ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

38 ASTRONOMY IN ANCIENT NATIONS after all not be very surprising if some learned Jews had been influenced by the opinion of Herakleides, since it is an established fact that the doctrines of the Kabbalists were intimately connec- ted with the later Greek philosophy. But any how nothing came of this isolated case, and the daily rotation of the heavens conti- nued to be universally accepted as a self-evident fact. Arabian astronomers and Ptolemaic system— Arabian astronomers who really wished to follow in detail the celestial motion were therefore obliged to adopt the Ptolemaic system altogether. New planetary tables had long been found to be a necessity, and this important work was at last undertaken by King Alfonso X. of Castille and several Jewish and Christian astronomers working under him at Toledo, who prepared the celebrated Alfonsine Tables, Apparently the King must have had his doubts about the physical truth of the system, judging from his well-known saying that if God had consulted him when creating the world, he would have given Him good advice. The tables were prepared under the direction of the Jew Ishak ben Said, called Hasan, and a physician, Jehuda ben Mose Cohen, and were finished in 1252, the year in which Alfonso ascended the throne of Castille. They continued in great repute for three hun- dred years as the best planetary tables; they were first printed in 1483, but had been spread all over Europe long before that time in numerous MS copies, many of which are still in existence, Twenty-six codices are counted up in the Libros del Saber de Astronomia del Rey D. Alfonso X.de Castella, Madrid, 1863-67 (5 vols. fol.). This compilation, a series of chapters on spherical and theoretical astronomy followed by tables, must have been made up from several codices, as there are numerous repetitions even of very elementary matters. In the third volume the theo- ries of the planets are dealt with, but one looks in vain for any improvement on Ptolemy; on the contrary, the low state of astro- nomy in the Middle Ages is nowhere better illustrated. In gene- ral the elements of the orbits are those of Ptolemy, though some- times only approximations are given, while different values are given in different chapters, though Ptolemy places the centre of the deferent midway between the centre of the equant and the Earth, the Libros del Saber places the centre of the equant (cerco del alaux.¹) midway between the Earth and the centre of the deferent

1. al is the Arbic article, aux (apside) is a corruption of the Arabic Oudj (Abu 'l Faraj, II. p. 25). The equant is called the cerco del Y guador.

ARABIAN ASTRONOMERS AND PTOLEMAIC SYSTEM 39 (cerco del levador¹), as in Ptolemy's theory of Mercury, which the authors would seem to have extended to the planets, omitting the motion of the centre of the deferent on a small circle; this, they have, however, correctly given in the case of Mercury.² There is a very curious figure³ of the deferent of Mercury in the form of an ellipse (the axes being as 6 to 5 nearly), with what looks like the Sun in the centre. This curve has been construc- ted from a number of small circular arcs,⁴ and it is obviously nothing but the curve described by the centre of the epicycle of Mercury in Ptolemy's theory. For according to the latter the centre of the deferent describes a small circle with radius = 1/21 of that of the deferent, in direction from east to west, in the same time which the centre of the epicycle takes to pass round the cir- cumference of the deferent from west to east. This makes the cen- tre of the epicycle describe a closed curve resembling an ellipse, the axes of which are in the ratio 11: 10, almost exactly the same as in the Spanish diagram, and there is therefore in the latter no antici- pation whatever of Kepler's great discovery, since in the case of the inferior planets it is the epicycle which is the real orbit.⁵ The small sun-like object in the centre of the ellipse represents the centre of Ptolemy's small circle, and it has either been inser- ted in the manuscript centuries after the essay had been written, or, more likely, it has been caused by a small blot on the place in the parchment where the stationary leg of the draughtsman's compasses had made a small hole. An oval deferent of Mercury occurs in several books published in the sixteenth and seven- teenth centuries.⁶

1. Vol. III. pp. 246—253.
2. Vol. III, pp. 253 and 278. In the the latter place the radius of the small circle is 1/21, as in the "Hypotheses" of Ptolemy.
3. Vol. III. p. 282.
4. See the lengthy description on pp, 278—280.
5. The editor, Don Manuel Rice y Sinobas, on p. xxxiii, of his preface, even goes so far as to suggest that Kepler may have known of this great discovery of Alfonso's or rather of Arzachel's, as the text attributes the construction to him. This and other similar diagrams were intended to be used instead of planetary tables in the manner afterwards adopted by Apianus.
6. First (about 1460) in Purbach's Theoricae novae Planetarum (ed. of Basle, 1573, p. 82) : "Ex dictis apparet manifeste, centrum epicycle Mercurij, (Continued on next page)

40 ASTRONOMY IN ANCIENT NATIONS Though the somewhat confused collection of essays entitled the Libros del Saber would not, if published in the thirteenth century, have advanced astronomical science, it cannot be denied that the Alfonsine Tables were very useful in their day. The actual elements are not given, nor is any thing said about any observations by which somewhat more correct values of the mean motions must have been found.¹ Arabs on motions of fixed stars. Thus we finish our review of the planetary theories of the Arabs. Now we shall say a few words about their ideas as to the nature and motion of the fixed stars. The exaggerated notion which prevailed before the invention of the telescope with regard to the apparent angular diameters of the stars natu- rally, led to erroneous estimates of their actual size, founded on the assumption that the sphere of the fixed stars (the eighth sphere) was immediately outside that of Saturn.² The stars of the first magnitude were supposed to have an apparent diameter equal to 1/20 of that of the Sun, from which it followed that their actual diameters were about 4 1/2 times that of the Earth, or about (Continued from previous page) propter motus supradictos non (ut in alijs planetis fit) circumferentiam deferentis circularem, sed potius figuræ, habentis similitudinem cumplana ovali peripheriam describere." Next by Albert of Brudzew in 1482 in his Commentariolum super theoricas novas, printed at Milan in 1495 (ed. Cracow, 1900, p. 124), where it is remarked that the centre of the lunar epicycle describes a similar figure. This is also stated by E. Reinhold in his commentary to Purbach, 1542, fol. p. 7 verso (ed. of Paris, 1558, fol. 78,) by Vurstisius in his Questiones novae in theoricas, & c., Basle, 1573, p. 233; and in Riccioli's Almagestum novum T. I, p. 564. The last three writers (who give a figure) also take the equable angular motion round the centre of the equant into account, which centre lies on the point of the circumference of the small circle nearest the Earth. The curve described by the centre of the epicycle thus becomes egg-shaped, and not like an ellipse.

1. The tables in vol. v. of the Libros del Saber are quite different from the Alfonsine Tables, and are apparently only intended for astrological purposes.
2. Al Battani (cap. 50) gives the greatest distance of Saturn = 18,094, and the distance of the fixed stars = 19,000 semidiameters of the Earth. Al Fargani (p. 82) puts them exactly equal. Al Kusgi gives the diameters in parasangs, of the concavity of the stellar sphere = 33,509,180 of the ninth sphere 33,524,309, of its convexity "no one but God knows" (Shah Cholgi, p. 97).

ARABS ON MOTIONS OF FIXED STARS 41 twice that of Mars.¹ As to the nature of the stars, they seem generally to have been assumed self-luminous, being condensed parts of the sphere, though Abraham ben Chija says that the eighth sphere does not shine with a uniform light, but has denser spots, which are illuminated by the Sun and appear to us as the fixed stars.² To account for the apparent slow motion of the stars para- lleled to the ecliptic, from west to east, whereby their longitudes increase while their latitudes remain unaltered, it became neces- sary to inroduce a ninth sphere (primum mobile), turning in twenty-four hours and communicating this motion to the eighth sphere, while the latter moved extremely slowly round its own axis forming an angle of 23° 35′ with that of the ninth.³ But the simple phenomenon of precession was by many Arabian astro- nomers complicated by being assumed variable. It may be mentioned that according to Theon and Proklus it had been assumed by some astronomers apparently before the time of Ptolemy, that the precessional motion of the stars was not pro- gressive, but was confined to an oscillation along an arc of 8°, along which the equinoctial points moved backwards and forwards on the ecliptic, always at the same rate of 1° in 80 years. The absurdity of the sudden change of direction must have become obvious as soon as astronomy began to be cultivated among the Arabs, for we find that one of the earliest astronomers, Tābit ben Korra, substituted a physically less objectionable theory.⁴

1. Al Fargani (p. 85, Golius) gives the cubic contents of the six spheres as 107, 90, 72, 54, 36, 18 times that of the Earth. Abu 'l Faraj, p. 199, gives a similar series from 93 to 15½ for the average star of each class. Shems ed-din of Damascus in his Cosmography (p. 3) merely says that the smallest fixed star is much larger than the Earth.
2. According to Suter, p. 77, a writer called Ibn Zura wrote a treatise "On the cause of the light of the stars, though they and the spheres consist of one single substance."
3. The outermost sphere is by the philosopher Ibn Sina (Avicenna) defined as a spherical, single (not composite) body, emanating directly from God and subject to dissolution, endowed innately with circular motion as an expression of its praise of the Creator (Mehren in Oversigt, K. Danske Vid. Selskab, 1883, p. 70).
4. The treatise "On the motion of the 8th sphere" has never been printed ; an abstract is given in Delambre's Hist. de l'astr. du Moyen Age, p. 73. Compare a quotation by Ibn Junis, Caussin, Notices et Extraits, VII, p. 116.

42 ASTRONOMY IN ANCIENT NATIONS He imagines a fixed ecliptic (in the ninth sphere) which intersects the equator in two points (the mean equinoxes) under an angle of 23° 33' 30", and a movable ecliptic (in the eighth sphere), attached at two diametrically opposite points to two small circles, the centres of which are in the mean equinoxes and the radii of which are = 4° 18' 43". The movable tropical points of Cancer and Capricorn never leave the fixed ecliptic, but move to and fro to the extent of 8° 37' 26", while two points on the movable ecliptic 90° from the tropical points move on the circumferences of the small circles, so that the movable ecliptic rises and falls on the fixed one, while the points of intersection of the equator and the movable ecliptic advance and recede to the extent of 10° 45' either way, This is a motoin of the eighth sphere, common to all stars, and the Sun will, therefore, sometimes reach its greatest declination in Cancer, sometimes in Gemini. Tabit does not say that the obliquity of the ecliptic is variable, and perhaps it did not occur to him that this would be a necessary consequence of his theory; he only notices the change in direction and amount of the motion of the equinoxes, which, he says, has increased since the days of Ptolemy, when it was only 1° in 100 years, while later observers have found 1° in 66 years. The erroneous value given by Ptolemy was, therefore, mainly responsi- ble for the continuance of the imaginary theory. It is to be observed that Tabit expresses himself with a certain reservation, and seems to think that further observations are necessary to decide if the theory is true or not. His younger and greater contemporary Al Battani was even more cautious, for though he repeats the account of the trepidation given by Theon (which he says that Ptolemy manifeste in suo libro declarat¹) he does not make use of it, but simply adopts 1° in 66 years (or 54".5 a year), which he finds by a comparison between his own observations and some made by Menelaus. In rejecting the erroneous value of Ptolemy, which Al Fargani alone had accepted,² Al Battani was followed by Ibn Junis, who came still nearer to the truth by adopting 1° in 70 years or 51".2 a year, and who does not allude to trepidation. It is greatly to the credit of several other Ara- bian writers that they were not led astray by this imaginary phe-

1. Cap. 52, (205). Plato's translation gives the period as 84 years, but Nallino's ed. has 80 (p. 127).
2. c. 13, p. 49. .
3. Schjellerup, Descr. des etoiles fixes, p. 43,

ARABS ON MOTIONS OF FIXED STARS 43 nomenon; among them are Al Sūfi, the author of the only urano- metry of the Middle Ages³, who followed Al Battani, also Abu 'l Faraj and Jagmini,¹ while Nasir ed-din mentions it but seems to doubt its reality.² By others it was willingly accepted, for instance by Al Zarkali, who made the period of oscillation of 10° either way equal to 2000 Muhammedan years (or 1940 Gregorian years, i.e. 1° in 97 years or 37" a year). The motion is in a circle of 10° radius; at the Hijra the movable equinox was it 40' in increasing precession, and in A: D. 1080 at 7° 25'.³. The diminu- tion of the inclination of the ecliptic, which the astronomers of Al Mamun had found=23° 33', no doubt lent countenance to the idea of trepidation, and the next step in the development of this curious theory was the combination of progressive and oscillatory motion. Al Betrugi, who gives a sort of history of the theory, beginning with a mythical H mes, makes out that Theon (or Taun Alexandrinus as he calls him) com- bined the motion of 1° in 100 years with the oscillation³. A century later this was actually done, and the theory received its last development by King Alfonso or his astronomers, who⁴ perceived that the equinoxes had receded much further than Tabit's theory allowed. The equioxes were now supposed to pass right round the heavens in 49,000 years (annual motion=26".45), while the period of the inequality of trepidation was 7000 years, so that in a sort of Great Jubilee year everything was again as it had been in the beginning.⁵ The progressive motion belongs to

1. Abu 'l Faraj, p. 12, simply says that the motion is 1° in 100 years according to Ptolemy, or 1° in 66 years according to others. But on p. 18 he says that if the ancient Chaldeans gave the tropical points a motion backwards and . and forwards, and if ancient astrologers adopted this, then the motion of the fixed stars must have been unknown to them. Jagmini (p. 229) says that most people adopt 1° in 66 solar years.
2. Spheres celestes, p. 347.
3. Sedillot, Memoire sur les instr. astr. des Arabes, pp. 31, 32. Abraham ben Chija (p. 196 of Munster's Sphaera mundi. Basle, 1546) gives the period as 1600 years without quoting any authority. He adds that the ancient Indians. Egyptians, Chaldeans, Greeks, and Latins first proposed the theory—Ptolemy neither approved nor disapproved of it, but Al Battani confuted it.
4. Alpetragius, f. 12a. He says that Al Zarkali did the same.
5. A later writer, Augustinus Ricius. De motu octave sphaere, Paris, 1521, who traces the theory back to Hermes, 1985 years before Ptolemy (!) credits (Continued on next page)

44 ASTRONOMY IN ANCIENT NATIONS the ninth sphere; the annual precession varies between 26″.45± 28″.96, or from + 55″.41 to −2″.51.¹ It was now necessary to assume the existence of a tenth sphere, which as primum mobile communicated the daily rotation to all the others, while the ninth produced the progressive and the eighth periodical motion on the small circles, which are situated "in the conca- vity of the ninth sphere." This was a nice and comfortable theory on account of the long periods involved and the slow changes it produced in the amount of annual precession; and oblivious of the fact that the theory had no foundation except the circumstance that the obliquity of the ecliptic was now about 20′ less than it had been stated to be by Ptolemy, and that he had given the amount of precession as 36″ a year instead of about 50″, and often shutting their eyes to several of the necessary consequences of it, such as the changes in the latitudes of stars which it ought to produce², astronomers continued to accept the theory until at last a real observer of the stars arose and wiped it out by showing that the obliquity of the ecliptic had steadily diminished, and that the amount of annual precession had never varied. We have in this place only alluded to it because it in- volved some rearrangement of the spheres and because it is eminently characteristic of the period during which no persistent observations were taken, and hardly an attempt was made to improve the theories of Ptolemy. The theory of trepidatio or titubatio, as it was sometimes called, was one attempt and it would have been better left alone. But it forms a not uninter- esting chapter in the history of astronomy. (Continued from previous page) this development to a Jew of Toledo, Isaac Hassan (see above. p. 38), adding that Alfonso four years after the completion of the tables became convinced of the futility of the theory by reading the book on the fixed stars by Al Sufi. Riccioli, Almag. novum, I. p. 166.

1. In the Alfonsine Tables the maximum took place at the birth of Christ. In Essler's Speculum astrologicum, p. 224 (appended to Purbach's Theoricae novae, Basle, 1573) the epoch is A.D. 15, diebus 137 completis. Reinhold in his commentary to Purbach (Paris, 1558, f. 163b) explains that 26″.45 is the space passed over by the Sun in 10 mins. 44 secs., by which amount the Alfonsine Tables made the tropical year smaller than 365¼ days.
2. Abraham ben Chija (p. 103, Schrackenfuchs) says that trepidation does not change the latitudes. Perhaps he refers to the earliest form of the notion, that described by Theon of Alexandria.

ARABS ON MOTIONS OF FIXED STARS 45 Here we finish our review of ancient astronomy. We have omitted as not coming within our province several valuable contributions to science which did not deal with cosmology or planetary theory. But even with this limitation enough has been said to show that when Europeans again began to occupy themselves with science they found astronomy practically in the same state in which Ptolemy had left it in the second century. But the Arabs had put a powerful tool into their hands by alter- ing the calculus of chords of Ptolemy into the calculus of sines or trigonometry, and hereby they influenced the advancement of astronomy in a most important manner. References

1. Peter Doig : A Concise History of Astronomy London
3. J.L.E. Dreyer : A History of Astronomy from Thales to Kepler Dover publications, 1953 (chapter XI reproduced).
4. Satya Prakash : Founders of Sciences in Ancient India, Delhi, 1965.

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CHAPTER II Personal References of Brahmagupta Sudhākara Dvivedī in his Gaṇaka Taraṅgiṇī, a small book on biographical skethes of astronomers and astrologers of this country gives a brief account of Brahmagupta thus : Brahmagupta was born in 520 Śaka (655 Vikramī or 589 A.D.) in the reign of King Vyāghramukha, belonging to the Cāpa family ; his father was Jiṣṇugupta ; and at the age of 30, he wrote in 550 Śaka (628 A. D.) his well known treatise on Astronomy known as the Brāhmasphuṭasiddhānta, which is corroborated by the statement in the Viṣṇudharmottara Purāṇa, (Chapter on the Brahma-siddhānta), His other treatise, entitled the Khaṇḍa- Khādyaka, which is a karaṇa book, was completed in 587 Śaka (665 A. D.). According to some authorities, Brahmagupta was the grandson of Viṣṇugupta, and the family suffix (Gupta) indicated that he belonged to the Vaiśya family, and he was in the service of the King of Rewah, known as Vyāghrabhaṭa. Brahmagupta was a great critic ; he did not spare any of his predecessors like Āryabhaṭa, Varāhamihira, Śrīṣeṇa, Viṣṇucandra and others. Later on, his influence on the writing of the succeeding generations has been immense. Bhāskarācārya II in his Bījagaṇita has acknowledged him as a great authority on algebra and has given him as the first place amongst the galaxy consisting of Brahmagupta, Śrīdhara, Padmanābha etc. The Eighteenth Chap- ter of the Brāhmasphuṭasiddhānta, known as Kuṭṭaka Chapter (on Pulveriser) has been translated by H. T. Colebrooke in English in 1817. The English translation of the Twelfth Chapter on Gaṇita or Calculations from the Brāhmasphuṭasiddhānta is also available in English. (See Colebrook's Algebra with Arithmetic and Mensuration from the Sanskrit of Brahmagupta and Bhās- kara, London, 1817). The Vāsanā Commentary on the Brāhmasphuṭasiddhānta by Pṛthudakasvāmī (860 A. D.) is also available though with diffi- culty (as indicated by Sudhākara Dvivedi) ; its incorrect manus-

48 PERSONAL REFERENCES OF BRAHMAGUPTA cript is available in the Library of the King of Banaras (Kāśīrāja) which has the colophony at the end as : श्री चापवंशतिलके श्री व्याघ्रमुखे नृपे शकनृपात् , पञ्चाशत्संयुक्तैर्वर्षशतैः पञ्चभिरतीतैः । ब्राह्मस्फुटसिद्धान्तः सज्जनगणितज्ञगोलवित्प्रीत्यै, त्रिंशद्वर्षेण कृतो जिष्णुसुतब्रह्मगुप्तेन ॥ Bhāskara II has written the famous treatise Siddhāntaśiromaṇi (1150), which is almost based on the Brāhmasphuṭasiddhānta. It has been edited by the author's own gloss (Vāsanābhāṣya) by Bāpu Deva Śāstrī (Vārāṇasī); by Murlidhar Jha with the com- mentaries, Vāsanāvārttika of Nṛsiṁha (1621) and Marīci of Munīśvara (1635), vol. I (containing chapter 1 of the Gaṇitā- dhyāya) (Vārāṇasī, 1917); by Girija Prasad Dvivedi with original commentaries in Sanskrit and Hindi, vols. I and II (Lucknow, 1911, 1926); English translation of the text only by Bāpu Deva Śāstrī and Wilkinson (Calcutta, 1861). In the very first Chapter (verse 2), Brahmagupta writes : The old calculations dealing with planets (i.e. the old astronomy), based on the system of Brahmā have become erroneous in course of past ages and therefore, I, the son of Jiṣṇugupta would like to clarify them. Brahmagupta was not a mere theorist, he based his calcula- tions on direct observations with the help of instruments or devices (nalikādi yantra); he was in favour of making correc- tions on the basis of these observations. He was himself an expert observer. In his Khaṇḍakhādyaka also he has emphasised the need of direct observation. At many places, Brahmagupta has severely criticised the Romaka and Pauliśa systems of astronomy which were introduced in this country by Lāṭadeva and Śrīṣeṇa. There are many passages where this criticism would be available with vehe- mance. Brhamagupta was opposed to the system of Āryabhaṭa I. He never spares the school of Āryabhaṭa which was regarded as the most authoritative then. Sudhākara Dvivedi says that as Brahmagupta was opposed to the system of Āryabhaṭa, so the Vaṭeśvara Siddhānta was opposed to that of Brahmagupta. The Institute has already published the Vaṭeśvara Siddhānta and now it has the privilege of publishing the Brahmasphuṭasiddhānta.

A NOTE ON BHILLAMĀLA 49 A Note on Bhillamāla It is said that Brahmagupta completed his Brāhmasphuṭa- siddhānta in Śaka 550, and he has come to be known as Bhilla- mālakācārya or a teacher residing in "Bhillamālaka." In this connection, therefore, it would be interesting to reproduce a note on Bhillamāla from G. Bhūler's article on Gurjara Inscriptions, No. III, published in the Indian Antiquary, July 1888, vol. 17, p. 192 : With a single exception all the complete inscriptions call the princes enumerated above, scions of the Gurjara race; and Khe I. and II. highly extol the greatness and wide extent of this family. Na. alone names the Mahārāja Karṇa as their ancestor. With respect to this personage it is for the present impossible to say whether the famous hero of the Mahābhārata may be meant, or some real historical king. But the name Gurjara makes it evident that this dynasty belonged to the great tribe which is still found in Northern and Western India and after which two provinces, one in the Bombay Presidency and one in the Pañjāba, have been named. The Gurjaras or Gūjars are at present pretty numerous in the western Himālaya, in the Pañjāba and in Eastern Rājputānā. In Kachh and Gujarāt their number is much smal- ler. It would, therefore, seem that they came into Western India from the north. Their immigration must have taken place in early times, about the beginning of our era or shortly after- wards. In Western India they founded, besides the kingdom of Broach, another larger state which lay some hundred miles further north. Hiuen Tsiang mentions in his travels¹ the kingdom of Kiu-che-lo and its capital Pi-lo-mi-lo. It has been long known that the former word corresponds to Gurjara. But the name of the town has been incorrectly connected by the French scholars with Bālmer in the Jēsalmīr territory, and this ident ification has been accepted in Mr. Beal's new transla- tion of Siyuki. As I have stated already formerly² following Colonel. J. Watson, Pilomilo corresponds exactly to Bhillamāla

1. Beal, Siyuki, Vol. II, p. 269f. Hiuen Tsiang assigns to the nor- thern Gurjara State an extent about double of that given for the kingdom of Broach.
2. Ante, Vol. VI. P. 63.

50 PERSONAL REFERENCES OF BRAHMAGUPTA one of the old names of the modern Bhīnmāl or Śrīmāl¹ in southern Mārwāḍ close to the northern frontier of Gujarāt. Another work, which was composed a few years before Hiuen Tsiang's visit to Gujarāt, contains likewise a notice of this northern kingdom of the Gurjaras. The astronomer, Brahma- gupta, who completed his Siddhānta in Śaka-Samvat 550 or 628 A.D. calls himself Bhillamālakakācārya², "the teacher residing in Bhillamālaka and is called so by his commentator Pṛthūda- kasvāmin. He further states that he wrote under king Vyāghra- mukha who was 'an ornament of the Cāpa race.' This family, whose name recurs in the Haḍḍāla grant of Dharaṇīvarāha³ prince of Vadhvān, thus seems to have been the reigning house of Bhillamāla. It is most probably identical with the Cāuḍas, Cāvōṭakas⁴ or Chāpōtkaṭas, who from 756 to 941 A.D. held Aṇhilvāḍ and still possess various small districts in northern Gujarāt. The Gurjara kingdom of Broach was without a doubt an offshoot of the larger State in the north, and it may be that its rulers, too, belonged to the Cāpa family.

1. Bhillamāla means etymologically 'the field of the Bhil' and Śrī- māla 'the field of Śrī'. The latter name must also be ancient, as the Śrī- mālī Brāhmaṇ as are called after it. The Jainas narrate various, of course incredible, legends, which explain how Śrīmāla came to be called Bhillamāla. Merutuṅga says that king Bhoja invented the latter name, because the people of Śrīmāla let the poet Māgha die of starvation. According to another authority, the town had a different name in each Yuga. It is in India very common for ancient towns to have two or even more names. Thus Kanauj was called, Kānyakubja, Gādhipura, and Mahodaya.
2. See Professor A. Weber, Die Sanskrit und Prakrit Handschriften der Berliner Bibliothek Vol. II. pp. 297-298. In the first passage the MSS. offers incorrectly Bhilamācārya; in the second which occurs in the commen- tary on the Khaṇḍakhādyaka, we have Bhillamālavakācārya, a slightly cor- rupt reading. This latter varia lectio occurs also in other MSS., see Weber, Indische Streifen. Vol. III, p. 90, and has given rise to erroneous suppositions regarding Brahmagupta's home. The Gujarātī Joshīs still pre- serve the tradition that Brahmagupta was a native of Bhinmāla.
3. Ante, Vol. XII. p. 190ff. The remark which I have made there that the Cāpas are not named elsewhere, of course requires correction.
4. The form Cāvoṭaka, which occurs in Dr. Bhagavanlal's grant of the Gujarāt Cālukya king Pulakeśin of Samvat 490, is the immediate prede- cessor of the word Cāuḍa. Its Sanskrit original is certainly not cāpotkaṭa which probably has been coined in comparatively speaking modern times, in order to explain the difficult Prakrit word, just as the bards of Rajputana have invented Rāstrauḍha as etymon for Rāṭhoḍ.

BRAHMAGUPTA'S OWN REFERENCES 51 Brahmagupta's own References In the Twenty-fourth Chapter (Sañjñādhyāya), of the Brāhmasphuṭasiddhānta, Brahmagupta has made a reference to his own biography : In the reign of Vyāghramukha belonging to the family of Cāpa, in the year 550 Śaka the treatise Brāhmasphuṭasiddhānta was composed for the benefit of benevo- lent astronomers by Brahmagupta, son of Jiṣṇugupta at the age of 30. (BrSpSi. XXIV. 7. 8) Then again he says : The Brāhmasphuṭasiddhānta has been written by Brahmagupta, son of Jiṣṇu, in 1008 verses of Āryāch- anda. (ibid 10 ) In the beginning of this Sañjñādhyāya, he refers to the differences in fundamental notions created by the various existing systems of astronomy as the Sūrya-siddhānta, Pulisa-siddhānta. Romaka-siddhānta, Vasiṣṭha-siddhānta and other Yavana-siddhā- ntas, which have caused anomalies in the calculations of eclipses. He also refers to the anomalies due to the calculations based on midnight day-reckoning and sunrise day-reckoning. From the point of view of own references, the following would be of interest : Brahmagupta, son of Jiṣṇugupta (Jiṣṇusuta-Brahmagupta) : BrSpSi. I. 2; XVI. 35. 37; XXIV. 8. 10; XXV. 73. It is strange that in the Khaṇḍakhādyaka, Brahmagupta has not given his name nor his father's name anywhere. At least the reading of the Khaṇḍakhādyaka as given by Pṛthūdakasvāmī does not contain this name. In the edition of Bhaṭṭotpala, there are three more chapters in the Khaṇḍakhādyaka (Chapters IX, X and XI). In the Chapter XI (known as Pātādhikāra), we have 21 verses and in the last 21st verse we find the name of Brahmagupta,¹ son of Jiṣṇu mentioned : Those who are eager to have the knowledge of the motion of stars and planets, for them and for the benefit of disciples in this field, Brahmagupta son of Jiṣṇu has composed this Khaṇḍakhādyaka.

1. खण्डखाद्यकमिदं तृप्त्यर्थं ग्रहगतिश्रुतात्तांनाम् । शिष्याणां हितार्थं प्रोक्तं जिष्णुसुतब्रह्मगुप्तेन ॥ — KK. XI. 21

52 PERSONAL REFERENCES OF BRAHMAGUPTA Reference to Āryabhaṭa We have said that it appears that Brahmagupta was a bitter opponent of Āryabhaṭa in his younger days (628 A. D.), but later on (in 665 A. D.), he climbed down to describe and teach one of the Āryabhaṭa's system of astronomy. Āryabhaṭa was universally revered. and it was difficult for Brahmagupta to have ignored him and thus he has to refer to this great authority some times to oppose some of his views and some times to expound his views. The following are the pasages in the Brahmasphuṭasiddha- nta, where the author has referred to the name of Āryabhaṭa. There are many other passages where the name "Aryabhaṭa" does not occur but where Brahmagupta indirectly means to quote the views of this great master. BrSpSi. I. 9. 12. 28. 32. 60. 61 ; II. 33. 46; V. 21.25; VI. 13; IX.10; XI. 4. 9. 10. 12. 25. 29. 33. 41. 42. 43. 44, 46, 47. 49, 62; XIII, 27; XIV. 45; XVI. 37. 46; XXI. 39. The largest references are in Chapter XI, where Brahmagupta has made an attempt to show the discrepancies of the Āryabhaṭī- aya system of reckoning astronomical observations and constants. In the Khaṇḍakhādyaka also there is a reference to the name of Āryabhaṭa but on very few occasions. In Chapter I, we have this reference at three places- Having made obeisance to God Mahādeva, who is the great cause of this world's rise (i.e. creation), existence and destruc- tion, I shall declare the Khaṇḍakhādyaka (i.e. a short treatise on astronomy, which is as pleasant as food prepared with sugarcandy), which will yield the same results as the great astronomical treatise of Āryabhaṭa. As in most cases calculation by the great work of Āryabhaṭa, for (the know- ledge of time and longitude of planets etc. at ) marriage, nativity and the like, is impracticable for common use every day, this smaller treatise is made so as to yield the same results as that. ¹

1. प्रणिपत्य महादेवं जगदुत्पत्तिस्थितिप्रलयहेतुम् । वक्ष्यामि खण्डखाद्यकमाचार्य्यार्य्यभट तुल्यफलम् ॥ प्रायेणार्य्यभटेन व्यवहारः प्रतिदिनं यतोऽशक्यः । उद्वाहजातकादिषु तत्समफललघुतरोक्तिरतः ॥ —KK. I, 1-2

REFERENCE TO ĀRYABHAṬA 53 These two verses show that Brahmagupta was not hostile to Ārya- bhaṭa when he wrote this Khaṇḍakhādyaka ; he merely intends presenting the subject matter on simple lines and furnishing the results obtained by Āryabhaṭa in a simpler way. In the following verse of the same chapter, he refers to Āryabhaṭa’s midnight day-reckoning system : The mean Saturn diminished by 3 seconds, the Śīghrocca of Mercury diminished by 22 seconds, the mean Mars increased by 2 seconds and the mean Jupiter increased by 4 seconds are equal to the respective mean planets of Ārya- bhaṭa's “midnight-system”.¹ In the Appendix of the Khaṇḍakhādyaka known as Khaṇḍa- khādyakottara, we again find a few verses where the name of Āryabhaṭa occurs; Three of these verses have been reproduced from the Brāhmasphuṭasiddhānta (BrSpSi I. 62, 6

54 PERSONAL REFERENCES OF BRAHMAGUPTA In the Appendix of the Khaṇḍakhādyaka, there is another verse which also speaks in the same strain against Āryabhaṭa (this verse does not occur in the Brāhmasphuṭasiddhānta) : As the process of finding the apparent places of pla- nets as given by Āryabhaṭa does not make them agree with observation, I shall, therefore, speak of this process. Of the Sun the apogee is at two signs and seventeen degrees (KK. IX. 4)¹ Brahmagupta opposed to Śrīṣeṇa— Viṣṇucandra, Lāṭa, Vijayanandī and others Brahmagupta was a great critic; he did not spare Āryabhaṭa, and along with him he was vehemently opposed to the doctrine of Śrīṣeṇa, Viṣṇucandra, Lāṭadeva, Vijayanandī and Pradyumna also. He was opposed to the Romaka and Pauliśa Siddhāntas, which were the systems of foreign astronomy, derived from Greece, Babylonia and other centres of learning. He did his best to resist the foreign influences on astronomy. The following are the verses in the Brāhmasphuṭasiddhānta, where Brahmagupta expressed his note of discord against the sys- tems or notions of Āryabhaṭa, Śrīṣeṇa and Viṣṇucandra : BrSpSi. I.60; II 46, 47; X 13, 62; XI.31, 46, 47, 48-50, 55, XVI. 36, 46; XXI.39; XXII. 2. In two of the verses, he refers to Lāṭasiṃha : BrSpSi. XI. 46, 48. In the following verse, he refers to Aṅkaciti, Vijayanandī, Prad- yumna and others: BrSpSi. XI. 58. In the Khaṇḍakhādyaka we do not find the names of these adversaries of Brahmagupta ; we, however, have a reference of Śrīṣeṇa, Āryabhaṭa and Viṣṇucandra in the verses already quoted, occurring in the Appendix of the Khaṇḍakhādyaka (Khaṇḍa- khādyakottara), Chapter IX. 2 and 3. As we have said before, these verses of the Appendix have been reproduced from the Brāhmasphuṭasiddhānta. (I.62 and II.47). Reference to Romaka and Pauliśa systems Brahmagupta speaks of his system as if expounded by Brahmā himself for the first time, later on deteriorated, and then revived by Brahmagupta himself. The very second verse of the Brāhma-

1. न स्फुटमार्यभटोक्तं स्पष्टीकरणं यतस्ततो वक्ष्ये । भानुमतो मन्दोच्चं राशिद्वयमंशकांश्च सप्तदश ॥ —KK. IX. 4.

REFERENCE TO ROMAKA AND PAULIŚA SYSTEMS 55 sphuṭasiddhānta (I. 2) substantiates this view : The science of astronomy (or the calculations of heavenly bodies) in course of long duration became ineffective or erro- neous; this was revived by Brahmagupta, son of Jiṣṇu.¹ (BrSpSi. I.2). The system of astronomy which goes by the name Brahma (Brahmasiddhānta) has been handed down to us in three forms : (i) one is as treated by the Śākalya Saṁhitā, (ii) one as described in prose in the Viṣṇudharmottara Purāṇa, and (iii) the one des- cribed by Varāhamihira in the Pañcasiddhāntikā, which re- cognises the yuga of the duration of five years. Which of these three was accepted by Brahmagupta is not clear. But from the measures of the number of revolutions performed by a planet in a given period (graha-bhagaṇa) etc., it is clear that Brahmagupta acknowledges the system as propounded in the Viṣṇudharmottara Purāṇa. In his chapter "Tantra-parīkṣādhyāya or Dūṣaṇā- dhyāya, he contradicts the notions of the Vedāṅga Jyotiṣa which accepts the yuga of five years. We may further emphasize the fact that Brahmagupta has not clearly detailed out the errors to which the Brahma-siddhānta succumbed in course of time, and how these errors were eradi- cated by him. During the days of Brahmagupta, Romaka and Pauliśa systems were getting currency in this country. Reference to these two are found in the Brāhmasphuṭasiddhānta at several places as follows : ROMAK : I.13; XI.50; XXIV.3 PAULIŚA : XIV.45; XXIV.3 In fact, BrSpSi. XXIV.3, we find the line "Sūryendu-Puliśa- Romaka-Vasiṣṭha-Yavanādyaiḥ", where we have a reference to all the then existing systems : Sūrya-siddhānta, Indu-siddhānta, Puliśa-siddhānta, Vasiṣṭha-siddhānta, and other Yavana-siddhā- ntas. Just as the Sun is one, so the astronomical system is also one ; this is a different thing that calculations in different systems may vary according to different sunrises in different places. Brahmagupta refers at one place to Varāhamihira in BrSpSi. XXI.39, where he has been spoken of in connection with a list of

१. ब्रह्मोक्तं ग्रहगणितं महता कालेन यत् खिलीभूतम् । अभिधीयते स्फुटं तज्जिष्णुसुतब्रह्मगुप्तेन ॥ — BrSpSi. I. 2.

56 PERSONAL REFERENCES OF BRAHMAGUPTA anti-authoritative versions of astronomical systems : Evam Varāhamihira-Śrīṣeṇācāryabhaṭa-Viṣṇucandrādyaiḥ Lokaviruddhamabhihitam Veda-smṛtisaṃhitābāhyam. At one place, he mentions the difference between the calculations based on the system of Āryabhaṭa and the Pañca-siddhāntas (Five systems) : Pauliśa, Romaka, Vasiṣṭha, Saura, and Paitāmaha. (BrSpSi. XIV.46). Brahmagupta was also familiar with the Jaina systems of astronomy; for example, at one place he uses the term "Jinoktam" (i.e., one propounded by the Jainas) : He repudiates the concept of two Suns and two Moons "Dvāvarkaindavau" (do canda do sujja) (BrSpSi. XI.3) as enunciated by the Jainas. At several places we find a reference to the Vasiṣṭha-sid- dhānta (BrSpSi. XI.49, 50; XXIV.3) Wherever, Brahmagupta has to press for his views in pre- ference to the views of others, he uses the words Brāhma or Brahmokta : BrSpSi. Brāhma : I.32; X.62; XI, 61; XVI.37 Brahmokta : II.31, 33; X.63, 69; XV.59; XVI.33 Reference Sudhākara Dvivedi : Gaṇaka Taraṅgiṇī G. Bühler : Gurjara Inscriptions, Indian Anti- quary, 1888. P. C. Sengupta : Khaṇḍakhādyaka, Calcutta, 1934. — . o : —

CHAPTER III Manuscripts of the Brāhmasphuṭa Siddhānta Sudhākara Dvivedī has given an account of some of the manuscripts of the Brāhmasphuṭasiddhānta in his Bhūmikā or Introduction appended to the edition published in the PAṆḌITA, Vol.XXIV, 1902 (New Series) : (i) available in the Library of the Government College, Kashi (Vārāṇasī) i. e. Kāśīka-Rājakīya-Pāṭha- laya,(ii) Dr.Thibaut's Manuscript,(iii) the Manuscript in possession with Yajñadatta Śarmā, the Chief Astronomer attached to the Prince of Ayodhyā. It is further mentioned that Dr. Thibaut's Manuscript was a copy of a Manuscript available in the Deccan College, Poona. The Manuscripts (ii) and (iii) were identical. The Manuscripts (iii) was very faulty and incorrect. The Manuscript from which Colebrooke translated out in English the Twelfth and the Eighteenth Chapters on Gaṇita or Mathematics and Kuṭṭaka (the Chapter on Pulveriser) respectively, appeared to be different from the three Manuscripts described above. The readings differed considerably. The Kuṭṭaka- Chapter of that book is, writes Sudhākara Dvivedī, still available in the India Office Library. (See Catalogue of the Sanskrit¹ Manuscripts in the Library of India Office, Part V.p. 995).¹

1. एतत्कृतस्यास्य सिद्धान्तग्रन्थस्यैका प्रतिः काशिकराजकीय पाठालयतो द्वितीया डा० थिबौ साहिब महाशयतस्तृतीया चायोध्यानरेशप्रधानज्योतिर्विच्छ्री यज्ञदत्तशर्मणो मया लब्धा । डा० थिबौमहा- शयस्य पुस्तकं कस्यचिद्दक्षिणदेशीय (Deccan College, Poona ) डेकन पुस्तकस्य प्रत्यन्तरम् । इदं पुस्तकं तथा पं० श्री यज्ञदत्त पुस्तकं चैकमातृकमेव । इदं पुस्तकत्रयमतीवाशुद्धं बहु च स्खलितं चास्ति । यत्पुस्तकानुसारेण व्यक्ताव्यक्तयोर्द्वादशाष्टादशसंख्ययोरंगलभाषायामनुवादः कोलब्रूकसा- हिबेन कृतस्तत्पुस्तकमेतत्त्रयतो भिन्नमित्यसंशयं विभाति पाठविभेदात् । तत्पुस्तकस्य कुट्टकाध्यायः संप्रति इण्डिया-आफिस-सरस्वती भवने वर्तते (See Catalogue of the Sanskrit Manuscripts in the Library of India Office, Part V, Page 995) पृथूदककृताऽस्य सिद्धान्तस्य टीका या कोलब्रूकसाहिबेन भारतवर्षे ह्युपलब्धा सा च संप्रति लण्डननगरे इण्डिया-आफिस-सरस्वती भवने वर्तते । तस्या एका प्रतिर्मद्द्वारेण श्री डा० थिबौ साहिबमहाशयैरुत्पादिता सापि संप्रति मन्निकटेऽस्ति । अहो इण्डिया-आफिस-सरस्वती (Continued on next page)