VOL. LXIX.] PHILOSOPHICAL TRANSACTIONS. 583 



and therefore I must believe that it is by mistake that one author of note entirely 

 omits so necessary a step which affects the conclusion by just one half. When a 

 body moves with any velocity in the direction am (fig. 7,) which would carry it 

 through the space ad in a small particle of time, and any force, which may be 

 reckoned constant for that time, urges the body through the space do perpen- 

 dicular to AM, the body at the end of that time will arrive at the point c ; but 

 joining ac, we are not to suppose that, if that force ceased to act, the body 

 would proceed in the direction acl : for take cm equal and parallel to ad, and 

 cd in CD produced equal to leu, then the direction of c at that point will be c/, 

 the diagonal of the parallelogram cdlm. 



Thus, when a body revolves in any curve by a centripetal force (fig. 8,) we 

 may, with Sir Isaac Newton, suppose the curve to be composed of an indefinite 

 number of right lines, and the body to move either in the chords or the tan- 

 gents of the curve ; but then we are to take care that we make not suppositions 

 inconsistent with each other. Let the curve be a circle, and ad a tangent at the 

 point A the direction of the body's motion when it arrives at that point, and let 

 DC, parallel to the diameter al, be the effect of the centripetal force : then, if 

 we suppose the body to move along the chord ac, and say, that the angle cad 

 measures the deflection of the path in the time of the body's moving through 

 the arc or chord ac, we shall mistake by one half of the true quantity ; for 

 draw the tangent at c, then since ac^ is equal to dc from the property of the 

 circle, the angle cc?d of deviation is equal to twice the angle cxd. The practice 

 of Newton in a similar case, where he is investigating the horary motion of the 

 lunar nodes in a circular orbit, is entirely consistent with this. See the Prin- 

 cipia, lib. 3, prop. 30. 



§ 15. M. D'Alembert has lately charged Simpson's account of the precession 

 of the equinoxes with some mistakes of this nature in his 2d lemma ; but, in 

 justice to Simpson, I must say, that, whatever other dei^cts there may be in 

 his paper, I am convinced, after the most diligent attention, that those alluded 

 to are without foundation. 



^ l6. Sir Isaac Newton first observed, that a homogeneous globe could not 

 possibly retain many distinct motions, without compounding them all into one, 

 and revolving with a simple and uniform motion about an invariable axis. When 

 2 forces impress on a globe 2 distinct circular motions, (see Principia, lib. I. 

 prop. 67, coroll. 22,) he briefly concludes in his way, from the laws of motion, 

 that it is the same thing as if those 2 forces were at once impressed in the 

 common intersection of the equators of those motions, and on this principle we 

 supposed Kt, in art. 7, to be the direction of the new equator. In order to re- 

 move any doubts that might arise about the justness of this mode of compound- 

 ing motion, Frisius has given a geometrical demonstration of the principle : but 



