If a body move uniformly, it is customary at once to lay down as the measure of the velocity the space described in a given time, usually the unit of time, a second, a minute, an hour, as the case may be. So far as the great object of calculation is concerned, this definition is perfect : by instituting measures of velocity, we can but want to answer one or other of these questions : Where will the moving point be at the end of a given time I or, In what time will the moving point pass over a given length! The body moves at the rate of v feet per second, it moves over rt feet in t seconds, and moves over the length a feet ins÷v seconds. Let us now take a point moving with a variable motion, that is, not describing equal lengths in equal times, say a particle descending by its own weight in a vacuum. In the first second it falls its feet ; but in the first half of this second it falls only 4 feet, and the remaining 12 feet in the second half-second. The space described in one second is therefore no measure of the rate of motion during that second, and it is now to be asked, What is the way of obtaining a measure of the speed after any interval has elapsed / What is velocity itself, when it cannot for want of uniformity be ascertained by the apace described in any given time f if the action of gravity were removed at the end of that time, so that the point would go on uniformly with its last acquired velocity, how much would it then describe in one second/ All these questions are the same, and the answer cannot be given without the introduction of the notion of a limit, whether with or without the forma of the differential calculus. At the end of the time t seconds, let the moving point be at A, distant by a feet from the fixed point o. During the ensuing fraction is of a second, let it deecribo the further space A (=k). The length k is then moved over in the time h and, if the velocity were uniform, that velocity would be ke-h feet in ono second ; for as is is to I (second), so (on the supposition of uniform velocity) is k to the space which would be described in one second. If As were very small, wo might reason (with tolerable exactness) as follows : In a very small time tho change of speed will be slight, and the motion of the point nearly uniform, though not absolutely so ; whence we easy say, without material error, that an is described as with a uniform velocity at the rate of k÷h feet per second. The process which the mathematician adds is the following :—The error of the preceding process, smell when h is small, becomes smaller when his still smaller, and may be diminished to any extent : that is, little as may be the departure from uniform motion in moving through a small length, it is less in moving through a smaller. If, then, instead of making is simply small, and then finding k. h, diminish h without limit, and find the limit towards which k÷h approaches, we find that uniform velocity which may be said to represent the speed of the point in passing through A, so far as any uniform velocity can be said to do so. Using such language as supposes the point to have volition, we have, in the limit of h, the length per second with which the point shows an intention of proceeding at the instant when it passes through A, though it does not preserve that intention wholly unaltered for any portion of time, however small.
Suppose for example that the point moves in such a way as to describe t feet in t seconds, for all values of 1, whole or frac tional. We have then a= t +1, + k= (t + h)+ + whence we obtain At the end bf three seconds, what is the velocity I Judging from the length described during the succeeding fraction is (and making t ee3), we should say that, k÷h being 7+ h, the limit of this, or 7, obtained by diminishing is without limit, is the velocity required; that is, the point is then moving 7 feet per second. If we suppose 7 feet per second, the length described in the fraction is of a second is the fraction 7h of a foot ; take any other uniform velocity p feet per second, and ph is the length described in the same time. Now what is really described ie 7 h+14=; so that the errors are /0 and 17—y) which are in the ratio of is to 7—p+h. Now if p differ (as we have supposed) from 7, the first error diminishes without limit as compared with the second, when is diminishes without limit : so that, of all uniform velocities, 7 feet per second is the one which best represents the motion of the point in any small time following the end of the third second ; and the better the smaller the time.
It appears then that we do not, properly speaking, undertake to say at what rate the point is moving at the end of three seconds, but what fictitious uniform rate best represents, at the instant, the variable rate at which it is moving. This will, for a moment, seem rather unsatis factory to the student who imaginer that lie has got an absolute idea of velocity, and here he should compare his notion on this subject with that of the direction of a point moving in a curve. [Dineeviox ; TANGENT.] What do we mean by saying that a point which moves in a curve has, at every instant, the direction of motion which is repre sented by the tangent of that curve? Answer, in nearly the same words as before, We do not, properly speaking, undertake to say in what direction the point is moving at any period of its motion, but what fictitious line of uniform direction. (straight line) best represents, at that instant, the line of variable direction (curve) on which it is moving. The study of these two things together, velocity and direc tion, is useful, as each throws illustration upon the difficulties of the other. In both cases the laws of matter agree in preferring that which is indicated as most simple by the laws of mind ; for if a point moving along a curve be suddenly relieved of the forces which keep it in a perpetual change of speed and direction, it will proceed with that very velocity which we have said it shows its intention to proceed with, uniformly; and will quit the curve for that straight line which we might equally well have said it showed a disposition to prefer to any other while moving on the curve, namely, the tangent of the curve.
If it should be said that we are reduced, in treating of variable velocities, to a necessity which does not occur in describing those which are uniform, namely, the use of limits, we altogether deny the fact : that is, we say that we are as much compelled to the use of limits in defining a uniform velocity as a variable one. For what does uniform velocity mean ? A point has uniform velocity when equal spaces, any equal spaces whatsoever, are described in equal times ; or when, k being described in the time h, k÷h is always the same. That is, k÷h must retain its value, however small h may be; or the limit of k÷-11 must also have that value. And we have seen that it would be impossible to declare, experimentally, the existence of uniform velocity, even if our senses had no imperfections, upon the experience of com parisons of any finite equal spaces, however small ; nothing but assurance of the limit of k-i-h being the same thing wherever the point A might be placed, would give mathematical evidence of the velocity being uniform.
In all cases, then, by the velocity of a point in motion, at any particular period of its motion, is to be understood the limit of the ratio which the increment of the length described bears to the incre ment of the time expended in the description of that inereineut of length. That is, if the length be measured in feet, and the time iu seconds, and if k he the fraction of a foot described in the fraction of a second h, the limit towards which the fraction k divided by the fraction is continually tends while is is diminished without limit, is the number of feet per second which, we may nay, expresses the rate of motion at the period in question. The student of the differential calculus will now have no difficulty in altering the preceding into the following form : if the length s be described in the time t, the velocity (r) at the end of the time t is thus expressed : If y be any function of x, and if .r represent the number of units of length described by a moving point in the time t, and y the same fur another moving point, and if y=rjor, we have by the rules of the differential calculus : dy dy do dx dt • Here dy : dt and dx : dt, represent y and Newton's FLuxioxs of y and x; and dy : dx is obviously the same thing as b : z. The term fluxion merely means velocity, and, after all, there can be formed no clearer notion of a differential coefficient than one which is formed from a consideration closely resembling the fluxional one. If y be a function of x, dy : dx is the rate at which y is increasing, as compared with that at which x increases. Thus if dy : which when s-+.-10 is 4000. What does this mean We say that nothing can answer more clearly than the following : If a number be imagined to be gradually increased [VARIABLE], by the time it becomes 10 its fourth power will be, at that instant, increasing 4000 times as fast as itself.