motion, and hardly different from the second of the propositions now enunciated.

The first demonstration, therefore, of the proposition on which the whole theory of the vis viva is actually founded, is the work of Newton himself; and whatever was his own opinion concerning the measurement of force, he was certainly in possession of the key to the solution of the difficulty, and of the true principle on which it was to be ultimately decided. The Newtonians who have engaged in this controversy, may not always have been aware, that the Principia contained the demonstration of all that was sound both in their own argument and in that of their adversaries; and may therefore, agreeably to Dr Wollaston's remark, have maintained propositions which the author of their philosophy would not have been inclined to support.

In considering the use made of the two different measures of force, in science or in art, Dr Wollaston has the following remark.

The former conception of a quantity dependent on the continuance of a given vis motiva for a certain time, may have its ufe, when correctly applied, in certain philofophical confiderations; but the latter idea of a quantity refulting from the fame force, exerted through a determinate Space, is of greater practical utility, as it occurs daily in the ufual occupations of men; fince any quantity of work performed is always appretiated by the extent of effect refulting from their exertions; for it is well known, that the railing of any great weight forty feet, would require four times as much labour as would be required to raife an equal weight to the height of ten feet; and that, in its flow defcent, the former would produce four times the effect of the latter in continuing the motion of any kind of machine. '

Now, with the judgment here given as to the respective utility of the two measures of the force of moving bodies, we cannot entirely agree; though we differ from Dr Wollaston with considerable diffidence; and the more, that his opinion is supported by one of the greatest authorities in practical mechanics of which this or any other country can boast-the late Mr Smeaton. That excellent engineer, in a paper on the Collision of Bodies, published in the Philosophical Transactions for 1782, and in a former one inserted in the same work for 1751, has given the name of Mechanical Power to that force in moving bodies, which is supposed proportional to their quantity of matter multiplied into the square of their velocities; and he has contended, that it is to this quantity that the effect of machines destined to produce motion is always proportional. In hydraulic engines, he endeavours to prove the truth of this maxim by a series of experiments, instituted with a degree of skill, ingenuity and exactness, altogether worthy of his high reputation. In deducing his conclusions

from

from them, however, he has not observed, that his measure of mechanical effect is one that involves the very principle he is objecting to, and virtually supposes that effect to be represented by the quantity of matter moved, multiplied, not into the square, but into the simple power of its velocity. He defines (Experimental Inquiry, p. 6.) the measure of the effect of a machine to be the weight raised, multiplied into the height to which it is raised in a given time; so that, if the time is not given, the measure of the effect is the product of the weight raised, into the height to which it is raised, divided by the time. Now, this is the same thing with the weight raised, multiplied into the velocity simply. For if W be the weight raised, h the height to which it is moved in the time t, and its velocity, the effect, by the W x b preceding definition, is being the velocity with which W x b W × vt

t

But

W moves, we have but, and therefore

t

=

t

=

many

W × v; and Wx v is therefore equal to the effect. Thus, in the very outset of the investigation, the principle of the vis viva, or mechanical power, is in fact abandoned, in consequence of including time in the measure of the effect, without which, however, that measure would be imperfect, and of no use for of the objects which mechanical contrivances have in view. This circumstance appears evidently not to have been observed by Mr Smeaton, Had he used the algebraic, instead of the ordinary language, such an oversight could not have happened. envelope which the latter affords for such pice ideas as those of force or of power, is not transparent enough, if we may say so, to allow all their relations to be clearly perceived; and hence one of those relations, though not far removed from the surface, has escaped the eye of a very sagacious and penetrating observer.

The

This remark being admitted, we shall easily discover that Mr. Smeaton's conclusions, which appear most hostile to what is call ed the Newtonian measure of force, are, in fact, perfectly consistent with it. This holds particularly as to the second of his general maxims, deduced from a comparison of experiments on undershot wheels, where the expense of water was the same, but the velocity different. That maxim is, that the expense of water being the same, the effect will be nearly as the height of the effective head, or (as it is expressed in maxim third) as the square of the velocity of the water. This conclusion seems, at first sight, quite in favour of the theory of mechanical force, as laid down by our author, and the other supporters of the vis viva; and yet we shall presently find, that it is perfectly conformable to the other theory, and to those reasonings of Desaguliers

and

and Maclaurin, which Mr Smeaton has censured, as leading to conclusions altogether wide of the truth.

บ

Let c be the velocity of the stream, that of the wheel, A the area of the part of the float-board immersed in the water, g the velocity which a heavy body acquires in one second when falling freely. Then c — will be the relative velocity of the stream and the wheel, or the velocity with which the water strikes the wheel; and if we take h, a fourth proportional to g2, (c — v)2 and g, h will be the height from which a body must fall to ac(cv). Where

2g

quire the velocity cv, and will be equal to fore, by a prop. well known in Hydraulics, the circumference of the wheel is urged by the weight of a column of water, of which (c — v)3, and of which the soli

the section is A, and the height

2g

(c—v)3. Thus far the investigation is 2g

dity is therefore A× applicable to all undershot wheels, and to all hydraulic engines of a similar construction. But to bring it to the case of those experiments in which the expenditure of the water was the same, let E be equal to that expenditure, that is, to the cubic inches

E

delivered in one second, then A== and so the pressure of the

с

E

water, or the intensity of the impelling power = (c — v)3.

2gc

Now, if the resistance overcome, or the weight raised, be = W, and its velocity u, then when the machine has attained a state of uniform motion, the momenta of the resisting and impelling forces must be equal; that is, W × u =

E

2gc

(c — v)'v. The quan

tity W x u will therefore be a maximum, when (c — v)3v is a

E

maximum, the coefficient being constant, and v alone being 2gc variable. But (cv)'v is a maximum, as is easily shown, when v=c, and therefore Wu is also a maximum in that case. Now, Wx u has already been shown to be the measure of the effect of the machine; therefore, the effect of the machine is greatest when ve, that is, when the velocity of the wheel is one third of the velocity of the water which impels it. In that

E

case, also, the quantity (cv)3v, which is always equal

2gc

E 4c2 с E 4c2

to the effect, becomes X X = X

2gc g 3 2g 27

therefore the

maximum of effect is proportional to or to the square of the

velocity

velocity of the stream; which is precisely Mr Smeaton's third maxim, as deduced from his own experiments. In the same way might the truth of his fourth maxim, that, when the aperture of the sluice is given, and when the height of the head varies, the effect is as the cube of the velocity of the water, be deduced from the received principles of Hydraulics; and yet Mr Smeaton evidently considers both these maxims as inconsistent with those principles. Here, therefore, this ingenious man has evidently fallen into an error; and this he has done, either from not having thoroughly considered the measure which he himself had assigned to the effect, or from not clearly perceiving that when a machine comes to its state of uniform motion, it is not because the resisting and impelling powers are equal, but because their momenta are equal; that is to say, the resisting power multiplied into its velocity, equal to the impelling power multiplied into its velocity.

We have gone over the above investigation with more minuteness than a matter of so little difficulty may seem to require; because we wished that it might appear quite plain from what sources all the reasoning was derived, and that no part of it involved the idea of mechanical power, or of the vis viva. The only part of it that can be suspected of doing so, is the hydraulic proposition concerning the impulse of a fluid; but it is certain that this theorem, though one of those about which some difference of opinion has arisen, can be deduced in a satisfactory manner from the nature of fluids, without any appeal to the doctrine of the vis viva.

At the same time that the theorems, derived in the manner exemplified above, bring out results that agree with Mr Smeaton's experiments as to the proportion between effects arising from different impulses, yet it must be acknowledged that they do not agree well as to the absolute quantities. Thus, the greatest effect is assigned by the theory, in the case of undershot wheels, to that velocity of the wheel which is one third of the velocity of the stream. Mr Smeaton found, that the velocity which gives the greatest effect, is between and, but nearer to the latter. The theory gives the maximum load of the impelling power; Mr Smeaton's experiments make these quantities approach very near to equality. Whatever be the reason of a difference that falls out on the opposite side to what is usual, the practical result being more favourable than the theoretical, it certainly does not lye in any thing which the introduction of the vis viva would cor

Were we, for example, to suppose that the forces of the single particles of the fluid, to put the wheel in motion, are as the squares of the velocities, we should then have, for the expression of the impelling force, (cv)3 multiplied into some con

stant

stant coefficient, so that the greatest effect would be when (cv)3v was a maximum, which happens when v=c, or when the velocity of the wheel is only one fourth of that of the On this supposition, also, the effect would be as the fourth power of the velocity: both of which conclusions are perfectly inconsistent with Mr Smeaton's experiments, and with all the best established maxims in hydraulics. To whatever cause, therefore, the imperfection of the theory of the machines moved by water is to be ascribed, it is not to any thing that would be corrected by the introduction of a measure of force different from that which is commonly in use. If we are right in this conclusion, it is evident that a large class of machines, all those, namely, that are moved by the impulsion of fluids, are taken from the list of those of which the effect is best estimated by what Mr Smeaton has called Mechanical Force. We believe that the same conclusion may be extended to many more; though we perfectly agree with Dr Wollaston, that when all that is to be determined is the quantity of effect that corresponds to a certain force, without any reference to time, the principle of the vis viva will afford the simplest and shortest way of determining that quantity.

There is, however, a consideration different from any that has been yet mentioned, which, if we mistake not, both in science and in art, will very much decide to what measure of force the mechanist must have recourse. The nature of the propositions on which those measures are founded, must be consulted; from whence it will appear whether the one or the other is most easily applicable to a given case.

The first method of measuring the force of percussion, is founded on this principle, that if the pressure or accelerating force, that acts uniformly during any interval of time, be multiplied into that time, and if the sum of all the products so formed be taken, that sum will be proportional to the simple power of the velocity communicated. Now, this theorem, in

order that it may be used readily, requires that the relation between the forces and the time should be known; or, in other words, that we should be able readily to express the force in terms of the time, or the time in terms of the force; in either case, the determination of the velocity is reduced to a problem in the summation of series, or in the quadrature of curves, more or less difficult as the relation between the time and the force is more or less complicated.

Again, the second method of finding the velocity communicated by the successive impulses of an accelerating force, is by multiplying each force into the length of the line over which the

body