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

32 ASTRONOMY IN ANCIENT NATIONS (the motion of the pole of the orbit being added to or subtracted from the motion of the planet), so that the epicycle is hereby rendered superfluous. The lengths of the radii of these small circles are not given, except in the case of Saturn, where the radius is 3° 3',¹ while the mean pole of the moon is 5° (the inclina- tion of the lunar orbit) distant from the pole of the ecliptic,² and the small circle is so exceedingly small as to produce no retrograde motion, which is also the case with the Sun. The periods of the poles of the outer planets are given by the following figures Saturn makes 57 revolutions in 59 years and 1½+¼ days, in which period the mean pole lags behind 2 revolutions 1½°+⅜°. Jupiter makes 65 revolutions in 71 years, the mean pole lagging behind 6 revolutions. Mars makes 37 revolutions in 79 years and 3¼+¹/₁₈ days, the pole lagging behind 42 revolutions and 3⅙°³. In other words, the motion on these small circles are com- pleted in the synodic periods of planets. Similarly, the pole of Venus makes 5 revolutions in the 8 years less 2¼d+¹/₂₀, lagging 1⅕ revo- lutions in one year ; and Mercury 145 revolutions in 46 years and 1¹/₃₀d⁴. It is curious that Alpetragius alters the order of the planets, placing Venus between Mars and the Sun, because the defectus (lagging) of Venus smaller than that of the Sun.⁵ He also says that nobody has given any valid reason for accepting the usually assumed order of the planets, and that Ptolemy is wrong in stating that Mercury and Venus are never exactly in a line with the Sun (a remark already made by Geber) ; and as they shine by their own light they would not appear as dark spots, if passing between us and the Sun. That they do not receive their light from the Sun is proved, he thinks, by the fact that they never appear crescent-shaped.⁶ There is no need to dwell any longer on this quaint theory

1. Fol. 16 a.
2. Fol. 25 a.
3. Fol. 16 a, 18 a, 19 b.
4. Fol. 21 b, 24 b.
5. "Nam reperimus defectum eius primum minorem defectu orbis solis et maiorem defectu orbis martis, et sequitur juxta radices nostras ut sit inter eos ambos."
6. Fol 21 a

NASIR ED-DIN AL TŪSI 33 of spiral motion, as it has been rather improperly called.¹ It represented a retrograde step of exceedingly great magnitude, totally unjustified as the theory could not seriously pretend to be superior to the Ptolemaic system, which had only become so very simple if one was content with representing only the princi- pal phenomena. We are told by the Jewish astronomer Isaac Israeli of Toledo, that the new system made a great sensation, but that it was not sufficiently worked out to be taken seriously, and that the system of Ptolemy, founded on the most rigorous calculations, could not be superseded by it.² Another Jewish author, Levi ben Gerson, in a work written in 1328, entered into a lengthy refutation of the hypotheses of Al Betrugi.³ But the latter certainly represented a general desire on the part of the Spanish Aristoteleans to overcome the physical difficulties in accepting the Ptolemaic system ; thus Averroes says that the astronomy of Ptolemy is merely a convenient means of compu- ting, and that he himself in his youth had hoped to prepare a work on the subject. Nasir ed-din Al Tūsi While ineffectual attempts were being made in the far west, to devise a new astronomical theory, the astronomers of the east did not remain blind to the desirability of finding a system, in which the planets were not supposed to move unsup- ported in space in such a wonderfully complicated manner; and in the thirteenth century we find one of the greatest astronomers, Nasir ed-din Al Tūsi, advocating a system of spheres which he supposed to be more acceptable than excentrics and epicycles.⁴ In addition to a review or digest of the Syntaxis of Ptolemy he wrote a shorter work entitled Memorial of Astronomy, in various

1. e.g. by Riccioli, Almag. Nov. T. I. p. 504, where Kepler's figure of the real motion of Mars in space from 1580 to 1596 (supposing the earth to be at rest) is copied, as if that had anything to do with the "Spirals" of Alpetra- gius.
2. He adds that he was not qualified himself to sit in judgment on the pro- posed system (Liber Jesod Olam, II, 9, p. XI.)
3. Munk, Melanges pp. 500 and 521.
4. "Les spheres celestes selon Nasir-Eddin Al tūsi. Par M. Carra de Vaux." Appendix VI. to Tannery's Recherches sur l'astr. anc. pp. 337—360. In- cludes a translation of the chapter in which the new theory is set forth.

34 ASTRONOMY IN ANCIENT NATIONS passages of which he shows his dissatisfaction with the Ptolemaic system. In the chapter on the Moon (to which we have already alluded ) he counts up the various anomalies, among which he mentions the anomaly of illumination, that is, the spots on the Moon, which he believes to be caused by other bodies moving in the lunar epicycle and unequally exposed to the Moon's light. He then says that we should expect in a simple theory to find the centre of the epicycle in equal times describing equal arcs on the deferent, and the diameter of the epicycle joining the pericentre and the apocentre pointing to the centre of the deferent. But neither of these conditions is fulfilled. In the theories of the planets he makes the same objections, which it must be said are very just, since the introduction of the equant was a very unna- tural arrangement. But this is nothing to the artificial machinery designed by Ptolemy to account for the motion in latitude of the five planets, especially of Mercury and Venus. Nasir ed-din des- cribes the marvellously complicated movements of the deferents and epicycles of these planets, and remarks that "these motions require the introduction of a system of guiding spheres, about which the ancients have not said anything". He next proceeds in the following chapter to explain a system of his own which allows us to discard these combinations. First he proves that if there are, two circles in one plane, one touching the other internally and of a diameter equal to half that of the other, and if the greater one rotates, and a point moves along the circumference of the smaller one in the opposite direction with twice the velocity and starting from the point of contact, then that point will move along a diameter of the greater circle.¹ These two circles may now be assumed to be the equators of two spheres, and for the point we may substitute a sphere representing the Moon's epicycle (1 in the figure), Nasir ed-din assumes another sphere (2) surrounding the epicycle and destined to keep the diameter from apogee to perigee in its place always coinciding with the diameter of the sphere (4) "let us give it a suitable thickness, but not too great, so as not to take up too much space." He next assumes two more spheres, one (3) which corresponds to the smaller sphere in the distance of the centre of the deferent in the Ptolemaic system from the

1. Compare Copernicus, De revolutionibus, III. 4 (Secular ed. 1873, p. 166).

NASIR ED-DIN AL TŪSI 35 centre of the Earth; and another sphere (4) with a diameter twice as great. Finally (4) is placed in the interior of a carrying sphere (5) concentric with the world and occupying the concavity of the Fig. 3.—Movements of deferents and epicycles of planets. The thickline is not a circle. All others are circles. sphere (6), the equator of which is in the plane of the lunar orbit. (2) and (4) and (5) revolve in the same period, that in which the centre of the epicycle performs a revolution; (3) revolves in half that time, while (6) revolves in the opposite direction with the same speed as the apogee of the excentric. The figure now shows how the epicycle moves to and fro along the diameter of (4) and during the revolution of the circle (5) describes a closed curve, about which Nasir ed-din justly says that it is somewhat like a circle but is not really one, for which reason it is not a perfect sub- stitute for the eccentric circle of Ptolemy. He estimates the greatest difference between the lunar places given by the two theories as one-sixth of a degree, half-way between syzygy and quadrature. Except for the action of the guiding sphere (2), it would not be the centre of the epicycle but the point of contact of circles (3) and (4), which describes the curve resemblng a circle. The same

36 ASTRONOMY IN ANCIENT NATIONS method may be adopted for Venus and the three outer planets, and Nasir ed-din promises to explain the new theory of Mercury in an appendix, but this appears to have been lost. Nasir ed-din also endeavours to improve on the machinery proposed by Ptolemy to illustrate the manner in which the epicy- cle remains parallel to the plane of the ecliptic. He mentions that the celebrated Ibn al Haitham (afterwards known in the west as Alhazen, author of a well-known book on optics) had written a chapter on this subject, adding to each epicycle two spheres to account for the inclination of the diameter perigee-apogee, and two additional ones for the inferior planets for the diameter at right angles thereto.¹ Nasir ed-din makes use of the same principle which guided him in his demonstration about the motion in longitude, and he shows how in this way we may by means of two spheres make the extremities of the diameter of the epicycle move backwards and forwards along an arc of a sphere.² He claims that this arrangement is superior to that of Ptolemy by not intro- ducing any error in longitude,³ but he acknowledges that he has not been able to get rid of the strong objection to Ptolemy's auxiliary circle, viz. that the irregular motion in longitude with regard to the centre of the deferent necessitates the introduction of a corresponding irregularity in the motion on the auxiliary circle by letting the motion be uniform with regard to an equant. It baffled Nasir ed-din's ingenuity to find an arrangement of spheres which could obviate the necessity of having recourse to this expedient. All the attempts at rebellion against the Ptolemaic system had thus turned out failures. And they deserved nothing else, since it was impossible to find anything better than what Ptolemy had produced, until it was perceived that where Ptolemy was wrong was not in his mathematical methods, which were perfect, but in the fundamantal idea of the Earth being at rest. The time

1. Ibn al Haitham said that by using discs instead of spheres one might com- plete the demonstration; but Nasir ed-din objects to the arrangement (about which he gives no details) that a non-spherical system is not in accordance with the principles of astronomy.
2. It is not quite clear whether this plan is his own or is the same as Ibn al Haitham's.
3. Due to disturbance of the position of the diameter from perigee to apogee, from which the anomaly is counted.

NASIR ED-DIN AL TŪSI 37 was apparently not ripe for a radical change with regard to this idea. Though the doctrine of the Earth's motion does not seem to have been mentioned by Arabian writers, we have evidence - that the hypothesis of the daily rotation of the Earth was not unknown among them, a natural consequence of their familiarity with the writers of antiquity. One of Nasir ed-din's fellow-workers at the Meragha observatory, Ali Negm ed-din al Katibi, who died in 1277, wrote a book, the Hikmat al-ain, on philosophy, in which he combats this opinion, which he attritutes to "some philos- phers." "I do not," he says, "advance as an argument against it that, if this were the case, a bird flying in the direction of the motion of the Earth would not be able to keep up with it, because the motion of the Earth would be much faster than that of a bird, inasmuch as it returns to its place in a day and a night. Such an argument is not conclusive, because it may be urged that the atmosphere which is close to the Earth partakes of its motion as the ether partakes of the motion of the heavenly sphere. But I reject this theory, because all terrestrial motions take place in a straight line, and therefore we cannot admit that the Earth should move in a circle."¹ What reformation of astronomy could be hoped for, as long as this kind of argument could be used ? We cannot see from this remark of Katibi's whether there really were any Arabian philosophers who believed in the rotation of the Earth. It is, however, stated in the Zohar, the great Kabbalistic work attribu- ted to Mosheh ben Shemtob of Leon (died 1305), that a certain Rabbi Hamnuna the Elder (otherwise unknown) taught that "the Earth turns like a sphere in a circle and that some people are above and others below."² Though this passage as well as others in the Zohar may have been interpolated much later, it would

1. A. Sprenger, "The Copernican System of Astronomy among the Arabs." Journ. Asiat. Society of Bengal, Vol. XXV. (1857), p. 189. Katibi's con- temporary, Abu 'l Faraj (II. p. 10) deems it necessary to prove that the Earth cannot be in motion, neither rectilinear nor circular, but his argu- ments (about birds and stones flung upwards) seem merely taken from Ptolemy, lib, 1. cap. 6. Kazwini (Kosmographie, p. 296) says that among the ancients there were some adherents of Pythagoras who maintained that the Earth continually moves round in a circle ; but whether these adherents were Greeks or Arabians cannot be seen from the context.
2. Sohar, Amsterdam, 1718, T. III. f. 10a ; Gunther, Studien Z. Gesch. d. math, Geogr., p. 113.

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