112 Hipparchus's mode of representing the Motion of the Apse 113 Substitution of the Elliptic for the Circular Orbit 114 Ptolemy's discovery of the Evection 120 Boulliaud, D'Arzachel, Horrocks consider it in a different manner 110 121 Tycho Brahé's discovery and representation of the Variation 112 123 Tycho Brahe's discovery and representation of the Annual 126 Tycho Brahe's discovery of the change of inclination and of the LUNAR THEORY. CHAPTER I. INTRODUCTION. BEFORE proceeding to the consideration of the moon's motion, it will be desirable to say a few words on the law of attractions, and on the peculiar circumstances which enable us to simplify the present investigation. 1. The law of universal gravitation, as laid down by Newton, is that "Every particle in the universe attracts every other particle, with a force varying directly as the mass of the attracting particle and inversely as the square of the distance between them." The truth of this law cannot be established by abstract reasoning; but as it is found that the motions of the heavenly bodies, calculated on the assumption of its truth, agree more and more closely with the observed motions as our calculations are more strictly performed, we have every reason to consider the law as an established truth, and to attribute any slight discrepancy between the results of calculation and observation to instrumental errors, to an incomplete analysis, or to our ignorance of the existence of some of the disturbing causes. Of the last cause of deviation there is a remarkable instance in the recent discovery of the planet Neptune, for our B acquaintance with which, as one of the bodies of our system,* we are indebted to the perturbations it produced in the calculated places of the planet Uranus. These perturbations were too great to be attributed wholly to errors of instruments or of calculation; and therefore, either the law of universal gravitation was here at fault, or some unknown body was disturbing the path of the planet. This last supposition, in the powerful hands of Messrs. Adams and Le Verrier, led to the detection of Neptune by solving the difficult inverse problem, viz:-Given the perturbations caused by a body, determine, on the assumption of the truth of Newton's law, the orbit and position of the disturbing body. Evidence so strong as this forces us to admit the correctness of the assumption, and we shall now consider how this law, combined with the laws of motion, will enable us to investigate the circumstances of the moon's motion, and to assign her position at any time when observation has furnished the requisite data. 2. The problem in its present form would be one of extreme, if not insurmountable difficulty, if we had to take into account simultaneously the actions of the earth, sun, planets, &c. on the moon; but fortunately the earth's attraction, on account of its proximity, is much greater than the disturbing force of the sun or of any planet;-these disturbing forces being so small compared with the absolute force of the earth, that the squares and products of the effects they produce may be neglected, except in extreme It had been seen by Dr. Lamont at Munich, one year before its being known to be a planet. "Solar System, by J. R. Hind." Since the sun attracts both the earth and moon, it is clear that its effects on the moon's motion relatively to the earth or the disturbing force will not be the same as the absolute force on either body. This will be fully investigated in Arts. (9) and (23). cases: and there is a principle, called the "principle of the superposition of small motions," which shows that in such a case the disturbing forces may be considered separately, and the algebraic sum of the disturbances so obtained will be the same as the disturbance due to the simultaneous action of all the forces. Principle of Superposition of Small Motions. 3. Let a particle be moving under the action of any number of forces some of which are very small, and let A be the position of the particle at any PP instant. Let two of these small forces m,, m, be omitted, and suppose the path of the particle under the action of the remaining forces to be AP in any given time. Let AP, be the path which would have been described in the same time if m, also had acted; AP, differing very slightly from AP, and PP, being the disturbance. Similarly, if m2 instead of m1 had acted, suppose AP, to have represented the disturbed path (AP, AP1, AP, are not necessarily in the same plane, nor even plane curves), PP, being the disturbance. Lastly, let AQ be the actual path of the body when both m, and m, act. Join P, Q. 1 Now, since the path AP, very nearly coincides with AP, the disturbances PQ and PP, produced in them by the action of the same small force m, will be very nearly parallel; and will differ in magnitude by a quantity which can be only a small fraction of either disturbance and which may be neglected compared with the original path AP. Therefore PQ is parallel and equal to PP Hence the projection of the whole disturbance PQ on any straight line, being equal to the algebraical sum of the pro |