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

श्रीब्रह्मगुप्ताचार्य विरचितः विभिन्नपाठान्तरसहितो ब्राह्मस्फुटसिद्धान्तः

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CHAPTER I Astronomy in Ancient Nations Brahmagupta's great works like the Khaṇḍakhādyaka and the Brāhma-sphuṭa-siddhānta took astronomy to Arabs through whom it spread to many countries of Europe. Al Beruni records this testimony in his great book on India.. It is doubtful that astronomy had its birth in Greece and China. From remote ages China, India, Greece, Arabia and Egypt developed the entire system in close cooperation. This knowledge must have spread from their common cradle home where man for the first time developed his culture and civilisation. In this chapter we pro- pose to give a review of astronomy as developed in many of these ancient lands, especially Arabia, people of which land came in close contacts with India much before any recorded time. Dawn of Astronomy The earliest man must have been the primitive astro- nomer. The striking spectacles presented to him by the varied appearances of a sky covered with thousands of twinkling and non-twinkling objects of different degrees of brightness, apparently revolving round the Earth, and the daily changing phases of the Moon must have raised strange feelings of the most primitive man also. Then he must have in course of time observed the bright morning and evening stars, and at a considerably late stage the comets and shooting stars and then on occasions eclipses of the Sun and the Moon. These phenomena not only raised feelings of admiration, but in different sections of human society often feelings of superstitious alarm. By and by stars became guides for the traveller by land and sea. In the midst of these observations, one discovered various cycles : cycle of day and night, cycle of seasons and cycle of other details. Then there was a striking observation of the tides in a sea changing with the phases of the Moon.

2 ASTRONOMY IN ANCIENT NATIONS Earliest Discoveries We shall briefly sketch out the order of astronomical discoveries. The first phenomena to be noted must have been the regularly recurring dawn (for this one may refer to the Uṣā Sukta of the Ṛgveda), the sunrise and sunset (which led to prātaḥ and sāyaṁ, i.e. morning and evening prayers of the Vedic times), daylight, twilight and night concerning which we have numerous Vedic hymns. Next it led to the measurement of a day (which was of a short duration in winters and of a long duration in summers). The Vedic Aryans also discovered the variations in the duration of the day along different latitudes, and the time of sunrise in places of different longitudes. In fact the idea of longitudes and latitudes came suffi- ciently afterwards. Man discovered month as related to the varia- tion of light with the Moon's phases. In temperate regions, where probably the first astronomical observations were systematically made, the changing length of the day or the direction of the Sun at rising or setting or the lengths of shadows cast at midday, would show that the Sun's daily path in the sky altered through- out the year, a time interval which was already marked by the changing vegetation. According to Sir W.C. Dampier, "attempts were made to determine the number of months in the cycle of the seasons in Babylonia about 4000 B. C. and in the China soon after. About 2000 B. C. the Babylonian year settled down to one of 360 days or twelve months, the necessary adjustments being made from time to time by the interposition of extra- months." In India, this concept is of even much earlier origin. The old inspired sages like Dīrghatamas discovered for the observing man the Vedic Era and intercalation. I have described this discovery in a special chapter on the subject in my book the Founders of Sciences in Ancient India and a reference may be made to the Āsya Vāmasya Suktam of the Ṛgveda. It is impossible to assign an age to these old traditions. Round the Yajña, developed the science of astronomy, mathematics, anatomy and medicine in this ancient land of ours, which in fact was the common heritage of a large number of people of the modern world. One might also say that a considerable period might have well elapsed before it was noticed that at a particular season of

ZODIAC 3 the year, the same stars are seen at corresponding hours of the night. Of course this circumstance was less conspicuous than the regular variation of the Sun's altitude in the sky as the year progresses. It is the surmise that the striking naked-eye cluster, the Pleiades, must have been one of the earliest noted star- groups, and it became the first star-group for providing the first fairly close determination of the length of the year as approxim- ately 365 days. The rising of this cluster in the evening was a mark of the coming winter to primitive man ; and the husband- man judged the time of reaping by its rising, and of ploughing by its setting in very ancient times ; Sirius, Arcturus, the Hyades and Orion were similarly equally useful to him. The passages in the Taittirīya Saṁhitā and in the Śatapatha Brāhmaṇa clearly indicate the confusion once created by following the concept of lunar months without further adjustments : "Now the seasons were desirous to have a share in the sacrifice among the gods and said, 'Let us share in the sacri- fice. Do not exclude us from the sacrifice ! Let us have a share in the sacrifice !' The gods, however, did not approve of this. The gods, not approving, the seasons went to the Asuras, the malignant, spiteful enemies of the gods. Those (Asuras) then throve in such a manner that they (the gods) heard of it, for even while the foremost (of the Asuras) were still ploughing and sowing, those behind them were already engaged in reaping and threshing : indeed even without tilling, the plants ripened forthwith for them. (ŚBr. I.6.1.1-3) The Zodiac It is difficult to say how much time it must have taken, but in fact, it was eventually noted that the Sun and Moon travel over very similar paths among the stars during their circuit of the sky. This led to the formation of the Zodiac and its constellations, the centre of this zone, a belt about 16° broad, being the annual path of the Sun or Ecliptic. The divi- sion into twelve parts, each corresponding to a month of the Sun's movement, was made ; and their connection with the solar course during the year was found by observations of heliacal risings or settings. These were the times of the year when certain bright stars would first be seen to rise before the Sun, or when

4 ASTRONOMY IN ANCIENT NATIONS they were last seen to set after sunset. In the case of Sirius, the brightest fixed star, these would happen when the Sun was about ten degrees below the horizon. For the less bright stars the angle would be a larger one. It must have been almost simultaneously observed that the Moon in going like the Sun round the heavens always in the same direction from west to east (i. e., opposite to the diurnal motion which she shares with the other bodies), kept in general to the same track in the sky. After a time, however, it must have been noted by careful observers that this path was not con- stant, but deviated from the centre line of the Zodiac, getting away from that line up to a maximum deviation on either side but slowly returning to it. In the course of a number of years, it must have become evident that the Moon's path among the stars does not lie always in the same line on the celestial sphere, but in a zone or band about twenty moon breadths (10°) wide, occu- pying the middle of the Zodiacal zone itself. Among the bright stars Mercury, Venus, Mars, Jupiter and Saturn (the first two of which are never seen very far from the Sun in the sky) soon must have been noted to be moving in the Zodiac with varying periods. The English name planet is de- rived from Greek planetes, meaning a wanderer, since the planets change their positions among the Zodiacal stars. There is a word Sṭr which in the Ṛgveda always occurs in the instrumental plural, Sṭrbhiḥ. The English word star is de- rived from this word. Parāśara and Gṛtsamada, I have shown elsewhere, were the first amongst the great observers, inspired by the Ṛgvedic hymns, and Vāmadeva identified Bṛhaspati or the Jupiter planet and Vena Bhārgava discovered tht planet Venus which still bears the name of its discoverer. Constellations Long before the Zodiacal belt was divided into "signs" (700 B. C.), a number of asterisms, or the configuration of stars in the sky had been arranged, the brighter stars of these configurations, thus identified, proved very useful in indicating the seasons of the year by the times of their rising or setting, and also in locating the positions on the celestial vault of such moving objects as planets, comets and shooting stars and in helping the traveller by land or sea to determine direction.

CONSTELLATIONS 5 These named constellations date back to very early period. In India, Gārgya is the name of an astronomer who is associated with a hymn of the Atharvaveda which for the first time enume- rates constellations. In many of these constellations, the stars form a well marked group, clearly separated from other groups, and the names given to these formations are supposed to have been suggested by a resemblance to the shapes of certain familiar objects. Of course, the resemblance is usually very slight, and depended merely on a fancy. It is remarkable that different countries devloped almost similar notions regarding these constellations. The late Dr. A. C. D. Crommelin considered that there is a reason to believe that the stars may have been grouped to some extent by the Egyptians as early as 4000 B. C., and he remarked on their use of the then Pole Star for orienting the Great pyramid, Again, Chinese are said to have mapped out the sky into many divisions of stars by 2500 B. C., if one can rely on their records. The idea of constellations takes us to a date much earlier than 2500 B. C. even. In total forty-eight have come down from extremely ancient times, but these do not cover the entire extent of the sky. The part not occupied by any of them evidently did not rise above the horizon where the early astronomers to whom we owe their naming lived; and the stars concerned were there- fore not included in their constellation schemes. The centre of this part (near the bright star Achernar) must have been near the South Pole of the heavens of the time, and its angular radius from the Pole gives us roughly the latitude of their homes. The date appears to have been about 2800 B. C., when, owing to the precession of the equinoxes, the South celestial pole was in the position indicated. The latitude seems to have been about 38° North. These are the findings of E. W. Maunder (Astronomy without a Telescope, p.5, 1902); but from the same considerations Dr. Crommelin assigns a latitude of 36° and a date 2460 B.C., and Proctor 2200 B. C. Maunder also suggested that the presence of the Lion and Bear among the stellar configurations and the absence of Elephant, Tiger, Camel and Crocodile seem to ex- clude India towards the East and the countries towards the West, the latitude and the longitude indicated being those of Asia Minor or Armenia. The suggestion that the blank area in

6 ASTRONOMY IN ANCIENT NATIONS the sky referred to gave an approximate date for the formation of the constellations appears to have beeen first put forward in 1807 by Carl Schwartz, for some time Swedish Consul at Baku.¹ Indo-Greek Contacts It is highly improbable that before Alexander, there had been absolutly no contacts between India and the distant nations. Even in pre-Alexandrian era, there had been such migration is clearly evinced by the philological and mythological studies. But we do not possess historic record of it. The conquests of Alexander the Great made the Greeks acquainted with the Eastern world, which had up to that time been visited probably by very few Europeans, and it likewise spread Greek culture to all the countries which the victorious Macedonian had been able to reach. The Indian province of his Empire became independent soon after Alxander's death, and though the spread of Buddhism in the third century B. C. checked the progress of Hellenism in Northern India, the rise of the Greek kingdom of Bactria and its gradual extension south and east continued for a long time to keep alive the connection between India and the West. Not only (as has beeen asserted) the Greek and Indian drama and architecture have been strongly influenced by Hellenistic and Indian contacts, it is beyond a doubt that the entire astronomy of the two great nations is the offspring of these mutual contacts. In earliest times astronomy had only been cultivated in India and in no other country. Some idea had been acquired during those days of the periods of the Sun and Moon and the planet Venus and Bṛhaspati (Jupiter), which were used for chrono- logical purposes, the lunar motions being specially connected with the proper times for sacrificial acts. The Vedic era was discovered during this period by Viśvāmitra, and Gārgya enumerated the Nakṣatras. Lagadha composed his Vedāṅga Jyotiṣa, which is the first book on astronomy written in human literature. India dev- eloped her geometry in connection with the construction of sacri- ficial altars, and its account is found in the Śulba Sūtras of Baudhāyana, Āśvalāyana and Kātyāyana. Āryabhaṭa laid the foundations of algebra. One might still say that there is no sign of

1. See, Peter Doig : A concise History of Astronomy, London, 1950.

INDO-GREEK CONTACTS 7 accurate knowledge of the planetary motions earlier than about the century of the Christian era. From thenceforth astronomy, hitherto confined to rituals, appears as a science, treated in the course of the next thousand years in a series of text-books, the Siddhāntas,¹ the contents of which, though supposed to be derived from divine sources are strongly influenced by Greek authors. Prior to the Greek influence, we had ceremonies like dvādaśāha (lasting for twelve days), ṣaḍaha (lasting for six days), tryaha (lasting for three days) besides darśa-pūrṇamāsa ceremo- nies connected with the New Moon and Full Moon. But the week of seven days (Saptāha) was unknown in India. This concept of week and the dedication of each day to the deity of one of the seven planets, now appears for the first time. It is difficult to say whether names of the planets were borrowed by Greeks from India or vice versa, but they became common, e.g., Aśvajit or Asphudit (Aphrodite), Dyugatiḥ or dyaus or Jīva (Zeus), Heli (Helios), &c., while the zodiacal signs have superseded the earlier but totally different twelve star-groups connected with the Sun's motion, and proclaim their origin by their names: Kriya, Tāvuri, Jituma, Karkin, Leya, Pāthena, Juka, Kaur- pya, Taukshika, Ākokera, Hridroga, Ittha, corresponding to Κριός, Ταῦρος, Δίδυμος, Καρκίνος, Λέων, Παρθένος, Ζυγόν, Σκορπίος, Τοξότης, Αἰγόκερως, Ὑδροχόος, Ἰχθύς². A great many other Greek terms connected with geometry, astronomy and astrology have also been transferred from Sans- krit works to Greek and vice versa. This conclusively shows the mutual influence on astronomy. Indian authors never failed to acknowledge the ideas they borrowed from Greeks, e.g. Varāha- mihira quotes the Yavanas or peoples of the west as authorities for some of the scientific statements he makes. The name of the Romaka Siddhānta (which is at least as old as A.D. 400) also points in an unmistakable manner to its origin in one of the pro- vinces of the Roman Empire.

1. The Romaka or Pauliśa Siddhānta (before 400 A. D.), See Varāhamihira's the Pañcasiddhāntikā (about 570 A.D., Varāhamihira died in 587 A.D.). The original Sūrya-siddhānta was prior to Varāhamihira, the modern edition is perhaps of the 13th century. See J. Burgess "Notes on Hindu Astronomy" J. R. A. S., October 1893, p. 742.

8 ASTRONOMY IN ANCIENT NATIONS Earth as a Sphere The astronomers of the Siddhāntas taught that the Earth is a sphere, unsupported in space, and they reject the ancient my- thological notion that it is supported by some animal like śeṣanā- ga (serpent), kacchapa (tortoise), or diggajas (elephants) which in turn rest on another, and so on, until the support of the last one after all has to be left unexplained. Bhāskara II, about A.D. 1150, who comments on the absurdity of this, also rejects the idea that the Earth is perpetually falling, since it would fall faster than an arrow shot upwards, on account of being heavier, so that an arrow could never again reach the Earth.¹ Round the Earth the planets are moving, all with the same linear velocity. The diameter of the Earth is 1600 yojanas, the distance of the Moon is 51,570 yojanas (or 64.5 times the radius of the Earth, nearly equal to Ptolemy's greatest distance, 64⅙), while the dis- tances of the other planets result from the assumption of equal velocities.² The equation of centre of the planets is found by an epicycle and to this arrangment the Hindus add one of their own invention, by assuming that the epi- cycle had a variable circumference, greatest when the planet is at apogee or perigee and least at 90° from these, when the equa- tion reaches its maximum. This contrivance of an oval epicycle was by some astronomers applied to all the planets, by others (Brahmagupta and Bhāskara) only to Mars and Venus, by others it was altogether rejected.³ Why they complicated the calculation in this way is not clear. Āryabhaṭa I of Kusumapura or Pāṭali- putra, born A.D. 476, made another deviation from the Alexand- rian doctrines, as appears in the Brahma-sphuṭa-siddhānta of Brahmagupta, wherein he quotes the following from Āryabhaṭa: "The sphere of the stars is stationary, and the Earth, making a revolution, produces the daily rising and setting of stars and planets." Brahmagupta rejects this idea, saying: "If the Earth moves a minute in a prāṇa, then whence and what route does it proceed? If it revolves, why do not lofty objects fall?" But his commentator Caturveda Pṛthudaka Svāmi replies: "Āryabhaṭa's

1. As. Res. XII. p. 229 (Essays, II, p. 394).
2. The distances are proportional to the orbital periods of revolution, but for Mercury and Venus to the periods in the epicycles.
3. For further details see As. Res. II. p. 251 (Davis) and XII. p. 236 (Colebrooke, also Essays, II. p. 401).

EARTH ROTATION 9 opinion appears nevertheless satisfactory, since planets cannot have two motions at once : and the objection, that lofty things would fall, is contradicted; for every way the under part of the Earth is also the upper; since wherever the spectator stands on the Earth's surface, even that spot is the uppermost spot.¹ Earth rotation by a current of aerial fluid It is very interesting to see the theory once advocated by Herakleides of Pontus transplanted on Indian soil, especially when we remember that Seleukus, the Babylonian, had adopted that theory. From Babylon the theory might easily find its way to India, though it is of course equally possible that Āryabhaṭa, quite independently of his Greek precursors, hit on the same idea. He appears to have accounted for the Earth's rotation by a wind or current of aerial fluid, the extent of which, according to the orbit assigned to it by him, corresponds to an elevation of little more than a hundred miles (114) from the surface of the Earth, or fifteen yojana's while he put the diameter of the Earth equal to 1050 yojanas (of 7.6. miles each²). This was in accordance with the general opinion of the Indians, that the planets are carried along their orbits by mighty winds with the same velo- city and parallel to the ecliptic (while one great vortex carries all stars round the Earth in twenty-four hours, but that the planets are deflected from these courses by certain invisible powers having hands and reins, with which they draw the planets out of their uniform progress. The power at the apogee, for. instance constantly attracts the planet towards itself, alternately with the right and left hand (like Lachesis in Plato's Republic), while the deity at the node diverts the planet from the ecliptic first to one side and then to the other. And lastly the deity at the con- junction causes the planet to move with variable velocity and to become occasionally stationary and even retrograde. This is gravely set forth in the Sūrya-śiddhānta, and even Bhāskara gives the theory in his notes, though he omits it from his text. Similarly Brahmagupta, although he gives the theory of eclipses, affirms the existence of an eighth planet, Rāhu, which is the immediate cause of eclipses; and he blames Varāhamihira,

1. Asiat. Res, XII, p. 227; Colebrooke's Essays, II, p. 392.
2. Colebrooke, Notes and Illustrations to the Algebra of Brahmagupta, p. xxxviii. Essays, II. p. 467.

10 ASTRONOMY IN ANCIENT NATIONS Āryabhaṭa and others for rejecting this orthodox explanation of the phenomenon.¹ Indian astronomy some times appears to be a curious mix- ture of old fantastic ideas and sober geometrical methods of calculation. But it is wrong to presume that these geometric calculations were derived from froeign contacts. Indians have always been fond of geometry (from the earliest times of the Vedic rituals), and they from the very beginning realised the impor- tance of applying geometry to astronomy, Side by side we find Greek contacts also. As remarked by Colebrooke, the absence of the most characteristic parts of Ptolemy's system, the equant and the details of the theories of the Moon and Mercury seems to indicate that Greek planetary theory must have been introduced in India between the times of Hipparchus and Ptole- my; and with the exception of the epicycle from the circular form, the Hindus did not modify the theory or perfect it in any way. The precession of the equinoxes they held to consist in a liberation within the limits of 27° (Āryabhaṭa says 24°) east and west of its mean position, but they came much nearer to the truth than Ptolemy did as regards the annual amount, as they supposed the space travelled over in a century to be 1½°. Contacts with Arabs Notwithstanding some isolation of India from Europe during the Middle Ages, her astronomy was destined to exercise an indirect influence on the progress of astronomy. Through the conquest of Persia in the seventh century, the Arabs, like the Greeks a thousand years earlier, came in contact with India, from whence physicians and astro- logers found their way to the court of the Caliph already before the reign of Harun al Rashid. We possess a detailed account of the manner in which the Indian astronomy was introduced at Baghdad, from the pen of the astronomer Ibn al Adami (who died before 920), confirmed by the celebrated memoir on India by Al Beruni, written in 1031². In the year 156 of the Hijra (A. D. 773), there appeared before the Caliph Al Mansur a man who had come from India; he was skilled in

1. Asiat. Res. XII, pp. 233, 241; Essays, II, pp. 398, 407.
2. Hankel, Zur Geschichte der Mathematik im Alterthum and Mittelalter, Leipzig, 1874, p. 229; Cantor, Gesch. d. Math. I, p. 656.

ARABS AND GREEKS 11 the calculus of the stars known as the Sindhind (i. e. Siddhānta), and possessed methods for solving equations founded on the kardagas (i. e. kramajyā, sines) calculated for every half degree, also methods for computing eclipses and other things. Al Mansur ordered the book in which all this was contained to be translated into Arabic, and that a work should be prepared from it which might serve as a foundation for computing the motions of the planets. This was accordingly done by Muham- med ben Ibrahim Al Fazari, whose works the Arabs call the great Sindhind, and from it an abstract was afterwards made for Al Mamun by Abu Giafar Muhammed ibn Musa al Kwarizmi, who made use of it to prepare his tables, which obtained great renown in the lands of Islam. But when Al Mamun became Caliph, he promoted these noble studies and called in the most learned men in order to examine the Almagest and make instru- ments for new observations. Arabs and Greeks The account of which the above is an abstract shows us clearly the origin of the study of astronomy and mathematics under the Abbasid Caliphs. But though the first impulse came from India the further development of Arabian science was to a considerable extent founded on that of Greece and Alexandria. It was through the court physicians from the flourishing medical school kept up by Nestorian Christians of Khusistan that a knowledge of Greek Philosophy and science was first spread among the subjects of the Caliphs; and by degrees the works of Aristotle, Archimedes, Euclid, Apollonius, Ptolemy, and other mathematicians were translated into Arabic. Fresh translations of Ptolemy were made from time to time in the various king- doms into which the vast empire of the Caliph was soon split up,¹ and a thorough knowledge of Ptolemaic astronomy was thus spread from the Indus to the Ebro. There were several special inducements for Muhamedans to pay attention to astronomy, such as the necessity of determining the direction in which the

1. The earliest is probably that of Al Haggag ben Jusuf ben Matar early in the ninth century. See Suter, Die Mathematiker und Astronomen der Araber und ihre Werke, Leipzig, 1900 (p. 9), which valuable bibliographical summary has been follwed by J. L. E. Dreyer as regards names and dates (J. L. E. Dreyer’s A History of Astronomy, 1953; we have reproduced this account from his chapter XI.)

12 ASTRONOMY IN ANCIENT NATIONS faithful had to turn during prayers, also the importance of the lunar motions for the calendar, and the respect in which judicial astrology was held all over the East. The Caliph Al Mamun, son of Harun Al Rashid (813-833) is the first great patron of science, although the Omayyad Caliphs had much earlier an observatory near Damascus, and the Jew Mashallah (who died about 815) had already before the reign of Al Mamun won a name as an observer and astrologer. But the Damascus observa- tory became quite eclipsed by that erected at Baghdad in 829 where continuous observations were made and tables of the pla- netary motions constructed while an important attempt was made to determine the size of the Earth. Among the astronomers of Al Mamun and his successors one of the greatest was Ahmed ben Muhammed Al Fargani (afterwards known in the West as Alfra- ganus), whose Elements of Astronomy were translated into Latin in the twelfth century and contributed greatly to the revival of science in Europe.¹ Tabit ben Korra (826-901) was a most prolific writer and translator, but is chiefly known in the history of astronomy as a supporter of the erroneous idea of the oscilla- tory motion of the equinoxes. A younger contemporary of his, Muhammed Al Battani (died 929), was the most renowned of all the Arabian astronomers and became known in the West in the twelfth century (under the name of Albategnius) by the trans- lation of the introduction to his tables.² Already in his time the power of the Caliphs had commenced to decline, and they soon lost all temporal power. The study of astronomy was, however, not influenced by this loss of patronage, as the Persian family of the Buyids, who in 946 obtained possession of the post of Amir-al-Omara (corresponding to the Frankish Major Domus) took over the role of patrons of science, so long and so honourably carried on by the Abbasid Caliphs. Sharaf al Daula built in 988 a new observatory in the garden of his palace, and among the astronomers who worked there was Muhammed Abu 'l Wefa al Buzjani (959-998), who wrote an Almagest in order to

1. First printed at Ferrara in 1493. See the edition of Golius, Amsterdam,
3. Translated by Plato of Tivoli. First Printed in 1537 after the book of Alfargani. Dreyer has used the edition of Bologna, 1645, and a new edition which is now being published by C. A. Nallino, of which the Arabic and a Latin translation of the text have already appeared (Pubbl. d. R. Osserva- torio di Brera in Milano, No. 40, 1899-1903).

WESTERN COUNTRIES UNDER ISLAM 13 make the contents of Ptolemy's work accessible to the less learned. In the nineteenth century this book gave rise to a long controversy, which we shall presently consider somewhat in detail. Western Countries under Islam In the eleventh and twelfth centuries we do not find any names of conspicuous astronomers in Muhammedan Asia. But the western countries under Islam had in the meantime become ready to do their share of the work of keeping the mathematical sciences alive. In the Fatimite kingdom of Egypt Ali ben Abi Said Abderrahman ben Ahmed ben Junis, generally called Ibn Junis (died 1009), was distinguished both as an astronomer and a poet. At Cairo a liberally equipped observatory enabled him to verify the planetary theories which had once been developed in the neighbouring Alexandria, and in token of his gratitude to the reigning sovereign, Al Hakim, he named his work the Hakemite Tables.¹ We have to pass to the farthest west to find the next astronomer of mark in the person of Ibrahim Abu Ishak, known as Al Zarkali (in Europe afterwards called Arzachel). He was a native of Cordova, lived about 1029-1087, and edited planetary tables called the Toledo Tables.² In the following century we find two celebrated astronomers of Seville, Gabir ben Aflah, known as Geber (died 1145, often mistaken for the great alchemist, Gabir ben Haijan, in the eighth century),³ and Nur ed-din al Betrugi (Alpetragius), both of whom raised objections to the planetary theories of Ptolemy, though they failed to produce anything better of their own. Spanish astronomy continued to flourish for a while, although the power of the Arabs in the Peninsula was rapidly declining, and it produced in the thirteenth century a very remarkable man, who, although a Christian king, must be included in this account of Arabian astronomy,

1. Caussin has published an extract in vol. vii of the Notices et Extraits des manuscrits (Le livre de la grande table Hakemite). Other Chapters, trans- lated by the elder Sedillot but never published, are reviewed by Delambre, Hist. de l'astr. du Moyen Age, p. 95 sqq.
2. Never published. Delambre, l. c. p 176, and Steinschneider, Etudes sur Zarkali, Bullettino Boncompagni T. xx. p. 1. 3 The word algebra has also sometimes erroneously been connected with his name.

14 ASTRONOMY IN ANCIENT NATIONS as he owed all he knew about the science to the example and the teaching of Muhammedans and Jews. King Alfonso X, of Castille named el Sabio (1252-1284), followed the example of the Caliphs and called astronomers to his court to assist in the preparation of the renowned Alfonsine Tables. With Alfonso the study of astronomy disappeared from Spain, but not before it had been revived in the East. In 1258 the still existing but shadowy Caliphate of Baghdad was swept away by the Mongol conqueror Hulagu Khan, grandson of Genghis Khan; but already in the following year this great warrior liste- ned to the advice of his new vazier, Nasir ed-din al Tusi (born at Tūs in Khorasan in 1201, died in 1274), and founded a great and magnificient observatory at Merāgha, in the north-west of Persia, In this observatory, which was furnished with a large number of instruments, partly of novel construction, Nasir ed-din and his assistants observed the planets diligently and produced after twelve years labour, the " Ilokhanic Tables," Among the astronomers of Merāgha seems to have been Juhanna Abu 'l Faraj, called Bar Hebrāyā, or the son of a Jew. He was a Christian, born in 1226 and from 1264, till his death in 1286 Maphrian or Primate of the Eastern Jacobites. He left a well- known chronicle and an astronomical work, both written in Syriac, as well as other writings.¹ The observatory at Merāgha had not a long life, and Asiatic astronomy had to wait a century and a half, until the grandson of another terrible conqueror erected another observatory. Ulug Begh, grandson of Tamerlan, drew learned men to Samarkand and built an observatory there about the year 1420, where new planetary tables and a new star catalogue, the first since Ptolemy's, were prepared. Ulug Begh died in 1449, he was the last great Asiatic protector of astro- nomy; but just as the Eastern countries saw the star of Urania setting, it was rising again for Europe. In this review of Arabian astronomers we have only men- tioned a few, omitting several names of distinction, whose ¹ Le livre de l' ascension de l' esprit sur la forme du ciel et de la terre. Cours d' Astronomie redige en 1279 par Gregoire Aboulfarag, dit Bar Hebraeus. Publie par F. Nau, Paris, 1899-1900 (2 parts, Syriac and French). His chronicle is the chief authority for the fable about the burning of the Alexandrian Library by order of the Caliph Omar. For a very thorough refutation of this see Butler, The Arab Conquest of Egypt. Oxford, 1902 pp 401-426.

FIGURE OF EARTH 15 owners devoted themselves to other branches of astronomy. Though Europe owes a debt of gratitude to the Arabs for keeping alive the flame of science for many centuries and for taking observations, some of which are still of value, it cannot be denied that they left astronomy pretty much as they found it. They determined several important constants anew, but they did not make a single improvement in the planetary theories. It will therefore be sufficient to enumerate the improvements attempted and the opinions held by Arabian astronomers with- out keeping strictly to the chronological order, although we are here dealing with a period of about six hundred years and men belonging to very different nations, who had little in common except their religion and the language in which they wrote. Figure of Earth Turning first to the question of the figure of the Earth, we find a remarkable contrast between Europe and Asia. In the world under Islam there was an entire absence of that hostility to science which distinguished Europe during the first half of the Middle Ages. Though we learn from Kazwini's Cosmo- graphy¹ that some of the earlier Arabs believed the Earth to be shaped like a shield or a drum, still there is no record of any Arabian having been persecuted for asserting that the Earth is a sphere capable of being inhabited all over. Whether this was in consequence of the warriors of the Caliphs having carried their arms to the centre of France on one side and to the borders of China on the other while their merchants travelled south- ward to Mazambique and northward to the centre of Asia, is another question : anyhow, the fact of the Earth being a sphere of very small dimensions in comparison to the size of the uni- verse was accepted without opposition by every Arabian scholar, and the very first scientific work undertaken after the rise of astronomy among them was a determination of the size of the Earth. It was carried out by order of the Caliph Al Mamun in the plain of Palmyra. According to the account given by Ibn Junis, the length of a degree was measured by two observers between Wamia and Tadmor and by two others in another loca- lity, we are not told where. The first measure gave a degree

1. Zakarija Ben Muhammed Ben Mahmūd El Kazwini's Kosmographie, deutsch von H. Ethe, Leipzig, 1868, p. 295.

16 ASTRONOMY IN ANCIENT NATIONS equal to 57, the second one equal to 56¼ Arabian miles of 4000 black cubits, and the approximate mean, 56⅔ miles, was adopted as the final result, the circumference of the Earth being 20,400 miles and the diameter 6500 miles. Another report, by Ahmed ben Abdallah, called Habash, an astronomer under Al Mamun (quoted by Ibn Jūnis), states that a party of observers (no names given) proceeded along the plain of Sinjar until they found a difference in meridian altitudes, measured the same day, equal to one degree, while the distance travelled over was found to be 56¼ miles¹. Probably two different determinations were made. If the "black cubit" is the Egyptian and Babylonian cubit of 525 mm.², the mile would be=2100 m. and 56⅔ miles=119,000 meters, rather a large result. The doctrine of the spherical earth remained undisputed in the Muhammedan learned world, though the curious error of assuming that the level of the sea was higher on some parts of the Earth than on others appears to have found some adherents among Arabian writers as well as in Europe.³ We may, there- fore, at once pass on to the motions of the heavenly bodies. Al Battani determined the longitude of the Sun's apogee and found it=82° or 16° 47¹ more than Ptolemy had given. As he believed

1. Caussin, Not. et Extraits. vii. pp. 94-96 ; Delambre, Hist. de l'astr du Moyen Age. pp. 78 and 97; Shems ed-din, Manuel de la cosmographie, traduit par Mehren, Copenhague. 1874 p. 6. Suter, p. 209, mentions a third report (from Ibn Challikān's Biographical Dictionary), according to which the sons of Musa first measured in the plain of Sinjar and afterwards as a test at Kufa, by order of Al Mamun. The eldest of the sons of Musa died 41 years after Al Mamun, and the names of the observers in the first report are different, so that the third report is not to be relied on. Al Fargani merely gives 56⅔ miles as the result of Al Mamun. According to Shah Cholgii Astronomica . . . . studio et opera Ioh. Gravii, Lon- don, 1652, p. 95, Ala ed-din Al Kusgi (one of the Ulug Begh's astronomers) gives the circumference of the earth=8000 parasangs. As a persian parasang =30 stadia (Hultsch; Griech u. Rom. Metrologie. p. 476) this would seem to be the value of Posidonius, 240,000 stadia. Kazwini (p. 298) gives the circu- mference=6800 parasangs on the authority of Al Beruni.
2. Hultsch p- 390.
3. It deserves to be mentioned that Shems ed-din of Damascus (1256-1327) explains the great Preponderance of dry land in the northern hemisphere by the attraction of the Sun on the water, which is the greatest when the Sun is in perigee, at which time it is nearly at its greatest south declination. That this accumulation of water would not be a permanent one does not occur to him (Cosmographie, p. 4).

MOON AND ITS ORBIT 17 that Ptolemy's value had been found by himself,¹ and as he adopted 54" (or 1° in 66 years) as the annual amount of pre- cession, there remained (assuming that 760 years had passed since the time of Ptolemy) an outstanding error of 79"—54"=25" per annum. In reality the annual motion of the solar apsides is 11½"; still we may say that the discovery of this motion is due to Al Battani, though he did not announce it as such ; in fact he merely gives his own value as an improvement on that of Ptolemy. Even Ibn Junis (who found 86° 10') did not suspect that the apogee was steadily moving but merely says that it must be corrected for precession (1° in 70 years), and remarks that the longitude of the apogee is very difficult to determine accurately.² On the other hand, Al Zarkali found a smaller value, 77° 50' and as he also found a smaller value of the eccen- tricity he thought it necessary to let the centre of the Sun's eccentric orbit describe a smaller circle, after the example set by Ptolemy in the case of Mercury.³ The inclination of the ecliptic which the Greeks had found—23° 51 20" was by the astro- nomers of Al Mamun found—23° 33' (in 830), by Al Battani (in 879), and by Ibn Jūnis 23° 35'⁴. When Al Zarkali found 23° 33',he, and afterwards Abu 'l Hassan Ali of Morocco, concluded that the obliquity oscillated between 23° 53 and 23° 33', an idea to which the prevailing belief in the "trepidation" of the equi- noxes lent countenance.⁵ Moon and its orbit If we now turn to the Moon, we do not find that the Arabs made any advance on Ptolemy. Several of them noticed that the inclination of the lunar orbit was not exactly 5°, as stated by Hipparchus. Thus, Abu 'l Hassan Ali ben Amagiur early in the tenth century says that he had often measured the greatest latitude of the Moon and found results greater than that

1. Scient. Stell. Cap. xxviii. Bologna, 1645, p. 72; Nallino. p. 44. At the end of Cap. xlv. he says the apogees of the Sun and Venus are both in 82° 14,' and Ibn Junis also gives 82° 14' as the value found by Al Battani (Caussin, p. 154).
2. Caussin, pp. 232 and 238. Abu'l Faraj gives 89° 28' for the year 1279 (p. 22).
3. Sedillot, Prolegomenes aux tables astron. d' Olough Beg (1847), pp. lxxx-lxxxii. Riccioli, Almag. Novum, I. p. 157.
4. Caussin. p. 56. For A.D. 900 Newcomb gives 23° 34' 54", with a diminution of 46" per century, so that the Arabian astronomers erred less then 1'.
5. Aboul Hassan Ali, Traite des Instruments astron. des Arabes: T.I.p. 175; Sedillot, Memoire sur les instr. astr. des Arabes, p. 32.

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.