Abstract
Joseph Ibn Naḥmias was a member of a distinguished and learned Spanish Jewish family, who lived in the late fourteenth-early fifteenth century; his one known work is dated to about 1400. This work, The Light of the World, has been edited and translated with a detailed technical commentary by Robert G. Morrison. It appears in two forms, each surviving in a single manuscript, the original Judeo-Arabic, a dialect of Arabic written in Hebrew letters, and a Hebrew translation. The two versions are here edited, the Judeo-Arabic translated completely, passages of the Hebrew that differ are translated, and both receive explanatory commentaries. The translations contain redrawn figures from the manuscripts, in which spheres are projected as plane circles, and the commentaries contain spherical figures, some very detailed, showing the circles and spheres in perspective and many additional details for analysis and computation. All of this is a considerable accomplishment because The Light of the World is extremely complex and in places very difficult to understand. Morrison has carried out an outstanding and exceedingly demanding undertaking for which we all must be grateful.
The principal subject of the work is descriptions of hypotheses, models, at length for the sun and moon, briefly for the fixed stars, and hypotheses for the planets, intended but not included or, it appears, not yet formulated, composed exclusively of homocentric spheres moving uniformly and centered on the earth, the center of the world. Although models of this kind were proposed in antiquity by Eudoxus and Calippus, those of concern here originate in the criticism of Ptolemy’s astronomy on the grounds that eccentric and epicyclic spheres violate the Aristotelian principle of strictly uniform celestial motions concentric to the earth. The earliest known appears to be by Ibn Bājja (Avempace d. 1139), although not completely because, as Maimonides had heard from his student, he excluded epicycles but permitted eccentric spheres. Ibn Rushd (Averroës d. 1198) wrote that in his youth he intended to formulate motions of the heavens of concentric spheres but did not do so. According to al-Biṭrūjī (Alpetragius late twelfth century), his teacher Ibn Ṭufayl (d. 1185) told him that he had discovered the correct principles for these motions without eccentric or epicyclic spheres, meaning strictly homocentric spheres, and intended to write a book about it, but neither did he do so. This set Biṭrūjī himself to thinking, and, taking his inspiration from Aristotle, he formulated a purely homocentric astronomy of the sun, moon, and planets that he described in On the Principles of Astronomy, which survives in Arabic and in Hebrew and Latin translations, the only book on the subject known to later writers. Finally, Maimonides (d. 1204) wrote a well-known and insightful analysis of the problem of the contradiction of mathematical astronomy of eccentrics and epicycles and physical astronomy of concentric spheres in The Guide of the Perplexed 2.24, although he mentions no one later than Ibn Bājja and knows nothing of al-Biṭrūjī.
Joseph Ibn Naḥmias knows everything there is to know about this subject and his own work is intended to do correctly what al-Biṭrūjī did incorrectly. If anything, his approach is guided by Maimonides’ careful analysis, although Maimonides only set out the problems, not the solutions, and is non-technical, while Ibn Naḥmias is entirely, indeed mercilessly, technical, and his hypotheses and explanations are the most complex I have ever seen by anyone. What he knows very well indeed is Ptolemy, and parts of his treatise are expositions of parts of the Almagest on, for example, the spherical astronomy of rising times and the method of computation for the sun and moon. Although never mentioned, he was evidently also a serious student of the Talmud because there are always at least two answers to every question and two ways of doing everything, which adds to the complexity. Here, we can only describe the models for the sun and moon briefly, and our explanation is very much “in principle” because the models do not always accomplish what is intended, although they are close.
First, and this is fundamental, following al-Biṭrūjī, he takes motions of spheres in the heavens to be in the direction of the first motion, the diurnal rotation from east to west, with the sun, moon, and planets “lagging” or “falling short” (taqṣīr, incurtare) depending upon their distance from the first motion, Saturn the least, the moon the most, so Saturn has the slowest motion in the direction of increasing longitude, to the east, and the moon has the fastest motion to the east. Biṭrūjī does not explain the geometry of this motion, which Ibn Naḥmias does. Because the diurnal rotation takes place parallel to the equator and the motions of the sun, moon, and planets take place in or near the plane of the ecliptic inclined to the equator, additional motions are required to keep the motion in its proper plane, so this turns out to be a rather complicated way of regarding the motions.
In the model for the sun, the pole of the “circle of the sun” moves in a circle about the pole of the equator in the direction of the diurnal rotation, east to west, “falling short” of a complete diurnal rotation of the heavens by some amount, say 2º to the east in one day, which displaces the circle of the sun, a quadrant from the pole, which initially coincides with the ecliptic, from the ecliptic. The sun is located on the circle of the sun, and so is also displaced from the ecliptic. In the same time, a second rotational motion of the circle of the sun about its pole, called the “motion in latitude,” moves the sun to the west one-half the first motion that displaces the circle of the sun from the ecliptic, that is, by 1º to the west in one day, which, in principle, returns the sun, “striving for perfection,” to the place on the ecliptic it would occupy if it simply moved 1º to the east in the ecliptic and the ecliptic was not displaced. Since these motions take place simultaneously, the sun never actually leaves the ecliptic. In this way, motions in the direction of the diurnal rotation to the west “falling short” can produce a seemingly independent motion in longitude in the opposite direction to the east, the conventional opinion, although Ibn Naḥmias considers his own, of motions in the same direction, to be preferable, indeed, to be the true physical hypothesis. However, it does not work, for Morrison shows by computation that if the sun is to return to, remain on, the ecliptic the “motion in latitude” does not equal the motion in longitude, and were the motions equal, the sun would not return to and remain on the ecliptic. But as it turns out, the sun is returned to the ecliptic in a completely different way in the model for the anomaly.
What is described so far is the model for mean, uniform motion. In order to produce nonuniform, anomalistic motion, for the sun Ibn Naḥmias uses a small circle, the equivalent of an epicycle, perpendicular to the plane of motion, carrying the sun, called “the circle of the path of the center,” which moves, not on the ecliptic, but on “the inclined circle carrying the circle of the path of the center” which intersects the ecliptic at an angle equal to the radius of the small circle. The radius of the small circle is equal to the maximum equation of the anomaly, which is given as 2º, closer to al-Battānī’s 1;59º or 2;10º of the Alfonsine Tables than to Ptolemy’s 2;23º, from which the correction to the mean motion is computed using plane triangles as in Ptolemy’s model, and he says that Ptolemy’s tables of mean motions and anomalies may be used. But now there is a problem because the motion on the small circle moves the sun, not only out of the plane of the inclined circle, but out of the plane of the ecliptic, and by different amounts at each position of the sun on the small circle. The solution, to return the sun to the ecliptic, is composed of two yet smaller circles, with radii equal to half the maximum equation, so1º, one on the small circle and the sun on the other, one form of the two circles, for there are two forms, using the same principle as Nasīr ad-Dīn aṭ-Ṭūsī’s well-known device, often called the “Ṭūsī couple,” to shift the sun itself in an arc from the small circle to the ecliptic.
For the moon, the mean motions must take into account both the motion of the moon in longitude and the regression of the nodes along the ecliptic. The resulting motion is not in the circle of the moon’s motion in latitude but on the “inclined deferent circle.” The moon is taken to have three anomalies producing (1) the equation at syzygy, conjunction and opposition; (2) the increase of the equation, which is greatest at quadrature; and (3) the correction of the anomaly, which is greatest near octants. The model for the first anomaly is, as for the sun, a small circle, equivalent to an epicycle, perpendicular to the plane of motion, called “the circle of the path of the moon’s center,” which carries the moon on it with the period of the anomalistic month and moves not on the circle of latitude of the moon but on the “inclined deferent circle” which intersects the circle of latitude at an angle equal to the radius of the small circle. The radius of the small circle is equal to the maximum equation of the anomaly at syzygy, the least maximum equation, and again the correction to the mean motion is computed using plane triangles as in Ptolemy’s model. The moon is to be kept on the circle of latitude using two circles, but these are not the same as for the sun and are located at the pole of the inclined circle. The supposed variation in the apparent diameter of the moon is attributed to changes in the air, that is, weather, or moisture, and to errors in measurement, as that Hipparchus takes the diameter of the moon to equal the sun at mean distance and Ptolemy takes the diameter to equal the sun at greatest distance, both of which cannot be correct. So since the distance of the moon is constant, so too is its apparent diameter.
Next are the two remaining anomalies of the moon. Remarkably, the model for these, replacing the previous model, is in essence a spherical form of Ibn ash-Shāṭir’s double-epicycle centered on and perpendicular to the plane of the “inclined deferent circle,” the larger epicycle with the period of an anomalistic month, the smaller, carrying the moon, with the period of half a synodic month, so two revolution each synodic month. The smaller epicycle increases the maximum equation of the anomaly at syzygy, which is not given but must be 5º, by 2½º at quadrature, the only number given, so 7½º, close enough to the maximum equations in the Alfonsine Tables of 4;56º and 7;34º and to Ptolemy’s 5;1º and 7;40º. The radius of the smaller epicycle is one-half the increase of 2½º, that is, 1¼º, which would mean the radius of the circle on which it moves is 6¼º, the least distance from the center at syzygy 5º, and the greatest distance at quadrature 7½º, all of these numbers approximate. The correction of the anomaly is, as in Ibn ash-Shāṭir’s model, the angle at the center of the larger epicycle between the center of the smaller epicycle and the location of the moon on the epicycle, and would be maximum, about 11½º, at the points of tangency to the small epicycle, which occur close to octants in the elongation of the moon from the sun. Finally, a small circle, with the same size and period as the smaller epicycle, carrying the pole of the deferent circle, serves to draw the inclined deferent circle, a quadrant from the pole, and with the deferent circle the double-epicycle and the moon itself, up and down through the length of the diameter of the small circle to place the moon on the circle of latitude.
We have reviewed briefly the sections on the sun and moon, which I realize may not be intelligible without figures that I cannot include in a review. These are preceded by sections on general principles, trigonometry, instruments, the beginning of spherical astronomy of rising times, and the solar year, and are followed by the motion of the fixed stars, incompletely the planets, apparently intended but not written, and then the completion of spherical astronomy of rising times. And my descriptions have barely penetrated the complexities of the models, which in all honesty I cannot say I have completely understood, at least in the time I have studied the work for writing this review. Ibn Naḥmias had a better head for these things than I do. Morrison’s treatment of all of this is very thorough, with detailed explanations and spherical trigonometric computations examining whether the models do what they are supposed to do in producing mean motions, equations and keeping the body on the correct circle. The results are usually close, which in some way Ibn Naḥmias must have known, although not by carrying out these computations even though the models were intended, at least in principle, for computation, which, I assume, no one ever did. As I wrote at the beginning, this is one of the most complex and difficult works I have ever seen, and we must be very grateful for Morrison’s study, which I believe he is uniquely qualified to write.