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

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

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 Unpublished chapters of Ibn Jūnis, reviewed by Delambre, Hist.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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

38 ASTRONOMY IN ANCIENT NATIONS
after all not be very surprising if some learned Jews had been
influenced by the opinion of Herakleides, since it is an established
fact that the doctrines of the Kabbalists were intimately connec-
ted with the later Greek philosophy. But any how nothing came
of this isolated case, and the daily rotation of the heavens conti-
nued to be universally accepted as a self-evident fact.
Arabian astronomers and Ptolemaic system—
Arabian astronomers who really wished to follow in detail
the celestial motion were therefore obliged to adopt the Ptolemaic
system altogether. New planetary tables had long been found to
be a necessity, and this important work was at last undertaken by
King Alfonso X. of Castille and several Jewish and Christian
astronomers working under him at Toledo, who prepared the
celebrated Alfonsine Tables, Apparently the King must have had
his doubts about the physical truth of the system, judging from his
well-known saying that if God had consulted him when creating
the world, he would have given Him good advice. The tables
were prepared under the direction of the Jew Ishak ben Said,
called Hasan, and a physician, Jehuda ben Mose Cohen, and
were finished in 1252, the year in which Alfonso ascended the
throne of Castille. They continued in great repute for three hun-
dred years as the best planetary tables; they were first printed in
1483, but had been spread all over Europe long before that time
in numerous MS copies, many of which are still in existence,
Twenty-six codices are counted up in the Libros del Saber de
Astronomia del Rey D. Alfonso X.de Castella, Madrid, 1863-67
(5 vols. fol.). This compilation, a series of chapters on spherical
and theoretical astronomy followed by tables, must have been
made up from several codices, as there are numerous repetitions
even of very elementary matters. In the third volume the theo-
ries of the planets are dealt with, but one looks in vain for any
improvement on Ptolemy; on the contrary, the low state of astro-
nomy in the Middle Ages is nowhere better illustrated. In gene-
ral the elements of the orbits are those of Ptolemy, though some-
times only approximations are given, while different values are
given in different chapters, though Ptolemy places the centre of the
deferent midway between the centre of the equant and the Earth,
the Libros del Saber places the centre of the equant (cerco del
alaux.¹) midway between the Earth and the centre of the deferent

1. al is the Arbic article, aux (apside) is a corruption of the Arabic Oudj
   (Abu 'l Faraj, II. p. 25). The equant is called the cerco del Y guador.

ARABIAN ASTRONOMERS AND PTOLEMAIC SYSTEM 39
(cerco del levador¹), as in Ptolemy's theory of Mercury, which
the authors would seem to have extended to the planets, omitting
the motion of the centre of the deferent on a small circle; this,
they have, however, correctly given in the case of Mercury.²
There is a very curious figure³ of the deferent of Mercury in the
form of an ellipse (the axes being as 6 to 5 nearly), with what
looks like the Sun in the centre. This curve has been construc-
ted from a number of small circular arcs,⁴ and it is obviously
nothing but the curve described by the centre of the epicycle of
Mercury in Ptolemy's theory. For according to the latter the
centre of the deferent describes a small circle with radius = 1/21 of
that of the deferent, in direction from east to west, in the same
time which the centre of the epicycle takes to pass round the cir-
cumference of the deferent from west to east. This makes the cen-
tre of the epicycle describe a closed curve resembling an ellipse, the
axes of which are in the ratio 11: 10, almost exactly the same as in
the Spanish diagram, and there is therefore in the latter no antici-
pation whatever of Kepler's great discovery, since in the case of
the inferior planets it is the epicycle which is the real orbit.⁵
The small sun-like object in the centre of the ellipse represents
the centre of Ptolemy's small circle, and it has either been inser-
ted in the manuscript centuries after the essay had been written,
or, more likely, it has been caused by a small blot on the place
in the parchment where the stationary leg of the draughtsman's
compasses had made a small hole. An oval deferent of Mercury
occurs in several books published in the sixteenth and seven-
teenth centuries.⁶

1. Vol. III. pp. 246—253.
2. Vol. III, pp. 253 and 278. In the the latter place the radius of the small
   circle is 1/21, as in the "Hypotheses" of Ptolemy.
3. Vol. III. p. 282.
4. See the lengthy description on pp, 278—280.
5. The editor, Don Manuel Rice y Sinobas, on p. xxxiii, of his preface,
   even goes so far as to suggest that Kepler may have known of this great
   discovery of Alfonso's or rather of Arzachel's, as the text attributes the
   construction to him. This and other similar diagrams were intended to
   be used instead of planetary tables in the manner afterwards adopted by
   Apianus.
6. First (about 1460) in Purbach's Theoricae novae Planetarum (ed. of Basle,
   1573, p. 82) : "Ex dictis apparet manifeste, centrum epicycle Mercurij,
   (Continued on next page)

40 ASTRONOMY IN ANCIENT NATIONS
Though the somewhat confused collection of essays entitled
the Libros del Saber would not, if published in the thirteenth
century, have advanced astronomical science, it cannot be denied
that the Alfonsine Tables were very useful in their day. The
actual elements are not given, nor is any thing said about any
observations by which somewhat more correct values of the
mean motions must have been found.¹
Arabs on motions of fixed stars.
Thus we finish our review of the planetary theories of the
Arabs. Now we shall say a few words about their ideas as to
the nature and motion of the fixed stars. The exaggerated
notion which prevailed before the invention of the telescope
with regard to the apparent angular diameters of the stars natu-
rally, led to erroneous estimates of their actual size, founded on
the assumption that the sphere of the fixed stars (the eighth
sphere) was immediately outside that of Saturn.² The stars of
the first magnitude were supposed to have an apparent diameter
equal to 1/20 of that of the Sun, from which it followed that their
actual diameters were about 4 1/2 times that of the Earth, or about
(Continued from previous page)
propter motus supradictos non (ut in alijs planetis fit) circumferentiam
deferentis circularem, sed potius figuræ, habentis similitudinem cumplana
ovali peripheriam describere." Next by Albert of Brudzew in 1482 in his
Commentariolum super theoricas novas, printed at Milan in 1495 (ed.
Cracow, 1900, p. 124), where it is remarked that the centre of the lunar
epicycle describes a similar figure. This is also stated by E. Reinhold in
his commentary to Purbach, 1542, fol. p. 7 verso (ed. of Paris, 1558, fol. 78,)
by Vurstisius in his Questiones novae in theoricas, & c., Basle, 1573,
p. 233; and in Riccioli's Almagestum novum T. I, p. 564. The last three
writers (who give a figure) also take the equable angular motion round
the centre of the equant into account, which centre lies on the point of the
circumference of the small circle nearest the Earth. The curve described
by the centre of the epicycle thus becomes egg-shaped, and not like an
ellipse.

1. The tables in vol. v. of the Libros del Saber are quite different from the
   Alfonsine Tables, and are apparently only intended for astrological
   purposes.
2. Al Battani (cap. 50) gives the greatest distance of Saturn = 18,094, and the
   distance of the fixed stars = 19,000 semidiameters of the Earth. Al Fargani
   (p. 82) puts them exactly equal. Al Kusgi gives the diameters in parasangs,
   of the concavity of the stellar sphere = 33,509,180 of the ninth sphere
   33,524,309, of its convexity "no one but God knows" (Shah Cholgi, p. 97).

ARABS ON MOTIONS OF FIXED STARS 41
twice that of Mars.¹ As to the nature of the stars, they seem
generally to have been assumed self-luminous, being condensed
parts of the sphere, though Abraham ben Chija says that the
eighth sphere does not shine with a uniform light, but has denser
spots, which are illuminated by the Sun and appear to us as the
fixed stars.²
To account for the apparent slow motion of the stars para-
lleled to the ecliptic, from west to east, whereby their longitudes
increase while their latitudes remain unaltered, it became neces-
sary to inroduce a ninth sphere (primum mobile), turning in
twenty-four hours and communicating this motion to the eighth
sphere, while the latter moved extremely slowly round its own
axis forming an angle of 23° 35′ with that of the ninth.³ But the
simple phenomenon of precession was by many Arabian astro-
nomers complicated by being assumed variable. It may be
mentioned that according to Theon and Proklus it had been
assumed by some astronomers apparently before the time of
Ptolemy, that the precessional motion of the stars was not pro-
gressive, but was confined to an oscillation along an arc of 8°,
along which the equinoctial points moved backwards and forwards
on the ecliptic, always at the same rate of 1° in 80 years. The
absurdity of the sudden change of direction must have become
obvious as soon as astronomy began to be cultivated among the
Arabs, for we find that one of the earliest astronomers, Tābit
ben Korra, substituted a physically less objectionable theory.⁴

1. Al Fargani (p. 85, Golius) gives the cubic contents of the six spheres as
   107, 90, 72, 54, 36, 18 times that of the Earth. Abu 'l Faraj, p. 199, gives
   a similar series from 93 to 15½ for the average star of each class. Shems
   ed-din of Damascus in his Cosmography (p. 3) merely says that the smallest
   fixed star is much larger than the Earth.
2. According to Suter, p. 77, a writer called Ibn Zura wrote a treatise "On
   the cause of the light of the stars, though they and the spheres consist of one
   single substance."
3. The outermost sphere is by the philosopher Ibn Sina (Avicenna) defined
   as a spherical, single (not composite) body, emanating directly from God
   and subject to dissolution, endowed innately with circular motion as an
   expression of its praise of the Creator (Mehren in Oversigt, K. Danske
   Vid. Selskab, 1883, p. 70).
4. The treatise "On the motion of the 8th sphere" has never been printed ;
   an abstract is given in Delambre's Hist. de l'astr. du Moyen Age, p. 73.
   Compare a quotation by Ibn Junis, Caussin, Notices et Extraits, VII, p. 116.

42 ASTRONOMY IN ANCIENT NATIONS
He imagines a fixed ecliptic (in the ninth sphere) which intersects
the equator in two points (the mean equinoxes) under an angle of
23° 33' 30", and a movable ecliptic (in the eighth sphere), attached
at two diametrically opposite points to two small circles, the
centres of which are in the mean equinoxes and the radii of which
are = 4° 18' 43". The movable tropical points of Cancer and
Capricorn never leave the fixed ecliptic, but move to and fro to
the extent of 8° 37' 26", while two points on the movable ecliptic
90° from the tropical points move on the circumferences of the
small circles, so that the movable ecliptic rises and falls on the
fixed one, while the points of intersection of the equator and the
movable ecliptic advance and recede to the extent of 10° 45'
either way, This is a motoin of the eighth sphere, common to
all stars, and the Sun will, therefore, sometimes reach its
greatest declination in Cancer, sometimes in Gemini. Tabit
does not say that the obliquity of the ecliptic is variable, and
perhaps it did not occur to him that this would be a necessary
consequence of his theory; he only notices the change in direction
and amount of the motion of the equinoxes, which, he says, has
increased since the days of Ptolemy, when it was only 1° in 100
years, while later observers have found 1° in 66 years. The
erroneous value given by Ptolemy was, therefore, mainly responsi-
ble for the continuance of the imaginary theory. It is to be
observed that Tabit expresses himself with a certain reservation,
and seems to think that further observations are necessary to
decide if the theory is true or not. His younger and greater
contemporary Al Battani was even more cautious, for though he
repeats the account of the trepidation given by Theon (which he
says that Ptolemy manifeste in suo libro declarat¹) he does not
make use of it, but simply adopts 1° in 66 years (or 54".5 a year),
which he finds by a comparison between his own observations and
some made by Menelaus. In rejecting the erroneous value of
Ptolemy, which Al Fargani alone had accepted,² Al Battani was
followed by Ibn Junis, who came still nearer to the truth by
adopting 1° in 70 years or 51".2 a year, and who does not allude
to trepidation. It is greatly to the credit of several other Ara-
bian writers that they were not led astray by this imaginary phe-

1. Cap. 52, (205). Plato's translation gives the period as 84 years, but Nallino's
   ed. has 80 (p. 127).
2. c. 13, p. 49. .
3. Schjellerup, Descr. des etoiles fixes, p. 43,

ARABS ON MOTIONS OF FIXED STARS 43
nomenon; among them are Al Sūfi, the author of the only urano-
metry of the Middle Ages³, who followed Al Battani, also Abu 'l
Faraj and Jagmini,¹ while Nasir ed-din mentions it but seems
to doubt its reality.² By others it was willingly accepted, for
instance by Al Zarkali, who made the period of oscillation of 10°
either way equal to 2000 Muhammedan years (or 1940 Gregorian
years, i.e. 1° in 97 years or 37" a year). The motion is in a circle
of 10° radius; at the Hijra the movable equinox was it 40' in
increasing precession, and in A: D. 1080 at 7° 25'.³. The diminu-
tion of the inclination of the ecliptic, which the astronomers of
Al Mamun had found=23° 33', no doubt lent countenance to
the idea of trepidation, and the next step in the development
of this curious theory was the combination of progressive and
oscillatory motion. Al Betrugi, who gives a sort of history
of the theory, beginning with a mythical H mes, makes
out that Theon (or Taun Alexandrinus as he calls him) com-
bined the motion of 1° in 100 years with the oscillation³. A
century later this was actually done, and the theory received
its last development by King Alfonso or his astronomers, who⁴
perceived that the equinoxes had receded much further than
Tabit's theory allowed. The equioxes were now supposed to pass
right round the heavens in 49,000 years (annual motion=26".45),
while the period of the inequality of trepidation was 7000 years,
so that in a sort of Great Jubilee year everything was again as it
had been in the beginning.⁵ The progressive motion belongs to

1. Abu 'l Faraj, p. 12, simply says that the motion is 1° in 100 years according
   to Ptolemy, or 1° in 66 years according to others. But on p. 18 he says that
   if the ancient Chaldeans gave the tropical points a motion backwards and .
   and forwards, and if ancient astrologers adopted this, then the motion of the
   fixed stars must have been unknown to them. Jagmini (p. 229) says that
   most people adopt 1° in 66 solar years.
2. Spheres celestes, p. 347.
3. Sedillot, Memoire sur les instr. astr. des Arabes, pp. 31, 32. Abraham
   ben Chija (p. 196 of Munster's Sphaera mundi. Basle, 1546) gives the
   period as 1600 years without quoting any authority. He adds that the
   ancient Indians. Egyptians, Chaldeans, Greeks, and Latins first proposed
   the theory—Ptolemy neither approved nor disapproved of it, but Al Battani
   confuted it.
4. Alpetragius, f. 12a. He says that Al Zarkali did the same.
5. A later writer, Augustinus Ricius. De motu octave sphaere, Paris, 1521, who
   traces the theory back to Hermes, 1985 years before Ptolemy (!) credits
   (Continued on next page)

44 ASTRONOMY IN ANCIENT NATIONS
the ninth sphere; the annual precession varies between 26″.45±
28″.96, or from + 55″.41 to −2″.51.¹ It was now necessary to
assume the existence of a tenth sphere, which as primum mobile
communicated the daily rotation to all the others, while the
ninth produced the progressive and the eighth periodical
motion on the small circles, which are situated "in the conca-
vity of the ninth sphere." This was a nice and comfortable
theory on account of the long periods involved and the slow
changes it produced in the amount of annual precession; and
oblivious of the fact that the theory had no foundation except
the circumstance that the obliquity of the ecliptic was now about
20′ less than it had been stated to be by Ptolemy, and that he had
given the amount of precession as 36″ a year instead of about
50″, and often shutting their eyes to several of the necessary
consequences of it, such as the changes in the latitudes of stars
which it ought to produce², astronomers continued to accept the
theory until at last a real observer of the stars arose and wiped
it out by showing that the obliquity of the ecliptic had steadily
diminished, and that the amount of annual precession had never
varied. We have in this place only alluded to it because it in-
volved some rearrangement of the spheres and because it is
eminently characteristic of the period during which no persistent
observations were taken, and hardly an attempt was made to
improve the theories of Ptolemy. The theory of trepidatio or
titubatio, as it was sometimes called, was one attempt and it
would have been better left alone. But it forms a not uninter-
esting chapter in the history of astronomy.
(Continued from previous page)
this development to a Jew of Toledo, Isaac Hassan (see above. p. 38),
adding that Alfonso four years after the completion of the tables became
convinced of the futility of the theory by reading the book on the fixed stars
by Al Sufi. Riccioli, Almag. novum, I. p. 166.

1. In the Alfonsine Tables the maximum took place at the birth of Christ. In
   Essler's Speculum astrologicum, p. 224 (appended to Purbach's Theoricae
   novae, Basle, 1573) the epoch is A.D. 15, diebus 137 completis. Reinhold in
   his commentary to Purbach (Paris, 1558, f. 163b) explains that 26″.45 is
   the space passed over by the Sun in 10 mins. 44 secs., by which amount
   the Alfonsine Tables made the tropical year smaller than 365¼ days.
2. Abraham ben Chija (p. 103, Schrackenfuchs) says that trepidation does
   not change the latitudes. Perhaps he refers to the earliest form of the
   notion, that described by Theon of Alexandria.

ARABS ON MOTIONS OF FIXED STARS 45
Here we finish our review of ancient astronomy. We have
omitted as not coming within our province several valuable
contributions to science which did not deal with cosmology or
planetary theory. But even with this limitation enough has
been said to show that when Europeans again began to occupy
themselves with science they found astronomy practically in the
same state in which Ptolemy had left it in the second century.
But the Arabs had put a powerful tool into their hands by alter-
ing the calculus of chords of Ptolemy into the calculus of sines or
trigonometry, and hereby they influenced the advancement of
astronomy in a most important manner.
References

1. Peter Doig : A Concise History of Astronomy London
3. J.L.E. Dreyer : A History of Astronomy from Thales to
   Kepler Dover publications, 1953 (chapter
   XI reproduced).
4. Satya Prakash : Founders of Sciences in Ancient India,
   Delhi, 1965.

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