VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 647 



in this inquiry, the tangent ought to be supposed to be equally inclined to the 

 two elements of the curve that terminate at the point of contact: and then the 

 subtense of the angle of contact will be found equal to half the second differ- 

 ence of the ordinate, which is its true value. 



Sir Isaac Newton, however, investigates the fluxions of quantities in a more 

 unexceptionable manner. He first determines the finite simultaneous incre- 

 ments of the fiuents, and, by comparing them, investigates the ratio that is 

 the limit of the various proportions which they bear to each other, while he 

 supposes them to decrease together till they vanish. When the generating 

 motions are variable, the ratio of the simultaneous increments that are gene- 

 rated from any term, is expressed by several quantities, some of which arise 

 from the ratio of the generating motions at that term, and others from the 

 subsequent acceleration or retardation of these motions. While the increments 

 are supposed to be diminished, the former remain invariable, but the latter de- 

 crease continually, and vanish with the increments; and hence the limit of the 

 variable ratio of the increments, or their ultimate ratio, gives the precise ratio 

 of the generating motions or fluxions. Most of the propositions in the pre- 

 ceding chapters may be more briefly demonstrated by this method, of which 

 several examples are given, and the author makes always use of it in the sequel 

 of this book. 



It is one of the great advantages of this method, that it suggests general 

 theorems for the resolution of problems, which may be readily applied as there 

 is occasion for them. Our author proceeds to treat of these, and first of such 

 as relate to the centre of gravity and its motion. In any system of bodies, the 

 sum of their motions, estimated in a given direction, is the same as if all the 

 bodies were united in their common centre of gravity. If the motion of all 

 the bodies is uniform and rectilineal, the centre of gravity is either quiescent, 

 or its motion is uniform and rectilineal. When action is equal to reaction, the 

 state of the centre of gravity is never affected by the collisions of the bodies, 

 or by their attracting or repelling each other mutually. It is not, however, the 

 sum of the absolute motions of the bodies that is preserved invariable in conse- 

 quence of the equality of the action and reaction, as they seem to imagine, 

 who tell us, that this sum is unalterable by the collisions of bodies, and that 

 this follows so evidently from the equality of action and reaction, that to en- 

 deavour to demonstrate it would serve only to render it more obscure. On this 

 occasion the author illustrates an argument which he had proposed in a piece 

 that obtained the prize proposed by the Royal Academy of Sciences at Paris in 

 1724, against the mensuration of the forces of bodies by the square of the ve- 

 locities, showing that if this doctrine was admitted, the same power or agent, 



