ON MOTION. . 25 



The joint result of any two motions is the diagonal of the parallelogram, of 

 which the sides would be described, in the same time, by the separate motions ; 

 that is, if we have two lines representing the directions and velocities of the 

 separate motions, and from the remoter extremity of each draw a line parallel 

 to the other, the intersection of these lines will be the place of the moving 

 body at the end of the given time. This is the necessary consequence of the 

 coexistence of two motions in the sense that has been defined; it is also ca- 

 pable of a complete illustration by means of the apparatus that has been de- 

 scribed. (Plate I. Fig. 7.) , 



Any given motion may be considered as the result of any two or more 

 motions capable of composing it in this manner. Thus the line described by 

 the tracing point of our apparatus will be precisely the same, whether it be 

 simply drawn along in the given direction, or made to move on the arm 

 with a. velocity equal to that of the arm, or, when the arm is in a different 

 position, with only half that velocity. (Plate I. Fig. 8.) 



This principle constitutes the important doctrine of the resolution of mo- 

 tion. There is some difficulty in imagining a slower motion to contain, as 

 it were, within itself, two more rapid motions opposing each other: but in 

 fact we have only to suppose ourselves adding or subtracting mathematical 

 quantities, and we must relinquish the prejudice; derived from our own feel- 

 ings, which associates the idea of effort with that of motion. When we 

 conceive a state of rest as the result of equal and contrary motions, we use 

 the same mode of representation, as when we say that a cipher is the sum of 

 two equal quantities with opposite signs; for instance, plus ten and minus 

 ten make nothing. 



«\> 



The law of motion here established differs but little in its enunciation 

 from the original words of Aristotle, in his mcclianical problems. He says, 

 that if a moving body has two motions, bearing a constant proportion to 

 each other, it must necessarily describe the diameter of a parallelogram, of 

 which the sides are in the ratio of the two motions. It is obvious that this 

 proposition includes the consideration not only of uniform motions, but also 

 of motions which are similarly accelerated or retarded: and weshould scarce- 

 ly have expected, that, from the time at which the subject began to be so clcar- 



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