CHAPTER XIII.

PTOLEMY'S PLANETARY THEORY.

Conditions required in investigating the orbits of the planets.-Determination of the orbits of the superior and inferior planets.—Investigation of the latitudes of the planets.-Imperfection and complexity of the system of Ptolemy.-The inferior planets supposed by Pliny and Vitruvius to revolve about the sun.-Hypothesis of Vitruvius concerning solar attraction.-Seneca's observation on the probable improvement of astronomy.

IN the ninth book of the Almagest, Ptolemy enters upon the theory of the planets which, as he has delivered it, consists in a combination of the preceding theories of the sun and moon, with some further modifications for Mercury. He informs us that the observations made by the more ancient astronomers were of three kinds; the times of their stationary appearances; of their risings and settings; and of their appulses to the moon; and he points out the inaccuracies to which they are subject: he remarks that the first kind of observation is uncertain, because the slowness of a planet's motion, when nearly stationary, does not permit the time to be marked with precision; he shews that the risings and settings of stars are affected by a serious cause of error arising from the impossibility of seeing them when they are exactly in the horizon, and from the apparent augmentation of their distances which then takes place, and which he attributes to the vapours of the earth; and, lastly, he observes that the phenomena of the appulses do not, from the errors in the computed place of the moon, afford means of determining the positions of the planets with sufficient correctness. On all these accounts it is evident that direct observations of the planets by means of the astrolabe are to be preferred; and from such, were the data obtained by which Ptolemy computes the elements of the orbits. In speaking of the apparent situations of celestial bodies when near the horizon, it is remarkable that this astronomer does not mention the refraction of light as a

cause of the augmentations of their altitudes, and distances from each other; and it is probable that, at the time of writing the Almagest, he had not made the discovery of the effect of the atmosphere in changing the directions of the rays of light, which, however, he must have found out soon afterwards, for he has introduced an account of the phenomena resulting from it, in his treatise on Optics, which, fortunately, is still in existence.

In the theory now to be described he supposes that the earth is at rest in the centre of the universe; that the planets, and the sun as one of them, revolve about it; he considers Mars, Jupiter and Saturn to be more remote from the earth than the sun is, but he observes, as we have before stated, that there were two opinions concerning the positions of Mercury and Venus; the more ancient philosophers supposing them to be situated between the earth and the sun and to revolve about the former while some, of later times, placed them beyond the sun with respect to the earth; and the first opinion is that which he adopts in his researches. He assumes the truth of the Aristotelian doctrine that the celestial motions are uniform and circular; but he observes that the movements of a planet cannot be explained, either on the hypothesis of a simple eccentric orbit, or on that of an epicycle moving upon a homocentric deferent, which probably constituted the theories of Apollonius and Hipparchus; and the following is an outline of the manner in which he has combined the two hypotheses, to satisfy the phenomena as far as they had then been observed.

Let E (Plate II. fig. 2.) be the centre of the earth and of the universe; and upon any line, as A P, take E C equal to the eccentricity of the planet's orbit; a distance corresponding to E C in the preceding description of the system of the moon. Bisect EC in z and about z as a centre describe the circle A MP which is that called the eccentric and which, in the solar theory, would be that on whose circumference the sun moves. The centre м of the epicycle L V V' moves on the circumference of this circle, and z is called the centre of mean, or constant distances, for м z is equal to the mean distance of the planet from the earth. The planet, if superior, moves on its epicycle in direct order, or from V towards T; if inferior, in retrograde order,

M.

or from V' towards T; but the motion of the centre of the epicycle is always direct, or from A towards м. The angle AC M is always produced by the mean movement of the planet; or the movement of the centre M, of the epicycle is such that, when seen from c, it appears equal to the planet's mean motion in longitude; hence c was called the centre of mean or equable movements, and a circle, as Htz, described about c as a centre with any radius was called the equant, because the point t, in the line C M, would appear from c to describe its circumference with uniform motion. The above description will serve for all the planets, superior and inferior, except Mercury, in whose system, z, the centre of the eccentric and of mean distances, moves upon the circumference of a circle described about c as a centre with a radius equal to cz, the half eccentricity; just as, in the moon's orbit, the centre of the eccentric moves upon a circle described about E as a centre.

The line ACEP is the line of the apsides of the eccentric circle; it was not considered as stationary in space but as having an angular movement about E with a velocity equal to that of the precession; LMC is the line of the apsides of the epicycle, and this tends constantly towards c for all the planets except Mercury. Now let a circle be described about E as a centre, with any radius, to represent the orbit of a fictitious sun, and on its circumference let such a sun be supposed to move uniformly with the mean motion of the real sun; then, as was shewn in the tenth chapter, the mean relative motion of the planet, or that performed in its epicycle with respect to the sun, being equal to the difference between the mean motion of the sun and that of the planet in longitude, if we suppose S' to be the place of the fictitious sun when the planet is at V' and in opposition, s' will be its place at the next time of opposition when the planet will be at v'; it must be observed, however, that E is not supposed to be the centre of the mean movements in the solar orbit, but the error arising from so considering it, is by Ptolemy disregarded in the planetary theory. The angles EMC and E M'C constitute, for the points м and M' respectively, what was called the anomaly of the eccentric, the planet's zodiacal, or proper, or first inequality, or the equation of the

centre; and this element corresponds to that, to which in the solar theory, the latter denomination was applied; the second inequality was supposed to depend upon the position of the planet in its epicycle, and was expressed by the variable angle M'ER, the planet being supposed at R.

The mean tropical revolution of a planet, and the mean revolution in its epicycle had been well ascertained, as we have shewn, before the time of Ptolemy, and the ancient values of these elements differ but little from those assigned to them in the modern tables: we cannot, however, say so much of the equations by which the mean, were reduced to the apparent movements, since the erroneous theory of the Greeks but badly represented the laws of the variations of planetary motion between the periods in which their inequalities are compensated; yet it will be both interesting and useful to shew in what manner Ptolemy determined the elements of the orbits of the planets on the hypothesis which has been just explained. For this purpose, it must be supposed that, besides a knowledge of the mean movements above mentioned, he possessed registers of many observed longitudes and latitudes of the planets at, or near, the times of the opposition of the superior, and the greatest elongations of the inferior planets with respect to the sun; and, from these, by the help of a table shewing the mean solar movements, he was enabled to ascertain the apparent places of the planets at the moments when those places were in opposition with the mean places of the sun. With these elements, by the rules of trigonometry, he found, as we shall explain, the places of the apogea, the eccentricity, the radii of the epicycles and the equations of the centres. The opposition of the true longitude of a planet to the mean longitude of the sun is made use of by Ptolemy because the planet, in this situation, is, according to his hypothesis, at the extremity of its epicycle, passing through the earth; and the second inequality, or that depending on the place of the planet in its epicycle, is, consequently, null. It may be observed here that, in the modern astronomy, the oppositions and conjunctions of the planets with the sun are, also, generally employed for investigating the elements of their motions in order to avoid the inequality caused by the distance

of the earth from the centre of the system; but the oppositions or conjunctions are those of the true geocentric longitudes of both sun and planet.

The data employed by Ptolemy in investigating the orbit of Mars were three longitudes of that planet observed by himself in the fifteenth and nineteenth years of Adrian and in the second year of Antoninus, when the apparent places of the planet were in opposition to the mean places of the sun. These longitudes were 2' 21°, 4° 48° 50′ and 8° 2° 34' respectively, and the intervals of time were 1530.8333 days, and 1557.0417 days; the mean movements of the planet in the two intervals, according to the tables and rejecting entire circumferences, were 81° 44', and 95° 28', respectively, while the observed movements, by taking the differences between the above observed places, were 67° 50′ and 93° 44'. Now let E (Plate III. fig. 1) be the earth, and M, M', M" be the three observed places of Mars, in the circle of mean distances whose centre is z, consequently we have ZMEM' 67° 50′ and M' E M"93° 44′ as above: let c be the centre of mean movements and describe the circle m m'm" about c as a centre with a radius equal to that of the circle of mean distances; draw cм m, C м'm', C м"m"; then m, m', m" become the places of Mars calculated by his mean movements, so that

mcm 81° 44′ and m'cm" 95° 28′ as above; and let it be required to find the eccentricity EC and the position AP of the line of the apsides. To obtain these, Ptolemy assumes, as known, the length of a line Ep, drawn through E and m", one of the mean places of the planet in the circle of mean movements; and, from this, with the given angles, he computes trigonometrically the value of the radius of the circle of mean distances and, subsequently, the value of the eccentricity in terms of that radius: from the same data he gets the angle made by the line of the apsides with a line drawn from c to one of the observed places of the planet, by which, and the observed longitude of that point, the longitude of the perigeum is determined.

The details of the investigation are as follow. The angles mEm', m'Em" do not differ much from ME M', M' E M"; therefore, as a first approximation, Ptolemy supposes them equal, and he produces m" E to p; the 4 m E p may be considered as