120 



embrace also the nature of their accumulation ; we have to ana- 

 lyse the component parts and consider them in connexion with 

 space and time. Speaking geometrically, the sides of the paral- 

 lelogram may be of equal length, and therefore contain an equal 

 number of parts ; but that is no reason that such parts should 

 be described in the same time. Time is another element, and 

 independent^ having no necessary connexion with pure geome- 

 try ; it belongs to physics, and if we combine them we must do 

 it legitimately. The laws of one are definite and unalterable, 

 whilst the other may be modified according to circumstances. 

 If we describe a circle, we find it in its usual exact proportion ; 

 if we apply motion and time to describe a circle, we must adopt 

 them for it, i. e. according to their own peculiar properties, and 

 not solely by geometry. We are told that the planets are re- 

 tained in their orbits simply by means of two forces, the tan- 

 gential and centripetal ; and that if one of the two forces were 

 taken away, these bodies would be either carried off in a tan- 

 gential direction, or fall to the focus of the centripetal action. 

 If a body put in motion by an uniform force in one definite 

 direction be met by another body moving at right angles to it 

 by a similar uniform force, they will form a compound at the 

 point of contact, which will be equal to the diagonal of the paral- 

 lelogram of the two forces, in direction and magnitude, and as 

 their respective forces were uniform, the compound will be uni- 

 form and straight. (See Plate XXIV.) 



Suppose the body set in motion by an impulsive or a vari- 

 able force be met by an uniform force, the diagonal will be 

 described by a variable velocity, and in a curvilinear direction. 

 If the two forces be variable in the same ratio, the diagonal of 

 the parallelogram will be straight, but described by a variable 

 motion. If we require the compound to form a circle, and to be 

 described in equal times and spaces, if the radius and veloci- 

 ties are to remain constant quantities, the centrifugal and cen- 

 tripetal must be constantly equal; but in order to form the 

 circle the tangential must possess the power of perpetually chan- 

 ging its direction, in a word, a circular motion ; without this the 

 orbit cannot be described by the compound. 



It must not be forgotten that an arc, although bounded by 

 the same extreme limits in a parallelogram, however small it may 



