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

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.

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.