests, cities; robbed of their all, without cattle, overwhelmed by the onslaught of hordes of robbers, the people wail miserably.’

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*abhyunnatā viyaty asaṁhatamūrtayo 'pi: muñcanti na kvacid*
*apaḥ pracuraṁ payodāḥ*
*sīmni prajātam api çoṣam upaiti sasyaṁ: niṣpannam apy*
*avinayād apare haranti.*

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‘The clouds aloft in the sky, though teeming, will not yield their abundant water; the grain that springs up in the fields withers away, or if it ripens is stolen by evil strangers.’ Very different is the case in happier times:

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*kṣatraṁ kṣitau kṣapitabhūtibalāripakṣam: udghuṣṭanaikajaya-*
*çabdavirāvitāçam*
*saṁhṛṣṭaçiṣṭajanaduṣṭavinaṣṭavargaṁ: gāṁ pālayanty avanipā*
*nagarākarāḍhyām.*

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‘The kingly power destroys on earth the vast power of the foe; the sky rings with many an echoing shout of victory; joyful the good, destroyed the wicked, and kings rule a land where cities and treasures abound.’ The sound effect of the first line is perfectly obvious, and it is very clear in the following line, where the intensives indicate the poet's grammatical knowledge:

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*pepīyate madhu madhau saha kāminībhir : jegīyate çravaṇahāri saveṇuvīṇam*
*bobhujyate 'tithisuhṛtsvajanaiḥ sahānnam : abde sitasya madanasya jayāvahoṣaḥ.*

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'In the spring month honey drink is drunk with loving maidens, to flute and lute many a sweet song is sung; with guests, friends, kindred, food is freely shared, and love reigns triumphant in the year of Sita.' A stanza cited in an anthology is effective and pointed :

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*lokaḥ çubhas tiṣṭhatu tāvad anyaḥ : parāṅmukhānāṁ samareṣu puṁsām*
*patnyo 'pi teṣāṁ na hriyā mukhāni : puraḥ sakhīnām iha darçayanti.*

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'However fair a face the world may turn to those who flee in battle, their wives in shame cannot bear to show their faces before their friends.'

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Still more interesting is the number of metres¹ used by Varāhamihira. In the *Bṛhatsaṁhitā* some sixty-four occur, eleven of which are of extreme rarity and of dubious form. The Āryā predominates; then come verses of Indravajrā type, then the Çloka, Vasantatilaka, Rathoddhatā, Çārdūlavikrīḍita, Çālinī, Vaitālīya, and Aupacchandasika: all else are sporadic, many occurring only in civ. They include Anavasitā, Aparavaktra, Kusumavicitrā, Kokilaka or Narkuṭaka, Tāmarasa, Toṭaka, Daṇḍaka, Dodhaka, Drutavilambita, Dhīralalitā, Puṣpitāgrā, Pṛthvī, Prabhāvatī, Pramāṇikā, Pramitākṣarā, Praharṣiṇī, Bhadrikā, Bhujaṅgaprayāta, Bhujaṅgavijṛmbhita, Bhramaravilasita, Maṇiguṇanikara, Mattamayūra, Mattā, Mandākrāntā, Mālatī or Varatanu, Mālinī, Meghavitāna, Meghavisphūrjita, Moṭanaka, Rukmavatī, Rucirā, Vaṁçapattrapatita, Vaṁçasthā, Vātormī, Vidyunmālā, Vaiçvadevī, Çikhariṇī, Çuddhavirāj, Çrīpuṭa, Suvadanā, Sragdharā, Svāgatā, Hariṇapluta, Hariṇī, and Udgatā, Drutapada, Vilāsinī, Sumānikā, Tūṇaka, and varieties of Vidyunmālā. The *Bṛhaj Jātaka* uses thirty-three metres, of which eight have irregularities. The skill of Varāhamihira is thus equal to that of the most accomplished Kāvya writers.

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¹ Stenzler, ZDMG. xliv. 4 ff.

9. Later Works on Astrology

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Little need be said of later exponents of this dubious science. Pṛthuyaças, son of Varāhamihira, wrote a *Horāṣaṭpañcāçikā*,¹ on nativities, on which Bhaṭṭotpala commented, as he did on all Varāhamihira’s works, the comment on the *Bṛhaj Jātaka* being dated in 966; he himself wrote a *Horāçāstra* in seventy-five verses. Bhaṭṭotpala is historically interesting for the citations he makes from lost works. The *Vidyāmādhavīya* (before 1350) claims to put in correct language as opposed to barbarisms (*apaçabda*) the precepts of Vasiṣṭha, Bṛhaspati, Gārgya, and others.

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¹ Ed. Calcutta, 1875.

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Other texts of dubious antiquity are not rare, including a *Vṛddha Vāsiṣṭha Saṁhitā* and the Jain *Jyotiṣasāroddhāra* of Harṣakīrti Sūri. The *Jyotirvidābharaṇa*² may be mentioned, because it is the source whence was derived the current doctrine of the nine jewels of the court of Vikramāditya; it is quite late, shows Arabic influence, and need not date before the sixteenth century; it was commented on in 1661. Many works were written also on the auspicious moments for ceremonies, marriages, journeys, and so forth, under the style of Muhūrta as the first part of the title; and, when Arabic and Persian influences became marked under the Mahomedan régime, Tājikas appeared, the name being derived from Persian Taiji, ‘Arabic’. Nīlakaṇṭha’s *Tājika*, in two parts, *Saṁjñā-* and *Varṣa- tantra*, was written in 1587 and exists in numerous manuscripts and editions.

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² Weber, ZDMG. xxii. 708 ff.; xxiv. 393 ff.

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On omens and prognostications there are also many late treatises; the *Adbhutasāgara*³ was begun in 1168 by Ballālasena of Bengal and finished by Lakṣmaṇasena, while the *Samudratilaka*, begun in 1160 under Kumārapala of Gujarāt by Durlabharāja, son of Narasiṁha, was finished by his son Jagaddeva, who also wrote the *Svapnacintāmaṇi*,⁴ explaining dreams; the similarity of dream motifs to Märchen is noteworthy. The *Narapatijayacaryā Svarodaya*⁵ was written at Aṇahillapaṭṭana under Ajayapāla of Gujarāt (1174–7) by Narahari, son of Āmradeva of Dhārā; it deals with the use of magic diagrams, marked with mystic letters or syllables, as a means of prognostication in respect of warlike operations and adventurous undertakings. From Persia appears to have been borrowed the art of geomancy, which is represented by the *Ramalarahasya*¹ of Bhayabhañjanaçarman and many other treatises of late date. Of much earlier date are two treatises on cubomancy under the style of *Pāçakakevalī* preserved as parts iv and v of the Bower manuscript² in bad Sanskrit with many signs of Prācritic influence; later tracts are known, ascribed to Garga,³ which show knowledge of the term Horā, and therefore postulate the period of Greek influence.

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³ Bhandarkar, *Report, 1887-91*, pp. lxxxii ff. Cf. IHQ. iii. 186-9.

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⁴ J. von Negelein, *Der Traumschlüssel des Jagaddeva* (1912); cf. WZKM. xxvi. 403 ff.

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⁵ Eggeling, IOC. i. 1110 ff. For Jagajjyotirmalla's comm. (1614) see Haraprasād, *Nepal Catal.*, p. lxiii. Cf. Keith, IOC. ii. 836 ff.

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¹ *Ibid.*, i. 1121 ff.

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² Hoernle, *Bower MS.*, pp. 84 ff

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³ J. E. Schröter, *Pāśakakevalī* (1900); Weber, *Ind. Streif.*, i. 274 ff. For astrology in general see also *Madras Catal.*, xxiv (1918).

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