Force of Percussion, is the name by which mechanicians distinguish that faculty of producing motion, or making other sensible mechanical impressions on bodies, by means of the stroke of a body in motion. It is nearly the same with impulse; only, it would seem that the very scrupulous and refined effect to limit the attention to the immediate cause of the motion, or other effect produced; to the something that is different, both from the force supposed to be inherent in the moving body (a hammer for example), and the subsequent motion and penetration of the nail which is driven by it. We may venture to say that it is needless to attempt any investigation of this object. It is hid, with all other causes of all other effects in the universe, in impenetrable darkness. If we reflect on the constitution of our own mind, so far as we can know it by experience and observation, and on the manner in which we draw conclusions, we must see that the knowledge Percussion, of the efficient cause of any effect is unattainable; for were the intervening something pointed out to us, and clearly conceived by us, we should find it just as necessary to find out why and how this something is connected with each of the events which we observe it invariably to connect.

But a knowledge of the force of percussion, in as far as it may or may not be distinguishable from other forces, is not unattainable. We can learn as much, and no more, concerning this, as concerning any other force; and we can contemplate that circumstance which, in our opinion, is common to it, with all other forces, and may perhaps discover other circumstances in which it differs from them. But in all this disquisition, it is plain that it is only events, which we conceive to be the characteristic effects of the cause, that we contemplate.

Percussion, considered as an effect, characteristic of a particular faculty of moving bodies, became an object of anxious research, almost as soon as philosophers began to think of motion and moving forces at all. The ancients (as has been observed in the article IMPULSION, Suppl.) contented themselves with very vague speculations on the subject. Galileo was the first who considered it as a measurable thing, the object of mathematical disquisition; being encouraged by his precious discovery of the laws of accelerated motion, and the very refined measure which these gave him of the power of gravity. It was a measure of the heaviness, not of the weight, of the body; and this was measured by its acceleration, and not by its pressure. Encouraged by this, he hoped to find some such measure of the force of percussion, which he saw so intimately connected with motion; whereas its connection with pressure was far from being obvious. He therefore tried to convert the terms; and as he had found a measure of the pressure of gravity in the acceleration of motion, he endeavoured to find in pressure a measure of the force of percussion arising from this acceleration. He endeavoured to find the number of pounds, whose pressure is equal to the blow of a given body, moving with a given velocity. The velocity was known to him with great precision, by means of the height from which the ball must fall in order to acquire it. It seems pretty clear that percussion may be measured in this way; for a body falling from a height will pierce an uniformly tenacious body to a certain degree, and no further; and experiment shows that this degree of penetration is very precise and constant. The same body, being merely laid on the tenacious body, will penetrate to a small depth by its weight. Laying more weight on it, will make it penetrate deeper; and a certain weight will make it penetrate as deep as the fall did, and no deeper. Thus, percussion seems very easily measurable by weight, or by any pressure similar to that of weight. It appears that Galileo made experiments with this view, and that he was disappointed, and obliged to acquiesce in the opinion of Aristotle, that percussion and weight are incomparable. He proposes, therefore, another experiment, namely, to drop a body into the scale of a balance from greater and greater heights, till at last the blow on the scale raises a weight that lies in the other scale. This offers itself so plausibly, that we are persuaded that Galileo tried it; but as he makes no mention... Neither of these experiments could give us a measure of the force of percussion, if this force be anything different from the forces which are excited or brought into action by percussion, in the manner described in the article IMPULSION, Suppl. When the ball comes into physical contact with the scale, it begins to compress it. This compression begins to stretch the strings by which the scale is supported. These pull at the arm of the balance, and cause it to press the centre-pin a little harder on its support, and to bend the balance a little, and cause it to pull at the cords which support the other scale. That scale is pulled upwards, diminishing a little its pressure on the ground, and pressing it harder to the incumbent weight. These forces are excited in succession from the one scale to the other, and a small moment of time elapses. The reaction of the scale diminishes, but does not instantaneously annihilate, the velocity of the falling ball. It therefore compresses the scale still more, stretches the threads, presses the fulcrum, and bends the balance still more (because the weight in the other scale keeps it down). The velocity of the falling ball is rapidly diminished; the balance is more bent, and pulls more strongly upwards at the threads of the other scale; and thus presses that scale more strongly against the incumbent weight, gradually communicating more and more motion to it, removing it farther from the ground, till, at last, the motion becomes sensible, or so considerable as to disengage some delicate catch as a signal. The experiment is now finished; and the mechanician fondly thinks that, at this instant, the pressure excited by the percussion, between the opposite scale and the under side of the incumbent weight, is just equal, or but a very little superior, to the pressure of the incumbent weight; and, since the arms of the balance are equal, and therefore the pressures on the two scales are equal, he imagines that that weight exerts a pressure equal to the percussion of the falling ball.

But all this is misconception, and also false reasoning. It is not percussion that we are measuring, but the pressures, excited by percussion, on the two scales. And these pressures are the forces of elasticity or expansion, belonging to, or inherent in, the particles of the balls and the scales; forces which are brought into action by the approach of those bodies to each other. This reasoning is also erroneous; and we should be mistaken if we think that the pressure actually exerted is equal to that of the weight in the opposite scale. It is greater than the mere pressure of that weight. The reaction of the opposite scale on its load was precisely equal to that weight before the ball was dropped from the hand; and, had the ball been equal to that weight, and simply laid into the scale on which it falls, it would have made no change on the mutual pressures of the scale and the other weight; it would only have relieved the ground from the pressure of that weight, and would have brought it on the threads which support its scale. The pressure of this scale upwards must be increased, before it can start the weight sensibly from the ground. How much it must be increased depends on the springiness of the scales, cords, and beam. By a proper adjustment of these particulars, the apparatus will give us almost any measure of percussion that we choose. For this reason, the improvements made on it by Gravefande Perus are of no value. The same reasoning, nearly, may be applied to the measurements of the force of percussion by means of the penetration of soft bodies.

Galileo mentions another very curious experiment, by which he thought that he had obtained a just measure of percussion. A vessel, filled with water, was suspended on the arm of a balance, with another vessel hanging from it, a great way below. All was exactly balanced by a weight in the opposite scale. By means of a suitable contrivance, a hole was opened in the bottom of the upper vessel, without disturbing the equilibrium. As soon as the water issued, and while it was falling through the air, that end of the balance rose; but when the water struck the lower vessel, the equilibrium was restored, and continued during the whole time of the efflux. Hence Galileo concluded, that the force of the stroke was equal to the weight of the falling water. But we apprehend that the observations made on this in the article IMPULSION, Suppl. will convince the reader that this conclusion is far from being legitimate. Besides, the stroke, in any one instant, is made by those particles only which strike in that instant, while the whole vein of water between the vessels is neither acting by its weight on the upper vessel, nor by its stroke on the lower; and we should conclude from the experiment, that the force of percussion is infinitely greater than the weight of the striking body. Indeed this is the inference made by Galileo. But if we have recourse to the experiments and reasonings of Daniel Bernoulli, in the article RESISTANCE OF FLUIDS, Encycl., we shall find that the seeming impulse on the lower vessel is really a most complicated pure pressure, and of most uncertain determination. The experiment is valuable, and gives room for curious reflections. We have repeated it, in a great variety of forms, and with great changes of impulse, and sometimes in such a manner that no impulse whatever can obtain, while at the same time a quantity of water was falling, unsupported by either vessel. In all the trials the equilibrium remained undisturbed. We were obliged to conclude, therefore, that the experiment afforded no measure of percussion. Indeed we were of this opinion before making the trial, for the reasons just now given.

We cannot say that the subsequent labours of philosophers have added much to our knowledge of this matter. Mr Leibnitz had contrived his whimsical doctrine of living and dead forces. The action of gravity, or of a spring, is a vis viva, when it actually produces motion in the body on which it acts; but when a stone lies on a table, and presses on it, this pressure is a vis mortua. Its exertion is made, and in the same instant destroyed, by an opposite vis mortua. Each of these exertions would have produced a beginning of motion (something different from any the smallest local motion); and the sum of all would, after a certain time, have amounted to a sensible motion and velocity. There seems no distinct conception to accompany, or that can accompany, this language. And, as a proof that Leibnitz had no distinct conceptions of the matter, he has recourse to this very experiment of Galileo in support of his genesis of a sensible motion from the continual exertions of the vis mortua; and he concludes that the force of percussion is infinitely, or incomparably, greater than pressure, because it is the sum total of an infinity of indivi- Nothing but the authority which Leibnitz has acquired on the continent, by the zealous efforts of his partizans, could excuse our taking up any time in considering this unintelligible discourse. Surely, if there is such a thing as a vis vis, it exists in the moving water, and its impulsions are not continual exertions of a vis mortua. Nor is it possible to conceive continual impulse, nor a beginning of motion that is not motion, &c. &c. It is paradoxical (and Leibnitz loved to raise the wonder of his followers by paradoxes) to say that percussion is infinitely greater than prelure, when we see that prelure can do everything that can be done by percussion. Nay, Euler, by far the most able supporter of the doctrines of Leibnitz about the force of bodies in motion, actually compares these two forces; and, in his Commentary on Robins's Artillery, demonstrates, in his way, that when a musket ball, moving with the velocity of 1700 feet per second, penetrates five inches into a block of elm, the force of its percussion is 127,760 times its weight. John Bernoulli restricts the infinite magnitude of percussion to the case of perfectly hard bodies; and, for this reason alone, says, that there can be none such in the universe. But, as this justly celebrated mathematician scouts with scorn the notion of attractions and repulsions, he must allow, that an ultimate atom of matter is unchangeable in its form; which we take to be synonymous with saying that it is perfectly hard. What must be the result of one atom in motion hitting another at rest? Here must be an instantaneous production of a finite velocity, and an infinite percussion. A doctrine which reduces its abettors to such subterfuges, and engages the mind in such puzzling contemplations, cannot (to say the best of it) be fished an explanation of the laws of Nature. The whole language on the subject is full of paradoxes and obscurities. In order to reconcile this infinite magnitude of percussion with the observed finite magnitude of its effects, they say that the prelure, or instantaneous effort, has the same relation to the force of percussion that an element has to its integral; and in maintaining this assertion, they continually consider this integral under the express denomination of a sum total, robbing Leibnitz's great discovery of the infinitesimal calculus of every superiority that it possessed over Wallis's Arithmetic of Infinites, and really employing all the erroneous practices of the method of indivisibles. We look upon the strange things which have been inculcated, with pertinacious zeal, in this doctrine of percussion and vis vis, as the most remarkable example of the errors into which the unguarded use of Cavalieri's Indivisibles, and of the Leibnitzian notion of the infinitesimal calculus, have led eminent mathematicians. It is not true that the prelure, and the ultimate force of percussion, have this relation; nor has the prelure and the resulting motion, which is mistaken for the measure of this ultimate force, any mathematical relation whatever. The relation is purely physical; it is the relation of pure cause and effect; and all that we know of it is their constant conjunction. The relation of fluxion and fluent is not a mathematical or measurable relation, but a connection in thought; which is sufficient for making the one an indication of the other, and the measures of the proportions of the one a mean for obtaining a measure of the proportions of the other. In this point of view, the relation of prelure to motion, as the measure of the percussion force of percussion, resembles that of fluxion and fluent, but is not the same.

Much has been said by the partizans of Mr Leibnitz about the incomparableness of prelure and percussion, and many experimental proofs have been adduced of the incomparable superiority of the latter. Buffinger says, that the prelure of many tons will not cause a spike to penetrate a block of hard oak half so far as it may be driven by a weak man with one blow of a mallet; and that a moderate blow with a small hammer will shiver to powder a diamond, which would carry a mountain without being hurt by its prelure. Nay, even Mr Camus, of the Academy of Paris, a staunch Cartesian, and an eminent mechanician, says that he beat a leaden bullet quite flat with a hammer of one pound weight, without much force; and that he found that 200 pounds weight would not have flattened it more than this blow; and he concludes from thence, that the force of the blow exceeded 200 pounds. These, to be sure, are remarkable facts, and justify a more minute consideration of a power of producing certain effects, which is so frequently and so usefully employed. But, at the same time, there are all very vague expressions, and they do not authorize any precise conclusions from them. Mr Camus saying "without much force," makes his pound weight, and his 200 pound weight, of no use for determining the force of the blow. He would have given more precise and applicable data for his decision, had he told us from what height the hammer should fall in order to flatten the bullet to this degree. But even then we should not have obtained any notion of the force in actual exertion during the flattening of the bullet; for the blow which could flatten the bullet in a longer or a shorter time, would unquestionably have been less or greater.

All the paradoxes, obscurities, and puzzling difficulties, in this subject disappear, if we leave out of our consideration that unintelligible force, which is supposed to preserve a body in motion or at rest; and if we consider both of these states of body as conditions which will continue, unless some adequate cause operate a change; and if we farther grant, that such causes do really exist in the universe, however unknown their nature may be by us; and, lastly, if we acknowledge, that the phenomena of elasticity, expansion, cohesion, gravity, magnetism, electricity, are indications of the agency of such causes, and that their actual exertions, and the motions and changes consequent on these exertions, are so invariably connected with particular bodies, that they always accompany their appearance in certain mutual relations of distance and position:—if we proceed thus, all the phenomena of collision will be explained by these causes alone, without supposing the existence and agency of a cause distinct from them all, and incomparable with them, called the force of percussion.

For it has been sufficiently demonstrated in the article IMPULSION (Suppl.), that that property of tangible coherent matter, which we call perfect elasticity, operates as a prelure during a certain small portion of time on both bodies, diminishing more and more the motion of the one, and augmenting that of the other, as the compression of one or both increases, till at last they separate with sensible velocities. In some very simple or perspicuous Perception. In such cases, we know what this pressure is in every instant of the action. We can tell how many pounds weight, at rest, will exert the same pressure. We can tell the whole duration of this pressure, and the space along which it is exerted; and, in such a case, we can say with precision what motion will be generated by this continued and varied pressure on the body which was at rest, and what diminution will be made in the motion of the other. All this can be done in the case

Plate XII. of a ball A (fig. 1.), moving like a pendulum with a small velocity, and striking a slender elastic hoop B, also suspended like a pendulum. We can ascertain by experiment, before the collision, what pressure is necessary for compressing it one inch, one-half, one-fourth, &c. Knowing this, and the weight of the hoop, and the weight and velocity of the ball, we can tell every circumstance of the collision—how long the compression continues—what is the greatest compression—how far the bodies have moved while they were acting on each other—and what will be the final motion of each:—in short, every thing that affords any mark or measure of a force of percussion. And we know that all this is produced by a force, familiarly known to us by the name of elasticity. Which of all these circumstances shall be called the percussion, or the force of percussion? Is it the ultimate or greatest pressure occasioned by the compression? This cannot be, because this alone will not be proportional to the final change of motion, which is generally taken as a measure of the percussion when a change of motion is its only observed effect.

We know that another perfectly elastic body, of the same weight, and struck by the same blow, and acquiring the same final velocity by the stroke, may not have sustained the tenth part of the pressure, in any one instant of the collision, if it has only been much more compressible. The greatest mutual pressure in the collision of a billiard ball is perhaps 1000 times greater than it is in a similar collision of a foot-ball of the same weight.

We also know what degree of compression will break this hoop, and what pressure will produce this compression. Therefore, should the fracture of the body be considered as the mark and measure of the percussion, we know what blow will just produce it, and be exhausted by so doing. In short, we know every mark and measure of percussion which this hoop can exhibit.

We can increase the strength of this hoop till it becomes a solid disk; and we see clearly, that in all these forms the mode of acting is the same. We see clearly that it is the same when, instead of the solid disk, it is an elastic ball; therefore everything that can indicate or measure the percussion of an elastic ball, is explained without the operation of a peculiar force of percussion, even when the ball is shattered to pieces by the blow.

Nor is the case materially different when the bodies are soft, or imperfectly elastic. When the struck body is uniformly tenacious, it opposes a uniform resistance to penetration, and its motion will be uniformly accelerated by the action of its own tenacity during the whole time of mutual action, except a trifling variation occasioned by the mere motion of the internal parts, independent of their tenacity. If we knew the weight necessary for merely penetrating this mass, and the weight and velocity of the penetrating body, we can tell how long it must be resisted by this force before its initial velocity will be annihilated, and therefore how far it will penetrate. We have tried this with deal, birch, willow, and other soft woods of uniform texture, and with nails having the body somewhat flatter than the end, that there might not be an irregularity occasioned by a friction on the sides of the nail, continually increasing as the penetration advanced. We made the hammer fall from a considerable height, and hit the nail with great accuracy in the direction of its length, by fixing it to the end of a long staff, moveable round an axis. The results corresponded with the calculation with all the precision that could be desired.

But it does not result from all this agreement, that the force, exertion, or effect, of a blow with a hammer is equal to the pressure of any number of pounds whatever. They are things that cannot be compared; and yet the force operating in the penetration by a blow is no way different from a pressure. It is a physical blunder to compare the area of the curve, whose abscissa is the depth of penetration, and the ordinates are the resistances, with any pressure whatever. This area expresses the square of a velocity, and its flips, bounded by parallel ordinates indefinitely near each other, are as the decrements of this square of a velocity, occasioned by a pressure, acting almost uniformly along a very small space, or during a very small time. It is an absurdity therefore to sum up these flips as so many pressures, and to consider the sum total as capable of expressing any weight whatever. Such a paralogism is peculiar to Leibnitz's way of conceiving his infinitesimal method, and it could have no place in the genuine method of fluxions. It is this misconception that has made Mr Leibnitz and his followers suppose that a body, accelerated by gravity, retains in it a sum total of all the pressures of gravity accumulated during its fall, and now forming a sit vice. Supposing that it requires a pressure of twenty pounds to press a five pound shot slowly through a mass of uniformly resisting clay; this pressure would carry it from the top to the bottom of a mountain of such clay. Yet this ball, if discharged horizontally from a cannon, would penetrate only a few yards, even though the clay should resist by tenacity only, independent of the motion lost by giving motion to its internal parts. In this experiment, the utmost pressure exerted during the motion of the ball did not much exceed the pressure of twenty pounds. In this comparison, therefore, percussion, so far from appearing infinitely greater than pressure, would appear much less. But there is perhaps no body that resists penetration with perfect uniformity, even though uniformly tenacious. When the ball has penetrated to some depth, the particles which are before it cannot be so easily displaced, even although they had no tenacity, because the particles adjoining are more hemmed in by those beyond them. We have always observed, that a ball impelled by gunpowder through water rises toward the surface (having entered horizontally through the side of the vessel at some depth), and this so much the more rapidly as it entered nearer to the surface. The reason is plain. The particles which must be displaced before the ball, escape more easily upwards than in any other direction. It is for this reason chiefly that a greater weight laid on the head of a nail will cause it sink deeper into the wood; and thus a great weight appears Also, while a bullet is battering more and more under a hammer during the progress of a blow, it is spreading under the hammer; more particles are resisting at once, and they find more difficulty in effecting their escape, being harder squeezed between the hammer and the anvil. The same increased resistance must obtain while it is battering more and more under the quiet pressure of a weight; and thus, too, a greater weight appears to be commensurable with a greater blow.

After all, however, a blow given by a falling body must excite a pressure greater than its mere weight can do, and this in any degree. Thus, suppose A.B (fig. 2.) to represent a spiral spring in its natural unconstrained dimensions, standing upright on a table. Let a b be the abscissa of a line a d k k, whose ordinates c d g b i k, &c., are as the elastic reaction of the spring when it is compressed into the lengths c b, g b, i b, &c. Suppose that, when it is compressed into the form C D, it will just support the weight of a ball lying on C. Then c d will be a reaction equal to the weight of the ball, and the rectangle a c d f will express the square of the velocity which this ball would acquire by falling freely through a c. If therefore the ball be gently laid on the top of the spring at A, and then let go, it will descend, compressing the spring. It will not stop when the spring has acquired the form C D, which enabled it to carry the weight of the ball gently laid on it. For in this situation it has acquired a velocity, of which the square is represented by the figure a d f' (See Dynamics, Suppl. p. 95.). It will compress the spring into the length c b, such that the area c g b d is equal to the area a d f'. If the ball, instead of being gently laid on A, be dropped from M, it will compress the spring into such a length i b, that the area a i k is equal to the rectangle m e d n; and, if the spring cannot bear so great compression, it will be broken by this very moderate fall.

Thus we see that a blow may do things which a considerable pressure cannot accomplish. The accounts which are given of these remarkable effects of percussion, with the view of impressing notions of its great efficacy, are generally in very indefinite terms, and often without mentioning circumstances which are accessory to the effect. It would be very unfair to conclude an almost infinite power of percussion, from observing, that a particle of sand, dropped into a thick glass bottle which has not been annealed, will shiver it to pieces. When Mr Bullinger says that a moderate blow will break a diamond which could carry a mountain, he not only says a thing of which he cannot demonstrate the truth, and which, in all probability, is not true; but he omits noticing a circumstance which he was mechanician enough to know would have a considerable share in the effect. We mean the rapidity with which the excited pressure increases to its maximum in the case of a blow. In the experiment in question, this happens in less than the millionth part of a second, if the velocity of the hammer has been such as a man would generate in it by a very moderate exertion. For the blow which will drive a good lath nail to the head in a piece of soft deal with an ordinary carpenter's hammer, must be accounted moderate. This we have learned by experiment to be above 25 feet per second. The connecting forces exerted between the particles of the diamond may not have time sufficient for their excitation in the remote parts, so as to share the derangement among them all, in such a manner that it may be moderate in each as not to amount to a diffusion in any part of the diamond. We see many instances of this in the abrupt handling of bodies of tender and friable texture. It is partly owing to this that a ball discharged from a pistol will go through a sheet of paper standing on edge without throwing it down, which it would certainly do if thrown at it by the hand. The connecting forces, having time to act in this last case, drag the other parts of the paper along with them, and their union is preserved. Also, when a great weight is laid on the diamond, it is gradually dimpled by it; and thus including many parts together in the dimple, it obliges them to act in concert, and the derangement of each is thus diminished.

We flatter ourselves that the preceding observations and reflections will contribute somewhat towards removing the paradoxes and mysteries which discredit, in some degree, our mechanical science. If we will not pertinaciously conjure up ideal phantoms, which, perhaps, cannot exist, but content ourselves with the study of that tangible matter which the Author of Nature has presented to our view, we shall have abundant employment, and shall perceive a beautiful harmony thro' the whole of natural operations; and we shall gradually discover more and more of those mutual adaptations which enable an atom of matter, although of the same precise nature wherever it is found, to act such an unspeakable variety of parts, according to the diversity of its situations and the scene on which it is placed. If a mind be "not captivated by the harmony of such sweet foundations," we may pronounce it "dark as Erebus, and not to be trusted."