ACCELERATION is the increase of velocity ; and in the article cited uniform acceleration has been considered, and its laws deduced, if not with the forms, yet on the principles, of the differential calculus. Precisely the same difficulties come before us in the development of the measure of acceleration as in that of velocity, and they are to be met in the same manner. In fact, by the acceleration is meant 'the rate of increase of the velocity, the velocity of the velocity. Suppose the velocity, first, to increase uniformly : that is to say, let b feet be added to it in every second, and in that proportion for all times elapsed; if then a be the initial velocity, that at the end of the time t is a +At, and we have if a be measured from the point of starting. Here at is the length due to the initial velocity a, and 6/ft= the effect of the continual acceleration. Now suppose, returning to the diagram, that the velocity at II is greater by 1 than that at a, and the fraction h of a second having elapsed between the two positions : that is, suppose that at A the point begins to move as if it meant (continuing our illustration) to describe v feet in the next second; but that by the time of corning to is it begins (from n), as if it would describe ro+1 feet in the next second. If this increase of velocity were uniformly given, that is, if in the time 6 A its velocity had become r +1,/, in A v+ Al, and so on for every fraction of lt, we might then infer that the acceleration at A, that is, the rate at which velocity is than increasing, measured by the quantity which would be gained in a second at the same rate, is 14.-A : for as It is to one second, so is I gained in the time A to what would be gained in a second at that rate. But if this supposition be not true, that is, if the speed receive unequal additions in equal times, we must then begin to reason as before, and to find what (pursuing the same illustration) we may call the intention of the velocity. If 1 be added to the velocity in the small time h, it will be added nearly uniformly ; if A be still smaller, still more nearly, and so on : in such manner that while, practically speaking, /444 is a sufficiently good representative of the current rate of acceleration, when A is small, the (uniform) rate of acceleration which best represents the state of things at A is the limit which is deduced by making A diminish without limit. And
here again, copying our own preceding words, we do not undertake to say at what rate the velocity is increasing when the moving point is at A, but what fictitious uniform rata of increase best represents, at the instant, the variable rate of increase which would be detected if the changes of velocity between A and n could be noted. And hence, if re be called the acceleration, the student of the differential calculus will easily deduce, do Ws rG to dl = .= 5, since v= and also vdc. weld.
Thus if the motion of the point be such that in t seconds there are described feet, we have as follows :— t, r =30+ to = 6t + At the end of two seconds, then, the state of things is this :—the point has advanced 8 +16 or 24 feet, and if allowed to move on without further change of velocity, would describe 12+32 or 44 feet in the next second, and has the velocity 44 (feet) in one second; while at the same time there is an acceleration taking place which would, if allowed to remain uniform for one second, add 12 +43 or 60 to the velocity, making it 44+60 or 104 at the end of the third second. But this rate of acceleration is itself increasing, since at the end of the third second the velocity is 27+324 or 351.
So far the subject, and all notions connected with it, fall within the province of pure mathematics : if there were no such thing as either matter or force, but only motion and a mind to conceive it, all that has been said might be intelligible. It is very much to be regretted that the connection between the mathematical doctrine of motion and the laws of matter is unduly made, and at too early a stage, by the appli cation of the term accelerating force, instead of simple acceleration, to the result d r : dt. Acceleration would be what we have described it — — to be, if matter were not inert, if it moved by its own volition, or on any supposition whatever, provided only that it moved. Why then should a theory be made to supply the name of a result antecedent to that theory, and which would be perfectly true even if that theory were false ? The consequence of this is, that when the laws of matter come to be applied to the mathematical expressions of motion, things are taken for granted which ought to be learnt.