340 Mr, J. J. Waterston on the Integral of Gravitation, 



a body conjointly with its quantity of matter, and it is evident 

 that, confining tlie word to this meaning, the correctness of the 

 'Principia' is no argument in favour of such definition applying 

 to the natural standard. There is no doubt, however, that both 

 Newton and Laplace believed it to be the natural standard. It 

 may assist in setting the subject in a clear light, if we take a 

 deliberate view of the arguments they have advanced. 



In Dr. Brewster's ' Life of Newton,' vol. ii. p. 379, there is 

 given what is stated to be Newton^s argument as opposed to 

 Leibnitz, who had asserted the force of falling bodies to be as 

 the square of their velocities. It is based on the assumption 

 that fine threads of silk may be broken with less and less force 

 as the velocity of the body that breaks through them increases. 

 These threads individually may be taken to represent the resist- 

 ance that gravity opposes to upward motioh through a constant 

 element of the vertical, as they are supposed to be arranged at 

 equal distances, and each requires the expenditure of an element 

 of force, or square velocity, to rend it. The retardation of the 

 motion of a globe through such threads horizontally, is pre- 

 cisely similar to the retardation of the motion of a body pro- 

 jected up a vertical. In an element of time there is a loss of an 

 element of velocity, or in an element of space there is the loss of 

 an element of square velocity. Thus the argument derived from 

 such an instance is not conclusive ; it is not, any more than the 

 laws of falling bodies, decisive of the question. 



Laplace's argument is stated in the Sysihnc du Monde, p. 300 

 (Pond's translation), as follows : — 



" It is observed upon the earth that a body solicited by any 

 force moves in the same manner, whatever be the angle which 

 the direction of this force makes with the direction of the 

 motion which is common to the body and to the part of the ter- 

 restrial surface to which it corresponds : the same thing takes 

 place in a vessel whose motion is uniform : a moveable body sub- 

 mitted to the action of a spring or of gravity, or any other force, 

 moves relatively to the parts of the ship in the same manner, 

 whatever be the velocity and direction of the vessel. It may 

 then be established as a general law of terrestrial motions, that 

 if in a system of bodies carried on by a common motion, any 

 force be impressed on one of them, its apparent or relative motion 

 will be the same, whatever be the general motion of the system 

 and the angle which its direction makes with the impelling 

 force. The proportionality of force to velocity residts from this 

 laiv, supposed rigorously exact; for if ice suppose two bodies moving 

 upon one straight line with equal velocities, and that by impressing 

 on one of them a force, which increases the jmmitive force, its 

 relative velocity to the other body remains the same as if both oj 



