ON MOTION. .19 



tions, 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 co-existence of two motions in the sense that has been denned ; it is 

 also capable of a complete illustration by means of the apparatus that has 

 been described. (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 

 motion. 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 mathe- 

 matical quantities, and we must relinquish the prejudice, derived from our 

 own .feelings, 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 I 

 1 from the original words of Aristotle, in his mechanical problems.* He ' 

 says, that if a moving body has two motions, bearing a constant propor- 

 tion to each other, it must necessarily describe the diameter of a parallelo- 

 gram, 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 we should scarcely have expected that, from the time at which the 

 subject began to be so clearly understood, two thousand years would 

 have elapsed before this law began to be applied to the determination of 

 the velocity of bodies actuated by deflecting forces, which Newton has so 

 simply and elegantly deduced from it. 



In the laws of motion, which are the chief foundation of the Principia, 

 their great author introduces at once the consideration of forces ; and the 

 first corollary stands thus : " a body describes the diagonal of a parallelo- 

 gram by two forces acting conjointly, in the same time in which it would 

 describe its sides by the same forces acting separately." It appears, how- 

 ever, to be more natural and perspicuous to defer the consideration of force 

 until the simpler doctrine of motion has been separately examined. 



We may easily proceed to the composition of any number of different 

 motions by combining them successively in pairs. Hence any equable 

 motions, represented by the sides of a polygon, that is, of a figure consist- 

 ing of any number of straight sides, being supposed to take place in the 

 same moveable body in directions parallel to those sides, and in the order 

 * Mech. Prob. c. 24. See also Galileo, Dial. 4, Prop. 2. 

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