Review of the Principia of Newton. 337 
force. — therefore, @ for the force, and s for, the space, 
» for the velocity, and ¢ for the time, in uniform motion, we 
shall have s x ¢ v, and in any kind of motion, because 3 and 
¢ are from their very definitions indefinitely small elements, 
uniformly generated, we shall have s « vtand v ae. but 
since is &y and v x S_ ; hereiore ® on 
t t, if we suppose 7 
constant, but when { varies @ « 2, if, therefore, instead of 
» we substitute = we shall have oc = This very useful 
formula has Bei finely illustrated by D’Alembert, and is 
identical in all its principles with what had been long before 
delivered in the 10th Lemma and the 39th proposition of the 
work now before us. In this proposition, as well as in many 
others of the Principia, we have examples of the advantage of 
the author’s metaphysique of fluxions, or the generation of 
quantities by motion, in its application to the solution of 
problems dependent on the laws and variation of motion ; 
since these are identical with the different orders of fluxions, 
of which the variable motion is susceptible. 
Since ¢ =—, and ¢ =— therefore — =", and @s =. 
vv, and # s =2 wv. This is precisely the first case of the 
39th Pr ais diss body fom th of 
If z be r distance of a e centre 
force, a epele ted ing or descending, and abe the 
from which it descended, or the utmost height to which it it 
would rise, by a force acting according to any law of dis- 
or according to any function of the distance, whose 
ee is n, then by hypothesis x will be the ordinate of the 
curve, whose area, by the 39th proposition, represents the 
square of the velocity and the fluxion of this area 28 
1 * 4c. T, nk beter oe 
ni 4 
whose fluent =~ 72 : 
t, which, when =a becomes a” + 1 and therefore 
ae at apy ¥é variable distance will be truly defined by 
Jva* + 1—a* +* which, if = substitute n—1, as the ex- 
VOL. XII. No. 2. 
