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

( २२ )
आग्नेये नैर्ऋत्ये वेष्टदिने संस्थितस्य योऽर्कस्य ।
शङ्कुच्छाये कथयति वर्षादपि वेत्ति सूर्य सः ॥
कथितोऽस्ति यदुत्तरं कोणाशङ्कोरानयनेन स्फुटमस्ति ।
संज्ञाध्यायकथनस्य किमपि प्रयोजनं नासीत् ॥
आचार्येण कथमेतस्याध्यायस्योल्लेखः कृत इति न जानीमः ।
ध्यानग्रहोपदेशाध्याये—चैत्रादौ मासगणानयनम् । चैत्रादौ दिनादिकं निधि
ध्रुवसाधनम् । इष्टमासादौ रव्यानयनम् । प्रतिमासं शशिकेन्द्रतिथिध्रुवक्षेपयोग-
नयनम् । प्रतिदिनचालनम् । चन्द्रसाधनमौदयिकरविसाधनञ्च । ज्याखण्डकानि
केन्द्रज्या साधनञ्च यथा -
त्रिंशत् सनवरसेन्दुर्जिनतिथिविषया गृहार्धचापानाम् ।
अर्धज्याखण्डानि ज्याभुक्तैक्यं स भोग्य फलम् ॥
गतभोग्यखण्डकान्तरदलविकलवधाच्छतैर्नवभिराप्तैः ।
तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम् ॥
इत्याचार्योक्तभोग्यखण्डस्पष्टीकरणमेव सिद्धान्तशिरोमणेर्ग्रहगणिता-
ध्यायस्य स्पष्टाधिकारे 'यातैष्ययोः खण्डकयोर्विशेषः शेषांशनिघ्नः' इत्यादिना
भास्कराचार्येण कथितम् । भास्कराचार्येण खार्क-१२० मिता त्रिज्या गृहीता,
अत्राचार्येण खतिथि १५० मिता त्रिज्या गृहीता । इत्यतीववैचित्र्यं यत्सर्वत्रैव
श्रीपतिनाऽऽचार्यश्लोकोक्तविषया एव छन्दोन्तरेण लिखिताः परं किं कारणं
यदपूर्वं भोग्यखण्डस्पष्टीकरणं न लिखितम् । चन्द्रे भुजफलसंस्कारः, तिथौ
फलसंस्कारश्च । इत्यादयः सर्वे विषया अपूर्वाः सन्ति । अत्राध्याये ये केचन
विषया लिखिताः सन्ति ते सूर्यसिद्धान्ते सिद्धान्तशेखरे, सिद्धान्तशिरोमणौ न
सन्ति ।
अनुगृहीतोऽस्म्यहं श्रीडाक्टर सत्यप्रकाश डी० एस० सी० महोदयानां
यैराङ्गलभाषायां कृपया ग्रन्थस्यास्य प्रस्तावना समरचि ।
सम्पादकमण्डलस्यान्ये सहयोगिनः ज्योतिषाचार्याः श्रीमुकुन्दमिश्राः,
श्रीविश्वनाथ झा, श्रीदयाशंकरदीक्षिताः, शास्त्रिणः श्रीओंदत्तशर्माणश्च सर्वे एव
मम धन्यवादस्य पात्रतामहंन्ति । एतेषाममूल्यसहयोगेनैवायं महान् ग्रन्थः सुचारु-
रूपेण सम्पूर्णतामगच्छत् ।
'पद्मश्रीप्रकाशनालयस्य' स्वामिने रमेशचन्द्रशर्मणोऽपि धन्याञ्जलिर्भवतु
यस्य महता परिश्रमेण ग्रन्थस्य प्रकाशनं कालेनैवाभवत् ।
अन्येभ्यः सर्वेभ्योऽपि धन्यवादान् प्रददे, यैरल्पमपि साहाय्यं विधायाहं
कृतार्थीकृतोऽस्मि, इति शम् ।
भृगु-आश्रमः } विदुषामनुचरः
३०-३-६६ } रामस्वरूप शर्मा

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.