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

18 ASTRONOMY IN ANCIENT NATIONS of Hipparchus, but varying considerably and irregularly. Ibn Jūnis, who quotes this, adds that he has himself found 5° 3' or 5° 8', while other observers are said to have found from 4° 58' to 4° 45'.¹ Want of perseverance and of accurate instruments caused them to miss a remarkable discovery, that of the variation of the lunar inclination. Abu 'l Wefa and his Almagest But an even more remarkable discovery has been claimed for an Arabian astronomer. In 1836 the younger Sedillot announced that he had found the third inequality, the variation, distinctly announced in Abu 'l Wefa's Almagest. A fierce controversy raged for a number of years as to the reality of this discovery, Sedillot alone defending his hero with desperate energy and refusing to listen to any arguments, while Biot, Libri and others as strenuously maintained that Abu 'l Wefa simply spoke of the second part of the evection, the prosneusis of Ptolemy. The fight had died out when, in 1862, Chasles suddenly took up the cudgels for Sedillot and pointed out what seemed to him to be some contradictions in Ptolemy's statement.² Nobody answered this until Bertrand did so in 1871; he called attention to several inaccuracies in the text of Abu 'l Wefa as we possess it now, and also showed that Abu 'l Wefa did not add his "mohazat" to the prosneusis, the latter not being included in his "second anomaly."³ It is unnecessary to enter into a more detailed account of the controversy; but to show that any weapon was considered good enough with which to defend Abu 'l Wefa, it may be mentioned that Sedillot and Chasles tried to prove that Tycho Brahe must have copied his discovery from Abu 'l Wefa, because he calls it hypothesis redintegrata. Tycho used this same phrase in speaking of his own planetary system, which he most emphatically claimed as

1. Sedillot, Prolegomenes, p. xxxviii. Materiaux pour servir a l' hist. des scienc- es chez les Grecs et les Orientaux, T. I. p. 283. The sons of Mūsā ben Sakir (about 850) seem to have been the first to find a value differing from that of the ancients. Abraham ben Chija, a Jewish writer who lived about A. D. 1100 says that Ptolemy found 5°, but that according to the opinion of the Ishmaelites it is 4½° (Sphaera mundi, Basle, 1546 p. 102).
2. Lettre a M. Sedillot sur la question de la variation lunaire, Paris, 1862, 15 pp. 4° and Comptes Rendus, vol. 54, p. 1002.
3. Comptes Rendus, vol. 73, pp. 581, 756, 889; Journal des Savants, 11 Oct.

ABU L' WEFA AND HIS ALMAGEST 19 an original discovery, and which he vigorously defended against other claimants. In future it will be hopeless for anybody to claim the discovery for Abu 'l Wefa as the matter has now been thoroughly sifted, both by mathematicians and orientalists. The Almagest of Abu 'l Wefa has never been published in full, but there are three translations of the chapters in question,¹ which only differ in some trivial points. In no part of the book does he make any advance on Ptolemy or claim to have made any new discovery, and in speaking of three inequalities he merely does what the other Arabian astronomers do.² He begins by describing the first (equation of the centre) and the second (evection) and states when they reach their maxima. He then says that we have found³ a third inequality, which takes place when the centre of the epicycle is between the apogee and the perigee of the eccentric, and which reaches its maximum when the Moon is about a tathlith or a tasdis from the Sun, while it is insensible in syzygy and quadrature. The maximum is ¾°. He explains that this is caused by a deviation of the line or apsides of the epicycle, and he describes quite correctly the construction adopted by Ptolemy (whose name he does not mention), letting the line of apsides be directed, not to the Earth but to another point on the line of apsides of the eccentric. It is difficult for an unbiassed reader to understand how anyone could fail to see that Abu 'l Wefa is simply copying Ptolemy. Sedillot maintained that the words tathlith and tasdis mean the octants (where the variation reaches its maximum); but every other orientalist who has expressed an opinion, states that by their roots the words correspond to the numbers 6 and 3, in other words, to elonga- tions 60° and 120° from the Sun. This is in accordance with

1. By Reinaud, Munk, and de Slane(for Biot)in the Journal des Savants, Mar- ch, 1845 (14 pp. the whole section on the Moon) ; by Sedillot, Materiaux, i. pp. 45--49 ; and by Carra de Vaux, “L’ almageste d’ Abu 'l Wefa Albūzdjani,” Journal asiatique, 8° Serie, T. xix. (1892), pp. 408--471 (translation on pp. 443-44). Most of the chapters on the planets are lost.
2. The unknown author of a short resume of astronomy (in the Bibl. Nationale) even calls the inequality of prosneusis the first equation (Carra de Vaux, 1.c. p. 460). This is not unreasonable, since the equation of the centre must be taken from the lunar tables, using as argument not the mean anomaly but the latter corrected for the effect of the prosneusis.
3. He uses exactly the same expression when speaking of the first and second inequalities.

20 ASTRONOMY IN ANCIENT NATIONS facts, as Biot has shown from the Ptolemy's numerical data that the deviation of the line of apsides reaches its maximum value of ± 13° 8′ .9 in elongations 90° ∓ 32° 57′.5.¹ But it must be acknowledged that the words in question are also used very vaguely, e. g. by Abu 'l Wefa himself, who says that the velocity of the superior planets after emerging from the Sun's rays dimi- nishes gradually till their distance from the Sun is about a tathlith, when they become stationary. It looks almost as if these words might be used to denote any elongation outside syzygy and quadrature.² If Abu 'l Wefa had made a new discovery, we should have expected later Arabian astronomers to have alluded to it. But not one of them gives anything but interpretations of the lunar theory of Ptolemy, and in expressions very similar to those employed by Abu 'l Wefa. Attention was at once called to this fact, and Isaac Israeli of Toledo (about 1310) and Geber of Seville were quoted as examples,³ though it would, of course, have been quite possible for these two writers to have remained ignorant of whatever progress astronomy might have made in the school of Baghdad. But this objection does not apply to Nasir ed-din al Tusi, in whose review of the Almagest and Mem- orial of Astronomy the inequalities known to Ptolemy and no others, are described and credited to Ptolemy⁴; not to Mahamud al Jagmini (about 1300) , who wrote a compendium (mulachchas)

1. Journal des Savants, 1843, p. 701 ("Sur un traite arabe relatif a l' astro- nomie," Reprint, p. 47). This deviation does not represent the amount of the correction to the Moon's place as seen from the Earth, so that there is not any contradiction in Ptolemy's account.
2. Carra de Vaux, l. c. p. 466. The Arabs had no word for "octants." Nasir- ed-din on one occasion wants to mention them, and has to call them "the points midway between syzygy and quadrature." 3 Isaac Israeli repeatedly speaks of these inequalities discovered by Ptolemy, two of which are not found at conjunction and opposition. Liber Jesod, Olam seu Fundamentum Mundi auctore R. Isaac Israeli Hispano, section III. ch. 8 and sect. v. ch. 16, Part I. p. xxiv Part II. p. xxxi (Berlin, 1848 and 1846 ; this publication is not mentioned by Carra de Vaux).
3. C. de Vaux, "Les spheres celestes selon Nasir Eddin Attusi", Appendix to P. Tannery's Recherches sur l' astr. anc. p. 342, and Journ. asiat. 1892, p. 459 : "The third anomaly is that of the prosneusis ; it is called the equation of the proper motion" (i.e. of the motion on the epicycle).

of astronomy.¹ Nor can any objection be raised to Abu'l Faraj (Bar Hebraeus), and it would be impossible to explain more clearly than he does the effect of the prosneusis. He says: "The third inequality is the angle formed at the centre of the epicycle by two lines which are drawn, one from the centre of the univ- erse and the other from the point called the prosneusis, at the end of which is the apogee of the epicycle, at which commences the proper motion, and which is called the mean apogee. The apogee which is at the end of the line drawn from the centre of the uni- verse is called the apparent one. The point prosneusis is on the side of the perigee of the eccentric, 10 parts 17 minutes from the centre of the world² which is itself at the same distance from the centre of eccentric. The maximum value of this angle is 13 parts 9 minutes when the Moon is a crescent or ¾ gibbous, that is, near the hexagon or trigon with the Sun. In fact, when the epicycle is four or eight signs distant from the apogee of the eccentric, the Sun is itself two or four signs distant from [the centre of the epicycle] because it is half way between this centre and the apogee. In the tables, this inequality of the two apogees is called the first angle and is included in the motion of the centre."³ While this describes the construction of Ptolemy as clrealy as possible, at the same time the agreement of the account with that of Abu'l Wefa is perfect. Abu'l Faraj even (like Nasir ed- din) describes as a fourth inequality in longitude that caused by the motion along an orbit inclined to the ecliptic, so that he would not have neglected to describe the variation, if it had been found by an astronomer of Baghdad. We may add that the Jewish writer Abraham ben Chija (A. D. 1100), in his Sphaera Mundi, also describes the "aberration" of the apside of the epi- cycle, chiefly "in sexta et tertia parte mensis."⁴

1. Translated by Rudloff and Hochheim, Zeitschrift der Deutschen Morgen- land Ges. XLVII pp. 213–275. He describes (p. 249) how the line of apsides is directed to a point called "the corresponding point," and gives its position correctly. The inequality he calls the deviation.
2. Nasir ed-din gives 10° 9′.
3. Le livre de l' ascension, & c. T. II. PP. 29-30. Two codices add after the word prosneusis : "This is the point mohazat."
4. Sphaera Mundi (1546, ed. Schreckenfuchs), p. 75. Munster's commentary to the Hebrew text (p. 116) has "cum centrum est in sextili aut trino aspectu [id est, quando abest a sole duobus signis aut quatuor]"; • the words in brackets are not in the Hebrew original. The words "sixth" and "third" are unmistakable (shithith and shelishith). Apparently no one has hitherto thought of consulting Abraham ben Chija.

22 ASTRONOMY IN ANCIENT NATIONS Abu 'l Wefa and Ptolemy Therefore, Abu 'l Wefa did not know a single thing about the motion of the Moon which he had not borrowed from Ptolemy. But the prosneusis of Ptolemy is not the variation discovered by Tycho Brahe. The latter depends solely on the elongation of the Moon from the Sun, as it is = +39' .5 sin 2ε, while it is beyond the power of mortal man to express the effect of the prosneusis without the anomaly. Ptolemy's exprssion for all the inequalities in longitude assumed by him, when developed analytically, found to contain, in addition to terms represening the equation of the centre and the evection, the latter being +1°19'.5 sin (2ε—m), a very considerable term +17.8 sin 2ε [cos (2ε+m) +2 cos (2ε—m)], where ε is the elongation and m the mean anomaly.¹ Obviously this term has nothing in common with the varia- tion, except that it disappears in the syzygies and quadratures. Tycho Brahe did not hang his new term on to the unaltered lunar theory of Ptolemy, and by doing that we should in fact only spoil the latter and make its maximum error rise to more than a degree.² Owing to the insufficiency of the observations at his disposal, Ptolemy could only perceive that there was some out- standing inequality after allowing for the evection, only appearing outside the syzygies and quadratures, but he was neither able to find the law which governed the phenomenon, nor was he aware what a large quantity it represented; he could only tinker up his constructions a little, and in this he was most faithfully followed by the Arabs, who added nothing to what he had done and left it to the reviver of practical astronomy to discover the third lunar inequality. Al Fargani and others on Planets Passing to the five planets, we find that, generally speaking, very few attempts were made to improve the work of Ptolemy. But the Arabs were not content to consider the Ptolemaic system P. Tannery, Recherches, p. 213. Another expansion of Ptolemy's lunar inequalities in a series was given by Biot, Journal des Savants, 1843, p. 703 (Reprint, p. 49). P. Kempf, Untersuchungen uber die Ptolemaische Theorie der Mondbewegung, Berlin, 1878 (Inaug. Diss.), p. 37.

AL FARGANI AND OTHERS ON PLANETS 23 merely as a geometrical aid to computation; they required a real and physically true system of the world, and had therefore to assume solid crystal spheres after the manner of Aristotle. Above the Moon is the Alacir, the fifth essence, which is devoid of light- ness and heaviness, and is not perceptible to the human senses; of this substance the spheres and planets are formed.¹ Already in the book of Al Fargani we find the principle adopted which we have seen dates from the fifth century (Proklus) and which became universally accepted in the Middle Ages, that the greatest distance of a planet is equal to the smallest of the planet immediately above it, so that there are no empty spaces between the spheres.² The semidiameter of the Earth is by Al Fargani given as 3250 miles, which corresponds very nearly to Al Mamun's 56⅔ miles to a degree, if we put π = 3 1/7. Starting from Ptolemy's distances of

1. Al Battani, cap. 50 (p. 195).
2. Al Fargani, cap. 21 (ed. Golius, p. 80). Much later, Maurolycus in his Cosmographia (Venice, 1543, f. 20a) proves that Mercury and Venus must be below the Sun, by pointing out that there would otherwise be a large vacant space between the Sun and the Moon.
3. pp. 189-191.
4. So in Nallino's ed. (Milan, 1903 p. 121) the ed. of 1645 has 1176.
5. The ed. of 1645 has 12, 420; obviously an error, as the ratio of greatest to smallest distance is given as 37:23 for Saturn 7:5 (misprinted 7:2), or "quan- titas unius et duarum quintarum ad unum" (p. 199). Nallino's ed (Milan

1903. has 12,924. Abraham ben Chija has 12,400.

24 ASTRONOMY IN ANCIENT NATIONS smallest distances being in substantial agreement with the theory of Ptolemy. Al Battani also gives a similar set of figures, though with some slight differences. He does not mention peculiar treat- ment given by Ptolemy to the theory of Mercury. The above table gives the distance expressed in semidiameters of the Earth. Al Kusgi and diameters of planets Al Kūsgī, one of the astronomers of Ulug Begh, gives a list of the semidiameters of the "concavities" of the planetary sphe- res (i.e. the smallest distances of the spheres) expressed in para- sangs, the diameter of the Earth being 2,545 parasangs.¹ Expres- sed in semidiameters of the Earth, the figures turn out somewhat differrent from those given above, e.g. the smallest distance of the Sun being 1,452 and the greatest of Saturn 26.332, but he does not supply any means of making out how these figures were found. Before leaving this subject, we shall also give the diameters of the planets according to Al Fargani, as they became known in Europe at an early date and were quoted by Roger Bacon and others.² With trifling variations the same values are given by Al Battani, Abu 'l Faraj, and Abraham ben Chija. Apparent True Diameter Diameter (Earth's=1) Moon in apogee ... ... ... 31½' ... 1 : 3½ Mercury, mean dist. ... ... ¹⁄₁₅ of Sun's ... ¹⁄₂₈ Venus " " ... ... ¹⁄₁₀ " ... 1 : 3⅓ Sun " " ... ... 31½' ... 5½ Mars " " ... ... ¹⁄₂₀ of Sun's ... 1⅙ Jupiter " " ... ... ¹⁄₁₂ of Sun's ... 4½ + ¹⁄₁₆ Saturn " " ... ... ¹⁄₁₈ " ... 4¼ Al Kazwini, Abu'l Faraj and Al-Jagmini on Excentric Spheres of the Sun The system of the spheres is set forth in greatest detail in three treatises of later date, the cosmograpy of Zakarija ben Muhammed ben Mahmud al Kazwini (about 1275), the astronomy

1. Astronomica Shah Chelgii, pp. 95-97.
2. There are some slight differences between the figures given in the various editions (J.L.E. Dreyer has compared those of 1493, 1546, and 1669), but those given above agree with the cubic contents according to Al Fargani. The figures of .Kazwini seem to have been greatly corrupted.

ALKAZWINI—EXCENTRIC SPHERES 25 of Abu 'l Faraj, written in 1279, and that of Mahmud ibn Muhammed ibn Omar al Jagmini, whose date and nationality are equally uncertain, but who probably wrote in the thirteenth or fourteenth century. We find in these text-books an elaborate system of spheres designed to account for every particular of planetary motion, in perfect agreement with each other as to the general arrangement of the spheres, and offering nothing new as to lunar or planetary theory. The accompanying figures (taken from Jagmini) will illustrate the ideas better than a lengthy des- cription.¹ The Sun is a solid sperical body, fitting between two excentric spherical surfaces, which touch two other surfaces, in the Fig. 1.—Planetary motions and system of spheres.

1. The Sun. 2. Excentric sphere. 3. The surrounding spheres. 4. The complement of the surrounding sphere.
2. Centre of the world. 6. Centre of the excentric sphere. common centre of which the Earth is situated, and which between them enclose a space (or intersphere, as Abu 'l Faraj calls it), named by Jagmini al-mumaṭṭal, or the equably turning sphere, which has the same motion from west to east as the fixed stars, i.e. precession. The spheres of the three outer planets and Venus are arranged on the same plan, except that the place of the body of the Sun is taken hy the epicycle-sphere of each planet, to the inner surface of which the planet (a solid spherical body) is attached or (as Abu 'l Faraj says²) "fixed like a pearl on a ring, touching the
3. Al Kusgi gives very similar diagrams of the spheres of the Saturn, Mercury, and the Moon.
4. Precession is supposed to be included in this, "the first motion." The second (Continued on next page)

26 ASTRONOMY IN ANCIENT NATIONS surface in one point." The axis of the excentric sphere is incli- ned to that of the mumaṭṭal sphere, which causes the motion in latitude. The lunar system comprises an additional sphere outside the others, the centre of which coincides with the centre of the world, and which is called al-gauzahar, signifying the constella- Fig. 2.—Spheres of Mercury

1. Upper Apsis. 2. Lower Apsis. Upper Apsis of deferent sphere.
2. Deferent sphere. 4 Lower Apsis of deferent sphere. 6. Epicycle.
3. Mercury. 8. Surrounding complement. 9. Surrounded part of Mumaṭṭal sphere. 10. Mudīr sphere. 11. Centre of the world.
4. Centre of Mudīr. 13. Centre of deferent sphere. tion Draco, as this sphere provides for the revolution of the lunar nodes ("the head and tail of the dragon") round the zodiac. The inner one of the two concentric spherical surfaces, between which the excentric sphere lies, surrounds immediately the fire sphere of the Earth. The system of Mercury is more complicated, as a space had to be provided for the revolution of the centre of the (Continued from previous page) one is that of the concentric oblique intersphere (called the mail sphere or the sphæra deflectens) round the centre of the world, 11°9′ per day, by which amount the lunar apogee moves towards the west. The third motion is that of the excentric, carrying the centre of the epicycle 24°22′ towards the east. The fourth is the motion on the epicycle. Abu 'l Faraj, p. 27.

ALKAZWINI—EXCENTRIC SPHERES 27 excentric sphere. The figure shows the excentric sphere enclosed in a sphere, al-mūdir or the turning one, which allows the upper apsis or apogee of the excentric or deferent sphere (3 in the figure) to move right round the outer surface of the mūdir. The inner surface of the mumaṭṭal sphere immediately surrounds the gauzahar sphere of the Moon. It was a necessary consequence of the large solar parallax of 3' accepted by Ptolemy, that Mercury and Venus must be very near the Earth, since they are assumed to be nearer than the Sun. Thus Abraham ben Chija says that the shadow of the Earth ex- tends beyond the orbit of Mercury but does not reach that of Venus.¹ Ptolemy never mentions the parallaxes of Mercury and Venus, as to which nothing was known, though they ought, of course, to be greater than 3'. But on the assumption that the smallest distance of Mercury is equal to the distance of the Moon at apogee, the parallax of Mercury ought to rise to 54', which must have been felt to be too large a quantity, though it does not seem to have struck Al Battani as anything surprising, per- haps because Mercury cannot be seen when in inferior conjunc- tion. It may have been this necessarily large parallax of Mercury, which induced Ibn Jūnis (without any explanation) to reduce the solar parallax from 3 to 2', or rather to 1' 57".² Geber³ blames Ptolemy for having said that the parallaxes of the planets are insensible, and remarks that he ought, therefore, logically to have placed Venus and Mercury above the Sun. He takes great pains to show that Venus may be exactly on the line joining the Sun and the Earth. Indeed, Geber neglects no opportunity of criticis- ing Ptolemy's methods of finding the elements of the orbits,⁴ and he is generally very unjust to him but he does not venture to

1. Sphaera mundi, ed. Osw. Schreckenfuchs, Basle, 1546 pp. 84, 86.
2. Unpublished chapters of Ibn Jūnis, reviewed by Delambre, Hist. de l'astr. du Moyen Age. p. 101.
3. Instrumentum primi mobilis a P. Apiano. Accedunt ijs Gebri filii Affla Hispalensis...libri IX. de astronomia. Norimbergæ, 1534, fol. (Introd. p. 3 and lib. VII. p. 104).
4. See the long indictment on pp. 2-3 of his introduction. He blames Ptolemy among other things for assuming that the centre of the deferent is half-way between the centres of the zodiac and of the equant, while he himself deduces this from the movements.

28 ASTRONOMY IN ANCIENT NATIONS substitute any other system and does not object to the general principles of the Ptolemaic system.¹ Three great names : Ibn Badja, Ibn Tofeil (Abubacer) and Abu Welid (Averroes) Geber's attempts to pick holes in the work of Ptolemy were, perhaps, not unconnected with the rapid rise of Aristotelean philo- sophy in Spain in the twelfth century, which, though not destined to last long, nevertheless exercised a considerable influence on the spread of knowledge of Aristotle in the Christian world, while it cast a halo round the Caliphate of Cordova, which at that time, under the enlightened rule of the Almohades, seemed to have reestablished the glory of the best days of the Moslem world. Three names are specially associated with this movement : (i) Abu Bekr Muhammed Ibn Jahya al Sayeg, called Ibn Badja (of Saragossa, died 1139), known as Avempace among the Scholastics ; (ii) his pupil Muhammed ben Abdelmelik Ibn Tofeil (of Granada, died 1185-1186), called Abubacer by the Scholastics ; (iii) and finally the greatest philosopher of Islam, Ibn Rosd Abu Welid, known as Averroes (1126-1198). In studying Aristotle they laid special stress on his scientific works, and did not, like their Christian successors, think of little but dialectics. The acceptance of the system of homocentric spheres or some modification of it must, therefore, have seemed a necessity to the Arabian philosophers and this, of course, led them to reject the theory of epicycles.. The little we know of the opinions of Ibn Badja on this subject is found in the famous work The Guide of the Perplexed of the great Jewish scholar Moses ben Maimun of Cordova, better known as Maimonides, who tells us that he had his information from a pupil of Ibn Badja. Like Geber (with whose son he had been familiar), Maimonides doubted that Mercury and Venus were nearer than

1. Copernicus possessed a copy of Geber's book; which is now in the Univer- sity library at Upsala. On the title page, after the author's name, he has written: "Egregii, Calumniatoris Ptolemaei," while a number of marginal notes show that he has read the book carefully. Curtze, Mittheilungen des Coppernicus Vereins, I, p. 37.

THREE GREAT NAMES 29 the Sun, though he would not venture to say how they actually moved.¹ But what is more important, he declared the motion of a planet on an epicycle to be contrary to physical principles, be- cause there are only three motions possible in this world : around its centre, or towards it, or away from it; while he also main- tained that according to Aristotle circular motion can only take place round a real, central body.² Though Aristotle in reality did not object to epicyclic motion with a mathematical point as centre, for the simple reason that it had not been proposed when he wrote, while, as we have seen, his moving principle had noth- ing to do with the centre of motion, it is easy to see that Ibn Badja's real difficulty was the same which afterwards produced so many obstacles to the advance of science in Europe : whatever could not be found in Aristotle's book must be unworthy of notice. According to Maimonides (who, however, makes the reservation that he had not heard it from disciples), Ibn Badja constructed a system of his own, in which he only admitted excentric circles but no epicycles. We are not given any particulars as to this system but there can hardly be any doubt that its author confined himself to generalities and did not attempt to represent phenomena like the lunar inequalities by it. Maimonides remarks that there is nothing gained by Ibn Badja's reform, since the excentric hypothe- sis is as objectionable as the epicyclic one, as it also supposes motion round an imaginary point outside the centre of the Earth. The centre of the excentric, on which the Sun is supposed to move, is outside the convexity of the lunar sphere and inside the concavity of that of Saturn's excentric is between the spheres of Mars and Jupiter. He adds that the revolution of a number of concentric spheres around a common axis is conceivable, but not the revolution round different axes inclined to each other, as the spheres would disturb each other unless there are other spherical bodies between them. This attempt to revive and modify the sys- tem of (movable ?) excentrics did, therefore, not mend matters.³

1. Rabbi Mosis Majemonidis Liber . . . Doctor Perplexorum. Basileæ, 1629, Pars II. cap. IX.
2. Ibid., Pars II. cap. XXIV.
3. Maimonides also remarks (in the same chapter) that the supposed inclina- tions of Mercury and Venus in the Ptolemaic system are difficult or impossible to comprehend or imagine as really existing. Therefore, if what (Continued on next page)

30 ASTRONCMY IN ANCIENT NATIONS Ibn Tofeil Ibn Tofeil, the second of the three Moslem philosophers of Spain, vizier and physician at the court of Jusuf ben Abd el Mumin of Morocco, seems to have walked in the footsteps of his master ; but the only extant work of his, a kind of religious mystic romance about the emancipation of a soul from the trammels of this material world, does not give any clue to his ideas as to the planetary system. But Averroes, who also objected to the excentrics and epicycles says in his commentary to Aristo- tle's Metaphysics that Ibn Tofeil possessed on this subject excellent theories ², and Ibn Tofeil's pupil, the astronomer Al Betrugi, in the introduction to his theory of the planets, says of him : "You know that the illustrious judge Abu Bekar Ibn Tofeil told us that he had found an astronomical system and principles of the various movements different from those laid down by Ptolemy and without admitting either excentrics or epicycles, and with this system all the motions are represented without error." Ibn Tofeil was therefore probably the real author of the fairly elabo- rate system, which his pupil worked out and handed down to us in a work on the planets, which was translated into Hebrew in the following century and from that again into Latin, and published in 1531 ³. The object of this system was to explain the constitution of the universe as it really is, and not merely to represent the motions of the planets geometrically, so as to be able to foretell their places in the heavens at any time ; and the author (be he Ibn Tofeil or Al Betrugi alias Alpetragius) specially disclaims any intention of testing the theory by comparing it with observations (Continued from previous page) Aristotle says is true, there is neither epicycle nor excentric, and every- thing turns round the centre of the Earth. 2. Munk ; Melanges de philosophie juive et arabe. Paris, 1859, p. 412. 3. Alpetragii Arabi Planetarum theorica phisicis rationibus probata, nuperrime latinis litteris mandata a Calo Calonymos, Hebreo Neapolitano, Venice 1531, 28 ff. folio (published with Sacrobosco's Sphaera). A translation by the famous Micheal Scot has never been printed, but is still extant in Paris (Munk, Melanges, p. 519). The principle of the system is described by Isaac Israeli, who, however, does not mention the author's name (Liber Jesod Olam, II. 9. Part I. p. XI

or of accounting for minor details of the motions.¹ The leading idea is that of the homocentric spheres, each star being attached to a sphere, and the motive power is the ninth sphere, the sphere outside that of the fixed stars. The Spanish philosopher ought, therefore, to have been content with the system of Eudoxus or its modification by Aristotle (whom he never mentions by name, but only as "the sage"), but unfortunately he became possessed with the notion that the prime mover must everywhere produce only a motion from east to west, and he had, therefore, to reject the independent motion of the planets from west to east, and revert to the old Ionian idea that the seven planets merely per- form the daily revolution with a speed slightly slower than that of the fixed stars. The true speed of the primum mobile is a little faster than this ; the eighth sphere performs a revolution in a slightly longer period (24 hours), and the effect of the prime mover is gradually weakened more and more, with increasing distance, until we find the sphere of the Moon, being furthest from the prime mover, taking nearly twenty-five hours to complete a revolution. This was the old primitive Ionian idea, but Al Betrugi (or his teacher) saw that this was not sufficient, as not only is the pole of the ecliptic different from that of the equator, which prevents the planets from moving in closed orbits, but the planets do not even keep at the same distance from the pole of the ecliptic but have each their motion in latitude, as well as variable velocity in longitude ; and all this had yet to be accounted for. The ninth sphere has but one motion, but the eighth has two, that in longitude (precession) and another which is caused by the pole of the ecliptic describing a small circle round a mean position, thereby producing the supposed oscillation or trepidation of the equinoxes.² Similarly, the pole of each planet describes a small circle round a mean position (i. e. the pole of the ecliptic), thereby producing inequalities in longitude and motion in latitude.³ Whenever the actual orbit-pole of a planet is on the parallel of the mean pole, it is obvious that the planet will perform its daily revolution with its mean velocity, while the velocity is increased or lessened when the actual pole is respectively at its minimum or maximum distance from the pole of the heavens

1. Fol. 8 b. 2. Fol. 9 b.
2. Fol. 14 b;. sq.

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.