THE DIRECTION OF MOTION. 255 



pirical generalization? or may it be established as a corollary 

 from a still deeper truth? The reader will anticipate the 

 answer. We shall find it dedncible from that datum of 

 consciousness which underlies all science. 



Suppose several tractive forces, variously directed, to be 

 acting on a given body. By what is known among mathe- 

 maticians as the composition of forces, there may be found 

 for any two of these, a single force of such amount and 

 direction as to produce on the body an exactly equal effect. 

 If in the direction of each of them there be drawn a straight 

 line, and if the lengths of these two straight lines be made 

 proportionate to the amounts of the forces; and if from the 

 end of each line there be drawn a line parallel to the other, 

 so as to complete a parallelogram; then the diagonal of this 

 parallelogram represents the amount and direction of a force 

 that is equivalent to the two. Such a resultant force, as it is 

 called, may be found for any pair of forces throughout the 

 group. Similarly, for any pair of such resultants a single 

 resultant may be found. And by repeating this course, all 

 of them may be reduced to two. If these two are equal and 

 opposite — that is, if there is no line of greatest traction, 

 motion does not take place. If they are opposite but not 

 equal, motion takes place in the direction of the greater. 

 And if they are neither equal nor opposite, motion takes 

 place in the direction of their resultant. For in either of 

 these cases there is an unantagonized force in one direction. 

 And this residuary force that is not neutralized by an oppos- 

 ing one, must move the body in the direction in which it is 

 acting. To assert the contrary is to assert that a force can 

 be expended without effect — without generating an equiva- 

 lent force ; and by so implying that force can cease to exist, 

 this involves a denial of the persistence of force. It 



needs scarcely be added that if in place of tractions we take 

 resistances, the argument equally holds; and that it holds 

 also where both tractions and resistances are concerned. 

 Thus the law that motion follows the line of greatest trac- 



