318 Review of the Principia of Newton. 



or defect of that which is necessary to retain a body in an 

 immoveable ellipse, and consequently what angular motion 

 in consequentia or antecedentia will be given to the apsides. 

 If there be a force, which is in any given ratio of the dis- 

 tance, and its index be denoted by n — 3, and M be to N in 

 the ratio of the angular motion of the body in the moveable, 

 to that in the fixed ellipse, we derive this proportion M : N : : 

 1 : \Z1T, or the angular motion in an ellipse, moveable about 

 the center of force, is to the angular motion in the same el- 

 lipse at rest, as one to the square root of a number, which 

 exceeds by 3 the index of the power, whose ratio the force fol- 

 lows. Therefore, from the force given, the angular motion 

 of the orbit, or apsides will be given, and vice versa, the mo- 

 tion of the apsides being given, the law of the centripetal 

 force may be found. The results, though not accurate by 

 this method, or such as can much improve practical astrono- 

 my, are sufficient for physical purposes, and the verification 

 of the Newtonian system of philosophy, for which they were 

 intended. They are, moreover, curious and instructive, as 

 more principles are employed than perhaps in any other iso- 

 lated problems of our celebrated author. His deductions, 

 however, embrace only the effects produced by forces in the 

 direction of the radius vector, which is a general problem in 

 Physico-Mathematics. The application of this to the lunar 

 orbit will be found to produce not more than one half of the 

 real motion of the moon's apsis, of which deficiency our au- 

 thor was fully aware, the other part of the motion is the ef- 

 fect of the sun's force acting perpendicularly to the radius 

 vector ; Mr. Clairaut was the first mathematician who in- 

 stituted a rigid analysis of the moon's motion. His calcula- 

 tion of the motion of the apsides brought out results very dif- 

 ferent from their true motion. Not suspecting that he had 

 made any material mistake, he began to question the accura- 

 cy of Newton's laws of gravitation. Similar doubts on this 

 point had also been entertained by Leonard Euler, the greatest 

 mathematician at that time in Europe. Mr. Clairaut, how- 

 ever, revised his calculations and detected a mistake, which, 

 when corrected, brought out the motion of the apsides, from 

 Newton's principles of gravity and motion, agreeing precise- 

 ly with their real motion, as ascertained by observation, and 

 established the Newtonian Celestial physics on the most im- 

 moveable basis. Euler, also, after very elaborate calcula- 

 tions, confirmed those of Clairaut ; his words are, " I have 



