But if the apse line was in quadrature at the same time as the moon, the second inequality vanished as well as the first. The mean value of the two inequalities combined was therefore fixed at 6° 201'. 115. To represent this new inequality, which was subsequently called the Evection, Ptolemy imagined an eccentric in the circumference of which the centre of an epicycle moved while the moon moved in the circumference of the epicycle. The centre of the eccentric and of the epicycle he supposed in syzygy at the same time, and both on the same side of R, the centre of the epicycle, would also be in syzygy. Now conceive c, the centre of the eccentric, to describe a small circle about E in a retrograde direction cc', while R, the centre of the epicycle, moves in the opposite direction, when she was in the nonagesimal, so that any error of longitude, arising from her yet uncertain parallax, would be avoided. Ptolemy, who records the observation, employs it to calculate the evection, and obtains a result agreeing with that of his own observations. (See Delambre, Ast. Ancienne.) in such a manner that each of the angles S'Ec', S'ER' may be equal to the synodical motion of the moon, that is, her mean angular motion from the sun; SES' being the motion of the sun in the same time. Now we have seen, Art. (110), that the first inequality was accounted for by supposing the epicycle RM to move into the position rm, r and R being at the same distance from E, and rm parallel to RM,* the first inequality being the angle rEm. But when the centre of the epicycle is at R', and R'M' is parallel to rm, the inequality becomes R'EM', and we have a second correction or inequality mEM'. 116. That this hypothesis will account for the phenomena observed by Ptolemy, Art. (114), will be readily understood. At syzygies, whether conjunction or opposition, the centres of the eccentric and epicycle are in one line with the earth and on the same side of it; the points r and R' coincide, as also m and M'. Hence mEM' = 0. At quadratures (figs. 1 and 2) c' and R' are in a straight line on opposite sides of the earth, and therefore R' and r at Fig. 1. Fig. 2. S R R R their furthest distance. If, however, M' and m be at the same time in this line, or, in other words, if the apse line * For simplicity we leave out of consideration the motion of the apse. be in quadratures (fig. 1), the angle mEM' will still be zero, or there will be no error in the longitude. But, if the apse line is in syzygy (fig. 2), the angle mEM' attains its greatest value.* Ptolemy, as we have said, found this greatest value to be 2° 39′, the angle mEr being then 5° 1′. 117. Copernicus (A.D. 1543), having seen that Ptolemy's hypothesis gave distances totally at variance with the observations on the changes of apparent diameter,† made another and a simpler one which accounted equally well for the inequality in longitude, and was at the same time more correct in its representation of the distances. Let E be the earth, OD an epicycle whose centre C describes the circle C'CC" about E with the moon's mean angular velocity. Let CO, a radius of this epicycle, be parallel to the apse *If Ptolemy had used the hypothesis of an eccentric instead of an epicycle for the first inequality of the moon, an epicycle would have represented the second inequality more simply than his method did. Dr. Whewell's History of the Inductive Sciences, vol. 1. p. 230. + See Delambre, Ast. Moderne, vol. 1. p. 116. Whewell's History of Inductive Sciences, vol. 1. p. 395. line EA, and about O as centre let a second small epicycle be described, the radii CO and OM being so taken that CO-OM = sin 5° 1', and CO+OM sin 7° 40'. The radius OM must now be made to revolve from the radius OC twice as rapidly as EC moves from ES, so that the angle COM may be always double of the angle CES. From this construction, it follows that in syzygies the angle CES being 0° or 180°, the angle COM is 0° or 360°; and therefore C and M are at their nearest distances, as in the positions C' and C'"'" in the figure. Then CM=CO—OM, and the angle CEM will range between 0° and 5° 1′, the greatest value being attained when the apse line is in quad rature. When the moon is in quadrature CES 90° or 270°, and therefore, COM=180° or 540° and C and M are at their greatest distance apart, as in the position C"; then, CM CO+ OM, and the angle CEM will range between 0° and 7° 40′, the former value when the apse line is itself in quadrature, and the latter when it is in syzygy. 118. Thus the results attained by Ptolemy's construction are, as far as the longitudes at syzygies and quadratures are concerned, as well represented by that of Copernicus; and the variations in the distances of the moon will be far more exact, the least apparent diameter being 28′ 45′′ and the greatest 37′ 33′′; whereas, Ptolemy's would make the greatest diameter 1°.* The values which modern observations give vary between 28' 48" and 33' 32". * Delambre, Ast. Moderne. 119. It will not now be difficult to shew that the introduction of this small epicycle corresponds with that of the term me sin ((2-2m-c) pt-2B+a} in our value of 0. For, referring to the preceding figure, we have - (moon's true long. - long. of apse)}, and OEM being a small angle whose maximum is 1° 19', we may write moon's mean longitude instead of the true in the argument, and also EC for OE; therefore, OM OEM: sin {2 (moon's mean longitude - sun's longitude) EC - (moon's mean longitude – longitude of apse)} = 791′ sin [2 { pt — (mpt + B)} — { pt− (1 − c) pt+a}] The value of the coefficient is from modern observations found to be 4589.61". 120. In Art. (70), we have considered the effect of this second inequality in another light, not simply as a small quantity additional to the first or elliptic inequality, but as forming a part of this first; and therefore, modifying and constantly altering the eccentricity and the uniform progression of the apse line. Boulliaud (A.D. 1645), by whom the term Evection was first applied to the second inequality, seems to hint at some |