Goldstein - The Arabic Version of Ptolemy's Planetary Hypotheses

The Arabic Version of Ptolemy's Planetary HypothesesAuthor(s): Bernard R. GoldsteinSource: Transactions of the American Philosophical Society, New Series, Vol. 57, No. 4 (1967),pp. 3-55Published by: American Philosophical SocietyStable URL: http://www.jstor.org/stable/1006040 .

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THE ARABIC VERSION OF PTOLEMY'S PLANETARY HYPOTHESES

BERNARD R. GOLDSTEIN *

CONTENTS

PAGE Introduction ................................ 3 Summary ................................ 4 Description of the manuscripts .................... 5 Translation .................... 5 Commentary .................... 9 Arabic Text .................... 13

INTRODUCTION

The Ptolemaic System is the name usually given to the world picture, current in the Middle Ages and the Renaissance, according to which the planetary 1 spheres are nested to fill exactly the space between the highest su1blunary element, fire, and the fixed stars. There is, of course, no trace of it in Ptolemy's Almagest and until now it has not been found in any other of his works. The earliest textual evidence for this system has been a passage in Proclus 2 (412-485) in which this hypothe- sis is mientionied olnly in an anonymous fashion.

In a recent paper,3 Willy Hartner concluded that the Ptolemaic System is in fact due to Ptolemy and ap- peared in his Planetary Hypotheses, of which only a part survives in the original Greek. Professor Hartner noticed that several Arabic authors ascribe values for planetary sizes anid distanices to a work by Ptolemy called kitab al-manshfir&t, which, he shows, is the Arabic title for the Planetarv Hypotheses. The published version of this treatise does not, however, include this passage, and Hartner was left with no alternative but that part of the text was lost.

The Planetary Hvpotheses was included by Heiberg in his edition of Ptolemy's minor astronomical works.4 There one finds an edited Greek text of what Heiberg took to be all of Book I and a German translation of the

* This research was supported by a National Science Founda- tion research grant (GS 821).

1 Following medieval usage, Sun and Moon are included among the planets.

2 Hvpotyposis, ed. Manitius, p. 220 if. 3 W. Hartner, "Medieval Views on Cosmic Dimensions"

MWlanges Alexandre Koyre (Paris, 1964), pp. 254-282. 4 J. L. Heiberg, Claiidii Ptolemaei opera qluae extant oninia,

Voiumnen II, opera astronomnica mninora (Leipzig, 1907), pp. 69-145.

Arabic version of Books I and II (the latter did not survive in Greek). Heiberg informs us in his introduction that the translation of the Arabic version was begun by Ludwig Nix, but that his early death left it to others (Heegaard and Buhl) to complete his task.5 As it turns out, the two Arabic manuscripts, from which they worked, contain a passage on planetary sizes and distances at the end of Book I, and yet Heiberg's edition gives us no inkling of it. Ironically, the omitted passage contains what is really of out- standing historical interest: that the Ptolemaic System is indeed the creation of Ptolemy.

Hartner's paper was the key to this discovery, for it led me to investigate whether there were any manuscripts of this text which had not been used in the preparation of the published edition. I soon found that Steinschneider 0 mentions a Hebrew manuscript in Paris of the Planetary Hypotheses. By chance, a microfilm of that manuscript was already in my posses- sion, and to my pleasant surprise, it contained the section on sizes and distances at the end of Book I. It seemed odd that the new passage belonged in the middle of the published version, rather than at the end as one might have expected, but the Arabic manuscripts con- firmed the Hebrew version.

The aim of this paper is to present a translation and commentary on the previously unpublished part (which I shall call part 2) of Book I of Ptolemy's Planetary Hypotheses. Moreover, it seemed desirable to include the Arabic text of the entire work with variant readings, because it. too. has never hben niihlidhdl A oz-nprn1

5 Heiberg, p. ix. The Arabic text is also discussed on pp. xvi and clxxiv.

O M. Steinschneider, Die Hebraeischen Uebersetzungen des llittelalters (Berlin, 1893) 2: p. 538.

3



4 GOLDSTEIN: PTOLEMY ON THE PLANETS [TRANS. AMER. PHIL. SOC.

summary of Book I, part 2, is presented below, but detailed comments on the numerical data are postponed to the commentary which follows the translation of this

passage. For ease of reference sections of the text have been numbered in the translation, although the Arabic text does not do so.

SUMMARY OF BOOK I, PART 2, OF PTOLEMY'S PLANETARY HYPOTHESES

The first part of Book I, published by Heiberg, includes a description of models for planetary motion and a set of parameters for these models.7 It ends with the description of Saturn's model, and this, in effect, is the end of a major section. The second part of Book I, published here for the first time, begins with a short Aristotelian introduction (see Section 1 of the translation below) on the nature of planetary motion followed by a general characterization of the planetary models.

Section 2 concerns the arrangement of the planetary spheres, and the place of the Sun in the order of the planets. Ptolemy remarks that the parallax of a planet is too small to be measured, and so its distance cannot be directly computed. We are then informed that a transit of the Sun would guarantee that a planet lay below the Sun, but Ptolemy was not aware of any such report. It is further argued that a transit might pass unnoticed because the greater part of the Sun would still be exposed, "for when the Moon eclipses part of the Sun equal to, or even somewhat greater than, the diameter of one of the planets, the eclipse is not perceptible." Nevertheless, the order agreed upon is: Moon, MIercury, Venus, Sun, Mars, Jupiter, Saturn, Fixed Stars. Two principles are now invoked (Section 3): (1) the ratio of the relative distances of a planet from the center of the earth, produced by its model, is equal to the ratio of the true distances of the planet from the earth, (2) the minimum true distance of a planet is equal to maximum true distance of the planet just below it, i.e. the planetary spheres nest inside one another. The minimum and maximum lunar distances are taken from the Almagest and rounded to 33 and 64 earth radii respectively. From the ratio of Mercury's maximum distance to its minimum distance, and likewise for VTenus, Ptolemy finds that the maximum distance of Venus is equal to 1,079 earth radii. Since the minimum solar distance was independently found to be 1,160 earth radii, Ptolemy notes that a space would exist between the spheres of Venus and the Sun contradicting the first principle mentioned above. But the space would be too small for the sphere of Mars, and so he argues that the Sun might be somewhat closer to the earth, for "when we increase the distance to the Moon, we are forced to decrease the distance to the Sun, and vice versa. Thus, if we increase the distance to the MIoon slightly, the distance to the Sun will be somewhat diminished and it will then correspond to the greatest distance of

7 For a comparison of these parameters with those in the Almagest and other works of Ptolemy, cf. e.g. B. L. van der Waerden, "Klaudios Ptolemaios," in Pauly-Wissowa, Realencvy- clopddie, 46 (1959).

Venus." The distances to the outer planets are then determined under the assumption that the minimum distance of Mars is equal to the maximum solar distance (1,260 earth radii).

Ptolemy states (Section 4) that the radius of the earth is equal to 2 ;52 myriad stades, and then converts the planetary distances from earth radii to stades.

Section 5 concerns the apparent sizes of the planets and includes a remark of Hipparchus not preserved in the Almagest. Ptolemy states that, in agreement with Hipparchus, he found the apparent diameter of Venus to be a tenth that of the Sun, when Venus is at mean distance. He then gives the apparent diameters of the other planets, but does not explicitly ascribe these values to Hipparchus.8 The apparent lunar diameter is given as 1, times the diameter of the Sun when the Moon is at mean distance (48 earth radii). Clearly this value is based on the model and not direct observation. From the apparent diameters of the planets, the true diameter of the Sun, and the planetary distances, Ptolemy derives the true diameters and volumes of the planets in relation to the earth. The celestial bodies in descending order of volume, then, are: Sun, first magnitude stars, Jupiter, Saturn, Mars, Earth, Venus, Moon, and Mercury.

Section 6 deals with the arcus visionis of the planets for their risings and settings and agrees with the values which Ptolemy used in the tables for planetary visibility in the Handy Tables.9 For acronychal risings of an outer planet (i.e. when the planet rises as the Sun sets), Ptolemy says that the arcus visionis is half of the value stated for its heliacal rising. Neither the Almagest nor the Handy Tables deals with acronychal risings. He further tells us that sometimes Mercury will fail to appear, because the elongation required for its appearance may exceed its greatest elongation from the Sun,

Bar Hebraeus (Le Liz're de l'ascension de l'espirit, ed. F. Nau [Paris, 1899], pp. 193-195) refers to this passage of Ptolemy but asserts that all of the apparent sizes were determined by Hipparchus. Nau identified K. da-nasire, the source mentioned by Bar Hebraeus for the apparent planetary sizes, as the Ce1ttiloquiitm, but Nallino (Al-Battani sive albatenii opus astrotomniculm. [Rome, 19071 2: p. xxvi) correctly pointed out that it is simply the Syriac translation of K. al-Manshiirat, i.e. Ptolemy's Planetary Hypotheses. Mr. Noel Swerdlow, who is engaged in a study of the medieval treatment of planetary sizes and distances, brought these passages to my attention. I wish to thank him for his assistance in preparing this paper.

9 Cf. B. L. van der Waerden, "Die handlichen Tafeln des Ptolmaios," Osiris 13 (1958) : p. 71. In the Handy Tables Ptolemy does not indicate an arcuts visionis for first magnitude stars on the ecliptic, but in the Planetary Hypotheses he does.



VOL. 57, PT. 4, 1967] DESCRIPTION OF MANUSCRIPTS 5

In Section 7 Ptolemy discusses an optical illusion which affects the estimation of apparent sizes at great distances. Such optical illusions are also treated in Ptolemy's Optics, though I have not found a passage there corresponding exactly to this one. 10 Nevertheless,

this passage in the Planetary Hypotheses reinforces the arguments for the authenticity of Ptolemy's Optics, which is only preserved in a Latin translation of an Arabic version.

DESCRIPTION OF THE MANUSCRIPTS

The manuscripts used for this edition are:

BM. British Museum, MS. arab. 426 (Add. 7473), fol. 81b -102b (Heiberg, cod. A).

L. Leiden, MS. arab. 1155 (cod. 180 Gol.), fol. 1a- 44a (Heiberg, cod. B).

Hebrew. Paris, MS. hebr. 1028 (ancien fonds 470), fol. 54b-87a.

The British Museum manuscript is dated (A.D. 1242),11 btut contains no information concerning the author of the Arabic text. The copyist tells us, however, that this copy was careftully collated with its prototype.

The Leiden manuscript is undated,12 but it informs us (fol. la) that the redactor of the Arabic version was Thlbit b. Qurra (d. 901) who is also known to have revised several other Arabic translationis of Greek scientific works.'1" The folios in MS. L are in disorder. The proper arrangement of the folios is: 1 to 21; 25 to 27; 22; 23a; 24b; 28 to 44a. Folios 23b and 24a are blank. No figures appear although blank spaces were left for them. A short note on fol. 44a, in the same hand as the rest of the mainuscript, informs us that the length of the sidereal year is about 3651/4 + 1/147 days, according to what Ptolemy proves in this treatise. This year length was computed by someone, other than Ptolemy, from the sum of Ptolemy's tropical year 3651/ - M300 days

and his value for precession, 10 per century, i.e. 3651/4 + 1/147 3651/4 - 1/3o + 1 ;0,55%oo

for 10 of solar motion takes 1 ;0,53 days. The Hebrew manuscript informs us in the colophon

(fol. 87a) that the translator was Kalonymos b. Kalonyimios. The catalogue entry for this manuscript indicates that the translation (from Arabic) was prob- ably completed in 1317 and that this unique copy is (lated 1342. 14 No figtires appear in the manuscript, although space was left for them.

In the margin of the Arabic text, presented below (pp. 13-55) which is a facsimile of MS. BM,15 I have indicated corresponding folios in MS. L (e.g. L lb),

and corresponding pages in the Heiberg publication (e.g. H 71). The variant readings of MS. L are presented below the facsimile of MS. BM.

The title of this treatise which appears on L 1 ais dis- played as the first footnote to BM 81b.

TRANSLATION OF BOOK I, PART 2, OF PTOLMEY'S PLANETARY HYPOTHESES

[BM 88a,4] 1. These are the models (hai'a) of the planets in their spheres. As we have said, there are anomalies in the motions of the (planetary) spheres not found in the sphere of the fixed stars, for the latter sphere's motion is very close to that of the universal motion, whose sphere, of necessity, has a simple nature, unmixed with anything, and containing no contrarity at

10 Cf. A. Lejeune, L'Optiquce de Claude Ptoleme'e (Louvain, 1956), pp. 124* ff., 74 ff. et passim; A. Lejeune, Euclide et Ptolemee (Louvain, 1948), p. 95 ff.

11 Cf. Catalogius codd. mss. orientalium, qui in Museo Brittan- nico asser-vantur (London, 1852), p. 205 ff. The catalogue quotes the beginning and end of Book I, as well as of Book II.

12 Cf. P. de Jong and M. J. de Goeje, Catalogus codicum orientalium Bibliothecae Academiae Lugduno-Batavae (Leiden, 1865) 3: p. 80.

13 H. Suter, Die Mathematiker und Astronomen der Araber ind ihre Werke, A bhandlungen zur Geschichte der mathe- nmatischen Wissenchaften X. Heft (Leipzig, 1900), p. 34ff.

all. The planets, all of which lie below the (prime) mover, move with it from east to west, and also move with another motion from west to east. They move forward and backward, and to the south and to the north, which are the directions of local (makdniya) motion. Local motion is the first of the remaining motions and things whose nature is eternal have only this kind of motion. The changes and opposition in quality' and quantity, and the coming-into-being of things which are not eternal are not like the changes apparent to us in the eternal, for these changes are in the thing itself and its substance.

The Sun, in our opinion, has but one anomaly in its 14 Catalogues des manutscrits hebreux et samaritains de la

Bibliotheque Impe'riale (Paris, 1866), p. 186. 15 r wish to thank the Trustees of the British Museum for

their permission to publish this manuscript. 1 Kaifiva: Hebr. eykh.




motion in the ecliptic, because there is nothing stronger than it to give it another anomaly in its motion. The remaining planets have two kinds of anomaly; the first anomaly, similar to the one we mentioned (for the Sun), depends on the place in the ecliptic, and the other depends on the return to the Sun. Each of the planets has one free motion, the other is determined of necessity. The motion of the planets in the two directions (north and south) takes place with respect to both the sphere of the fixed stars (i.e. the equator) and the sphere of the Sun (i.e. the ecliptic). The first variety of this motion is simply due to the inclination of the ecliptic to the equator. The Moon has two such (motions), the one just mentioned, and the other due to the inclination of its orbit to the ecliptic. The five planets have three such (motions), and three is the greatest number of variations which occur; two of them have already been mentioned, and the third is due to the inclination of the deferents, which rotate about the earth, to the epicycles. The characteristics ('amr) of these spheres (i.e. the epicycles) are similar to those of the rest of the inclined spheres (i.e. the deferents). But one may imagine differences between the two kinds of spheres because (the epicycles) do not go around the earth, for the earth lies outside of them. Moreover, motion on the inclined spheres brings about motion in the two opposite directions (north and south of the ecliptic), whereas motion on the epicyclic spheres takes place on planes parallel (muwze7zsh) to the ecliptic. The inclination of epicycle to deferent is fixed, like that of the [BM 88b] ecliptic to the plane of the equator.

If we imagine the intersection of (the ecliptic with) the meridian al)ove the earth as the apogee, and that undler the earth as the perigee, then the horizon in both dlirectiolis serves as the mean distance. The inclination of the ecliptic is one and the same and does not change. The motion of this sphere, inclined to the equator, takes place about its poles. The northern limit of this sphere is called the summer solstice, and it is sometimes on the intersection analogous to the apogee (i.e. iiidheavein), somletimes at the poinlt in the east, and solmietimes at the point to the west. Similarly, the southern limit is the winter solstice. The vernal point is analogous to the ascending node; it too may lie in the direction of apogee (i.e. midheaven), or of perigee (i.e. lower midheaven), or to the east, or to the west. The same is true for the autumnal point which is analogotus to the descending node. In similar fashion, we may imagine all of the conditions of the inclined sphere that surrounds the earth. The sphere of the Moon has characteristics similar to those mentioned, as do the eccentric spheres which incline to the epicycles.

When we wish to turn our atttention from the first type (of inclination) to the second type (of inclination), we need do no more than replace the equator by the ecliptic, and the ecliptic by the deferent. In the third type of inclination, which takes place outside the earth, the role of the equator is taken on by the fixed epicycle

and the role of the ecliptic is taken on by the deferent; and the inclination varies in the way I shall describe.

We see that the spheres surrounding the earth, on which move the Sun, the centers of the epicycles, the Moon, and the planets, return according to their periods. The epicycles return with the return of the centers of the epicyclic spheres, not with the return of the planet that moves on them-this is the condition (h.1) for each of one of the spheres.

2. The arrangement of the spheres has been a subject of some doubt up to this time. The sphere of the Moon is certainly the closest sphere to the earth; the sphere of Mercury closer to the earth than the sphere of Venus; the sphere of Venus closer to the earth than the sphere of Mlars; the sphere of Mars than the sphere of Jupiter; the sphere of Jupiter than the sphere of Saturn; and the sphere of Saturn than the sphere of [BM 89a] the fixed stars. It is clear from the course of the planets that this sphere is closer to the earth and that sphere further away, along a straight line from the eye. But with respect to the Sun, there are three possibilities: either all five planetary spheres lie above the sphere of the Sun just as they all lie above the sphere of the Moon; or they all lie below the sphere of the Sun; or some lie above, and some below the sphere of the Sun, and we cannot decide this matter with certainty.

The distances of the five planets are not as easy to determine as those of the two luminaries, for the distances of the two luminaries were determined, mostlv, on the basis of combinations of eclipses. A similar proof cannot be invoked for the five planets, because no phenomenon allows tIs to fix their parallax with certainty. Moreover, up to this time we have not seen an occultation of the Sun (-by any of the planets), and therefore it is possible for one to assert that all five planetary spheres lie above the sphere of the Sun. But the argument so far does not permit one, whose inten- tion is to seek the truth, to draw a conclusion. Firstly, the occultation of a large body (the Sun) by a small one (a planet) may not be perceptible on account of the remainder of the solar body which would still be exposed, for when the Moon eclipses part of the Sun equal to, or even somewhat greater than, the diameter of one of the planets, the eclipse is not perceptible. Moreover, such events could only take place at long intervals, for (an inner planet) is closest to the Sun (in longitude) when it is at the apogee and perigee of its epicycle; but (the planet) is found in the plane of the ecliptic only twice in every revolution on the epicycle, when it passes from the north to the south, and when it passes from the south to the north. When the center of the epicycle is at one of the nodes, and the planet is also at that node, and the planet is also at the apogee or perigee (of its epicycle), then the planet may hide (part of the Sun). According to those who report observations and examine them carefully, a long time (ntiqddr al-zaman)



VOL. 57, PT. 4, 1967] TRANSLATION 7

must elapse before the return (of the center) of the epicycle and the planet in conjunction (with the Sun) above the earth. With these conditions, it is clear that one cannot judge with certainty for the two (inner) planets, nor even for the planets on which it is agreed that they lie above the sphere of the Sun, i.e. Mars, Jupiter and Saturn.

3. We began our inquiry into the arrangement of the spheres with the determination, for each planet, of the ratio of its least distance to its greatest distance. We then decided to set the sphere of each planet between [BM 89b] the furthest distance of the sphere closer to the earth, and the closest distance of the sphere further (from the earth). Let us assume that only the spheres of MIercury and Venus lie below the sphere of the Sun, but that the others do not. We have explained in the Almagest (K. al-sitaksis) 2 that the least distance of the Moon is 33 earth radii, and its greatest distance 64 earth radii, dropping fractions. Moreover, the least distance of the Sun is 1,160 earth radii, and its greatest distance 1,260. The ratio of the least distance of Mer- cury to its greatest distance is equal to about 34:88, and it is clear from the assumption that the least distance of Mercury is equal to the greatest distance of the Moon, that the greatest distance of Mercury is equal to 166 earth radii, if the least distance of Mercury is 64 earth radii. The ratio of the least distance of Venus to its greatest distance is equal to about 16:104. It is clear from the assumption that the greatest distance of Alercury is equal to the least distance of Venus, that the greatest distance of Venus is 1,079 earth radii, and the least distance of Venus 166 earth radii. Since the least distance of the Sutn is 1,160 earth radii, as we mentioned, there is a discrepancy between the two distances which we cannot account for: but we were led in- escapably to the distanices which we set down. So much for the two (planetary) spheres which lie closer to the earth thani the others. The remaining spheres cannot lie between the spheres of the Moon and the Sun, for even the sphere of Mars, which is the nearest to the earth of the remaining spheres, and whose ratio of greatest to least distance is about 7:1, cannot be accommodated between the greatest distance of Venus and the least distance of the Sun. On the other hand it so happens that when we increase the distance to the Moon, we are forced to decrease the distance to the Sun, and vice versa. Thus, if we increase the distance to the Moon slightly, the distance to the Sun 3 will be somewhat dimiinished and it will then correspond to the greatest distance of Venus.

The argument which forces the above-mentioned order of spheres is not entirely based on the distances, but on the differences in their motions as well. The most compelling argument is that the further from the hy-

2 Hebrew: ha-magisti. 3 With L and Hebrew; BM reads: Moon.

pothesis of the Sun, which is in the middle from all standpoints, the further (the sphere must be) from the Sun. Thus the sphere of Mercury is adjacent to the sphere of the Moon, for both the spheres of Mercury [BM 90a] and the Moon are eccentric, and the eccenter moves about the center of the universe in the direction of the daily rotation, in contrast to the motion of (the centers of) their epicycles; and it follows that these centers lie at apogee and perigee twice in every revolution. The spheres nearest to the air move with many kinds of motion and resemble the nature of the element adjacent to them. The sphere nearest to universal motion is the sphere of the fixed stars which moves with a simple motion, resembling the motion of a firm (mnathbuit) body whose revolution in itself is eternally unchanging.

The distances of the three 4 remaining planets may be determined without difficulty from the nesting of the spheres, where the least distance of a sphere is con- sidered equal to the greatest distance of the sphere below it. The ratio of the greatest distance of Mars to its least distance is, again, 7:1. When we set its least distance equal to the greatest distance of the Sun, its greatest distance is 8,820 earth radii and its least distance 1,260 earth radii. The ratio of the least distance of Jupiter to its greatest distance is equal to the ratio 23 :37.5 When we set the least distance of Jupiter equal to the greatest distance of Mars, its greatest distance is 14,187 earth radii and its least distance 8,820 earth radii. Similarly, we set the ratio of the least distanice of Saturn to its greatest distanice equal to the ratio 5 :7, and the least distance of Saturn equal to the greatest distalnce of Jtipiter. Therefore, the greatest distance of Satturn, which is adjacent to the sphere of the fixed stars, is 19,865 earth radii, anid its least distance is 14,187 earth radii.

In short, taking the radius of the spherical surface of the earth and the water as the unit, the radius of the spherical surface which surrounds the air and the fire is 33,6 the radius of the lunar sphere is 64, the radius of Mercury's sphere is 166, the radius of Venus' sphere is 1,079, the radius of the solar sphere is 1,260, the radius of Mars' sphere is 8,820, the radius of Jupiter's [BM 90b] sphere is 14,187, and the radius of Saturn's sphere is 19,865.

4. The radius of the spherical surface of the earth and water is two myriad stades (al-astadhiya)7 and half and third and one part in thirty myriad stades [2 ;52 myriad stades], for the circumference (of the earth) is 18 myriad stades.

The boundary that separates the fiery and the lunar

4 With L and Hebrew; BM reads: fixed. 5 Hebrew 23 :38. 6 With L and Hebrew; BM reads: 63. 7 Hebrew: ris.



8 GOLDSTEIN: PTOLEMY ON THE PLANETS [TRANS. AMER. PlIIL. SOC.

sphere lies at a distance of 94 8 myriad stades and a half and a tenth myriad stades [94 ;36 myriad stades]. The boundary that separates the lunar sphere from the sphere of Mercury lies at a distance of 183 9 myriad (stades) and a third and a tenth and one part of thirty myriad stades [183;28 myriad stades]. The boundary that separates the sphere of Mercury from the sphere of Venus lies at a distance of 475 myriad stades and a half and a third and one part in thirty myriad stades [475; 52 myriad stades]. The boundary that separates the sphere of Venus from the sphere of the Sun lies at a distance of 3,093 myriad (stades) and a tenth of a myriad (stades) and one part in thirty myriad stades [3,093 ;8 myriad stades]. The boundary that separates the solar sphere and the sphere of Mars lies at a distance of 3,612 myriad (stades). The boundary that separates the sphere of Mars and the sphere of Jupiter lies at a distance of 2 myriad myriad and 5,284 myriad stades. The boundary that separates the sphere of Jupiter from the sphere of Saturn lies at a distance of 4 myriad myriad and 4,769 10 myriad and a third and one part of thirty myriad stades [44,769 ;22 myriad stades 11]. The boundary that separates the sphere of Saturn from the sphere of the fixed stars lies at a distance of 5 myriad myriad and 6,946 myriad stades and a third of a myriad stades.

If (the universe is constructed) according to our description of it, there is llo space betweeni the greatest anid least distances (of adjacent spheres), and the sizes of the surfaces that separate one sphere from another do not differ from the amounts we mentioned. This arranigement is most plausible, for it is not conceivable that there be in Nature a vacuum, or any meaningless and useless thing. The distances of the spheres that we have mentioned are in agreement with our hypotheses. But if there is space or emptiness between the (spheres), then it is clear that the distances cannot be smaller, at ally rate, than those menitioned.

5. It is now possible to determine the diameters of the celestial bodies in relation to one another. To determine these sizes, we need the apparent diameters of the planets, the models for their motions, and the scale of these models [lit.: bodies], which are given by the aforementioned distances. The procedure which allows Us to determine the sizes is described below.

Hipparchus said that the apparent diameter of the Sun is 30 times as great as that of the smallest star, and that the apparent diameter of Venus, which appears to be the largest star, is about a tenth the apparent diameter of the Sun. The diameters which are seen do not misrepresent (tughadiru) the vision of their true diameters perceptibly [?]. In this statement, Hipparchus

8 With L and Hebrew; BM reads: 74. 9 With L and Hebrew; BM reads: 133. 10 With L and Hebrew; BM reads: 4,999. 11 This number is corrupt; see commentary.

(also) said that he determined minimum values for the sizes of the celestial bodies, and that he used a common [BM 91a] distance in relation to which the earth is a point. Hipparchus did not make clear at which distance of Venus its diameter takes on the value quoted, but we consider this amount to be its apparent diameter at mean distance where the planet is usually seen, for at apogee and perigee it is hidden by the rays of the Sun. We too find that the apparent diameter of Venus is a tenth that of the Sun, as Hipparchus stated. Moreover, we find the diameter of Jupiter to be 1/12 the diameter of the Sun; Mercury's 1/15 the diameter of the Sun; Saturn's 1/18 the diameter of the Sun; and the diameter of Mars, and of first magnitude stars, 1/20 the diameter of the Sun. The diameter of the Moon at mean distance on its sphere, and mean distance of the eccentric sphere, is equal to 11/3 times the diameter of the Sun.

If all the diameters subtended the same apparent angle at their mean distances, the ratio of one diameter to another would equal the ratio of their distances, because the ratio of the circumferences of circles, as well as of similar arcs, one to another, is equal to the ratio of their radii. In the measure in which the diameter of the Sun is 1,210, the diameter of the Moon is 48; the diameter of Mercury 115; the diameter of Venus 6221?4; the diameter of Mars 5,040; the diameter of Jupiter 11,504; and the diameter of Saturn 17,026. The diameter of the first magnitude stars in this measure, as- suming that their (sphere) is adjacent to the furthest distance of Saturn, is 19,865, or about 20,000; and the amount is surely not less than 20,000. But the diameters do not subtend equal angles, for the diameter of the Moon subtends an angle 11/3 times that of the Sun, and the diameters of the planets subtend angles smaller than the-Sun in the ratios mentioned. It is clear that in the measure where the diameter of the Sun is 1,210, the diameter of the Moon is 64 because it is 11/3 times 48; the diameter of Mercury is 8 because it is about 1/is of 115; the diameter of Venus is 62 which is about 'ln of 6221/?; the diameter of Mars is 252 which is [BM 91b] 'Ao of 5,040; the diameter of Jupiter is 959 which is about 1/12 of 11,504; the diameter of Saturn is 946 which is about Y18 of 17,026; the diameters of the first magnitude stars is 1,000 which is 1k0 of 20,000, and they are certainly not smaller.

We have already explained in the Almagest (K. al- sitaksis)'2 that the solar diameter is 5? in the measure where the diameter of the earth is 1. This 5?k is to 1,210 as 1 part in 220. If we take this amount with the values previously set down, we find that in the measure where the diameter of the earth is 1, the diameter of the Moon is 1/4 and 1?4; the diameter of Mercury '27; the diameter of Venus 1/4 + 1k2; the diameter of the Sun 512; the diameter of Mars 114; the diameter of Jupiter 41/3 + 1/4o; the diameter of Saturn 4Y4 + %o; and

12 Hebrew: ha-magisti.



VOL. 57, PT. 4, 1967] COMMENTARY 9

the diameters of the fixed stars of first magnitude at least 41/2+ /2o. In the measure where the volume of the earth is 1, the volume of the Moon is 1/4o; the volume of Mercury is 149, 683 ;13 the volume of Venus is 1/44; the volume of the Sun 166Y3; the volume of Mars 1A; the volume of Jupiter 82Y2+Y?+Y '/o; the volume of Saturn 791/2; the volume of first magnitude stars at least 94Y' + 'A. Accordingly the Sun has the greatest volume, followed by the fixed stars of first magnitude. The third in rank is Jupiter, the fourth Saturn, the fifth Mars, the sixth earth, the seventh Venus, the eighth the Moon, and lastly Mercury.

We now repeat that, if all the distances have been given correctly, the volumes are also in accord with what we have said. If the distances are greater than those we described, then these sizes are the minimum [BM 92a] values possible. If their distances are correctly given, Mercury, Venus, and Mars display some parallax. The parallax of Mars, at perigee, is equal to that of the Sun at apogee. The parallax of Venus at apogee is close to that of the Sun at perigee. The parallax of Mercury at perigee is equal to that of the Moon at apogee, while the parallax of Mercury at apogee is equal to that of Venus at perigee. The ratio of each of them to the lunar and solar parallax is equal to the ratio of the distances that we have mentioned to the distances of the Sun and the Moon.

6. The first appearance of a star and its disappearance unider the rays of the Sun, takes place when the star is on the horizon, risilng or setting, and the Sun is near the horizon. The arcus visionis is measured on the great circle through the center of the Sun and the zenith. For first magnitude stars which lie on the orb of the zodiac, it is about 150; for Saturn about 13?; for Jupiter 90; for Mars 141/20; for Venus at morning setting and evening rising 70, but at evening setting and morning rising 50; for Mercury 120. For acronychal risings of the outer planets, the Sun must be below the earth (i.e. the horizon) by about half the above-

mentioned arc. (Two) different (values) were noted for the solar distance (arcus visionis) of Venus, but not for the other planets. The three outer planets, Saturn, Jupiter, Mars, appear anld disappear from under the rays of the Sun onlly when they are near the apogee of their epicycles. Mercury, however, may appear near both apogee and perigee, but in either case it disappears and appears near its mean distance. Sometimes the elongation required for appearance is greater than its greatest elongation so that on occasion it fails to appear altogether. Venus disappears and appears near apogee and perigee, and its magnitude ('uzm) at the time of its appearance varies owing to the difference in its distance (from the earth) at the times of its heliacal risings and settings.

7. Let us now consider the reason that our imagina- tion ascribes magnitudes to these celestial bodies which are not in the same ratio as their distances. We should recognize that this effect is an optical illusion in [BM 92b] accordance with (the principles of) optics (ikhtildf al-manazir). We shall explain this discrepancy in everything which is seen at a great distance. The eye cannot estimate such great distances, and similarly it cannot estimate the difference in the relative sizes of things of diverse magnitudes, for the eye (merely) gathers (the visual rays) which are then interpreted in terms of what is more familiar. Hence, the planets seem closer to us than they truly are, for the eye (naturally) compares them to things at more familiar distances, as we have explained. The (estimated) magnitude varies according to the distance, but at a smaller ratio (than geometric rules would require) on account of the weakness of our visual perception to dis- cern quantity of either kind (i.e. distance or magnitude), as we have mentioned.

End of Book I of Ptolemy's Planetary Hypotheses

COMMENTARY ON THE PLANETARY DISTANCES AND SIZES

Section 3: Planetary Distances in Earth Radii

In the Almagest (V, 13), Ptolemy states that, in earth radii, the mean distance to the Moon at syzygy is 59, at quadrature 38 ;43, and the epicyclic radius is 5 ;10. Hence the maximum distance to the Moon is 64;1O earth radii (59 + 5;10), and the minimum distance is 33 ;33 earth radii (38 ;43 - 5 ;1O). Dropping fractions we get the values here: minimum lunar distance, 33 earth radii; maximum lunar distance, 64 earth radii.

13 Hebrew: ?9,663-

The solar distance is given in the Almagest (V, 15) as 1,210 earth radii, but it is not explicitly stated that this is the mean distance. Here 1,210 earth radii is the mean solar distance, and the minimum and maximum solar distances are then correctly computed as 1,160 earth radii and 1,260 earth radii, for the solar eccentric- ity is 21' parts in 60.

Mercury presents a problem here for the ratio of its least to greatest distance cannot be 34:88, as the text states. With the parameters of the Almagest, the ratio




is 33 ;4 to 91 ;30 as Hartner has already noted.' In the first part of the Planetary Hypotheses,2 Ptolemy changes the parameters of his Mercury model 3 (the radius of the epicycle is changed from 22;30 to 22;15, and the radius of the circle on which the center of the deferent moves is changed from 3 to 2 ;30) but these parameters still do not yield the ratio 34:88 but rather about 34 to 90;15 (i.e. 60 + 3 + 2;30 + 2;30 + 22;15). From the ratio 34:88 and the minimum distance of 64 earth radii, Ptolemy derives the maximum distance of Mercury equal to 166 earth radii.

For Venus, Ptolenmy states here that the ratio of minimunm distance to maximum distance is 16 :104. From the model for Ventis in the Almagest,4 the ratio

D

S /~~~

Nr

p R~

x

FIG. 1

1 Hartner, "Medieval Views on Cosmic Dimensions," p. 267 ff. Hartner argues that the ratio 34 :88 is rounded from the ratio 64:166 (the minimum aind maximum distances in earth radii), and that the maximum distance in earth radii was computed with Almagest parameters for Mercury's relative minimum and maximum distances but with the erroneous value, 60 e.r., for the minimum distances in earth radii, i.e. 60- (91;3 %3.4) 166.

2 Heiberg, p. 87, 89. 3 For a description of Ptolemy's Mercury model, see 0.

Neugebauer, Exact Sciences in Antiquity (Providence, 1957), pp. 200, 207, and Hartner, p. 266 ff.

4 Cf. Hartner, p. 271.

should be 15 ;35 to 104;25 and this ratio was then rouinded to 16:104. (Note that both for Mercury and Venus the ratios have not been reduced to lowest terms.) From the ratio of minimum distance to maximum distance and a minimum distance of 166 earth radii, Ptolemy computes the maximum distance of Venus equal to 1,079 earth radii.

To eliminate the space between the maximum distance of Venus and the minimum distance of the Sun, Ptolemy suggests that the lunar distance be increased slightly. The solar distance was determined in the Almagest (V, 15) from three conditions: (1) the Mfoon at its maximum distance exactly covers the Sun arid they both subtend an angle of 0;31,20?; (2) the maximum lunar distance is 64;10 earth radii; and (3) the ratio of the shadow diameter to the lunar diameter is 2% to 1. As Ptolemy remarks here, the derivation of the solar distance is quite sensitive to small changes in the lunar distance. Thus, we can compute that for a maximum lunar distance of 65 earth radii, the mean solar distance is reduced to 985 earth radii. To demon- strate this, consider figure 1, where s (GD) is the radius of the Sun, m (HT) the radius of the Moon, r (MN) the radius of the earth, and n (RP) the radius of the shadow. The distance to the shadow is equal to the distance to the Moon, i.e.

TN=NP=d (1) Let a be the angle subtended by the radius of the Sun and the Moon, i.e.

angle DNG = angle TNH = a = 0;15,40? (2) Moreover we are given that

a=/a 2- (3) where p is the angle subtended by the radius of the shadow. Thus

n _ 2 5 (4) But

M = d sin a (d/2) crd 2 a = d x 0;0,16,24 (5) and

2r=ST+RP =SH+m+23 m (6)

or, letting r = 1 SH= 2 - 34 m (7)

In triangles MNG, and DGN SH GH DT 1 GN DN

Therefore DN-DT d

1-SH= (9) DN DN or

DN = d l (10)



VOL. 57, PT. 4, 19671 COMMENTARY 11

If we let d=65 (instead of 64;10) m =0;17,46 (instead of 0;17,33) (11)

and DN = 985 earth radii (instead of 1,210 e.r.) (12)

It is clear that the increase in the lunar distance affects the solar distance to a much greater extent than it affects the distance of Venus.

For Mars, Ptolemy here takes the ratio of its maxi- nmum distance to its minimum distance to 7:1. Using the parameters for Mars from the Almagest the ratio should be 105 ;30 to 14 ;30 which is only approximately 7 :1. Setting the minimum distance of Mars equal to the mlaximium distance of the Sun (1,260 earth radii), he finids the maximum distance of Mars equal to 8,820 earth radii. Note that, if the Sun is drawn in to eliminate the space between the maximum distance of Venus and the minimum distance of the Sun, one should no longer take 1,260 earth radii as the maximum solar distance.

For Jupiter, Ptolemy here takes the ratio of minimum distance to maximum distance equal to 23 :37. Using the parameters of the Almagest, the ratio is 45 ;45 to 74;15, btit here Ptolemy has rounded this ratio to 46:74 and reduced it to 23 :37. Ptolemy then computes the maximum distance of Jupiter and finds it equal to 14,187 earth radii, but my recomputation with the same data yields 14,188.7 earth radii. Since the words for seven and nine are easily confused in Arabic manuscripts, the Greek text may have read: 14,189.

For Saturn, Ptolemy here takes the ratio of minimum distance to maximum distance equal to 5 :7. Using the parameters of the Almagest, the ratio is 50 ;5 to 69 ;55. Ptolemy then computes the maximum distance of Saturn and finds it equal to 19,865 earth radii. But 5 of this amount is 14,189 earth radii, which seems to con- firm the suggested emendation for the maximum distance of Jupiter.

Section 4: Planetary Distances in Stades.

The circumference of the earth is given here as 18 myriad stades, the same value found in Ptolemy's Geography (VII, 5).5 Ptolemy takes the earth's radius equal to 2 ;52 myriad stades, from which it is clear that the amount of the circumference was rounded off from a more precisely computed figure, for 9,2:s2 (i.e. 3.139) is too low a value for 7.

The distances to the boundaries between the planetary spheres are now computed in stades from the values previously stated in earth radii and the value for the radius of the earth in stades. The numbers in the text are derived from the minimum distance of the Moon, 33 earth radii; the maximum distance of the Moon, 64 earth radii; the minimum distance of Venus, 166 earth

5 Cf. M. R. Cohen and I. E. Drabkin, A Source Book in Greek Science (Cambridge, Mass., 1958), p. 180.

radii; the maximum distance of Venus 1,079 earth radii; the minimum distance of Mars 1,260 earth radii; and the maximum distance of Mars, 8,820 earth radii. The distance to the boundary between Jupiter and Saturn presents a difficulty, and I see no way to correct the corrupt number in the text. If we take the distance in earth radii as stated in the text, 14,187, this boundary would lie at a distance of 40,669;24 myriad stades, whereas if we accept the emendation that the distance in earth radii is 14,189, the boundary would lie at a distance of 40,675 ;8 myriad stades. The distance to the boundary between the spheres Saturn and the fixed stars agrees with the maximum distance of Saturn, 19,865 earth radii.

Section 5: Relative Si-es of the Planets.

The apparent diameters included here, which be- came canonical values in the Middle Ages,6 are given in terms of the solar diameters when the planet is at its mean distance. Table I lists the mean distances, the apparent diameter, the true diameter, and true volume for each planet as stated in the text.

TABLE I

Mean Apparent True Distance Diameter Diameter Volume

in compared compared compared Earth to the to the to the

Planet Radii Sun's Earth's Earth's Moon 48 1 /3 1/4 + 1/24 Y40 Mercury 115 'A5 7 1/19,683 Venus 622? MO YA + 1/20 1/44 Sun 1,210 1 5 166% Mars 5,040 'Ao 1 /h 112 Jupiter 11,504 142 43 + Y40 821/4 + 1'o Saturn 17,026 'As 4Y4 + 'Ao 79'2 1. Magn.

Stars 20,000 Mo 41A + o20 94'/6 + Y8

The mean distances were computed by taking one-half the sum of the maximum and minimum distances already mentioned. The apparent diameter given for the Moon indicates that Ptolemy has taken his lunar model as accurately measuring the size as well as the distance of the Moon, i.e. if at maximum lunar distance (64 earth radii) the lunar diameter equals the solar diameter, then at % of that distance, the Moon must appear 4A times as large as the Sun.

The true diameter of each planet is derived from the mean distance and the apparent diameter as follows. Let d be the mean distance of the planet, D the mean distance of the Sun, a the apparent angular diameter of the planet, ,B the angular diameter of the Sun, k the ratio of the planet's apparent diameter to that of the Sun, P the true diameter of the planet, and S the true diameter of the Sun.

6 Cf. J. L. E. Dreyer, A History of Astronomy from Thales to Kepler (reprinted by Dover Publications, 1953), p. 258.




Then, P da (1)

and S- D: (2)

where a a=k (3)

Thus

p Sdk (4) D

The true diameter of the Sun, S, is taken from the

Almagest (V, 16) as equal to 5? earth radii, and the other diameters are computed from equation (4). For the true diameter of Venus, all of the Hebrew and Arabic manuscripts give the value entered in the table (i.e. 0;18). But according to my computation 1/4+ %o (i.e. 0;17) should be the result. This emendation is supported by the volume of Venus which was certainly computed by cubing 0;17.

The volumes are computed by cubing the true diameters, and both the diameters and the volumes were rounded in the computation as well as in the result. In the Almagest (V, 16) the volume of the earth is given as 391/4 times that of the Moon, whereas the volume of the Sun is given as 170 times that of the earth.



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18 GOLDSTEIN: PTOLEMIY ON THE PLANETS [TRANS. AMER. PHIL. SOC.

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Goldstein - The Arabic Version of Ptolemy's Planetary Hypotheses

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