PTOLEMAEUS. the earth, as was done with the stars, by circles drawn from the pole perpendicularly to the equator, that is, by latitudes and longitudes. His method of eclipses was long the only one by which difference of meridians could be determined ; and it is by the projection of his invention that to this day we con- struct our maps of the world and our best geogra- phical charts." We shall now proceed to give a short synopsis of the subjects treated in the Almagest : the reader will find a longer and better one in the second vo- lume of the work of Delambre just cited. The first book opens with some remarks on theory and practice, on the division of the sciences, and the certainty of mathematical knowledge : this preamble concludes with an announcement of the author's intention to avail himself of his pre- decessors, to run over all that has been sufficiently explained, and to dwell upon what has not been done completely and well. It then describes as the intention of the work to treat in order: — the relations of the earth and heaven ; the effect of position upon the earth ; the theory of the sun and moon, without which that of the stars cannot be undertaken ; the sphere of the fixed stars, and those of the five stars called planets. Arguments are then produced for the spherical form and motion of the heavens, for the sensibly spherical form of the earth, for the earth being in the centre of the heavens, for its being but a point in comparison ■with the distances of the stars, and its h:i,ving no motion of translation. Some, it is said, admitting these reasons, nevertheless think that the earth may have a motion of rotation, which causes the (then) only apparent motion of the heavens. Admiring the simplicity of this solution, Ptolemy then gives his reasons why it cannot be. With these, as well as his preceding arguments, our readers are familiar. Two circular celestial motions are then admitted : one which all the stars have in common, another ■which several of them have of their own. From several expressions here used, various writers have imagined that Ptolemy held the opinion maintained by many of his followers, namely, that the celestial spheres are solid. Delambre inclines to the con- trary, and we follow him. It seems to us that, though, as was natural, Ptolemy was led into the phraseology of the solid-orb system, it is only in the convenient mode which is common enough in all systems. When a modern astronomer speaks of the variation of the eccentricity of the moon's orbit as producing a certain effect upon, say her longitude, any one might suppose that this orbit was a solid transparent tube, within which the moon is materially restrained to move. Kad it not been for the notion of his successors, no one would have attributed the same to Ptolemy: and if the literal meaning of phrases have weight, Copernicus is at least as much open to a like conclusion as Ptolemy. Then follows the geometrical exposition of the mode of obtaining a table of chords, and the table itself to half degrees for the whole of the semi- circle, with differences for minutes, after the man- ner of recent modern tables. This morsel of geometry is one of the most beautiful in the Greek ■writers: some propositions from it are added to many editions of Euclid. Delambre, who thinks as meanly as he can of Ptolemy on all occasions, mentions it with a doubt as to whether it is his own, or collected from his predecessors. In this. PTOLEMAEUS. 575 as in many other instances, he shows no attempt to judge a mathematical argument by any thing except its result : had it been otherwise, the unity and power of this chapter -would have established a strong presumption in favour of its originality. Though Hipparchus constructed chords, it is to be remembered we know nothing of his manner as a mathematician ; nothing, indeed, except some re- sults. The next chapter is on the obliquity of the ecliptic as determined by observation. It is followed by spherical geometry and trigonometry enough for the determination of the connection between the sun's right ascension, declination, and longitude, and for the formation of a table of de- clinations to each degree of longitude. Delambre says he found both this and the table of chords very exact. The second book is one of deduction from the general doctrine of the sphere, on the effect of po- sition on the earth, the longest days, the determi- nation of latitude, the points at which the sun is vertical, the equinoctial and solsticial shadows of the gnomon, and other things which change with the spectator's position. Also on the arcs of the ecliptic and equator which pass the horizon simul- taneously, Avith tables for different climates, or parallels of latitude having longest days of given durations. This is followed by the consideration of oblique spherical problems, for the purpose of calculating angles made by the ecliptic with the vertical, of which he gives tables. The third book is on the length of the year, and on the theory of the solar motion. Ptolemy in- forms us of the manner in which Hipparchus made the discovery of the precession of the equinoxes, by observation of the revolution from one equinox to the same again being somewhat shorter than the actual revolution in the heavens. He discusses the reasons which induced his predecessor to think there was a small inequality in the length of the year, decides that he was wrong, and produces the comparison of his own observations with those of Hipparchus, to show that the latter had the true and constant value (one three-hundredth of a day less than 365| days). As this is more than six minutes too great, and as the error, in the whole interval between the two, amounted to more than a day and a quarter, Delambre is surprised, and with reason, that Ptolemy should not have detected it. He hints that Ptolemy's observations may have been calculated from their required result ; on which we shall presently speak. It must be re- membered that Delambre watches every process of Ptolemy with the eye of a lynx, to claim it for Hipparchus, if he can ; and when it is certain that the latter did not attain it, then be might have attained it, or would if he had lived, or at the least it is to be matter of astonishment that he did not. Ptolemy then begins to explain his mode of ap- plying the celebrated theory of excentrics, or revo- lutions in a circle which has the spectator out of its centre ; of epicycles, or circles, the centres of which revolve on other circles, &c. As we cannot here give mathematical explanations, we shall refer the reader to the general notion which he probably has on this subject, to Narrien's History of Astronomy, or to Delambre himself. As to the solar theory, it may be sufficient to say that Ptolemy explains the one inequality then known, as Hipparchus did before him, by the supposition that the circle of the sun is an excentric ; and that he does not
