mentioned which, as the theory has since verified, were real onward 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. St. John's College, Cambridge, April, 1853. In the present edition, besides the change of form and the incorporation of the figures with the text, which it is hoped will render the work more commodious, very few alterations have been thought necessary; and, except in one or two instances, where additional paragraphs have been introduced, nothing but the wording of some of the sentences has been altered. October, 1859. CHAPTER III. DIFFERENTIAL EQUATIONS, 19 Formation of the Differential Equations. 22 Orders of the Small Quantities introduced 23 Values of the Forces P, T, S 24 Order of the Sun's disturbing Force 19 22 24 26 . CHAPTER IV. INTEGRATION OF THE DIFFERENTIAL EQUATIONS. 25 General Process described 27 Terms of a higher order which must be retained 30 Solution to the first order 35 Introduction of c and g 28 30 32 35 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. . 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 84 86 87 89 91 92 93 94 97 HISTORY OF THE LUNAR PROBLEM BEFORE NEWTON. 109 Description of the Eccentric and the Epicycle 112 Hipparchus's mode of representing the Motion of the Apse . 103 113 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 110 121 Tycho Brahé's discovery and representation of the Variation 112 123 Tycho Brahé's discovery and representation of the Annual 124 Tycho's Table for the Reduction 125 Inclination of the Moon's orbit and motion of the Node cal- 126 Tycho Brahé's discovery of the change of inclination and of the want of uniformity in the motion of the Node |