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body has moved while that force acted on it; and the sum of all these products will be proportional to the square of the velocity, and, of course, the square root of the said sum, to the velocity itself.

Now it is obvious, that in order to apply this theorem to any case, we must be able to express the forces in terms of the distances at which they act; for then the sum of the products described in the theorem will either be found by the summation of series, or the quadrature of curves; so that the thing wanted will be determined. The circumstance, therefore, which distinguishes the one of these kinds of dynamical problems from the other, is, whether the forces that produce the motion can be most easily expressed in terms of the time reckoned from a given instant, or in terms of the distance reckoned from a given point. Instances of both cases are easy to be given. Suppose it required to determine the velocity of a body accelerated or retarded by the action of a constant force, as heavy bodies are in their descent or ascent at the surface of the earth ;– In this case, either of the two methods may be employed indifferently. The force being given, if it be multiplied into the time during which it acts, the product will be proportional to the velocity, according to the first proposition. And in the same way, if the given force be multiplied into the distance passed over, the square root of the product will be proportional to the velocity; and thus, in either way, may the velocity, with nearly equal facility, be determined. It must be determined in both ways to make the investigation complete; and it is a matter of indifference with which we begin.

But it is not so if the accelerating force is variable, and expressed by some function of the distance from a given point, (as gravitation really is when we take in a considerable range): the first step in the inquiry must be made by help of the second proposition, that is, by multiplying the force into the fluxion of the distance from the said point, and making the fluent (which will easily be found) equal to the square of the velocity. The velocity being thus expressed in terms of the distance, the time required for moving over a given distance will next be found. It is in this way that Newton has resolved the very problem here proposed, in the 39th proposition of the first book of the Prin cipia, before referred to. It is therefore according as the data in any problem furnish means for integrating one or other of the formulas derived from the propositions above mentioned, that the one or the other must be employed in the solution of that problem.

In the use of this second method, however, there is a circumstance

stance that must be attended to that makes the theorem a little more complex than in the enunciation just given, and a little more embarrassing in the application. If the velocity treated of does not begin or end with the distance at which the action of the force begins or ends, it is not the square of the velocity generated that is proportional to the sum or area found by this theorem, but it is the difference between the square of the ini tial velocity, and the square of the velocity ultimately acquired that is proportional to that quantity.

Dr Wollaston has very distinctly pointed out those cases in practical mechanics, where the second method of estimating power is peculiarly applicable. They are those where the total effect to be produced, while a certain space is travelled over, is all that is required to be found, and where there is no question about the time. There are no doubt cases of this sort, and to such this method of investigation is well accommodated. When the artillerist would compute the effect of his shot, he looks only to the total amount; he is, in most cases, quite unconcerned about the time; and if he knows that, by doubling the velocity of the ball, he can sweep away four times as many men as before, he is nowise interested to discover by how many millionths of a second one of the victims of his destructive art may. be destined to survive another. But it is not always that such indifference about time can accompany the exertions of human power. In most instances, time is a very material element in the estimation of an effect, or an event of any kind; and is, of all our resources, that which it most behoves us to economize. In the case of all engines which move with a moderate velocity, the time of producing the effect is of great consequence to be known; and whenever the effect is estimated, as Mr Smeaton has supposed, by the space over which the load is raised in a given time, that is, as we have shown, by the resisting force, or the weight raised, multiplied into its velocity, there the ordinary supposition, that force is proportional to velocity, is necessarily introdu ced. It is here, as we observed before, that the skilful engineer just named has been led into mistake, and has supposed that some of his experimental conclusions were contrary to a theory with which they are in fact perfectly consistent. As the experiments them selves are extremely valuable, and made with a scrupulous attention to every circumstance that could secure their accuracy, it were to be wished that they should be subjected to a complete theoretical examination.

Another remark which we must be permitted to make is, that, even in the cases when it would seem to be sufficient to know that the power employed can produce a certain effect without any re

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gard to time, yet some other circumstance must be tacitly taken into account, otherwise the information would be too vague and unlimited to be of much practical utility. When it is faid, for example, that a bufhel of good coals will give to a steam engine the power required to grind eleven bufhels of wheat, this must always imply a rate of burning included within certain limits; for the fuel might be applied fo flowly, that the steam generated would not be of ftrength futhcient to work the mill; or it might be made to burn fo faft, that very little effect would be produced. In the fame way, when Mr Smeaton fays that if 1000 tons of water be let out on an overfhot wheel, and defcend through twen ty feet, it will grind the fame quantity of corn, at whatever rate it be expended, (Experimental Examination, p. 90.), the extreme cafes of very great flownefs, or very great rapidity, must furely be excepted. But if the extreme cafes must be excepted, it is a proof that, even in the intermediate cafes, the effect is not conftant or invariable in its magnitude, though the differences may be inconfiderable: this, at leaft, is what one would be difpofed to infer from that continuity in the variation of caufes and effects, to which there is perhaps no exception, either among the works of nature or of art.

These are fome of the difficulties that feein to ftand in the way of the application of the principle of the vis viva, or of mechanical force, as the fole measure of the effect produced by machinery or power of any kind; and on account of them, the judgment pronounced by Mr Smeaton, and fupported by Dr Wollaston, must be admitted, as appears to us, with confiderable limitations. Another subject, of which Dr Wollaston takes notice in his lec ture, is the incompatibility that fome philofophers have believed to exist between the third law of motion, and what concerns the prefervation of the vis viva; or, in other words, between the fact, that in all phyfical action the quantity of motion generated in any one direction is just equal to that which is loft in that fame direction; and the other fuppofed fact, that after the action has taken place, the quantity of the vis viva (ariling from multiplying each body into the fquare of its velocity) is the fame that it was before. The fuppofition, however, that fuch an inconfiftency exifts, is entirely a misconception; and we have no doubt that though fome of the followers of Newton fell into this mistake, it is im poffible that he himself should have done fo. If the two fuppofitions juft ftated be reduced into equations, it will be found that, in order that they may be compatible with one another, in the cafe of two bodies, a third condition muft belong to the mo tion of these bodies, viz. that their relative velocity, after their mutual action, must be the fame that it was before it. This conVOL. XII. NO. 23.

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dition

dition does not neceffarily take place in all inftances of phyfical action; it does fo, however, in many, as in the collifion of elaftic bodies; and then, of course, the quantity of the vis viva remains unchanged.

Lafily, we muft join with Dr Wollafton in recommending the ufe of different terms for expreffing the different modifications of power that are concerned in the production of mechanical effects. He has fuggefted two; but the purposes both of science and of art feem to require that there fhould be three, and each exclufively applied to its peculiar object. The controverfy to which we have fo often referred, concerning living and dead forces, arose in a great measure from the want of appropriate language; and though the difpute was not wholly verbal, it did moft ftrongly illuftrate Bacon's maxim, Credunt homines rationem fuam verbis imperare; fed fit etiam ut verba vim fuam fuper intellectum retorqueant & reflectant. Power, when of the fimple kind that is immediately comparable to preffure, or to the weight of a quiefcent body, we would call force, and would be fcrupulously exact never to use the latter term but for this purpofe. This would create little innovation in the language of mechanics: the terms centripetal force, centri fugal force, force of gravity, force of elafticity, &c. would all remain as they now are. Next, to denote the power of Percuf fion or of a body in motion, when we speak relatively to the effect produced by that power in a given time (which is proportional to the quantity of matter multiplied into the velocity), we must have a term different from the preceding. Dr Wollafton proposes the word Momentum; but as that term has been employed by many mechanical writers, to denote what, by operative men, is called purchafe, or power relative to its effect in producing angular motion, it would perhaps be wrong to risk the ambiguity arising from that circumstance. This modification of power might therefore be called Energy, at least till a better word shall be discovered.

The third and laft modification of power, that which is meafured by the force acting, and the length of the line which the. body moves over while it is acted on, and which, as we have seen, is proportional to the quantity of matter multiplied into the fquare of the velocity, Dr Wollafton propofes to call Impetus, a term that is perfectly unexceptionable.

Thus the generic term POWER would have its three principal modifications or fpecies denoted by the words FORCE, ENERGY, and IMPETUS; and, by a rigorous adherence to this very simple nomenclature, there can be no doubt that the science of Dynamics would be materially improved.

ART.

ART. VIII. Poems. By the Reverend George Crabbe. 8vo. pp. 260. London, 1807.

WE

E receive the proofs of Mr Crabbe's poetical existence, which are contained in this volume, with the same sort of feeling that would be excited by tidings of an antient friend, whom we no longer expected to hear of in this world. We rejoice in his resurrection; both for his sake, and for our own but we feel also a certain movement of self-condemnation, for having been remiss in our inquiries after him, and somewhat too negli gent of the honours which ought at any rate to have been paid to his memory.

It is now, we are afraid, upwards of twenty years since we were first struck with the vigour, originality, and truth of description of The Village;' and since we regretted that an author, who could write so well, should have written so little. From that time to the present, we have heard little of Mr Crabbe; and fear that he has been in a great measure lost sight of by the public, as well as by us. With a singular, and scarcely pardonable indifference to fame, he has remained, during this long interval, in patient or indolent repose; and, without making a single movement to maintain or advance the reputation he had acquired, has permitted others to usurp the attention which he was sure of commanding, and allowed himself to be nearly forgotten by a public, which reckons upon being reminded of all the claims which the living have on its favour. His former publications, though of distinguished merit, were perhaps too small in volume to remain long the objects of general attention, and seem, by some accident, to have been jostled aside in the crowd of more clamorous competitors.

Yet, though the name of Crabbe has not hitherto been very common in the mouths of our poetical critics, we believe there are few real lovers of poetry to whom some of his sentiments and descriptions are not secretly familiar. There is a truth and a force in many of his delineations of rustic life, which is calculated to sink deep into the memory; and, being confirmed by daily observation, they are recalled upon innumerable occasions, when the ideal pictures of more fanciful authors have lost all their interest. For ourselves at least, we profess to be indebted to Mr Crabbe for many of these strong impressions; and have known more than one of our unpoetical acquaintances who declared they could never pass by a parish workhouse, without thinking of the description of it they had read at school in the Poetical Extracts. The volume before us will renew, we trust, and extend many such impressions. It contains all the former productions of the author, with about double their bulk of new

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