steps. The perusal of this chapter will shew to what extent we are indebted to our great philosopher; at the same time we cannot fail being impressed with reverence for the genius and perseverance of the men who preceded him, and whose elaborate and multiplied hypotheses were in some measure necessary to the discovery of his simple and single law. I take this opportunity of acknowledging my obligations to several friends, whose valuable suggestions have added to the utility of the work. HUGH GODFRAY. Cambridge, April 16th, 1853. CONTENTS. 1 Newton's Law of Universal Attraction 3 Principle of Superposition of Small Motions 19 Formation of the Differential Equations 22 Orders of the small Quantities introduced INTEGRATION OF THE DIFFERENTIAL EQUATIONS. 27 Terms of a higher order which must be retained 30 Solution to the first order 43 Solution to the second order 51 True Longitude in terms of the Mean obtained by Reversion of Series 60 First Method. Theoretical Values 61 Second Method. Values deduced by the Solution of a large number of simultaneous Equations 62 Third Method. Independent determination of each Coefficient ib. 51 64 Discussion of the terms in the Moon's Longitude 77 Discussion of the terms in the Moon's Latitude . 91 The Moon is retained in her orbit by gravity 92 The Moon's orbit is everywhere concave to the Sun 93 Effects of Central and Tangential Forces separately considered 94 The value of c to the third order 95 The value of g to the third order 100 Inequalities depending on the figure of the Earth 109 Description of the Eccentric and the Epicycle 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 115 His manner of representing it 117 Copernicus's Hypothesis for the same purpose 120 Boulliaud, D’Arzachel, Horrocks consider it in a different manner 96 121 Tycho Brahé's discovery and representation of the Variation 123 Tycho Brahé's discovery and representation of the Annual Equation 99 124 Tycho's Table for the Reduction 125 Inclination of the Moon's orbit and motion of the Node calculated by 126 Tycho Brahé's discovery of the change of inclination and of the want of uniformity in the motion of the Node 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 acquaint B ance 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 must 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 disturbingt 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 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 * 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). |