60 LECTURE VIII. 



proportion to it ; and some have considered it as the true measure of the 

 quantity of motion ; but although this opinion has been very universally 

 rejected, yet the force thus estimated well deserves a distinct denomina- 

 tion. After the considerations and demonstrations which have been pre- 

 mised on the subject of forces, there can be no reasonable doubt with 

 respect to the true measure of motion ; nor can there be much hesitation in 

 allowing at once, that since the same force, continued for a double time, is 

 known to produce a double velocity, a double force must also produce a 

 double velocity in the same time. Notwithstanding the simplicity of this 

 view of the subject, Leibnitz,* Smeaton,t and many others have chosen to 

 estimate the force of a moving body by the product of its mass into the 

 square of its velocity ; and though we cannot admit that this estimation 

 of force is just, yet it may be allowed that many of the sensible effects of 

 motion, and even the advantage of any mechanical power, however it may 

 be employed, are usually proportional to this product, or to the weight of 

 the moving body, multiplied by the height from which it must have fallen, 

 in order to acquire the given velocity. Thus a bullet, moving with a 

 double velocity, will penetrate to a quadruple depth in clay or tallow : a 

 ball of equal size, but of one fourth of the weight, moving with a double 

 velocity, will penetrate to an equal depth : and, with a smaller quantity of 

 motion, will make an equal excavation in a shorter time. This appears at 

 first sight somewhat paradoxical : but, on the other hand, we are to con- 

 sider the resistance of the clay or tallow as a uniformly retarding force, 

 and it will be obvious that the motion, which it can destroy in a short 

 time, must be less than that which requires a longer time for its destruc- 

 tion. Thus also when the resistance, opposed by any body to a force tend- 

 ing to break it, is to be overcome, the space through which it may be bent 

 before it breaks being given, as well as the force exerted at every point of 

 that space, the power of any body to break it is proportional to the energy 

 of its motion, or to its weight multiplied by the square of its velocity. 



In almost all cases of the forces employed in practical mechanics, the labour 

 expended in producing any motion, is proportional, not to the momentum, 

 but to the energy which is obtained ; since these forces are seldom to be 

 considered as uniformly accelerating forces, but generally act at some dis- 

 advantage when the velocity is already considerable. For instance, if it 

 be necessary to obtain a certain velocity, by means of the descent of a 

 heavy body from a height to which we carry it by a flight of steps, we 

 must ascend, if we wish to double the velocity, a quadruple number of 

 steps, and this will cost us nearly four times as much labour. In the same 

 manner, if we press with a given force on the shorter end of a lever, in 

 order to move a weight at a greater distance on the other side of the ful- 

 crum, a certain portion of the force is expended in the pressure which is 

 supported by the fulcrum, and we by no means produce the same mo- 



* Acta Erudit. Lips. 1686. 



t Ph. Tr. 1776, p. 450, and 1782, p. 337. See Desaguliers's Exp. Ph. ii. 92 ; 

 and Ph. Tr. 1723, xxxii. 269, 285. Eames on the Force of Moving Bodies, Ph. Tr. 

 1726, xxxiv. 188. Clarke in Ph. Tr. 1728, xxxv. 381. Zendrini, Sulla Inutilita 

 della Questione Intorno alia Misura delle Forze Vivi, 8vo, Venezia, 1804. 



