Let BAM be a circle, CA a radius, E a point in AC near C; CB, ED two parallel lines making an angle a with CA. Suppose a body M to describe this circle uniformly with an angular velocity p, the time being reckoned from the instant when the body was at B, and the longitude as seen from E being reckoned from the line ED; therefore EC DEM=0, B D C BCM=pt, AEM-0-a, ACM=pt-a. Now is a small fraction, and if we represent it by e, CM this would give M, and then e sin (pt — a) by the formula 0=pt+ M. This was called an eccentric, and the value of e was called the eccentricity, which, for the moon, Hipparchus fixed at sin 5° 1'. 110. Another method of considering the motion was by means of an epicycle, which led to the same result. A small circle PM, with a radius equal to EC of previous figure, has its centre in the circumference of the circle RPD (which has the same radius as that of the eccentric), and moves round E with the uniform angular velocity p, the the body M being carried in the circumference of the smaller circle, the radius PM remaining parallel to itself, or, which is the same thing, revolving from the radius PE with the same angular velocity p, so that the angle EPM equals PEA. Now, when the angle AEP equals the angle ACM of the former figure, it is easily seen that the two triangles EPM, ECM are equal, and there A M fore the distance EM and the angle AEM will be the same in both, that is, the two motions are identical. 111. The value of e being small, we find, rejecting e, &c., M=e sin (pt- a), therefore 0=pt+e sin (pt — a). If we reject terms of the second order in our expression for the longitude, and make c=1, we get, Art. (51), which will be identical with the above if we suppose the eccentricity of the eccentric to be double that of the elliptic orbit. Ptolemy (A. D. 140) calculated the eccentricity of the moon's orbit, and found for it the same value as Hipparchus, viz. The eccentricity in the elliptic orbit is, we know, about. These values will pretty nearly reconcile the two values of given above, and this shews us, that for a few revolutions the moon may be considered as moving in an eccentric, and her positions in longitude calculated on this supposition will be correct to the first order. Her distances from the earth will not however agree; for the ratio of the calculated greatest and least distances would be 1+11/2 1 12 or 1, while that of the true ones would be or 2, which differ by 13. 1+28 1-20 It would, therefore, have required two different eccentrics. to account for the changes in the moon's longitude and in her radius vector. Changes in the latter could not, however, be easily observed with the rude intruments the ancients possessed, and it was very long before this inconsistency was detected. 112. We have said that the moon's longitude, calculated on the hypothesis of an eccentric, will be pretty accurate for a few revolutions. The data requisite for this calculation are, the mean angular motion of the moon, the position of the apogee, and the magnitude of the eccentricity. But it was known to Hipparchus and to the astronomers of his time, that the point of the moon's orbit where she seems to move slowest, is constantly changing its position among the stars. Now this point is the apogee of Hipparchus's eccentric, and he found that he could very conveniently take account of this further change by supposing the eccentric itself to have an angular motion about the earth in the same direction as the moon herself, so as to make a complete revolution in about nine years, or about 3° in each revolution.* This motion of the apsidal line follows also from our expression for the longitude, as shewn in Art. (66). It is there, however, connected with an ellipse instead of an ec On the supposition of an epicycle, this motion of the apse could as easily be represented by supposing the radius which connects the moon with the centre of the epicycle to have this uniform angular velocity of about 3° in each revolution, and also in the same direction. centric; and though the discovery that the elliptic is the true form of the fundamental orbit was not the next in the order of time after those of Hipparchus, yet, as all the irregularities which were discovered in the intervening seventeen centuries are common both to Hipparchus's eccentric and to Kepler's ellipse, it will be as well for us to consider at once this new form of the orbit. Elliptic Form of the Orbit. 113. We need not dwell on the steps which led to this great and important discovery. Kepler, finding that the predicted places of the planet Mars, as given by the circular theories then in use, did not always agree with the computed ones, sought to reconcile these variances by other combinations of circular orbits, and after a great number of attempts and failures, and eight years of patient investigation, he found it necessary to discard the eccentrics and epicycles altogether, and to adopt some new supposition. An ellipse with the sun in the focus was at last his fortunate hypothesis, which was found to give results in accordance with observation; and this form of the orbit was, with equal success, afterwards extended to the moon as a substitute for the eccentric: but the departures from elliptic motion, due to the disturbing force of the sun, are, in the case of the moon, much greater than the disturbances of the planet Mars by the other planets. In Kepler's hypothesis, then, the earth is to be considered as occupying the focus of an ellipse, in the perimeter of which the moon is moving, no longer with either uniform linear or angular velocity, but in such a manner that the radius vector sweeps over equal areas in equal times. This agrees with our investigation of the motion of two bodies, Art. (10). Evection. 114. The hypothesis of an eccentric, whose apse line has a progressive motion, as conceived by Hipparchus, served to calculate with considerable accuracy the circumstances of eclipses; and observations of eclipses, requiring no instruments, were then the only ones which could be made with sufficient exactness to test the truth or fallacy of the supposition. Ptolemy (A.D. 140) having constructed an instrument, by means of which the positions of the moon could be observed in other parts of her orbit, found that they sometimes agreed, but were more frequently at variance with the calculated places; the greatest amount of error always taking place at quadrature and vanishing altogether at syzygy. What must, however, have been a source of great perplexity to Ptolemy, when he attempted to investigate the law of this new irregularity, was to find that it did not return in every quadrature,-in some quadratures it totally disappeared, and in others amounted to 2° 39', which was its maximum value. By dint of careful comparison of observations, he found that the value of this second inequality in quadrature was always proportional to that of the first in the same place, and was additive or subtractive according as the first was so: and thus, when the first inequality in quadrature was at its maximum or 5° 1', the second increased it to 7° 40', which was the case when the apse line happened to be in syzygy at the same time.* * It would seem as if Hipparchus had felt the necessity for some further modification of his first hypothesis, though he was unable to determine it; for there is an observation made by him on the moon in the position here specified when the error of his tables would be greatest; and at a time also |