VELOCITY. This word, rendered into English, is simply swiftness or quickness, and would be soon disposed of, if it were not that various circumstances connected with its measure and calculation render its consideration one of the most useful exercises which the student can have, not only in mechanics, but also in pure mathematics. And since the views which must be developed in treating properly of this word are almost identical with those which arise in explaining the meanings of other words nearly as important, we have made references from all quarters to this article, which, though they will increase its length, will upon the whole save room.
The difficulty in the way of a beginner, which he meets with in acquiring a clear notion of the measure of velocity, is the tendency to confound the velocity and its measure ; a tendency which is increased by any elementary work which hastens too rapidly to the mathematical treatment of the word. The consequence of this confusion is (since the measure of velocity. must be a length described, or rather a length capable of being described) a walit of power to distinguish between the space which a body does describe in a given time, and that which, judging from its velocity, it seems to be going to describe at the beginning of that time. Hence arise many notions mathematically false : these might perhaps be prevented by attributing volition to the moving particle, and distinguishing between its apparent intention at the beginning of the given time and that which it actually accomplishes in the given time. Such an illustration would probably receive no approbation ; but the errors to which it would lead would nut be of the least consequence in mathematics.
A point is in motion, and during a certain second it moves over ten feet : if the same thhig should happen in preceding and succeeding seconds, there is a presumption that the body is moving uniformly at the rate of ten feet a second ; that is to say, there is a presumption that, in any portion of time whats 'ever, during its motion, there is a length described which bears to ten feet the same proportion as that portion of time bears to one second. But this is a presumption only. It does not follow, because ten feet are described in one second, that five feet are described in each half of a second, and one foot in each tenth of a second. If the second could be divided into a million of parts, and it could be shown that the millionth part of ten feet is described in each and all of therm parte, it would be no doubt a very strong presumption that the motion is really uniform, but still not amounting to certainty; for it is possible that in each of those parts of time there may be a variation of speed : for instance, the moving point may do all its work in the first half of the email interval, and rest during the remainder. Something of this kind takes place in the
motion of the minute-hand of a clock, which is propelled once in each second during a portion of the second, and rests during the remainder. But so rapidly do the small propulsions follow one another, and so small are their individual effects, that, even when the hand has been watched until its motion is ccrtaiu, there is no irregularity discoverable by ordinary eyes. And, speaking with reference to common purposes, there is no occasion to deny uniformity of motion so long as the lengths described in those times which are convenient to be mentioned are Neal or nearly equal. It would be useless, in speaking of the pace with which a man walks four miles an hour, to remind the hearer that no person walks uniformly,anil that in every step the centre of gravity of the body moves up and down, advancing most rapidly when it is at the highest, and most slowly when it is lowest. But for mathematical purposes a correct measure of speed must be obtained, and the pre ceding account would at first seem to lead to the inference that it is impossible to have such a measure. Nor indeed has velocity yet received its definition in this article, at least not its measure : wo have spoken of velocity and of its changing, but without alluding to any mode of estimating the quantity of change. But there is that about the word which needs no definition : when we say that the railroad carriage moves "faster" than the old stage-coach, or that two bodies which net oat together and keep together are always moving "at the same rate," there is no need of explanation of the words which are in marks of quotation. And we must now refer to the considerations in II:aeons, as a constituent part of this article, showing that we may have a perfect idea, both of velocity that it is a magnitude, and that there is such a thing as uniform velocity, previous to any definite ideas of the most proper mode of measuring even that uniform velocity, to go no farther.