Review of the Principia of Newton. 319 



renewed my inquiries on this affair, (the motion of the apo- 

 gee), and after most tedious calculations, I have at length 

 found to my satisfaction, that Mr. Clairaut was in the right, 

 and that this theory is entirely sufficient to explain the motion 

 of the apogee of the moon. As this inquiry is of the great- 

 est importance and difficulty, and as those who have hitherto 

 pretended to have proved this nice agreement of the theory 

 with the truth, have been much deceived : it is to Mr. Clair- 

 aut, that we are obliged for this important discovery, which 

 gives quite a new lustre to the theory of the great Newton, 

 and it is but now that we can expect good astronomical ta- 

 bles of the moon." The anticipated improvements in as- 

 tronomy deducible from our author's philosophy, have been 

 realized to an astonishing extent, principally by the research- 

 es of Euler, and Laplace. The most minute irregularities in 

 the motions of the moon and other planets, which observa- 

 tions could never detect, are now like truth in the abstract 

 mathematics, derived directly by calculation, from their phys- 

 ical causes. Newton's philosophy could have no higher evi- 

 dence of its truth, than the exact coincidence of these mathe- 

 matical results in innumerable complicated actions, with the 

 real motions of the celestial bodies. The practical utility of 

 these great refinements in astronomy, is now well known, 

 and explained in navigation, geography, and other useful ap- 

 plications. 



To pursue inquiries through the different corollaries of this 

 great problem of the motion of the apsides, would lead us in- 

 to a field of speculations too extensive for our object. It may 

 be proper, however, to state, summarily, a few of the deduc- 

 tions not less curious than important in the theory of orbicu- 

 lar and trajectory motion. 



1. The species of the curve which the moving body will 

 describe, depends principally on the law or variation of the 

 central force, except the circle, which may be described by 

 an uniform force of any kind, it having no variation of curva- 

 ture will want no variation of force, with which it is always 

 commensurate* 



2. To revolve in an accurately immoveable ellipse, the 

 force being in the focus, its variation must be according to 

 the law of the inverse duplicate ratio of the distance, and if 

 the force be in the centre, it must vary directly as the distance 

 from it. In the first case, the body will come to an apsis in 

 a semi-revolution, or to the same apsis in an entire revolution 



