the planetary theory of ibn al-shatir: latitudes of the planets

The Planetary Theory of Ibn al-Shatir: Latitudes of the PlanetsAuthor(s): Victor RobertsSource: Isis, Vol. 57, No. 2 (Summer, 1966), pp. 208-219Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/227960 .

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The Planetary Theory of Ibn

al-Shatir: Latitudes of the Planets

By Victor Roberts *

I. INTRODUCTION

T HIS IS THE concluding paper of a series of four describing the models for predicting planetary motion which were originated by an astron-

omer of fourteenth-century Damascus, Abft al-Hasan 'Ali b. Ibrahim al- Ansari, known as Ibn al-Shatir. The three preceding studies 1 showed the

following:

1) Ibn al-Shatir's planetary models for longitude predictions are non-Ptolemaic to the extent that they are expressible as combinations of uniform circular motions. Hence they are not subject to the criti- cisms directed toward Ptolemy in ancient and medieval times.

2) Several of the models (except for their being geostatic) are identical with those of Copernicus, who lived a century and a half after his Arab predecessor.

3) In reducing the geometric models to numerical longitude tables both Ibn al-Shatir and Copernicus made use of Ptolemy's elegant interpolation procedure but performed independent computations.

Subsequent investigations, as yet unpublished, have shown that the basic devices and the impetus for Ibn al-Shatir's work go back to a group of scientists headed by Nasir al-Din al-Tuisi, director of the Maragha observa-

tory, founded by the Mongol Ilkhanid dynasty of Iran. Moreover, Profes- sor Otto Neugebauer has discovered in a Byzantine manuscript (Vat. Gr. 211, fol. 116r) -which was in Italy at the time of Copernicus- a diagram of a solar model which includes a second epicycle. Thus there has been found the first textual evidence for the transmission of these theories from the Near East to Western Europe.

To return to the task at hand, it is noted that in the models the planets of the solar system move in plane orbits. The planes of all these orbits

*The American University of Beirut and Theory of Ibn al-Shatir," Isis, 1957, 48:428- the Faculty of Sciences, University of Baghdad. 432; E. S. Kennedy and Victor Roberts, " The This work was supported by the National Planetary Theory of Ibn al-Shatir," Isis, 1959, Science Foundation and by a grant from the 50:227-235; Fuad Abbud, "The Planetary Research Committee, Faculty of Arts and Sci- Theory of Ibn al-Shatir: Reduction of the ences, American University of Beirut. Geometric Models to Numerical Tables," Isis,

1Victor Roberts, "The Solar and Lunar 1962, 53:492-499.

Isis, 1966, VOL. 57, 2, No. 188. 208



THE PLANETARY THEORY OF IBN AL-SHATIR

contain the sun, but they intersect each other at small angles. In particular, the plane of the earth's orbit, the ecliptic, is intersected by any other orbital plane along a fixed line in space, the line of nodes. In general, the vector from the earth to the planet makes a nonzero angle with the ecliptic. This angle, f, is the planet's geocentric latitude, zero only when the planet is passing through its nodal line. For a particular planet, P is a function of time.

Any serious planetary theory must include a definition of these functions, and it is to Ibn al-Shatir's definitions that we address ourselves. Before doing so, however, it will be necessary to examine the actual situation in space in some detail and to review Ibn al-Shatir's solution for the simpler problem of longitude prediction. This is done in Sections II and III below. Section III also sets up the conventions and notation adopted for the rest of the paper; these are necessarily elaborate, since the configuration in space is quite complicated. In Section IV Ptolemy's two latitude mechan- isms are described and compared. The following two sections present the constructions for the superior and inferior planets, respectively, again for comparison. The concluding Section VII draws such inferences as seem warranted by the evidence.

As with the previous three papers, the main source is Bodleian MS Marsh 139, a copy of Ibn al-Shatir's treatise (called Nihayat al-Suil) describing his planetary theory. The latitude theory of the superior planets is handled in Chapter 24, that of the inferior planets in Chapter 25. This material is found on folios 30v-35v. References to the text will give the folio and line numbers, separated by a colon. Gratitude is again acknowledged to the Keeper of Oriental Books at the Bodleian Library for making a micro- film of the manuscript available for study.

II. THE PROBLEM

It is convenient to think of the line from the earth to a planet as being the sum of two vectors, one from the earth to the sun, the second from sun to planet. This has several advantages. It facilitates a description of the planet's complicated geocentric motion, for each of the vectors is of almost constant length and rotates with almost constant angular velocity in or parallel to the planes of the ecliptic and the planetary orbit, respectively. Moreover, the resulting configuration is very close to the Ptolemaic planetary model, provided that in all cases the larger of the two vectors is placed first in the linkage. The path its endpoint traces is called the deferent (in Arabic, al-falak al-kharij al-markaz). The smaller makes up the epicycle (falak al-tadwir). The orbit of an inferior planet is smaller than that of the earth; hence for it the epicycle of the Ptolemaic model is its own orbit, and the deferent is congruent to the earth's orbit about the sun (or the sun's orbit about the earth, to take the ancient point of view). With a superior planet, whose orbit is larger than the earth's, the situation is reversed: the epicycle is the earth's orbit; the deferent, the planet's.

A planet's longitude at any time is the angle, x, from some fixed reference

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VICTOR ROBERTS

line in the ecliptic to the projection of the earth-planet vector on the

ecliptic. Since the fixed angle between the ecliptic and the plane of the

planet's orbit is in all cases small, very little precision is lost in longitude calculations if this angle is made zero. There is a great gain in simplicity, for then all action takes place in the ecliptic plane, and the problem is two-dimensional. This was done by all three astronomers discussed here:

Ptolemy, Ibn al-Shatir, and Copernicus. Since the latitude is the measure of the extent to which the planet is not

on the ecliptic, the latitude problem is inevitably three-dimensional - hence its difficulty. For an inferior planet it is accurate to think of its deferent as being in the ecliptic while its epicycle plane slides along, displacing parallel to itself, the line of nodes passing always through the epicycle center and also moving parallel to itself. For a superior planet the deferent is fixed at a small angle to the ecliptic, the intersection between the two

planes being the line of nodes. The epicycle plane now displaces parallel to the ecliptic.

III. NOTATION

The remarkably close approximation to observation which Ptolemy's model achieved was obtained by placing the earth away from the center of the deferent and adopting the so-called equant hypothesis. These meas- ures, admirable from the scientific point of view, had the effect of making elements in the two-vector linkage vary slightly in length and rate of rotation. Thus Ptolemy violated the Aristotelian dictum that any celestial motion must be made up of uniform circular rotations.

Ibn al-Shatir (like the scientists of the Maragha school before him and

Copernicus after him) removed the Aristotelian objection by introducing two additional vectors into the linkage. For him the resulting vector chain was ordered as follows (of course, none of the ancient or medieval texts make explicit mention of the vector concept, but it greatly facilitates description if this is done):

ri al-ma'il (" the inclined ") is the deferent radius, of length 1,0 = 60 and rotating with the mean angular velocity of the planet (X.).

r2 al-hamil (" the carrier") of length one and a half times the Ptolemaic eccentricity, displaces parallel to the line of apsides; hence has zero angular velocity.

r3 al-mudir (" the rotator ") of length half the Ptolemaic eccentricity, has an angular velocity of 2kX.

r4 al-tadwir, the epicycle radius, equal in length to the Ptolemaic epicycle radius, and rotating with the mean anomalistic motion of the planet (&m).

There are a few variations in detail, for which the reader may consult an earlier article.2 In particular, Mercury is so eccentric as to require the

2 Kennedy and Roberts, op. cit.

210




addition of two more vectors. But for defining the latitude functions, the four named above will suffice.

However, it is now necessary to elaborate the apparatus because of the increase in dimensionality. Let P1, P2, P3, and P4 denote the planes of rotation of ri, r2, rs, and r4 respectively, and let ni, n2, n3, and n4 be normals to these planes erected at the initial points of the respective vectors. (In an absolute sense r2 does not rotate at all, but with respect to rl, say, it does rotate, and it will be convenient to associate with it a plane and a normal, the behavior of which will be described later.) The subscript zero will

designate the ecliptic plane when P1 does not coincide with it. We designate by Z rr2, Z n3n4, .. ., the angles between the respective entities involved.

The term sinusoidal will be used to describe displacements of the form

0 = c sin kt,

where c and k are arbitrary constants and t is time.

IV. THE PTOLEMAIC LATITUDE MODELS

Ptolemy in his Planetary Hypotheses 3 (al-Iqtisdi) effected improvements

over the arrangements made in the Almagest.4 In both documents and for all three superior planets L PoP, - no,n is held fixed (since there are only two vectors in these models, subscripts 0, 1, and 4 serve for ecliptic, deferent, and epicycle, respectively). In the Almagest the motion of P4 is such that whenever the epicycle center is in the nodal line L nn= -O. Hence,

regardless of the value of the anomaly at the nodes, the latitude is identically zero. In between, /n,n4 varies sinusoidally in such a manner that its maximum exceeds the fixed no,n, and the maximum latitudes occur when the planet is nearest the earth. In the Hypotheses Po and P4 are maintained parallel; hence Z n,n,1 Z nn4.

For the inferior planets the Almagest has P1 seesawing about the nodal line. The motion of P4 is broken up into two components. Consider the

plane determined by no and r1; then the angle between n4 and this plane varies sinusoidally, being zero when r, is along the nodal line and maximum

(or minimum) when r, is a quadrant away from the nodal line. To describe

the second component, consider a plane normal to r,. The angle between

n4 and this second plane also varies sinusoidally, being zero when ri is

midway between the nodal line and maximum (or minimum) when ri is at either one of the nodes. In the Hypotheses ZPoP1 and ZPoP4 are

both constant and equal to each other. Hence P4 moves parallel to itself,

as should be the case.

3 Ptolemy, Opera quae exstant omnia .... Heiberg (2 vols. Leipzig: Teubner, 1898-

II. Opera astronomica minora, ed. J. L. Hei- 1903). Also, German translation by K. Mani-

berg (Leipzig: Teubner, 1907). tius (2 vols., 2nd ed. Leipzig: Teubner, 1963). 4 Ibid., I. Syntaxis mathematica, ed. J. L.

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VICTOR ROBERTS

V. IBN AL-SHATIR ON LATITUDES OF THE SUPERIOR PLANETS

The following is an abstract of Chapter 24 of the text. Where the author's statements are clear we give an abridged equivalent; when, however, perti- nent parts of the text are ambiguous, we present a literal translation, the next best thing to the Arabic original. In all cases symbols with the sub-

script notation of Section III have been substituted for the verbal nomen- clature of the text.

The chapter commences (fol. 30v:13-15) with a statement of the values of Z PoP1 shown in Table 1 entered in the column headed c1. These are identical with the values of the Almagest, although Ibn al-Shatir states that he has found them by observation. In the Ptolemaic models they are fixed, and presumably this is the case here also.

TABLE 1

Cl C2 C3 C4

Almagest and Planet Ibn al-Shatir Ibn al-Shatir

ZP1P3 PaP4 Z PoP1 Z PP, (= Z PsP4) (= Z PAP4)

Saturn 2?0 4?0 31? 1?

Jupiter 1i 2 2 Mars 1 24 1 + +

Next (fol. 30v:16-3 r:11) is a passage describing the behavior of Z PP4. Its variation is said to be smooth, reaching a maximum midway between the nodes of one side, at which point the epicyclic perigee is farther from Po than the epicyclic apogee (see Fig. 1). As r1 reaches the nodal line, P4 coincides with P0. Thence ZP1P4 increases, again reaching a maximum

midway between the nodes at the point opposite the first maximum. In folio 31r:12-22 are given the nodal positions and maximum latitudes dis-

played in Table 2 below.

int epicyclic periqee ns

Fll^ 7 T MODEM 4: C ~

p '--'

--............ =_.. . ................ . ..

C^ \~~~~~~~~~ * epcyclic up.ofe

P.

FIGURE 1

Next (fol. 31v:1- 32r:1) is a long allegation claiming that the latitude

problem has never been properly handled by previous astronomers. Hip- parchus and Ptolemy of the ancients, and Ibn al-Haytham, Nasir al-Din al-Tfsi, Mu'ayyad al-Din al-'Urdi, and Qutb al-Din al-Shirazi (the latter three of the Maragha school) are all mentioned by name. The crux of the matter seems to be Ibn al-Shatir's feeling that no mechanism has yet been

212




provided to insure that P4 will fold down into PO as r1 passes through the node.

Finally (fol. 32r: 1 - 33r: 19) he gets around to expounding his own solution, giving at the same time the values for LP1P2 and ZP2P4 shown in Table 1 under the c3 and c4 columns. He says

... then the orbs (afldk) move with steady motions about their centers at the rates assumed for them and in the directions assumed. If r1 moves through a quadrant and r2 moves a quadrant too (negatively, with respect to r,), r3 moves a half turn, the inclination turning over to the other side. So the direction of P4's inclination comes to be opposite the direction of P2, and P4 merges with P0.

To understand what he must have had in mind, note that he has chosen C3 and c4 so that in all cases,

C3 C4 = C1

and Cs + C4 = C2,

as Ibn al-Shatir himself explicitly points out. Refer also to Figure 1, which is a schematic profile view looking along the line of nodes at a time when

TABLE 2

Extremal latitudes

Computed Longitude of A Imagest from ascending

and Ibn al-Shatir node Ibn al-Shatir model (Ibn al-Shatir)

Saturn N Epicyclic apogee 2;4? 2;4,20? Epicyclic perigee 3;3 3;4,36

S Epicyclic apogee 2;2 1;58,20 Epicyclic perigee 3;5 2;58,20

Jupiter N Epicyclic apogee 1;7 1;9 Epicyclic perigee 2;4 2;6,40


Mars N Epicyclic apogee 0;7 a 0;12 Epicyclic perigee 4;21 4;32


aAlmagest has 0;8?.

P4 has maximum inclination. Then all the planes involved project as

straight lines. The text says nothing about any angle between P2 and P3;

hence we take them as coinciding. Note that Equations (1) are satisfied. Now consider that P4 is rigidly attached to r3. It is therefore true, as Ibn al-Shatir states, that according to his longitude model, when r1 rotates

213

(1)



VICTOR ROBERTS

through 90? from the midnodal position shown down into PO at the node, r3 will rotate through 180?. It will carry with it P4 into the position shown dotted on Figure 1, parallel to Po. Meanwhile, r, having come down into P, P4 also is supposed to merge with P0. It is clearly implied that P2 and P3 move parallel to themselves.

The idea is rather neat. It is necessary, of course, to think of the rotation as involving the plane of the vector r4 and not the vector itself, except to the extent that the latter is constrained to lie in its own plane. It is as though the vector had a slippery turntable sliding underneath it. The arrangement is not impaired by the fact that in general the apsidal line is not located a quadrant's distance from the nodal line, again because it involves primarily rotation of the planes.

There is, however, one vitiating circumstance: P2 and P3 never change their common angle with PO; hence r2 and r3 never are contained in P0. In particular, the endpoint of r3 is not in P0 when r1 passes the node. For this reason P4 will then be close to Po, but not actually in it as Ibn al-Shatir asserts. In the original there is an illustration along the lines of Figure 1, but it is drawn as though r2 and r3 had shrunk to zero.

At folio 32v:8 he remarks: " The modified center, not the absolute, must be used for the arrival of the epicycle center at the two nodes." The word " center " (markaz) in this case means the longitude of the epicycle center measured from the apsidal line as base. The author is simply saying that the event in question does not occur with the arrival of r, at the nodal line (the direction of r1 being the mean or absolute, mutlaq, center) but when the endpoint of rs reaches the node. The direction of the endpoint of r3 is the modified (mu'addal) center.

At the very end of the chapter (fol. 33v: 1), almost as an afterthought, the following statement appears: " If we assume P2 to be in P1, and we assume Pa to be inclined to P2 at the amount of the assumed inclination of P2 to P,, and we assume the inclination of P4 to P3 (left) as it is, the consequence is the same." This is the situation shown in Figure 2. The same remarks apply to it as to the first case. In fact, when r, is at the node, P4 is closer to P0 than before, as can be seen from a comparison of Figures 1 and 2.

.SECOND Mt, OD L (P, - .) ..........................................

FIGURE 2

In order to test the effect on the extremal latitudes, computations have been made and the results reported in Table 2, juxtaposed with the externals called for by the observations of Ptolemy. The method of com- putation, though tedious, is a straightforward matter involving the solution

214




of a succession of plane triangles and need not be described here. As was to be expected, the divergences between the two models are not in general large, the vectors r2 and r3 being small in comparison to r1.

VI. THE INFERIOR PLANETS

Chapter 25 of the Nihdyat al-Sul is divided into two sections, one on the latitude of Venus (beginning on fol. 33v:4), the second (starting with fol. 34v:12) on the latitude of Mercury. In turn, each section has two parts: the A lmagest model and the Hypotheses model are described for each planet. Thus there are four cases, and for each Ibn al-Shatir works out a mechanism which (sometimes with slight modification) satisfies the Ptolemaic require- ments. Again the essence of the solution seems to be the same exploitation of the double angular velocity of r3 with respect to r1 which is peculiar to his own longitude model and which was utilized for the superior planets.

For all planets Ptolemy determines /PoP4 by prescribing the inclination of a pair of orthogonal diameters on the epicycle. Let us call them dc and dp, the former being the diameter whose projection on the ecliptic would pass through the deferent center. In the case of the superior planets, dp is maintained parallel to Po. With the inferior planets, however, the angles made by both with P1 oscillate sinusoidally with the period of the mean planet, and in such a manner that one is 90? out of phase with the other. For both planets the apsidal and nodal lines are separated by a quadrant, and the inclination of da to P1 is maximum at the node, that of dp along the apsidal line. These maxima are shown in Table 3. The whole

TABLE 3

Cl C2 C'2 Ca C4 C'8 C'4

At At At nodal nodes apogee At nodes quadrature

z P,1 Max Z PP4 L PPA Z PP4 L P2P3 Z PaP4

Venus Almagest 0;10? 2?0 310 30? ? 0;5? 0;5?

Hypotheses 0;10 31 3j 0 3+ 0;5 0;5

Mercury Almagest - 0;45 a 61 7 61f -0;5 -0;5

Hypotheses -0;10 6+ 61 0 6+ -0;5 -0;5

a Ibn al-Shatir has - 0;10.

arrangement is a complicated way of insuring that the epicycle plane, P4, shall move more or less parallel to Po.

Following is a translation of the passage (fol. 34r:9-34v:3) in which Ibn al-Shatir gives his solution for a model satisfying the Almagest para- meters for Venus:

Neither Ptolemy nor anyone else up to this time has set down elements that (can) move the epicycle and the deferent (according) to the (above-) mentioned description because of the difference of P4's inclination in the two cases and the difference in (the inclination of) the two (epicycle)

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VICTOR ROBERTS

diameters, one of which is inclined and the other lies either in P0 or P1. But it has been facilitated (for us) by the grace of Almighty God. It is that we assume the maximum inclination of P1 to the north to occur at the apogee of Venus. Its maximum is one sixth of a degree (0;10?), and it is an inclination of a constant magnitude to the north and the same to the south.... And we assume the centers of the epicycle and the rotator (i.e., the initial points of r4 and r3) to be on the line passing through the apogee, and we assume P2 to be in Pi, and we assume the mean epicyclic apogee to be inclined to the south of P1 at an angle whose magnitude is five minutes. The half of P3, the beginning of which is from the deferent apogee and which is called the first, is north of P1 by an angle at its center whose magnitude is three degrees. We assume that the epicyclic apogee is inclined to the apogee of r3 to the north at five minutes. So the epicycle diameter through the apogee remains in Pt, and we assume the first half of P4, the beginning of which is at the apogee, to be inclined also to the north side at half a degree with respect to the half of P3 which is inclined to the north side.

That having been decided, (let) the orbs rotate; i.e., r1 rotates a quadrant and r2 a quarter circle also, and meanwhile r3 turns through a semicircle. Then the inclination of P4 turns over and the (epicycle) diameter perpen- dicular to the (diameter through) the apogee (d,) merges with Po and the other diameter (d^) becomes inclined to P0 at an angle whose magnitude is two and a half units, on the assumption that the difference of the epicycle inclination is the same as that mentioned in the Almagest. But what the moderns (al-muta'dkhirun, as opposed to " the ancients," al-mutaqaddimin) have amended is that the inclination of the epicycle at the apogee and perigee is three and a half units. (Now) we illustrate his view in the Hypotheses.

The same description, in slightly different words, and with the numbers shown in the second, third, and fourth lines of Table 3, is given also for the Hypotheses Venus model, and for the A Imagest and Hypotheses Mercury models, respectively. The formulation is very obscure, but if any sense at all is to be made from Ibn al-Shatir's latitude theory for the inferior planets, it must be deduced from this passage, for there is nothing else to go on. As stated in the next section, he did not compute tables on the basis of these models, and the obscurity of this explanation may well reflect lack of clarity in his own mind.

Be that as it may, the passage contains two elements which indicate his

approach to the problem. The first is his emphasis that when r, rotates

through 90?, r3 turns through 180?. The second is the fact that for all four models, relations analogous to Equations (1) of Section V subsist, namely

C3 -

C4 = C2,

(2) c3 + C4 = Cf2

and Ct3 - C4 = 0,

(3) C't + C4 = Cl;

~~~so~~ca = c4 = C2 c\3= cP4 = c1/2.

216




These observations jointly lead us to reformulate Ibn al-Shatir's basic latitude device, already encountered in Section V, as follows.

It is desired, by a rotation of 180? about some axis, to change by an arbitrary angle 0 the inclination of some plane P with respect to a fixed plane Q without changing the direction of the inclination. That is, if P' is the rotated position of P, the lines of intersection between P and Q and between P' and Q should be parallel.

A solution is obtained by choosing an axis in a plane normal both to P and to Q and which is inclined to P at an angle of 90? - 0/2. The situation is portrayed in Figure 3; there, since P and Q are seen in profile, all the angles involved appear in their true sizes. The required axis is n, and a plane normal to it is N, inclined at 0/2 to P and to P'.

\n

X /,P' \90o'- yP

' Q-tQ

FIGURE 3

We submit that Ibn al-Shatir attempts to satisfy the limiting conditions given in the columns headed c1, c2, and c'2 in Table 3 (all of which are of Ptolemaic origin) by two simultaneous applications of the device illustrated in Figure 3. He changes the Almagest model to the extent of making 1PoP1 constant at 0;10?, instead of being oscillatory with 0;10? as a maximum.

One requirement is to bring the epicycle diameter dc from an inclination of zero with respect to P1 (hence an inclination of 0; 10? with respect to P,) at the apogee position into an inclination of 0;10? with respect to P1 at the node. So now 0, the required change of inclination, is 0;10?, and Equations (3) are invoked to give c'- = 0;5. Let P3 (corresponding to N in Fig. 3) be inclined to P1 and P2 (the latter two coinciding) at a constant angle of 0;5?. This should also be the inclination of dc to P3. Now, in going from apogee to node, r3 with P3 rotates through 180? about n3 (the

217



VICTOR ROBERTS

n of Fig. 3) carrying d, with it and changing its inclination with respect to P1 by 0;10?. Hence the angle that do makes with r1, now in P0, is also to be 0;10?, as required. Figure 4 illustrates the idea schematically.

The second requirement (for the Almagest Venus model) is to bring the other epicycle diameter, dp, from an inclination of 3?? at the apogee to 21? at the node. Now Equations (2) apply, and 0 is 1?. The same rotation of 180? about n3 is to accomplish this also. Figure 5 illustrates the second application.

\nl \

d (oat node)

_ _ _~~~~~

FIGURE 4

There is no particular gain in describing in detail the arrangements for the other three models. The basic apparatus is the same; only the para- meters are changed as shown. The object is always to produce a machine which will produce the Ptolemaic latitudes at apogee, perigee, and nodes, with smooth transitions in between. The text speaks only of the quadrant's motion from apogee to node, but the presumption is clear that the process is to continue around the rest of the ecliptic.

FG rz -u

FIGURE 5

If our interpretation of the text is correct, it follows that the arrangements are unsatisfactory for several reasons. For one thing, the validity of the transformation shown in Figure 3 depends upon the fact that the lines accurately measure the inclinations of the planes they represent. Ibn al-Sihatir attempts the same thing with dc and dp, but neither can be taken

218

- '

?' \i




as measuring the variable inclination of the plane P4 in which they lie. Moreover, as the epicycle center moves from apogee to node, although r3 indeed rotates through 180?, dc and dp turn through a quadrant, which also vitiates the device which worked reasonably satisfactorily with the

superior planets. VII. CONCLUSIONS

From an examination of Ibn al-Shatir's zij (astronomical handbook) it is clear that he did not take the ultimate step (which he did for longitudes 5)

of reducing the latitude theory to numerical tables for the calculation of

ephemerides. In the Bodleian copy of the zij (MS Seld. A. inf. 30, fol. 28r: 19-22) he says

The latitudes of these planets (i.e., the entries in the latitude tables) are taken from the Almagest, except for the first latitude of Venus, which is according to what the successors (of Ptolemy) have corrected. The whole of these latitudes is what the successors have agreed upon in their books. And if God permits (me) time, I will work them out as I have (already) worked out the mean motions and the planetary equations.

The latitude tables of the zij are indeed those of the Almagest, modified

slightly in the manner indicated above. For whatever reason, whether it was indeed a matter of time, or whether he sensed that his constructions would not perform quite as he asserted, the matter was carried no further.

The situation with Copernicus is somewhat different. An examination of the De revolutionibus,6 the Commentariolus,7 and principally a set of unpublished notes on the Copernican theory by Neugebauer and made available by him, has shown that Copernicus proceeded to calculate his own latitude tables on the basis of his models.

Thus, in contrast to the planetary longitude theory, insofar as the latitude

theory and tables are concerned we find no evidence of an influence of Ibn al-Shatir upon Copernicus.

5 Abbud, op. cit. Copernican Treatises: The Commentariolus, 6 Copernicus, Nikolaus Kopernikus Gesamt- The Letter Against Werner, The Narratio

ausgabe. Band II: De revolutionibus orbium Prima of Rheticus (2nd ed. New York: Dover, caelestium (Munich: R. Oldenbourg, 1939). 1959).

7 Edward Rosen (ed. and trans.), Three

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the planetary theory of ibn al-shatir: latitudes of the planets

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