793 



MOTION. 



MOTION. 



may be to conventional ideas o rigour, have never failed to introduce 

 perplexity and obscurity to the beginner. It may be right to remind 

 the student that the change of place introduced by Euclid (i., prop. 4 

 and other propositions)" has not necessarily all the concomitants of the 

 idea of motion ; geometry would not interfere to prevent the super- 

 position from being made without the notion of the triangle, whose 

 place is changed, passing through the intervening parts of space. It 

 was the introduction of the idea of time which the parties who objected 

 to the doctrine of fluxions repugned. 



But if we consider matter' in motion, we must inquire into the 

 external causes of motion, and the capabilities of matter with respect 

 to motion; this we shall do in the next article [MOTION, LAWS OF], 

 confining ourselves hi the present one to the first-mentioned branch of 

 the subject. 



Next to the idea of motion comes that of swiftness, rate of motion, 

 or velocity (see also the latter word), suggested by observing different 

 motions, or different changes of place in the same time. But here we 

 must observe, that we are rather indebted to motion for our measure of 

 time than to time for our measure of motion. If sentient beings, like 

 ourselves, had lived in perpetual day, without any recurrence of 

 periodical phenomena in nature, or any mechanical means of gene- 

 rating equable motion, we have no right to suppose that they would 

 ever have learned to consider time as a measurable magnitude. They 

 might admit that it might be more or less, as we do of industry, courage, 

 or any other moral qualities [MAGNITUDE], but we cannot be more 

 destitute of measures for those qualities, than they would be of means 

 for measuring time. Since however we have obtained, though by 

 means of equable motion, a distinct idea of successions of duration, 

 equal in magnitude, we u=e this idea in the definition of motion, just 

 as in geometry we consider the line before the surface, though we 

 have no certainty that we ever should have a distinct notion of a line, if 

 we had not formed lines by the intersection of surfaces. We say, 

 though we have no certainty, but we do not forget that many philo- 

 sophers are of opinion that such ideas as those of time and of a line are 

 fundamental notions, resulting from our rational organisation, and (if 

 we do not mistake them) anterior to observation, or, at least, not 

 derived from it. This question is here immaterial, as we suppose all 

 parties ready to start with a definite notion of time. Considering 

 the motion of a simple point, which describes a line, it is called 

 uniform when the lengths described in successive equal times are 

 equal, whatever each time may be. It is important to remember this, 

 since different successive motions may be uniform in some respects 

 and not in all. Thus successive revolutions may be performed in 

 equal times, as to whole revolutions, but equal fractions of one revolu- 

 tion may not be performed in equal times. In uniform motion, an 

 arbitrary unit of time is chosen, and the length described in that time 

 is called the velocity, which is simply the Latin for quickness. If 

 extreme verbal correctness were required, this length should be called, 

 not the velocity or swiftness, but the measure of the velocity. For the 

 length described in (say) one second is not the velocity or swiftness, 

 but something by which we judge of it. The word velocity is an 

 abstraction from the comparison of motions ; of two moving points, 

 that one which described the greater length in a given time moved the 

 quicker : and swiftness is the absolute substantive by which we 

 express the existence of the obvious relation, just as magnitude is that 

 by which we express the existence of the relation of greater and less. 



When equal spaces are not described in equal times, we can imagine 

 the rate of motion to change either gradually or discontinuously. Thus 

 it can be imagined that a body which moves for some seconds uniformly 

 at the rate of 10 feet in a second, may at once, without any intermediate 

 state, take a velocity of 20 feet. But such a conception cannot be 

 realised on any material body, though there may be all the appearance 

 ol it. [IMPULSE.] When the rate of motion is changing perfectly 

 gradually, there seems to be no direct method of obtaining the rate at 

 any one instant ; for no successive equal spaces are described in equal 

 times. This difficulty will be discussed in the article VELOCITY : for 

 the present, it may be considered sufficient to take a length so small 

 that the change of rate undergone in passing through it is insensible, 

 and to consider the point as moving uniformly through that length. 

 Let the very small portion of a foot represented by 8 be described in 

 the small fraction of a second, represented by S t. Let r be the number 

 of feet which would be described in one second, at the same rate. 



Then, f :8 = 1 : St.'.v =- ; and 5 and St being taken as small as 

 8 1 



we please, may be taken for the velocity at any instant. 

 9 1 



The existence of motion is detected either by a change of the distance 

 of an object, or of its direction, or both ; but it is not necessarily the 

 object which moves. The spectator himself may be in motion uncon- 

 sciously, and it is matter of common experiment that every motion of 

 the spectator of which he is not conscious, and every rapid motion, 

 whether he be conscious of it or not, causes surrounding objects to 

 appear in motion. In walking, the effort necessary to maintain motion 

 perpetually reminds us that it is ourselves who move ; in a carriage, at 

 an ordinary pace, we can always destroy the illusion of surrounding 

 motions by a moment's thought. But if the attention drop, and we 

 ook at objects with the mind intent on other things, they soon take 



the motion of the carriage in a contrary direction. In the smooth 

 motion of a boat, no effort of thought will enable the spectator to 

 realise his own motion, and destroy that of the shore or a neighbouring 

 vessel. We state that which we find to happen to ourselves ; perhaps 

 the experience of other persons may be different. 



It may also happen that the object is in motion as well as the 

 spectator, in which case the latter motion will be transferred to the 

 former, in the manner in which we shall describe. The whole motion 

 of the object, compounded of that which it has of its own, and that 

 which it appears to have from the motion of the spectator, is called 

 the apparent or relative motion. 



The method of ascertaining the relative motion is as follows : Since 

 we only determine the positions of bodies by their distances and 

 directions, and since we suppose the motions both of the spectator and 

 the object to be given, let a fixed point be taken to represent the 

 position in which the spectator imagines himself to remain, and, laying 

 down the real distances and directions of the object at the end of 

 successive times, set off those distances from the fixed point in the 

 proper directions. The relative positions of the object being thus 

 secured, the line passing through these positions will be that in which 

 the object appears to move. For instance, let the spectator move 

 through 12 3... 8 9 while the object moves through A BO... HI, so 

 that when the first is at 1, the second is at A; when the first is at 2, 

 the second is at B, and so on, the last positions being 9 and i. Take 

 o for a fixed point, at which the spectator fancies himself to be, and 

 having joined 1 and A, 2 and B, &c., draw o a parallel and equal to 1 A, 

 6 parallel and equal to 2 B, &c., and o parallel and equal to 9 i. 

 Hence the spectator, fixed at o, will see the object successively at the 

 same distances and in the same directions as a, b, &c., and i ; whence 

 the line abc...hi will be that of its apparent motion. 



When both motions are rectilinear and uniform, the apparent motion 

 may be more simply obtained, as follows : Let the spectator move 

 uniformly from o to A, while, in the same time, the object moves from 

 B to c. Take the following method of fixing the spectator : As he 

 moves forward from o to A, let the paper on which the figure is drawn 

 move backward in the direction contrary to o A, so that by the time 

 the spectator has reached A, the point A shall have receded to where o 

 was. He will therefore never have changed his place, his progression 

 on the paper having been always compensated by the retrogression of 

 the paper itself. Take c D parallel and equal to A o, whence the point 

 c will, by the motion of the paper, be at the end of the motion, where 



Fig. 2 







D was at its beginning. Consequently, the spectator, who imagines 

 himself at rest, will give to B that motion which is compounded of a 



