in architecture, a capital or plinth, to a pillar, or pilaster, or pier, that supports an arch, &c.

IMPEL, is the term employed in the law doctrine of guare of mechanical philosophy, for expressing a supposition peculiar exertion of the powers of body, by which a moving body changes the motion of another body by hitting or striking it. The simplest case of this action is when a body in motion hits another body at rest, and puts it in motion by the stroke. The body thus put in motion is said to be impelled by the other; and this way of producing motion is called impulse, to distinguish it from pression, thrusting, or protrusion, by which we push a body from its place without striking it. The term has been gradually extended to every change of motion occasioned by the collision of bodies.

When speculative men began to collect into general History of classics, the observations made during the continual exertions of our own personal powers on external bodies, in order to gain the purposes we had in view, it could not be long before they remarked, that as we, by the strength of our arm, can move a body, can stop or any how change its motion; so a body already in motion produces effects of the same kind in another body, by hitting it. Such observations were almost as early and as interesting as the others; and the attention was very forcibly turned to the general facts which obtained in this way of producing motion; that is, to the explication of the general laws of impulse. We do not find, however, in what remains of the physical science of the ancients, that they had proceeded far in this classification. While mechanics, or the science of machines, had acquired some form, and had been the subject of successful mathematical discussion, we do not find that anything similar had been done in the science of impulse. Yet the artillery of ancient times was very ingenious and powerful. But although Vegetius, and Ammianus Marcellinus, and Hero, describe the mechanism of these engines with great care, and frequently with mathematical skill, we see no attempts to ascertain with precision the force of the missile weapon, or to state the efficacy of the battering ram, by measures of the momentum, and comparison of it with the resistance opposed to it. The engineers were contented with very vague notions on these points.

Aristotle, in his 20th Mechanical Question, and Galen in some occasional observations, are the only authors of antiquity whom we recollect as treating the force of impulse as a quantity susceptible of measure. Their observations are extremely vague and trivial, chiefly directed, however, to the discrimination of the force of impulse from that of pressure.

In more modern times, great additions had already been made to the affluence we had derived from the impulsive efficacy of bodies in motion. Water-mills and wind-mills had been invented, and had been applied to such a variety of purposes, that the engineers were fast acquiring more distinct notions of the force of impulse. Naval construction was changed in such a manner, that there hardly remained any thing of the ancient rigging. The oblique action of wind and water were now found even more effective than the direct; and ships could could now fail with almost any wind. All these things fixed the attention of the engineers and of the speculators on the numberless modifications of the force of impulse.

But it soon appeared that this was a refined branch of knowledge, and required a more profound study than any other department of the science of motion. At the same time, it was equally clear, that it was also of superior importance. Mills worked by cattle, or by men's hands, were everywhere giving place to wind and water-mills; and a ship alone appeared to every intelligent mechanician to be the greatest effort of human invention, and most deserving his careful study. All these improvements in the arts of life derived their efficacy from the impulse of bodies. The laws of impulse, therefore, became the objects of study to all who pretended to philosophical science. But this is a branch of study wholly new, and derives little assistance from the mechanical science already acquired; for that was confined to the determination of the circumstances which regulated the equilibrium of forces, either in their combined action on bodies in free space, or by the intervention of machines. But in the production of motion by impulse, the equilibrium is not supposed to obtain; and therefore its rules will not solve the most important question, "What will be the precise motion?"

Galileo, to whom we are indebted for the first discoveries in the doctrine of free motions, was also the first who attempted to bring impulsion within the pale of mathematical discussion. This he attempted, by endeavouring to state what is the force or energy of a body in motion. The very obscure reflections of Aristotle on this subject only served to make the study more intricate and abstruse. Galileo's reflections on it are void of that luminous perspicuity which is seen in all his other writings, and do not appear to have satisfied his own mind. He has recourse to an experiment, in order to discover what pressure was excited by impulsion. A weight was made to fall on the scale of a balance, the other arm of which was loaded with a considerable weight; and the force of the blow was estimated by the weight which the blow could thus start from the ground. The results had a certain regularity, by which some analogy was observed between the weights thus started and the velocity of the impulse; but the anomalies were great, and the analogy was singular and puzzling; it led to many intricate discussions, and science advanced but slowly.

At last the three eminent mathematicians, Dr Wallis, Sir Christopher Wren, and Huyghens, about the same time, and unknown to each another, discovered the simple and beautiful laws of collision, and communicated them to the Royal Society of London in 1668 (Phil. Trans. n° 43—46.). Sir Christopher Wren also invented a beautiful method of demonstrating the doctrine by experiment. The bodies which were made to strike each other were suspended by threads of equal length, so as to touch each other when at rest. When removed from this their vertical situation, and then let go, they struck when arrived at the lowest points of their respective circles, and their velocities were proportional to the chords of the arches through which they had descended. Their velocities, after the stroke, were measured, in like manner, by the chords of their arches of ascent. The experiments corresponded precisely with the theoretical doctrine.

In the mean time, this subject had keenly occupied the attention of philosophers, who found it to be of a very abstruse nature; or, which is nearer the truth, they indulged in great refinement in prosecuting the study. The first attempts to measure the impulsive force of bodies, by setting it in opposition to pressures, which had long been measured by weights, gave rise to some very refined reflections on the nature of these two kinds of forces. Aristotle had said that they were things altogether disparate. If so, there can be no proportion between them. Yet the analogy observed in the experiments above mentioned of Galileo, showed that impulse could be gradually augmented, till it exceed any pressure. This indicates famelics in kind, according to Euclid himself. A curious experiment of Galileo's, in which the impulse of a vein of water was set in equilibrium with a weight, seemed not only to establish this identity beyond a doubt, but even to show the origin of pressure itself. The weight in one scale is sustained as long as the stream of water continues to strike the other scale. In this experiment, therefore, pressure is equivalent to continual impulse. But continual impulse is not conceivable; we must consider the impulse of the stream as the effective impulse of the different particles of water, at intervals which are altogether indistinguishable.

From these considerations were deduced two very momentous doctrines: 1. That pressure is nothing but repeated impulse; 2. That although pressure and impulse are the same in kind, they are incomparable in magnitude. The impulse is equal to the weight of a column of water, whose length is the height necessary for communicating the velocity. Now this is insufficient; and the weight is sustained during any the smallest moment of time, by the impulse, not of the whole column, but of the infensible portion of it which is then making its stroke. Impulse, therefore, is infinitely greater than pressure.

These abstruse speculations have a charm for certain ingenious speculative minds; and when indulged, will lead them very far. Accordingly, it was not long before the only fame of the most ingenious philosophers of Europe taught that impulse was the sole origin of pressure. There is but one moving power (said they) in mechanical nature: This is impulse.—Nihil movetur (says Euler) nisi a contiguo et moto. Moreover, having been long and familiarly conversant with the actions of animals, and the actions of moving bodies, and conceiving, with sufficient distinctness, that impenetrable bodies cannot move without moving those with which they are surrounded and in contact, they imagined that they fully understood how all this displacement of bodies is carried on; and therefore they maintained, that any motion is fully explained when it is shewn to be a case of impulsion. But they saw many cases of motion where this impulsion could not be exhibited to the senses. Thus, the fall of heavy bodies, the mutual approach or recess of magnetic and electric bodies, exhibited no such operation. But even here their experience helped them to an explanation. Air is an invisible substance, and its very existence was for a long time known to us only by means of its impulse. As we see that pressures are generated. generated by the impulse of water and of air, may there not be fluids still more subtle than air, by whose invisible impulse bodies are made to fall, and magnets are made to approach or avoid each other? The imposibility of this cannot be demonstrated, and the laws of impulse had not as yet been so far investigated as to show that they were incompatible with those productions of motion. It was therefore an open field for diffusion; and the philosophers, without further hesitation, adopted, as a first truth, that all motion whatever is produced by impulsion. The bulwarks of the philosopher, therefore (say they), is to investigate what combination of invisible impulsive forces is competent to the production of any observed motion; such as the fall of a heavy body, the elliptical motion of a planet, or the polarity of a magnetic needle. The curious disposition of iron-filings round a magnet encouraged this kind of speculation: it looks to like a stream of fluid; but it is a number of quiescent fragments of iron. This does not hinder us from supposing such a stream, not of iron-filings, but of a magnetic fluid, which will arrange (say the atomists) those fragments, just as we see the flotsam in a brook arranged by a stream of water.

Fluids, therefore, moving in streams, vortices, and a thousand different ways, have been supposed, in order to explain, that is, to bring under a general known law of mechanical Nature, all those cases of the production of motion where impulsion is not observed by the senses.

As we have gradually become better acquainted with the laws of the production of motion by impulsion, we have been able to explode many of those proffered explanations, by shewing that the genuine results of the supposed invisible motions, that is, the impulsive forces which they would produce, are very unlike the motions which we attempt to explain. It has been shewn, that the vortices supposed by Descartes, or by Leibnitz, or by Huyghens, cannot exist; and they have been given up. But, it is answered to all those demonstrations of futility, that full the axiom remains. Motion is produced only by impulse; but we have not yet discovered all the possibilities of impulsion; and we must not despair of discovering that precise set of invisible motions, and consequent impulsive forces, of which the phenomenon before us is the necessary result.

But this is by no means sufficient authority for deferring the rule of philosophizing, to prudently and judiciously recommended by Sir Isaac Newton; namely, not to admit as the cause of a phenomenon anything that is not seen to operate in its production. The prudence of this restriction is evident; and it has also been sufficiently shewn (Philosophy, Encyclop. n° 48. &c.), that true philosophical explanation, or extension of knowledge, is unattainable, if this rule be not strictly adhered to. We therefore require a cogent reason for a practice that opens the door to every absurdity, and that cannot give us the knowledge which we are in quest of. What, then, is the reason that always induces philosophers to have recourse to impulsion for the explanation of a phenomenon, and to rest satisfied in every case where it can be clearly proved that the phenomenon is really a case of impulsion? We say that we inquire into the reason why a body falls, and that we will be satisfied if it can be shewn us that it has received a number of impulsive downward. Do we inquire why a body in motion puts another body in motion by hitting it? And if we do, have we discovered the reason?

We believe that none of the philosophers, who have recourse to invisible impelling fluids, ever ask a reason for motion by impulsion. Indeed they should not, otherwise it would cease to be a first principle of explanation. Other philosophers, indeed (namely, such as ask no reason for the weight of a body, but the fiat of the Almighty), require an explanation of motion by impulse, and think that, in almost every case, they have found it out.

If the philosophers ask no reason for this production of motion, they must (that it may serve as a principle of explanation) say that impulsion is an original property of matter, either contingent or essential. Accordingly, we believe that this, or something like this, has been assumed as a principle by the greater part of mechanicians. It has been assumed, as we have observed in the article Dynamics, Suppl., that a moving body possesses the power of producing motion in another body by hitting it; and they call it the impulsive force of moving bodies—the force inherent in a moving body. The reader will have observed, in our manner of treating that article, and also in several passages of different articles of the Encyclopedia Britannica, that we do not consider this assumption as very clearly authorised by observation, or deducible by abstract reasoning, from the first principles of philosophy. There is no branch of natural philosophy on which so many ingenious dissertations have been written; and perhaps there is none that has been more successfully prosecuted: Yet this is the only part of the science of motion that has given rise to a serious dispute; a dispute that has divided, and still divides, the mechanicians of Europe.

Some may think it presumptuous in us, in a Work of this kind, which only aims at collecting and exhibiting in one view the existing science of Europe, to pretend to give new doctrines, or to decide a question which has called forth all the powers of a Leibnitz, a Bernoulli, a Jurin, a MacLaurin, &c. But we make no such pretensions; we only hope that, by separating the question from others with which it has, in every instance, been complicated, and by considering it apart, such notions may be formed, in perfect conformity to the principles adopted by all parties, that the mystery which has gradually gathered like a cloud, may be dispelled, and all cause of difference taken away. We apprehend that this requires no very extensive knowledge, but merely a strict attention to the conceptions which we form of the actions of bodies on each other, and a precision in the use of the terms employed in the discussion.

We trust that our philosophical readers perceive and approve of our anxiety to establish (in the article Dynamics, Suppl.) the leading principles of mechanical philosophy, from which we are to reason in future on acknowledged facts, or laws of human thought. It is not too much the question, What is the essence of material Nature, from which all the appearances in the universe proceed? as it is, What do we know of it? how do we come by this knowledge? and what use can we make of it? The tenant knows nothing of the solar system, and man is ignorant of the cause of impulsive nets. Other intelligent creatures may have senses, of which this is the proper object; and others, of a still more more exalted rank, may perceive the operations of mind as clearly as we perceive those of matter, while they are equally ignorant of ourselves of the causes which connect the conjoined events in either of those operations. But "known unto God, and to Him alone, are all His works!"

To accomplish this purpose, we directed the reader's attention to what passes in his own mind when he thinks on the mechanical phenomena of Nature; on what he calls body; on the perceptions which bring it into his view, and which give him all the notions that he can form of its distinguishing, its characteristic properties. How does he learn that there is matter in a particular place? He has more than one mean of information; and each of these informs him of peculiar qualities of the thing which he calls matter. Many appearances suggest to his mind the presence of a body. Show a monkey or a kitten (and even sometimes a human infant) a mirror, and it will instantly grope round it to find a companion. Why does the creature grope about so? It is not contented with the first indication of matter, and nothing will satisfy it but touching or grasping what is behind the mirror. It is by our sense of touch alone that we get the irresistible conviction that matter or body is perceived by us, and it never fails to give us the perception; nay, we have the perception even in some cases where the experienced philosopher thinks himself obliged to doubt of its truth. Some sensations, arising from spasm, cannot be distinguished from the feeling of touch; and the patient infers that something presses on the diseased part, while the physician knows that it is only a nervous affection. Every person will think that a cobweb touches his face when an electrified body is brought near it, and will try to wipe it off with his hand. But the modern philosopher sees good reason for asserting, that in this instance our feeling gives us very inaccurate, if not erroneous, information. He shows, that the fact, of which our feeling truly informs us, is the bending of the small hairs or down which grow on the face, and that these only have been touched; and the followers of Epicurus deny that even this has been demonstrated.

The philosopher adopts this mode of perception as unquestionable, and allows that, and that alone, to be matter, which invariably produces this sensation by contiguity. But engaged in speculations which fix his attention on the external object, he neglects and overlooks the instrument of information, and its manner of producing the effect, just as the astronomer overlooks the telescope, and the union and decussation of the rays of light which form the picture by which he perceives the satellite of Jupiter travel across its disk. The philosopher finds it convenient to generalize the immense variety of touches which he feels from external bodies, and to consider them as the operations of one and the same discriminating quality, a property inherent in the external substance itself; and he gives it a name, by which he can excite the same notion in the minds of his hearers. It is worth while to attend to what has been done in this matter, because it gives much information concerning the first principles of mechanism. An exquisite painting has sometimes such an appearance of prominence, that one is disposed to draw the finger along it, and we expect to feel some roughness, some abrasion, something that prevents the finger from going over the place. Perhaps we doubt, and impulsive want to be assured. We press a little closer; but feel no obstruction; and we desist. The very first appearance, therefore, which this indicating quality, viewed as the property of external matter, has in our conceptions, is that of an obstruction, an obstacle, to the exertion of one of our natural powers. The power exerted on this occasion is familiarly and distinctively known by the name of PRESSURE. This is the name of our own exertion, our own action; and, in this instance, and (we think) in this alone, the word is used purely, primitively, and without figure. When we say that a stone presses on the ground, we speak figuratively, as truly as when we say that the candlestick stands, and the snuffers lie, on the table. It is a personification, authorized by the similarity of the effects and appearances. Further, when we speak of our pressure on any thing, with the intention of being precise in our communication, we speak only of what obtains in the touching parts of the finger and the thing pressed, paying no attention to the long train of intermediate exertions of the mind on the nerves, the nerves on the muscular fibre, the fibre on the articulated machine, and the machine on the touching part of the finger. And thus the exertion of the sentient and active being is attributed to the particles of lifeless inactive matter at the extremity of the finger, and these are said to press immediately on the touching parts of the external body. And, lastly, as this our exertion is unquestionably the perceived employment of a faculty in us, which we call force, power, strength, distinguishing it from every other faculty by these names; we say (but figuratively), that force or power is exerted at the tips of the fingers, and we call it the FORCE OF PRESSURE.

By far the greatest part of our actions on external and preliminary bodies is with the intention of putting them out of their present situations; and we can hardly separate the thought of exerted pressure from the thought of motion produced by it. Therefore, almost at its first appearance in the mind, pressure comes before us as a motion of MOVING POWER. Nay, we apprehend, that the more we speculate, and the more we aim at precision in our conceptions, we shall be the more ready to grant that we have no clear conception of any other moving power. No man will contend that he has any conception at all of the power exerted by the mind in moving the body. It is of importance to reflect on the manner in which this notion is extended to all other productions of motion. We think that this will show, that in every case we suppose pressure to be exerted.

The philosopher proceeds in his speculations, and observes, that one man can press on another, and can push him out of his place, in the same way as he removes any other body; and he cannot observe any difference in his own exertions and sensations in the two cases. But the man who is pushed has the same feelings of touch and pressure. By withdrawing from the pressure, he also withdraws from the sensation; by withdrawing and resisting it, he feels the pressure of the other man; and what he feels is the same with what he feels when he presses on the other person, or on any piece of matter. The same sensations of touch are excited. He attributes them to the pressure of the other person. Therefore he attributes the same sensations to the counter pressure of any other body that excites All these things are distinctly and invariably felt; but they require attention, in order to be subjects of recollection and after-consideration. From this, and no other source, are derived all our notions of corporeal prelure, of counter-prelure, of action, re-action, of resistance, and of inactivity or inertia. Our notions of moving power, of the mobility of matter, and of the necessity of this power to produce motion in matter, have the same origin. Our notions also of the resistance of inanimate matter, indicated by the expenditure of actual pressure, are formed from the same premises: the counter-prelure, or what at least produces the same feelings in the person who is the mover, is considered as the property of dead matter; because we feel, that if we do not exert real force, we are displaced by the same prelure that would displace a lifeless body of the same bulk.

These direct inferences are confirmed as we extend our acquaintance with things around us. We can exert our force in bending a spring; and we feel its counter-prelure, precisely similar to that of another man. We feel that we must continue this prelure, in order to keep it bent; and that as we withdraw our prelure, the spring follows our hand, still producing similar feelings in our organs of touch, and requiring similar exertions of our strength to keep it in any state of tension. These phenomena are interpreted as indications of pressures actually exerted by the spring, and quite different from what we should feel from its mere resistance to being moved. This action resembles our own exertion in every particular; it produces all the effects of prelure; it will squeeze in the soft flexible parts of our body with which we act on it; it will compress any soft body, just as we do ourselves; it will put bodies in motion. Farther, we can feel the action of one spring in opposition to that of another, and observe that each is bent by bending the other; and we see that their touching parts exert prelure, for they will compress any soft body placed between them.

Thus, then, in all those cases, we have the same notion of the power immediately exerted between the two bodies, animated or inanimated. It is always prelure. If indeed we begin to speculate about the modus operandi in any one of these instances, we find that we must stop short. How our prelure excites the feeling of prelure in the other person, or how it produces motion, eludes even conjecture—So it is—Nay, how our intention and volition causes our limb to exert this prelure, or how the springing of a spring produces similar effects, remains equally hid from our ken. Unwearied study has greatly advanced our knowledge of these subjects in one respect. It has pointed out to us a train of operations, which go on in our animal frame before the sensible prelure is produced: we have discovered something of their kind, and of the order in which they proceed; we have gone farther, and have discovered, in some of the pressures exerted impulsion, by lifeless matter, similar trains of intervening operations. In the case of a spring, we have discovered that there is a certain combination of the properties of all its parts necessary for the visible exertion. But what is the principle which thus makes them co-operate, we cannot tell, any more than in our own exertions of prelure. Such being the origin of our notions on these subjects, it is no wonder that all our language is also derived from it. Force, power, prelure, action, re-action, resistance, impulsion, are, without any exception, words immediately expressive of our own exertions, and applied metaphorically to the phenomena of matter and motion.

Lastly, when we see a body in motion displace another body by hitting it, and endeavour to form a notion of the way in which this motion is immediately produced, fixing our attention on what passes in the very instant of the change, we find ourselves still obliged to suppose the thing we call prelure. We can have no other conception of it; and there is no violence in this act of the imagination. For we know, that if we are jolted from our place, and forcibly driven against another person, we put that person in motion without any intention or action of our own; and we experience, in doing this, that the very same feelings of touch and prelure are excited as in the instances of the same motions produced by exerted prelure. We also see, that when a body strikes another, and puts it in motion, it makes an impression or dimple in it if soft, or breaks it if brittle; and in short, produces every effect of prelure. A ball of soft clay makes a dimple in the ball of soft clay which it displaces, and is dimpled by it. Springy bodies compress each other in their collisions, and reflect from each other. In short, in every case of this class, mutual prelure, indicated by all its ordinary effects, appears to be the intermediate by which the changes of motion are immediately produced; and the previous motion of the striking body seems to be only the method of producing this prelure.

From this copious induction of particulars, and prelure is careful attention to the circumstances of each, we think it plain, that prelure is the only clear notion that a mind, not familiar with scrupulous discussion, forms of moving moving power; and therefore that it is very singular to power, think of excluding it from the list, and saying that impulsion is the only power in nature, and the source of all prelure.

It may perhaps be said, that the mutual immediate action to which the vulgar, and many philosophers, have erroneously given the metaphorical name prelure, is, indeed, the real cause of motion or change of motion; but still it is now properly called impulsion, because it is occasioned only by the previous motion of the impelling body. We conceive clearly, (they may say) how this previous motion produces the impulsion. Since matter is impenetrable, we see clearly that a solid body, or a solid particle, cannot proceed without displacing the bodies with which it comes into contact; we have notions of this as clear as those of geometry; whereas, how prelure is produced, is inconceivable by us. If we press a ball ever so strongly against another, and remove the obstacle which prevented its motion, it will not move an inch, unless we continue to follow it, and press it forward; but we see a moving body pro-

5 G 2 duce compression, bend springs, make pits in soft bodies, and produce all the effects of real animal pressure. Impulse, therefore, is the true cause of motion, and the solicitation of gravity is nothing but the repeated impulse of an invisible fluid.

But, in the first place, let it be observed, that both parties profess to explain the phenomena of mechanical nature, that is, to make them easier conceived by the mind. Now it may be granted, that could we have any previous conviction of a fluid continually flowing toward the centre of the earth, we could have some notion of the production of a downward motion of bodies, but not more explanation than we have without it, because impulsion is as little understood by us as pressure.

But there are thousands of instances of moving forces where we cannot conceive how they can be produced by the impulse of a body already in motion. There appear to be many moving powers in nature, independent of, and inexplicable by, any previous motion; these may be brought into action, or occasions may be afforded for their action, in a variety of ways. The mere will of an animal brings some of them into action in the internal procedure of muscular motion; mere vicinity brings into action powers which are almost irresistible, and which produce most violent motions. Thus a little aquafortis poured on powdered chalk contained in a bombshell, will burst it, throwing the fragments to a great distance. A spark of fire brings them into action in a mass of gunpowder, or other combustibles. And here it deserves remark, that the greater the mass is to which the spark is applied, the more violent is the motion produced. It would be just the contrary, if the motion were produced by impulse. For in all cases of impulsion, the velocity is inversely proportional to the matter that is moved. When a spring is bent, and the two ends are kept together by a thread, a pressure is excited, which continues to act as long as the thread remains entire. What contrivance of impelling fluid will explain this, or give us any conception of the total cessation of this pressure, when the thread is broken, and the spring regains its quiescent form?

We can explain, in a most intelligible manner, why the hardest pressure produces no sensible motion in the case referred to above. We can conceive, with sufficient distinctness, a tube filled with steel wires, coiled up like cork screws, and compressed together into 1/8th of their natural length. A tube of 10 inches long will contain 100 of them. While in this state, compressed by a plug, we can suppose each of the springs to be tied with a thread. Suppose now that the thread of the spring next the piston is burnt or cut; it will press on the piston, and force it out, accelerating its motion till it has advanced one inch; after this, the piston will proceed with a uniform motion. It is plain, that the velocity will be moderate, perhaps hardly sensible, because the pressure acted on it during a very short time. But if two springs have been set at liberty at the same instant, the pressure on the piston will be continued through a space of two inches, and the final velocity will be greater, because the same (not a double) pressure will be exerted through a double space. Unbending four springs at once, will give the piston a double velocity (See Dynamics, Suppl. n°95). Now the effect of the motion of the second spring is to keep the pressure of the first in action during a longer time, by following it, and impelling, keeping it in a state of compression. There is nothing supposed of this kind in the case of strong pressure alluded to; and therefore no motion is produced when the obstacle is removed, except what the intense compression produces by accelerating the body along an inflexible space. If all the 100 springs are discharged at once, the piston will be accelerated through 100 inches, and will acquire ten times the velocity that one spring can communicate. (N.B. The force expended in moving the springs themselves is not considered here).

It is in this way only that the previous motion of the impelling body acts in producing a considerable motion. The whole process will be minutely considered by and bye.

We may now ask, how it is so clear a point, that a solid body in motion must displace other bodies? This seems to be the very point in question. Is the affirmative clearly deduced from our notion of solidity? What is our notion of solidity, and whence is it derived? We apprehend that even this primary notion is derived from pressure. It is by handling a thing, and finding that we cannot put our hand into the place where it is without displacing it, that we know that it is material. All this is indicated to us by the feeling excited by our pressure. We feel this property always as an obstacle; and therefore say, that by this property it resists our pressure. Nay, there are cases where even the philosopher prefers this quality to impulsion as a test of matter. To convince another that the jar out of which he has poured the water that filled it is not empty, but full of matter, he dips the mouth of the jar into water, and shows, that although he presses down till the surrounding water is above the bottom of it, the water has hardly gotten half an inch into the jar; there is something there which keeps it out; there is matter in it. He then opens a hole in the bottom of the jar; the water immediately rises on the inside of the jar, and fills it. He says that the pressure of the water has driven the matter out by the hole; and he confirms the materiality of what is expelled by holding a feather above the hole. It is agitated, showing that the expelled thing has impulsion, another property (he says) of matter: what filled the jar was air, and air in motion is wind. The philosopher can exhibit some new cases, where something like impulsion appears. A slender magnet may be set on one end, the south pole, for instance, and will stand in that tottering situation. If a person bring the north pole of a powerful magnet hastily near the upper end, it will be thrown down, just as it may be blown down by a puff of wind; therefore (says the philosopher) there may be appearances of impulsion, and I may imagine that there is impelling matter; but nothing but matter excludes all other matter from its place: this property, therefore, is the surest test of its presence.

Thus we see, that our notion of solidity or impenetrability (a name still indicating an obstacle to pressure), gives us no clearer conception of the productions of motion by impulsion than pressure does; for it is the same, or indicated by the same sensations.

The question now seems to be reduced to this—Since impulsion by the strongest pressure of a quiescent body does not produce motion, or excite that kind of pressure which is inherent in the immediate cause of motion, while a body in motion does not tion, exciting but a very moderate pressure (as may be seen by the trifling compression or dimpling), produces a very considerable motion; how is the previous motion conducive to this purpose? The answer usually given is this: A body in motion (by whatever cause), perseveres in that motion by the inherent force; when it arrives at another body, it cannot proceed without displacing that body. The nature of the inherent force is such, that none of it is lost, and that a portion of it passes into the other body, and the two bodies instantly proceed with the same quantity of motion that was in the impelling body alone. This is an exact enough narrative of the general fact, but it gives no great explanation of it. If the impelling body perseveres in its motion by means of its inherent force, that force is exerted in performing its office, and can do no more. The impelled body seems as much to possess an inherent force; for the same marks and evidences of pressure on both sides are observed in the collision. If both bodies are soft or compressible, both are dimpled or compressed. We are as much entitled, therefore, to say, that part of the force by which it perseveres at rest, passes into the other body. But the rest, or quiescence of a body, is always the same; yet what passes into the impelling body is different, according to its previous velocity. We can form no conception how the half of the inherent force of the impelling body is expended by every particle, passes through the points of contact, and is distributed among the particles of the impelled body; nay, we cannot conceive this halving, or any other partition of the force. Is it a thing sui generis, made up of its parts, which can be detached from each other, as the particles of salt may be, and really are, when a quantity of fresh water is put into contact with a quantity of brine? We have no clear conception of this; and therefore this is no elucidation of the matter, although it may be an exact statement of the visible fact.

Let us take the simplest possible case, and suppose only two particles of matter, one of which is at rest, and the other moves up to it at the rate of two feet per second. The event is supposed to be as follows: in the instant of contact, the two particles proceed with half of the former velocity. Now this instant of time, and this precise point of space, in which the contact is made, is not a part of either the time or space before collision, or of those after collision; it is the boundary between both; it is the last instant of the former time, and the first instant of the latter time; it belongs to both, and may be said to be in both. What is the state or condition of the impelling particle in this instant? In virtue of the previous motion, it has the determination, or the force, or the power, to move at the rate of two feet per second; but, in virtue of the motion after collision, it has the determination or power of moving at the rate of one foot per second. In one and the same instant, therefore, it has two determinations, or only one of them, or neither of them. And it may, in like manner, be said of the impelled body, that, in that instant, it was both at rest, and moving at the rate of one foot per second. This seems inconceivable or absurd.

It is not perhaps very clear and demonstrable, nor is it intuitively certain, that the moving body or particle must displace the other at all. All that we know of it, that matter is moveable, and that causes of this motion exist in nature. When they have produced impulsion, this motion, they have performed their task, and the motion is their complete effect: the particle continues in this condition for ever, unless it be changed by some cause; but we do not see anything in this condition, that enables us to say what causes are competent to this change, and what are not. Is it either intuitive or demonstrable, that the mere existence of another particle is not a sufficient or adequate cause? Is it certain that the arrival at another particle is an adequate cause? Or can we prove that this will not stop it altogether? The only conclusion that we can draw with any confidence is, that "two particles, or two equal bodies, meeting with equal velocities, in opposite directions, will stop." But our only reason for this conclusion is, that we cannot assign an adequate reason why either should prevail. But this form of argument never carries luminous conviction, nor does it even give a decision at all, unless a number of cases can be specified which include every possible result. This can hardly be affirmed in the present case.

We apprehend that the next case, in point of time, but an obliquity, has full less intuitive or deductive evidence; several facts, namely, when bodies meet in opposite directions with equal quantities of motion. It is by no means easy, if it be at all possible, to show that they must stop. The proof proceeds on some notion of the manner in which the impulsion, exerted on one particle, or on a few of each body, namely, those which come into contact, is distributed among all the particles. A material atom is moved only when a moving force acts on it, and each atom gets a motion precisely commensurate to the force which actuates it. Now, it is far from being clear, how a force impressed on one particle of a solid body occasions an equal portion of itself to pass into every particle of that body, and impel it forward in the same direction, that the very authors who affirm the present proposition as an elementary truth, claim no small honour for having determined with precision the moving forces that are exerted on each particle, and the circumstances that are necessary for producing an equal progressive motion in each. It was by no means an easy problem to show, that the motion of the body (eliminated by an average taken of the motions of every particle) is precisely that which is announced by this proposition. We must also consider how this investigation is conducted. It is by assuming, that whatever force connects a particle \(a\) with a particle \(b\), or whatever force \(a\) exerts on \(b\), the particle \(b\) exerts an equal force on \(a\) in the opposite direction—Surely no logician will say that this is an intuitive truth. The contrary is most distinctly conceivable. It was a discovery of the astronomers, that every deflection toward the sun is accompanied by an equal deflection of the sun. It was a discovery, that a piece of iron attracts a lodestone; and it was a discovery (and we dare not yet affirm it to be without exception), that every action of bodies is accompanied by an equal and contrary reaction. But this is by no means a first principle. It is the expression of a most generally observed fact, a fundamental of knowledge. When received on this authority, it is fully competent to solve every case of impulsion, independent of all obscure and illogical doctrines of force inherent in moving bodies, of force of inertia, of communication of motion, &c.

The impossibility of conceiving the detachment of part of the force inherent in A, and transferring this part into B, and the similar impossibility of conceiving the imparting to B some of the motion that was in A, should make us reject any proposition involving such conceptions, and refuse its admission as an elementary truth. Much more should we reject a proposition that obliges us to suppose that a particle of matter has two determinations, forces, motions, or call them by any other name, in one and the same instant. One of these necessarily excludes the other. Indeed this was so evident, even to the most eminent partizans of the doctrine of the transmutation of inherent force, and others consequent on it, that they found themselves obliged to deny that there was such a thing in the world as a perfectly hard body, in which the motion must be instantaneously changed into another, differing from it by any sensible quantity. The existence of perfectly hard bodies is positively denied by the celebrated mathematician of Basle, John Bernoulli, in his Dissertation on the Communication of Motion, which contended for the prize given by the Academy of Sciences at Paris 1716. His reason for this rejection is singular, and somewhat amusing. "In the collision of perfectly hard bodies, the conservatio virium vivorum, demonstrated by the most eminent mathematician (Mr. Leibnitz), to be a law of nature, would be broken without any effect being produced. He does not observe, that it is as completely broken by elastic bodies in the instant of greatest compression. A British philosopher, nullius additius jurare in verba magistris, asked, What will be the case of two encountering atoms of matter? Without calling them hard, we must conceive that they acquire their changes of motion in the instant of mutual contact, and that they acquire them totally, being indivisible. No answer has been given, or indeed can be given, but what implies the same difficulty. From all that has been said, we must conclude, that this branch of mechanical philosophy is not put, by those philosophers, into the condition of an elementary foundation of clear and demonstrative science; that the transmutation, or transference, either of force or motion, is not a thing of which we have a distinct conception; and that it necessarily leads us into very untenable doctrines. Far less does it seem safe for us to confide so much in its clearness and certainty, as to affirm, that impulsion is the sole moving force in mechanical nature, and the source of what we call pressure.

All this difficulty and obscurity has arisen from our arrogant notion, that we are competent judges of first principles; whereas we must acknowledge, that we can only perceive such as are properly related or accommodated to our intellectual powers; these powers, being specific and peculiar, cannot judge of principles of the first class, but of those only that are suitably compounded. We can never know or comprehend any essential property of matter—we can only know the relative properties of such matter as we see.

Therefore let us quit entirely the barren and trackless fields of abstraction, and rest satisfied with contemplating what the Author of Nature has exhibited to our view, and such as he has been pleased in his wisdom to exhibit it. We grant that there are no bodies open to our inspection which are perfectly hard, receiving finite changes of motion in an instant. It has not pleased God to put any such within our reach. When God created matter, it was with the purpose of forming a beautiful universe of this matter. He therefore gave it properties which fitted it for this purpose. It is this matter only that he has exposed to the wondering view of man. Thanks to his bounty, he has also given us properties of mind, by which this adaptation, when perceived by us, becomes a source of dignified pleasure to the observer.—A Newton, to whom "Jovis omnipotens," a Daniel Bernoulli, were rapt almost into ecstacy by a single atom, when they observed how its properties, and only such properties, fitted it for making part of a world, which

Unwearied, and from day to day, Should its Creator's power display.

Let the unhappy La Place consider these properties, which ensure the permanency of the solar system through ages of ages, as proofs of fatalism, as qualities essential to matter. But this Gallic torch effaces the bloom of life from the universe, the expression of the Supreme Mind which shines from within; and it spreads over the countenance of Nature the ghastly pallor of universal death. But let us Britons rather follow the example of our illustrious countryman, and solace ourselves with every discovery which tends to quicken our perception of Nature's animated charms. Let us listen to the conjectures of him who had already discovered so many, and who endeavoured to remove the veil which concealed the rest.

Newton, in his maturity of judgment, after having collected much information from his unwearyed experiments in magnetism, in chemistry, in optics, &c., said, all matter, that "he strongly suspected, that, in the same manner as the bodies of the solar system were connected conjunctly by gravitation, so the particles of sublunary bodies were connected together, and affected each other, by means of forces which acted at small, and, in many cases, insensible distances; producing the phenomena of cohesion, in all its forms of hardness, elasticity, ductility, softness, fluidity, by which their mechanical actions on each other were modified and regulated."

Father Bozovich, one of the first mathematicians of Europe, was the first who gave this conjecture of Newton's the attention that it so highly deserved. Other writers indeed, such as Keill, Freind, Boerhaave, &c., took occasional notice of it, and even made some use of it in their attempts to explain some complicated phenomena of nature. But they were so careless in their employment of Newton's conjecture, so completely neglected his cautious manner of proceeding, indulged so wantonly in hypothetical assumptions, and reasoned so falsely from them, that they brought his conjecture into discredit. Bozovich, on the contrary, copied Newton with care, and secured his progress as he advanced by the aid of geometry; establishing a set of uncontrovertible propositions, which must be the inevitable results of the premises adopted by him. He then proceeded to compare these with the phenomena of nature; and he shows that the coincidence is as complete as can be desired. All this is done in his Theoria Philosophiae Naturalis, first published at Vienna in 1759. We have given a very short account of it in the article Bosovich, Suppl.; but it hardly goes beyond the enunciation of the general principle, and the indication of of its applicability to the purposes intended. His application to the production of motion by the collision of bodies, is peculiarly satisfactory. But as the work is written chiefly with the view of gaining the approbation of persons well instructed in natural philosophy, it can hardly be called an elementary work, or be employed for the instruction of persons entering on the study.

We shall attempt to explain this important law of mechanism in a way that will give our readers a distinct notion (and, we apprehend, a just one) of the procedure of Nature in all the cases of impulsion that we can observe. We hope to do this, by considering the changes of motion produced by moving bodies in a certain series of familiar cases, where the procedure of nature may be distinctly observed, and where it is uniformly conceived by every spectator; and which will gradually lead the mind to those cases where the procedure is not observed with distinctness: but the similarity to the former case is concluded by so fair analogy, that we imagine no person will controvert it.

We shall begin by attending to the manner in which two magnets in motion affect each other's motions; a phenomenon that is familiarly known in the general, although, perhaps, few persons have attended to it minutely.

Let us, therefore, suppose two magnets, A and B (fig. 1.) equal in weight (in the first instance). Let them be made to float on water, by placing them on pieces of cork. Let them be placed with their north poles touching each other. Let A be held fast, and let B be at liberty to move. We know that it will gradually recede from A, with a motion that would continually accelerate, were it not for the resistance of the water.

What is the inference drawn from this appearance? Surely this, that either a moving power, inherent in A, repels B, or that B avoids A, by an evasive power inherent in itself. It is immaterial for our purpose which opinion we adopt. Let us say that A repels B. This admits more concise language than the other. If we prevent this motion of B by means of a very slender spring applied to its remote end, we shall observe that the spring is bent back a little, just as if we were pushing away the magnet gently with the finger; and we observe, that the bending of the spring is so much the greater as B is nearer to A. We can judge of the intensity of the force by which B is actuated, by the bending of the spring—This force is equal to the weight of any body that will bend the spring to the same degree. This force is analogous, therefore, to the weight, the pressure of gravity, and we may call it a pressure, and measure it by grams weight. Every force that can bend a spring will move a body. This is a well-known fact. Therefore it is next to certain, that it is this force which causes B to recede from A; nay, if we compare the motion of B with what would result from the action of a force having this very intensity, and varying in the same manner by a change of distance from A, taking in the diminution which the resistance of the water must occasion, we shall find the motions precisely the same. All this can be discovered by Dynamics, no. 95, &c. Therefore we must conclude that this, and no other, is the cause of the recedes of B.

If, instead of placing B in contact with A, we place it at a distance from it, and push it toward A with an initial velocity, somewhat less than it would have acquired in that place by its recedes from A, we shall find that it will approach A with a motion gradually retarded, till it stop at a small distance from A; and will now recede from it again with an accelerated motion. In short, we shall find that its whole motion to and from A is precisely the same with what results from a similar computation by no. 95, of Dynamics.

The whole of this phenomenon is conceived by every beholder, who has not inhibited some peculiar theory of a stream of impelling fluid, as the indication and effect of a repulsive force exerted by A on B, or of a quality of B, by which it recedes from A.

If now B be held fast, and A be set at liberty, it is observed to be repelled by B, or to recede from B, in the same manner, and with the same force.

Thus, the two magnets appear to affect each other's motions, and are thought, and said, by all to repel each other. The effect appears curious, but excites no farther thought in most minds: it is only the speculative that begins to suspect that he has not conceived it properly.

Now, let us suppose that B is afloat on the surface of the water, and at rest; and that A is pushed towards it, by a single stroke, causing it to move so moderately that it shall not strike B, but have its motion destroyed by the repulsion before it reaches it; and let us further suppose, that the initial velocity of A was exactly measured—the fact will be as follows. As soon as A comes within a certain distance of B, its motion begins to be affected; it gradually diminishes, and at length it ceases entirely, and A remains ever after perfectly still. But it is also observed, that in the instant that A slackens its motion, B begins to move; that it gradually accelerates in its motion, and at last acquires the initial velocity of A, with which it proceeds, till the resistance of the water brings it to rest, perhaps at a considerable distance from A. This experiment is very amusing, and the initial velocity of A may be increased in each succeeding trial, till at last it strikes B. Even then the general appearance remains the same: A is brought to rest and remains at rest, neither resisting nor advancing forward; and B moves off with the initial velocity of A. What we wish to be particularly noticed is, that as long as the initial velocity of A is less than a certain quantity (depending on the strength of the magnets), the motion is communicated to B; or, to express it more cautiously, motion is produced in B, without anything happening that can get the name of impulsion with propriety.

In the ordinary conceptions and language of mankind, impulse always supposes actual contact; and impulsion is equivalent to a blow or a stroke. Both of these are indeed metaphorical terms, as well as impulsion. Perhaps the word "to hit," expresses this particular case more purely, and it is perhaps without any figure, and is the appropriate word. We do not speak at present of the conception and language of philosophers, but of persons taking an unconcerned view of things, without any intention of speculating farther about the matter.

Appearances perfectly similar are observed in electrified bodies. If we hang two equal bunches of very light downy feathers by two equal linen threads, so as to hang close by each other like pendulums without touching, touching; and if, after having electrified them so that they repel each other to some distance, we draw one of them, which we shall call A, considerably aside from the perpendicular, and then let it go swinging like a pendulum; we shall observe, that instead of accelerating till it reach the lowest point of its vibration, its motion will be retarded; it will stop entirely when its thread is perpendicular, and will remain at rest. In the meantime, the other bunch B will acquire motion, which will gradually increase till it equal the motion of A in its maximum state; and with this it would proceed forever, were it not rising like a pendulum in the arch of a circle. The general fact is the same as in the case of the magnets. The moving body is brought to rest, in which state it continues, and the quiescent body moves off with an ultimate velocity, equal to the initial velocity of the other; and all this happens without contact or impulsion, but is produced by the mutual repulsion of the electrified bodies.

If this general fact be compared with what happens in the collision of two billiard balls, it will be found perfectly similar in every respect, but that of the contact and the impulsion, properly so called. The impelling ball is brought to rest, and remains at rest; and the impelled ball moves off with the velocity of the impelling ball.

This being the case, it is plain that we may derive some information from the motion of the magnets, that must greatly assist us in our conceptions of what passes in the rapid, if not instantaneous, production of motion in a billiard ball, by hitting it with another. In the case of the magnets, we perceive, and can discriminate, a progressive train of changes, which terminate in a final change, perfectly similar to the change in the impulsion of the billiard ball. This will justify a very minute attention to, and statement of, all the circumstances.

Let us attend to the process of this operation, and the production of motion in the magnet originally at rest, and the abolition of it in the one originally in motion; and let us reflect on what passes in our minds when we try to explain it to ourselves. The trials mentioned at first, when one magnet was held fast, shew us that each magnet repels or avoids the other, and that this action is found to be equal on both sides, producing equal compression of the spring employed for attaining the intensity of this repulsion when the distances are the same. This is the fact. It is no less a fact, that equal moving forces, such as equal pressures must be supposed to be, produce equal changes of motion in their own direction. Therefore, as soon as A comes to such a distance from B that the mutual action takes place, both magnets are affected, and equally affected; that is, equal changes of motion are produced on each, but in opposite directions. The motion of A is diminished, perhaps \( \frac{1}{2} \)th part, in \( \frac{1}{4} \)th of a second, and (let it be carefully remembered) while A passes over a certain space, suppose the \( \frac{1}{10} \)th of an inch. During this small portion of time, B acquires as much motion as A loses. This is not the motion lost by A. This is inconceivable; for motion is not a thing, but a condition. But it is an equal degree of motion. B has passed over a small space during this time, perhaps the \( \frac{1}{50} \)th part of an inch, with an almost imperceptible motion, that is gradually accelerated from nothing. Since A is moving faster than B, it must still gain upon it; and therefore the mutual repulsion will increase; and in the next \( \frac{1}{10} \)th of a second this force will take another and greater portion of A's original velocity from it, and will add a greater velocity to that already acquired by B. And thus, in every succeeding minute portion of time, the motion of A will be more and more diminished, and that of B as much increased, by the equal, though continually increasing, simultaneous repulsions acting in opposite directions. It is evident, that it is possible that the velocity of A may be so much diminished, and that of B so much increased, that the remaining velocity of A shall be just equal to the acquired velocity of B. Till this happens, the distances of the magnets have been continually diminishing; for A has been moving faster than B; and gaining on it. If the operation of the mutual repulsions could be stopped at this instant, both magnets would move forward forever with equal velocities.

It is of particular importance to know what this common velocity is. This is determined by our previous knowledge, that the magnets repel or avoid each other with equal forces. These forces may vary by a variation of distance; but the force acting on A is always equal and opposite to the force acting at the same time on B. This is the uncontroverted fact (the authority for which shall soon be considered). These equal forces must therefore produce equal and opposite changes of motion. The motion acquired by B is equal to that lost by A. But the magnets being supposed equal, and moving with equal velocities, they have equal quantities of motion. Therefore the motion acquired by B, or that lost by A, is equal to what remains in A; that is, A has lost half of its motion, and therefore half of its velocity; or the common velocity is half of the primitive velocity of A.

It was for the sake of a somewhat easier discussion that we supposed the magnets to be of equal weights. But it is almost equally easy to ascertain what this common velocity will be in any other proportion of the quantities of matter in A and B. It is a matter of unexpected experience, that whatever be the weight or strength of two magnets, their actions on each other are always equal. Therefore the simultaneous force must always produce equal changes of motion in the two bodies. But the change of motion is expressed by the product of the quantity of matter and the change of velocity. Therefore let A and B represent the quantities of matter in the magnets; and let \( a \) be the primitive velocity of A, and \( x \) the velocity which obtains when both are moving with one velocity. The velocity lost by A is \( a - x \). Therefore we must have \( Bx = A \times a - x \), \( = Aa - Ax \); and \( Aa = Ax + Bx = A + B \times x \), and \( x = \frac{Aa}{A + B} \). The common velocity is therefore obtained by dividing the primitive quantity of motion by the sum of the quantities of matter.

This may be conceived more commodiously in another way. Since B acquires as much motion as A loses, the whole quantity of motion is the same as before; Therefore the common velocity must be had by dividing this quantity of motion by the whole quantity of But we wished to make the reader keep his attention fixed on the steps of procedure, and see the connection of each with the causes.

We shall find that this period of the whole process, namely, the moment when both bodies have acquired a common velocity, and the precise magnitude of this velocity, are points of peculiar importance in the doctrine of impulsion; indeed they almost comprehend the whole of it.

But this is a state that cannot continue for a moment in the example before us. The repulsive or evasive forces are still acting on both magnets, and still diminish the motion of A, and equally increase the motion of B. Therefore the velocity of A, in the very next moment, must be less than that of B; and B has, during this moment, gained on A, or has removed farther from it. This continues; A is still retarded, and B is accelerated; and therefore gains more and more upon A, or separates farther and farther from it. This must continue as long as the mutual repulsions are supposed to act. If we suppose that the sensible action of these forces is limited to some determinate distance, the mutual action will cease when B has got to that distance before A. We may call it the inductive distance. After this, A and B will proceed with the velocities which they have at that instant. Let us inquire into these final velocities; and thus complete our acquaintance with the process.

We see (and it is important) that the magnets are in their state of greatest proximity at the instant of their moving with a common velocity, and that after this they gradually separate, till they are again at their inactive distance. During this separation they attain distances from each other equal to what they had during the period of their mutual approach. At these distances the repulsions are the same as before, and act in the same direction. Therefore, in each moment of separation, and at each distance, A sustains the same diminution, and B gets the same augmentation of its motion, as when they were at the same distance in the period of their mutual approach. The sums total, therefore, of these equal augmentations and diminutions must be equal to the augmentation and diminution during the approach. Therefore the whole diminution of A's motion must be double of the diminution sustained during the approach; and the whole augmentation of B's motion must, in like manner, be double of that acquired during the approach of A. Hence we easily see, that when the magnets are supposed equal, A must be brought to rest; for in the period of approach it had lost half of its velocity. It must now have lost the whole. For similar reasons B must finally acquire the primitive velocity of A; for in the instant of greatest proximity, it had acquired the half of it.

Thus we see, that the equal mutual repulsions are precisely adequate to the production of the changes of motion that are really observed; and must therefore be admitted as the immediate causes of these changes.

It is equally easy to ascertain the final velocities when the magnets are of unequal sizes; for the equality of their mutual repulsions is not affected by any inequality of their magnitudes. Their separations, and consequently the changes of motion during these separations, will be the same with their approaches and the corresponding changes of motion; and the whole change on each will be double of the change sustained at the instant of

\[ A \times a + B \times b = A \times x + B \times x, \quad \text{or} \quad A + B \times x = A \times a + B \times b, \quad \text{and} \quad x = \frac{Aa + Bb}{A + B}. \]

Therefore Therefore the common velocity is had by dividing the sum of the primitive quantities of motion by the sum of the quantities of matter.

But the repulsive forces continue to act as in the former case. The motion of A is still more diminished, and that of B augmented; therefore the velocity of B must now exceed the velocity of A, and the magnets must separate. Reasoning in the same way as in the former case, it is evident that the mutual action does not cease till the magnets have separated to their inactive distance from each other, and that the whole change of motion in each is double of the change that it had sustained when they were in their greatest proximity, and moving with a common velocity. These considerations enable us to ascertain the final state of each. The common velocity is \( \frac{Aa + Ab}{A + B} \). Therefore the change made on the velocity of A, at the instant of greatest proximity, is \( a - \frac{Aa + Ab}{A + B} \), or \( = \frac{B \times a - b}{A + B} \), and the final velocity of A is \( a - \frac{2B \times a - b}{A + B} \).

In like manner, the change produced on the velocity of B is \( \frac{Aa + Bb}{A + B} - b \), or \( = \frac{A \times a - b}{A + B} \), and the final velocity of B is \( b + \frac{2A \times a - b}{A + B} \). We may also obtain the final velocity of each, by taking its initial velocity from twice the common velocity.

If, in this example of two magnets in motion, we suppose them of equal weight, we shall find that they will finally proceed with exchanged velocities. For when \( A = B \), it is plain that \( a - \frac{2B \times a - b}{A + B} \) is \( = a - 1 \times a - b \), \( = a - a + b \), \( = b \); and \( b + \frac{2A \times a - b}{A + B} \) is \( = b + 1 \times a - b \), \( = b + a - b \), \( = a \).

This case is easily subjected to experiment, and will be found fully confirmed, if we take into account the retardations occasioned by the resistance of the water to the motions.

Let us, in the next place, suppose the magnets to be moving in opposite directions with the velocities \( a \) and \( b \); and (in order that the magnets may not strike each other) let the sum of \( a \) and \( b \) be less than the sum of \( a \) and \( b \), which the repulsions of the magnets would produce by repelling them from contact to their inactive distance.

As soon as the magnets arrive at their acting distance, their mutual and equal repulsions immediately begin to diminish both of their motions; and in any minute portion of the period of their approach, equal quantities of motion are taken from each. It is evident, that if the primitive quantities of motion have been equal; that is, if A and B have been moving with velocities reciprocally proportional to their quantities of matter, then, when the motion of one of them has been annihilated by their mutual repulsion, the motion of the other will be destroyed at the same time, and both will be brought to rest. Were the repulsions annihilated at impulse, this instant, they would remain at rest. But because those forces continue their actions, the magnets will separate again, regaining, at every distance, the velocity which they had, when at that distance, during their mutual approach; and when they have reached their inactive distance, they will have regained each its original momentum and velocity, but in the opposite direction. This needs no further comment; but must be kept in mind, because this case has a precise counterpart in the collision of solid bodies, meeting each other in opposite directions with equal momenta. But if the momentum of one exceed that of the other, thus, if \( A \times a \) be greater than \( B \times b \), then, when the magnet B is brought to rest, A has still a momentum remaining equal to \( Aa - Bb \). Having therefore a certain velocity, while B has none, it must approach still nearer to B, and a still greater repulsion will be exerted on B than if A had also been brought to rest, but still repelling B. Since B is now acquiring motion in the direction opposite to its former motion, and A is still losing motion, a time must come when the motion of A is so much diminished, and that of B so much augmented, that they are moving with a common velocity in the direction of A's primitive motion. The reasoning employed in the foregoing examples shew us, that, in the present case also, this state of common velocity is also the state of the greatest proximity, and that the magnets separate again, till they attain their distance of inaction, and that the total change in each is double of what it was in their state of greatest proximity.

To find this common velocity, recollect, that when the momentum of B was extinguished, that of A was common \( = Aa - Bb \). From what has been already said on the other cases, we know that when the common velocity obtains, the whole momenta are still equal to set the \( Aa - Bb \). Therefore the common velocity \( x \) must be change in doubled by the subsequent separation.

The velocity lost by A must therefore be \( a - \frac{Aa - Bb}{A + B} \), and the final velocity will be \( a - \frac{2B \times a + b}{A + B} \). The final motion of A will be in the same direction as at first, if \( a \) be greater than \( \frac{2B \times a + b}{A + B} \), otherwise it will be in the opposite direction.

In like manner, the change of velocity in B is \( b + \frac{Aa - Bb}{A + B} \), because the former velocity \( b \) is destroyed, and the new velocity is \( \frac{Aa - Bb}{A + B} \) in the opposite direction. This is \( = \frac{Aa - Bb}{A + B} \), and the final velocity of B is \( b - \frac{2A \times a + b}{A + B} \).

Thus we have shewn, in the case of magnets acting on each other by repulsive forces, or actuated by forces equivalent to repulsive forces, how changes of motion are produced, which have a great resemblance to those which are seen in the collision of solid bodies. The motions which obtain in the instant of greatest proximity die. proximity are precisely similar to what are observed in the collision of unelastic bodies. Their common velocity after collision is always \( \frac{Aa + Bb}{A + B} \), or \( \frac{Aa - Bb}{A + B} \), according as the bodies were moving in the same or in opposite directions. The final motions of the magnets are also precisely similar to what are observed in the collision of perfectly elastic bodies. We took the instance of magnets, because the object is familiar; but we can substitute, in imagination, an abstract repulsive force in place of magnetism, and we can assign it any intensity, and any law and limits of action we please. We can imagine it so powerful, that although its action be limited to a very small, and even insensible distance, it shall always reduce the meeting bodies to a common velocity before they come into actual contact; and therefore without any real impulsion, as impulsion is commonly conceived.

There are some farther general observations that may be made on those motions which are of importance:

1. We see that the changes of motion, and consequently the actions, are dependent on the relative motions only, whatever the absolute motions may be: For changes are always as \( a - b \) when the bodies are moving in one direction, and as \( a + b \) when they are moving in opposite directions. Now \( a = b \) is the relative motion.

2. The change of velocity in each of the two bodies is inversely as its quantity of matter, or is proportional to the quantity of matter in the other body. The changes in A and B are \( \frac{B \times a - b}{A + B} \) and \( \frac{A \times a + b}{A + B} \). The changing forces being equal on both sides, produce equal changes in the quantities of motion; and therefore produce changes of velocity that are inversely as the quantities of matter.

3. During the whole process, the sum of the momentums, or quantities of motion, remains the same, if the bodies are moving in one direction: if they are moving in opposite directions, it is the difference of momentums that remains the same; for in every instant of the process equal changes of momentum are made in opposite directions. When the motions are in the same direction, as much is taken from the one as is added to the other; and therefore the sum remains unchanged. When the motions are in opposite directions, equal quantities are taken from both; and therefore the difference remains unchanged. This is called the conservatio momentorum; and it is usually enunciated by saying, that the quantity of motion, estimated in one direction, is not changed by the equal and opposite actions of the bodies. This is a particular case of a general law affirmed by Descartes, that the quantity of motion in the universe remains always the same when estimated in any one direction.

4. When the whole process is completed, the sum of the products made by multiplying each body by the square of its final velocity, is equal to the sum of the products made by multiplying each body into the square of its initial velocity. For when the process is completed, the two bodies are at the same distance from each other as when the mutual action began. Therefore, during the process, each body has passed over an equal space, and in every similar point it has been acted on by an equal force (although this force be different in different points of this space). Therefore, in every instant, the simultaneous products of the quantity of matter by the momentary variation of the square of the velocity are equal on both sides; and therefore the products of the quantity of matter by the whole change of the square of the velocity are also equal on both sides.

See Dynamics, Suppl. no. 95, and 110, where \( v = \frac{f}{m} \); and therefore \( m \times v = f \times i \), and \( m \times V^2 - v^2 \), or \( n \times v^2 - V^2 = f \times i \). Now, since these changes are in opposite directions, as much is added to one product as is taken from the other, and the sum of the products of the quantities of matter by the squares of the final velocities, is equal to the sum of the products of the same quantities of matter by the squares of the initial velocities.

This is a particular case of the famous conservatio theorem of virium vivarum, claimed as a mighty discovery by the partisans of Leibnitz, and ascribed to him; but principles, he has no claim whatever to the discovery. It was first communicated to the Royal Society of London in 1668 by Huyghens, as one of the general laws of impulsion, obtaining in what he calls hard bodies. Several of the Leibnitzian school, indeed, extended it farther than Huyghens had done; some of them indeed very lately. The observation of this general law was soon applied to many excellent purposes in the solution of very intricate problems; because it often saved the trouble of tracing the intermediate steps of a complicated process. Affirmed that these products were invariable, the mathematician found it an easy matter to state what conditions of the question insured this equality of products; and thus the problem was solved. In this manner Daniel Bernoulli gives most elegant solutions of some, otherwise almost intractable, problems in Hydraulics. For such reasons, as a mighty aid in mechanical investigation, the discovery of Huyghens is extremely valuable. Its merit in this respect is perfectly similar (though perhaps somewhat greater) to Descartes's observation of the conservatio momentorum. It is also like the observation or discovery of Maupertuis, which he calls the law of smallest action (indeed it is the same under a different aspect), or La Grange's law of virtual velocities, or D'Alembert's law of equilibrium of action—all of these are general facts, laws by which the changes of motion are observed to proceed. But their authors have vaunted them as principles, as causes, from which to conclude effects; whereas they are really inductions from particular instances. We must also observe, that this law of conservatio virium vivarum was not deduced either by Huyghens or any of the Leibnitzian school, by reasoning from more general principles. It was an explication of familiars in events, diversified by other circumstances. We do not recollect any author who has given what can be called a demonstration of it, deducing it from principles or laws still more general. We apprehend, that the present case of its truth has been so demonstrated by us. The principle is, that "a moving force is to be measured by the change of motion produced by it?" And the law to which this principle is applied is, that "the mutual repulsions of magnets..." are equal and opposite;” and the application is made by means of the “39th proposition of the first book of Newton’s Principia.” Our principle, which is the same with Sir Isaac Newton’s second law of motion, is really an axiom of human thought. The proposition is the consequence logically drawn from this axiom; and the law of magnetism is an observed fact. We hope to show by and bye, that this proposition, which is our no 95 of Dynamics, is found to obtain in every instance that has been or can be given of the conservatio virium vivarum, and that this conservatio is only another way of expressing the proposition. Having done this, we shall not think ourselves chargeable with vanity when we say, that we have given the first demonstration of this famous law. We cannot refuse ourselves some satisfaction at having done this; because it has been so highly esteemed, chiefly for the support derived from it for the Leibnitzian measurement of the force of moving bodies by the square of the velocity which it communicates; whereas it is the logical consequence of the force being proportional to the simple velocity. We have only taken a weapon out of the hands of a plunderer, and restored it to its lawful owner, Sir Isaac Newton. Non ita certandi cupidus, quam propter amorem: For we must say,

Tu pater et rerum inventor, tu patria nobis Suppeditas praecpta, tuisque ex, inclute, chartis Floriferis ut apes in salutis omnia libant, Omnia nos itidem depastae aurea dirita Aurea, perpetua semper dignissima vita.

We trust that our reader will not think that this minute discussion of the mutual actions of magnets or other repelling bodies, in which we have engaged him, has been thrown away, since it has enabled us to apprehend clearly a case of two such general laws as the conservatio momentorum, and the conservatio virium vivarum.

In the moment of greatest vicinity and common velocity, there is a certain determinate loss of the vis viva, or products of the matter by the square of the velocity; and this loss is proportional to the square of the relative motion. The vis viva, at the commencement of the mutual action, are \( A^2 + B^2 \) (I.). In the moment of greatest proximity, the quantity of matter \( A + B \) is moving with the common velocity \( \frac{Aa + Bb}{A + B} \); therefore the vis viva are \( \frac{Aa + Bb}{A + B} \times \frac{Aa + Bb}{A + B} = \frac{Aa + Bb}{A + B} \times \frac{Aa + Bb}{A + B} = \frac{A^2a^2 + B^2b^2 + AB \times 2ab}{A + B} \) (II.).

\( I. \times A + B = A^2a^2 + B^2b^2 + AB \times a^2 + b^2. \)

\( II. \times A + B = A^2a^2 + B^2b^2 + AB \times 2ab. \)

Difference \( AB \times a - b^2. \)

\( \text{Loss of vis viva} = \frac{AB}{A + B} \times \frac{a - b}{a - b}, \) a quantity that is proportional to \( a - b, \) the square of the relative velocity \( a - b. \)

Had the bodies been moving in opposite directions, then (II.) \( \times A + B \) would have been \( A^2a^2 + B^2b^2 - AB \times 2ab, \) and the difference from \( A^2a^2 + B^2b^2 \times A + B \) would have been \( = AB \times a + b^2, \) proportional to the square of the relative velocity \( a + b. \)

Such is the fact; and we shall find it of importance in the great debate about the force of moving bodies. Let us inquire into the physical or mechanical cause of it. In the moment of common velocity, the bodies are nearer to each other than they are at the beginning and at the end of their mutual action. Therefore (when they are moving in one direction) the body \( A, \) which follows, has been retarded through a space which is greater than the space along which the preceding body \( B \) has been accelerated. But, because the simultaneous forces acting on the bodies along these unequal spaces are always equal, the area which measures the diminution of the square of \( A \)'s velocity (Dynamics, no 95,) must exceed the area which expresses the augmentation of the square of \( B \)'s velocity, and there must be a loss of vis viva.

Now, we learned above, that the mutual action is the same when the relative velocity is the same; and therefore the approximation, which is the occasion of this action, must be the same. And it is demonstrated in Dynamics, no 95, that the area, whose abscissa is the space described, and ordinates the forces, expresses the square of the generated or extinguished velocity. This is evidently the relative velocity of the bodies, because they are brought to a common velocity in the instant of greatest proximity; that is, their relative velocity is destroyed.

6. During the whole process, the common centre of the composition or gravity (\( A \)) is moving uniformly with the common velocity \( \frac{Aa + Bb}{A + B}. \) For the motion of the centre of gravity is the average of the motion of every particle of matter in both bodies. \( Aa \) is the sum of the motions of every particle of matter in \( A, \) and \( Bb \) is the sum of the motions of every particle in \( B, \) before the mutual actions began. Therefore \( Aa + Bb \) is the whole motions when the bodies are moving in the same direction with their different velocities. The number of particles is \( A + B. \) Therefore, if the whole motions be equally divided among all the particles, the velocity of each must be \( \frac{Aa + Bb}{A + B}. \) This is the average motion, or the motion of the centre of position, deduced from the notion we wish to impress of the character of this centre, as the index of the position and motion of any assemblage of matter. This velocity may be deduced more easily from its geometrical property.

(a) See the article Position in this Supplement; where it will be demonstrated, that the centre of gravity (determined in the usual manner) is the point by whose situation and motion we estimate with the greatest propriety the situation and motion of the assemblage, of which it is the centre: it is therefore called the centre of position. The reader is only desired at present to recollect, that the centre of gravity, or position of two bodies, is situated in the line joining their centres; and that its distance from each is inversely as their quantities of matter; and that the distance and motion of the centre is the medium or average of all the distances or motions. It is a point so situated between A and B, that its distance from each is reciprocally proportional to the quantities of matter in A and B, as is well known of the centre of gravity. It is equally plain, that when the bodies are moving in opposite directions, the average velocity \( x \) must be

\[ x = \frac{Aa - Bb}{A + B} \]

Thus we see, that the motion of the centre of position, before the magnets have begun to act on each other, is the same with its motion when their mutual repulsion is the greatest; namely, at the moment of their greatest vicinity. It has continued the same during the whole process: For we have already seen, that the sum or difference of the momenta, or \( Aa - Bb \), remained always the same; consequently

\[ \frac{Aa - Bb}{A + B} = x \]

or \( x \), the motion of the centre, remains always the same. Therefore the proposition is demonstrated. It is, indeed, a truth much more general than appears in the present instance.

If any number of bodies be moving with any velocities, and in any directions, the motion of the centre of position is not affected by their mutual, equal, and opposite actions on each other.

7. During the whole motion, the motion of the bodies relative to each other, is to the motion of one of them, relative to the centre of position, as the sum of the bodies is to the other body: For when they were moving with a common velocity, this velocity was the same with that of the centre; and they are then at rest, relative to each other, and relative to the centre. And because their distances from the centre are inversely as the bodies, their changes of distance, that is, their motions relative to the centre, are in the same proportion; and the sum of their motions relative to the centre is the same with their motions relative to each other. Therefore \( A + B : A = a - b : \text{motion of } B \text{ relative to the centre}. \)

Indeed we saw, that in their mutual action, the change of B's motion was

\[ \frac{Aa - Bb}{A + B} \]

and the change of A's motion was

\[ \frac{Bx - a - b}{A + B} \]

Hence we learn, that while the centre moves uniformly, the bodies approach it, and then recede from it, with velocities reciprocally proportional to their quantities of matter. This will be found a very useful corollary. We may also see that their final velocity of mutual recedes is equal to that of their first approach, or, their relative motions are the same in quantity after the action is over as before it began, but in opposite directions.

All these general facts, which are distinctly appreciable, and very perceivable, in this example of magnets, or electrified bodies, are equally appreciable in all cases of mutual repulsions, however strong they may be; and although the space through which they are exerted should be so small as to elude observation, and though the whole process should be completed in an insensible moment of time.

It scarcely needs any comment to make it clear that the very same changes of motion must take place, if a solid body A should come up to another solid body B, at rest, or moving more slowly in the same direction, or moving in the opposite direction; provided that there be a spring interposed between them, which may hinder A from striking B; for, as soon as A touches the impulsion, spring, it begins to press it against B, and, therefore, to compress the spring. It cannot carry the spring before it, without the spring's pushing B before it. Pressure on B is required for this purpose. This is supplied by that natural power which we call elasticity, which is inherent in the spring, whether it be in motion or at rest. It is not in action, but in capacity, faculty, capability, power, or by whatever name we may choose to express the possession. The occasion required for its exertion is compression. This is furnished by the motion of A; for A cannot advance without compressing it. This inherent force of the spring is known to act with perfect equality at both ends, in opposite directions. It exerts equal and opposite pressures on A and on B; it diminishes the motion of A, and equally augments the motion of B (if both are moving that way). A is retarded, and B is accelerated; A is still moving faster than B; and therefore the compression and the consequent reaction of the spring increases, and still more retards A and accelerates B. After some time, both bodies, with the spring compressed between them, are moving with equal velocities; the spring, however, is strongly resisting on both, and must now cause them to separate; still retarding A, and accelerating B—They must separate more and more, till the spring regain its quiescent form, and its elastic reaction cease entirely. During its restitution, its pressures are the same as during its compression; therefore, the whole change produced on each of the bodies must be double of what it was when the spring was in its state of greatest compression, and the bodies were moving with a common velocity. In short, the whole process in this example must be precisely similar to that of the magnets in every circumstance relating to the changes of motion in A and B. The common velocity must be

\[ \frac{Aa - Bb}{A + B} \]

The final velocity of A must be

\[ \frac{a - b}{A + B} \]

and that of B must be

\[ \frac{2Aa - Bb}{A + B} \]

The motion of the common centre must be unaffected by the action of the spring, and the motion of each body, relative to the centre, must be reciprocally as its quantity of matter, &c. &c.

We apprehend that this process can scarcely be called the changed impulsion; A has not struck B. The changes of motion can scarcely be ascribed to forces inherent in A or B, in consequence of their being in motion. Any influence, not already warped by a theory, will (we cannot forget) ascribe them to a force inherent in the spring; i.e., which is inherent in it, whether at rest or in motion, and only requiring a continued compression as the proper opportunity for its continued exertion. This spring may be supposed to make a part of B, or of A, or of both; and then, indeed, the force may be said to be inherent in either, or in both. But it is not the peculiar force inherent in motion, or in moving bodies only—it is the force of elasticity, inherent in part of the body, but requiring a continued compression for the production of a continued repulsion. The effect of this reaction is modified by the very occasion of the compression. This may be the elasticity of another spring. In this case it will only compress that spring—it may be the advance of a body in motion; the reaction produces a retardation. impulsion of that motion; it may be the obstacle of a quiescent body—it will give it motion; or, it may be the obstruction by a body moving more slowly away than the spring is pressed forward—it will accelerate that motion. Thus, in all these cases, we cannot help distinguishing the immediate cause of these changes of motion from the supposed force of a moving body. Nay, the process of motion is similar, even when we suppose that the spring is not a thing external to the body, although attached to it; but that the whole body, or both bodies, are springy, elastic, and therefore compressible. As soon as the bodies come into sensible contact, compression must begin; for we may suppose the bodies to be two balls, which will therefore touch only in one point. The mutual pressure, which is necessary in order to produce the retardation of A, and the acceleration of B, is exerted only on the foremost particle of A, and the hindmost particle B; but no atom of matter can be put in motion, or have its motion any way changed, unless it be acted on by an adequate force. The force urging any individual particle, must be precisely competent to the production of the very change of motion which obtains in that particle. Except the two particles which come into contact in the collision, all the other particles are immediately actuated by the forces which connect them with each other; and the force acting on any one is generally compounded of many forces which connect that particle with those adjoining. Therefore, when A overtakes B, the foremost particle of A is immediately retarded—the particles behind it would move forward, if their mutual connection were dissolved in that instant; but, this remaining, they only approach nearer to the foremost striking particles, and thus make a compression, which gives occasion for the inherent elasticity to exert itself, and, by its reaction, retard the following particles. Thus each stratum (so to conceive it), continuing in motion, makes a compression, which occasions the elasticity to react, and, by reacting, to retard the stratum immediately behind it. This happens in succession: the compression and elastic reaction begin in the anterior stratum, and take place in succession backward, and the whole body gets into a state of compression. Things happen in the same manner in B, but in the contrary direction, the foremost strata being the last which are compressed. All this is done in an instant (as we commonly, but inaccurately speak), that is, in a very small and insensible moment of time; but in this moment there is the same gradual compression, increase of mutual action, greatest compression, common velocity, subsequent restitution, and final separation, as in the case of bodies with a slender spring interposed, or even in the case of the mutually repelling magnets. In all the cases, the changes of motion are produced by the elasticity or the repulsion, and not by the transmutation of the force of motion. The changing force is indeed inherent in the bodies, but not because they are in motion; the use of the motion is to give occasion, by continued compression, for the continued operation of the inherent elasticity. The whole process may be very distinctly viewed, by making use of bodies of small firmness, such as foot-balls, or blown bladders. If blown bladders are used, each loaded with sand, or something that will require more force, and consequently more compression to impel it forward; we shall observe the compression of both to be very considerable, and that a very sensible time elapses during the process of collision. This may even be observed very distinctly in a foot ball, which is always seen to roll a little on the toe before it flies off by the stroke. When one ball is strongly driven against another, they plainly adhere together for some time, and then the frictional ball flies off.

If we return to the example of the two balls with the spring interposed, we may make some farther useful observations. When the spring is in its state of greatest compression, and the balls are moving with a common velocity, we can suppose that the spring is arrested in that situation by a catch. It is evident that the two bodies will now proceed in contact with this velocity, which we have shown to be:

$$\frac{A}{A+B} = \frac{B}{A+B}$$

Now, in the constitution of such masses of tangible matter as we have the opportunity of subjecting to our perfect experiments, we find a state of aggregation which very much resembles this. Some bodies are almost perfectly elastic, that is, when their shape is changed by external pressure; and that pressure is removed, they recover their former shape completely, and they recover it with great promptitude. Glass, ivory, hard steel, are of this kind. But most bodies either do not recover it completely, or they recover it very slowly—some hardly recover it at all. A rod of iron will, when considerably bent, not nearly recover its shape; a rod of lead will let; and a rod of soft clay will hardly recover it in any degree. These, however, are but gradations of one and the same quality: if the quiescent form of a body is very little disturbed, it will recover it again. Thus, a common soft iron wire of No. 6. and 12 inches long, if twisted once round, will return completely to its original form, and will allow this to be repeated for ever; but if it be twisted 1½ turns, it will untwist only 1; and in this new form, it will twist and untwist one turn as often as we please. Even a rod of soft clay, ¼th of an inch in diameter, and 7 feet long, will bear one twist as often as we please; but if twisted 4 times, will untwist itself only one turn, and will do this as often as we choose. In short, it appears that the particles of bodies, usually called unelastic, will admit a small change of distance or situation, and will recover it again, exhibiting perfect elasticity, in opposition to very small forces; but if they are forced too far from this situation, they have no tendency to return to it completely, but find intermediate situations, in which they have the very same connections with the surrounding particles; and in this new situation, they can again exhibit the same perfect elasticity, in opposition to very small forces. Mr Coulomb conceives such bodies to consist of elastic particles: they manifest perfect elasticity, so long as the forces employed to change their shape do not remove the particles from their present contacts; but if they are removed from these, they slide on to other situations, where they again exhibit the same appearances. To understand this fully, the reader may consult the article Bosovich of this Supplement. The fact is sufficient for our present purpose. Now, in this variable constitution, where the particles may take a thousand different situations, and still cohere, it is plain, that when a body If a body has been dimpled by compression, the particles have nothing to bring them back to their first situation when the compressing force is removed; the utmost elasticity to be expected, is that which will not extend to one shift of situation; therefore, the restitution is altogether impossible. This is the case with all soft bodies, such as clay—the same quality is manifested in all ductile bodies, such as lead, soft iron and steel, soft copper, soft gold.

Now let one of these bodies strike another. The compression, or the sliding of the particles over each other, requires force, or mutual pressure—this being accompanied by a reaction perfectly equal, must operate, during the compression, precisely as the equal repulsive forces did. It will take as much momentum from A as it gives to B; so that \( \frac{A}{A+B} = \frac{B}{A+B} \). The compression can proceed no farther, and the two bodies must now proceed in contact with this velocity.

And thus we see, that in the case of compressible, but unelastic bodies, the changes of motion are produced by the cohesive forces inherent in the bodies; but not inherent in them because they are in motion. We see clearly in this way, how the pendulum used by Robins and his followers gave a true measure of the velocity of the ball. All the while that it was penetrating into the pendulum, overcoming the cohesion as it went in, this cohesion was acting equally in both directions. While the fibre was breaking, it was pulling both ways; it was holding back the ball which was breaking it, and it was pulling forward the parts to which it still adhered; and when it broke at last, it had produced equal effects on the ball and on the pendulum in opposite directions. By such a process, the pendulum was gradually accelerated, and acquired its utmost velocity when the ball had ceased to penetrate:

\[ \text{Therefore, this velocity must be } \frac{A}{A+B}. \]

What should we now expect to happen in the collision of bodies? Such bodies as exhibit perfect elasticity, when examined by bending, or other similar trials, should have their motions changed precisely like the magnets, or bodies which repel or avoid each other at sensible distances. Bodies which exhibit no elasticity whatever, should continue in contact after collision. The common velocity in these should be

\[ \frac{A}{A+B}. \]

The perfectly elastic bodies should sustain changes of motion which are precisely double of the changes sustained by unelastic bodies, and should separate after collision with a relative velocity of recoil or separation, precisely equal to their relative velocity of mutual approach. And bodies possessing imperfect elasticity, should sustain changes of motion, which differ from the changes on unelastic bodies, precisely in proportion to the degree of elasticity which they are known to possess. And, lastly, if the changes of motion which obtain in the collision of bodies, are precisely those which would result from the operation of those inherent forces of elasticity and cohesion, NO OTHER FORCE WHATSOEVER CONCURS IN THEIR PRODUCTION: For we know that those forces do operate in the collision; impelling us to see the compression and restitution which are their effective causes, and their immediate effects. If any other force were superadded, we should see its effects also, and the motions would be different from what they are.

Now the fact is, that we have never seen a body that is not, in some degree, compressible. It has not pleased the Almighty Creator to make any such here below. Afforded He has not found such to be of use for the purposes He had in view in this our sublunary world. We know of no body that is perfectly unchangeable in its shape and dimensions. It is therefore no loss whatever to us, although we should not be able to say a priori what their motions will be in collision. We cannot even fairly guess them, by reasoning from what we observe in other bodies: For it is just as likely that their motions may resemble those of perfectly elastic bodies as those of unelastic bodies; for we find that bodies of the most extreme hardness are generally highly elastic. Diamond, crystal, agate, quartz, and such like, are the most elastic bodies we know. Philosophers, however, rather think that the motions of perfectly hard bodies will resemble those of unelastic bodies; because elasticity supposes compression. We do not pretend to say with confidence, what would be the motion of a single atom of matter (which cannot admit of compression), which is hit by another in motion. We see all the particles of terrestrial matter connected with each other by certain modifications of the general force of cohesion, so as to produce various forms of aggregation; such as aerial fluidity, liquid fluidity, rigidity, softness, ductility, firmness or hardness; all of which are combined with more or less elasticity. These tangible forms result from certain positive properties of the material atoms of which the particles are composed; and, in all the cases which come under our observation, these properties produce pressures of one kind or another; all of which are moving forces. They are inherent in the particles and atoms; therefore when such atoms are in motion, these forces are in a condition which affords occasion for a continuation of this pressure that is competent to the production of motion in another particle. But what would be the event of the meeting of atoms divested of such forces, we profess not to know, or even to conceive.

The fact also is, that all the changes of motion, commonly called impulsions, which have been observed are effects precisely such as have been described. Unelastic bodies of collision are perfectly elastic bodies separate after collision, and each produces double of the change that is sustained by an unelastic body. Bodies of imperfect elasticity differ from the two simple cases, precisely in the proportion of the elasticity discoverable by other trials. The mutual actions are observed to be in the proportion of their relative motions, whatever the real motions may be. For not only are the changes of progressive motion exactly in this proportion, but the compressions and changes of figure, which we consider as the immediate occasions of those actions, are also observed to be in the same proportions, in all cases that we can observe and measure with accuracy. Impulsion, accuracy. All these things can be ascertained with great precision by means of the collision of pendulous bodies in the way pointed out by Sir Christopher Wren (a method attributed by the French to their countryman Mariotte, but really invented by Wren, and exhibited to the Royal Society of London the week after he communicated his theory of impulsion).

We must also infer from these facts, that the actions of bodies on each other are mutual, equal, and opposite. This is really an inference from the phenomena, and not an original or first principle of reasoning. The contrary is conceivable, and therefore not absurd. In the same way that we can conceive a magnet repelling iron, without imagining that the iron repels the magnet, we may conceive a golden ball capable of impelling a leaden ball before it, without conceiving that the leaden ball will impel the golden ball. We do not find this easy indeed; because the contrary is so familiar, that the one idea instantly brings the other along with it. We apprehend it to be impossible to demonstrate, that a leaden ball will not stop as soon as it hits the golden ball, or vice versa. But all our experience shows us, that the pressures exerted in contact are mutual, equal, and opposite. The same thing is observed in the forces which connect the parts of bodies. A quantity of sand or water balanced in a scale will remain in equilibrium in whatever way it is stirred about; its parts always exert the same pressure on the scale; so does a body suspended by a string or resting on the scale, by whatever points it is supported. This could not be if the particles did not exert mutual and equal forces; nor could the phenomena called impulsions be what they are, if the pressures occasioned between the particles by the compressions and dilatations were not mutual and equal.

This law of action and reaction must be admitted as universal, though contingent, like gravity. Doubtless it results from the properties which it has pleased the great Artificer to give to the matter of which He has formed this world. There is one way in which we can conceive, most distinctly, how this may be a universal property of matter. If we grant the reality of attractions and repulsions e distanti, and suppose that every primary atom of matter is precisely similar to every other atom in all its properties, and that this assemblage of properties constitutes it a material atom; it follows, that every atom exerts the same attractions and repulsions, or has the same uniting and evasive tendencies, and then the law of action and equal reaction follows of course. This is surely the very notion that any person is disposed to entertain of the matter. And if mechanical force and mobility are the qualities which distinguish what is material from mind or other immaterial substances, the law of equal and contrary reaction seems nearly allied to the laws of first principles.

Of all the phenomena that indicate this perfect equality of action and reaction, the most susceptible of accurate examination is the sameness or equality of action when the relative motions are equal. Now there is no phenomenon more certain than this. In consequence of the rotation of the earth round its axis, and its revolution round the sun, it is plain that all our experiments and observations are on relative motions only. Now, we not only find that the actions of two bodies subjected to experiment are equal when the relative motions are equal, but we find that all our measures of action impulsive on a single body are proportional to the apparent motions which they produce. It requires precisely the same force to impel a ball eastward, westward, south, or north, at 12, or 3, or 6, or 9 o'clock; yet the real motions are immensely different in all these cases, and it is only the relative motions that have the proportions which we observe. Another very important point deducible from our experiments is, that the same pressure produces the same change of motion, whatever may be the velocity. We know this by observing, that when the mutual dimpling or compression is the same, the change of motion is the same, whatever be the hour of the day. This could not be if it required a greater pressure to change the velocity 10000 into 100001, than to change the velocity 1 into the velocity 2. Yet this is one of Leibnitz's great metaphysical arguments for proving that the force accumulated, and now inherent, in a moving body, is proportional to the square of its velocity. We beg that this may be kept in remembrance.

It must be granted, that what we have already said on the subject of impulsion may be called an explanation; for it deduces the phenomena from general and unquestionable principles, and from acknowledged laws of Nature. The only principle used is, that a moving force is indicated, characterized, and measured, by the motion which it produces. It is an acknowledged law of Nature, that pressures are moving forces; also, that moving forces appear in cases where we observe neither pressures nor impulsions, and which we call repulsions or evasive tendencies; that these are mutual and equal; and we have shewn, how a certain set of changes of motion result from them, and have stated distinctly the whole process; we shewed, that these phenomena are similar to those of common impulsion; and we then shewed in what manner the motion of a body gives occasion to the exertion of various moving forces, called elasticity, cohesion, &c., and that this exertion must produce motions similar to those produced by repulsions e distanti; and, lastly, we inferred, from the perfect sameness of those results with the actual phenomena of impulsion, that those corporeal forces are the immediate and only causes of the changes called impulsions, and commonly ascribed to a peculiar force inherent in a moving body.

From a collective view of the whole, we think it clear, that the opinion that impulsion is the sole cause of motion is unwarranted. We see that the phenomena of other attempts at impulsion are brought about by the immediate operation to explain of pressure; and we see numberless instances of pressure, in which we cannot find the smallest trace of impulsion. It is therefore a most violent and unwarranted opinion, which ascribes to repeated unperceived impulsions all those solicitations to motion by which, or in consequence of which, the motions of bodies are affected by distant bodies, or bear an evident relation to the situation and distance of other bodies; as in the examples of planetary deflection, terrestrial gravitation, magnetic and electrical deflexions, and the like. There is nothing in the phenomenon of the pressure of gravity that seems to make impulsion more necessary or more probable than in the pressure of elasticity, whether that of a spring or of an expansive fluid. The admission of an unperceiv- ved fluid to effect those impulsion is quite unwarranted, and the explanation is therefore unphilosophical, even although we should perceive intuitively that an atom in motion will put another into motion by hitting it. We apprehend that this cannot be affirmed with any clear perception of its truth.

On the whole, therefore, we must ascribe that contented acquiescence in the explanations of gravitation, and other attractions and repulsions, by means of impulse (if the acquiescence be not pretended), to the frequency and familiarity of impulsion, and perhaps to the personal share and interest we have in this mode of producing motion. We know that it is always objected that nothing is explained, when we say that A repels B, or that B avoids A; but we may say in return, that nothing is explained, when we say that A impels B by hitting it, or that B flies away from the stroke. Why should it not be allowed to use the term repelling power, when it is allowed to use the term impelling power, the force of impulse, inertia? All these terms only express phenomena. Does the word body express any more?

The maxim, that a body cannot act where it is not, any more than when it is not, is a quaint and lively expression, and therefore has considerable effect: It may be granted; for we apprehend that we understand too little about when and where, that we cannot demonstrate the affirmative or negative in either case, and that they are on a par with respect to our knowledge of them. We can have no doubt, however, of the fact, that our mind can be affected by an external object that is merely recollected. And we apprehend, that we know nothing of the difference between body and mind but what we have learned by experience. Body, for any thing that we affirmed know to the contrary, may affect, or be affected by, a distant body, as well as mind may be. It is therefore worth while to pay some farther attention to the phenomena, in order to see whether this experience is so universal and unexpected as is believed. As Mr Cotes, and many of Newton's disciples, are accused of explaining many phenomena by attraction and repulsion which their opponents affirm to be cases of impulsion; it is not impossible but that ordinary observers, who have no preconceived theories, may imagine impulsion to obtain in cases where a more accurate inspection would convince them that no impulsion has happened.

When we kick away a foot-ball, we consider it as a sort of solid continuous body; yet we know that it must be filled with compressed air. It may not be impossible to have it of its round shape without being so filled: but we know that, in this condition, it would not fly away from our foot by the stroke; we should only force in the side which we kick, and the flaccid skin would lie at our feet. But when it is filled with strongly compressed air, we can form to ourselves a pretty distinct notion how it is made to move off. Our foot presses on a part of the skin: this compresses the air against the anterior part of the bag, and forces it away. If we reflect more seriously on the process, we can still conceive it clearly enough, by thinking on a row of aerial particles, reaching from the part struck by the foot to the anterior part, each touching the other, and therefore forcing the anterior part forward. The air is conceived to consist of a number of little spherules in contact, each of which is compressible; and we think the operation illustrated by supposing each to be like a little vehicle or bladder. This we believe to be the usual way of conceiving the constitution of expansive fluids: But this will not agree at all with the known properties of air; for it can be strictly demonstrated, that if such a collection of elastic vehicles be compressed into the half of their ordinary bulk, every vehicle will be changed from a sphere into a perfect cube, touching the adjoining cubes in every point of its six sides, and strongly pressed against them. It can also be demonstrated, that if a leaden cube of one inch be included in the box, and placed with its sides parallel to the sides of the box, and the compression be then made, all the little cubic vehicles will acquire the same position. If the box be now turned upside-down, it can be demonstrated that the weight of this leaden cube will not be sufficient for overcoming the resistance of the compressed cubes. This compressed fluid will not be fluid, but will require a very considerable force to press the leaden cube through it, just as we find such a force necessary for moving a body through melted glass: the particles no longer slide on each other like uncompressed spherules; each will require about half of the compressing force, in order to overcome the friction, or obstruction like friction, produced in sliding along the surface of the contiguous cubes. But we know that air remains perfectly fluid, although vastly more compressed than this. This, therefore, cannot be like the constitution or form of air. Moreover, it is well known that air has been made ten times denser than its ordinary state, and is then perfectly fluid. It has also been made a hundred times rarer, and it still remains perfectly fluid. In this state its particles must be ten times farther removed from each other than in the former state, of a thousand times greater density. Yet we know that this rare air is compressed with a force equal to the weight of a stratum of mercury $\frac{1}{4}$d of an inch in thickness, and that if $\frac{1}{4}$d of this pressure be removed, it will expand till it is 150 times rarer than common air; that is, there is some force which pushes the particles still farther from each other. This force evidently extends beyond the tenth particle of air that is made ten times denser than common air. Therefore the elasticity of air does not arise from the contact of particles, which are elastic like blown up bladders, but from some force which extends beyond the adjoining particles. There is no greater reason, therefore, for supposing, that the particles of air touch each other, than for supposing that the two magnets touch each other because they repel. A row of magnets floating on quicksilver, and placed with their similar poles facing each other, and very near, will tend to separate, and they require to be held in by a stop put at each end of the canal; and if one stop be gradually withdrawn, the magnets will all separate, and exhibit the general mechanical effects of a row of aerial particles separating by the removal of pressure. There seems, therefore, to be the same necessity for the operation of an intervening impelling fluid for producing this separation or elasticity of the aerial mass, as for separating the magnets.

The result of these remarks seems to be, that the very impulsion of a foot-ball is not brought about in the way that is commonly imagined, by the excitement of corporeal Inspiration corporeal pressure at the points of contact of the two foot-balls. For we see it almost demonstrated, that the progressive motion of the anterior part of one of the balls has been produced without contact, or, at least, by the intervention of repulsions acting at a distance. May not this obtain, even in the points in which we suppose the two balls actually to touch, in the act of impulsion?

But farther—Every person has observed the brilliant dew drops lying on the leaves of plants. Every person acquainted with Newton's optical discoveries, must be convinced that the dew-drop is not in mathematical contact with the leaf; if it were, it could have no brilliancy. Most persons have observed the rain drops of a summer shower fall on the surface of water, and roll about for a few seconds, exhibiting the greatest brilliancy. They cannot, therefore, be in mathematical contact with the water. There must be a small distance between them, and therefore some force which keeps them sundered, and carries the weight, that is, counteracts the downward pressure of the rain drop.

We know that some insects with long legs can run about on the surface of water; and if we lift them carefully, and set them on glass, their feet do not wet it. Put a little spirit of wine into this water, and make it lukewarm, and the insect instantly sinks up to the belly, and cannot move about as before: Its feet will now wet a glass. A well-polished steel needle, even of considerable size, if perfectly clean and dry, will float on water without being wetted: It is observed to make a considerable depression on the surface of the water, just as a heavy bar of iron would make when laid on a featherbed—the needle displaces a quantity of water equal to itself in weight, yet does not touch it, for it is not wetted. If it be previously wetted, it will not displace any water, and will not float. There is something, therefore, which keeps the water at a distance from the feet of the insect, and from the needle, exerting a certain upward pressure on them. The pressure and the reaction are indeed very small; but they would produce a very sensible motion if continued sufficiently long in proper circumstances. Here would be a production of motion, which most persons would call an impulsion—yet there would be no stroke, no contact, and therefore no true impulsion.

We now beg the reader to attend minutely to Newton's famous experiment with the object glasses of long telescopes, which we have mentioned circumstantially in the article Optics, Encycl. n° 63—68.

When the upper glass is very thin and light, no colour appears at the point of contact: but by pressing it down with sufficient force, we shall have a black or unreflecting spot in the middle, surrounded by a very ring, and then by a series of rings of various colours, according to the distance between the parts of the glasses where the colours appear. Newton has counted 50 of these rings. He shews, by a careful computation from the known figure of the glasses, that the differences between the distances which exhibit these colours are all precisely equal, and that each is about \( \frac{1}{20} \) of an inch. Therefore, supposing that the glasses are in mathematical contact where the unreflecting spot appears, making one continuous mass of glass, their distance at the outermost ring must not be less than \( \frac{1}{20} \) of an inch, or \( \frac{1}{20} \) of an inch. Therefore, when one glass carries the other, without any appearance of colour at the middle, we must conclude that there is a repulsion exerted between the nearest parts, at a distance not less than \( \frac{1}{20} \) of an inch, sufficient for supporting the upper glass. It requires an increase of pressure to produce the first appearance of colour; and when the pressure is still more increased, new colours appear in the middle, and the colour formerly there is now seen in a surrounding ring; these multiply continually, by new ones spreading from a central spot. A great pressure at last produces the unreflecting spot in the centre, which, unlike to all the coloured spots which had emerged in succession, is sharply defined, and never round, but ragged, and it is immediately surrounded by a bright silvery reflection. The shape of this spot depends on the figure of the surfaces; for, on turning the upper lens a little round its axis, the inequalities of the edge of the spot turn, in some degree, with it. This seemingly trifling remark will be found important by the mechanician: A still farther increase of pressure enlarges the unreflecting spot, and the dimensions of all the rings—When the pressure is gradually withdrawn, the rings shrink in their dimensions, the unreflecting spot disappears first, and each ring in succession contracts into a spot, and vanishes. Here we have, by the way, an explanation of the brilliancy of dew-drops: they come so near, perhaps, that the nearest point reflects the silvery appearance—but they do not touch; the instant that they touch a wetted part, making one mass of transparent matter, all brilliancy is gone.

Here then are incontestible proofs of a force, be its origin what it may, which keeps the glasses asunder, and even causes them to separate; which manifests itself by withstanding pressure; and therefore is, itself, a pressure, or equivalent to a pressure—it varies in its intensity by a change of distance; but we have not been able to ascertain by what law. It must not be measured by the simple variation of the external pressure; for since we see that, even before any colour appears in the centre, the weight of the upper lens is supported, we must conclude that the glasses are exerting at least an equal force all around the circumference of the outermost ring. It is evident, that the computation of the whole force, exerted over all the coloured surface, must be difficult, even on the simplest hypothesis concerning the law of repulsion: we can only say that it increases by a diminution of distance. It is very easy to compute the increase of external pressure, which would suffice if the repelling force were equal at all distances; or if it varied according to any single power of the distances. We have tried the inverse simple, duplicate, and triplicate ratio; but the fact deviated widely from them all. The repulsion does not change nearly so much as in the simple inverse ratio of the distances, if the glasses be supposed to touch in the whole surface of the unreflecting spot. But we found, that if we suppose them separated, though at a distance equal to forty times the difference of distance at which the colours change, that is, \( \frac{1}{20} \) of an inch, the pressures employed in the experiment accord pretty well with a repulsion inversely as the distance, but still with a very considerable deviation in the great pressures. In the course of a number of experiments with a favourite pair of lenses, we broke the uppermost by too strong Now, what is the consequence of all this in the doctrine of impulsion? Surely this:—If a lump of this glass strike another lump, and put it in motion, and if the mutual pressure in the act of collision do not exceed 700 pounds on the square inch, the motion has been produced without mathematical contact, and the production can no more be called impulse than the motion of the magnet in our first experiment. The changes of motion have been the operation of moving forces, similar to the force of magnetism; and if a stream of truly impelling fluid be necessary for producing the motion of the magnet, it is equally necessary for producing the motion of the piece of glass.

It may be said here, that we cannot compare impulse and pressure. A slight blow will split a diamond which could support a house. A slight blow may therefore be enough for exciting all the pressure necessary for producing mathematical contact. We must here appeal to what every man feels on this occasion. We doubt exceedingly whether any person will think that, when one piece of glass gives another a gentle blow, and puts it into motion, with the velocity of a few inches per second, a blow which he distinctly hears, there has been exerted a pressure at all approaching to 800 pounds per square inch.—We have suspended a pair of lenses, by an apparatus so steady and firm, that they could touch only at the centres of each surface; and, having placed ourselves properly, we could see, with sufficient distinctness, the momentary appearance of the coloured spot at the instant of collision. We saw this, with the fullest confidence that it was of no considerable breadth in moderate stroke, and that it was very sensibly broader when the stroke was more violent. We did not trust our own eye alone, but viewed it to persons ignorant of philosophy, and even to children, often without telling them what to look for, but asking them what they saw. From all the information that we could gather, none of the pressures came near to what must have been necessary for producing the black spot. This could not be mistaken: for although the outer rings are but faint, there are five or six near the centre which are abundantly vivid for affecting the eye by the momentary flash. Besides, the dimensions of the lenses, and the weight of the metal cells in which they were fixed, were such as must have caused them to split before the black spot could be produced in the centre.

These things being maturely considered, we imagine that few persons will now doubt the justice of our assertion, that in all these examples, the motions have been produced without mathematical (or rather geometrical) contact.—And we imagine also, that few will refuse granting that this is not peculiar to glass, but obtains also in the collision of other bodies. We have not thought of any method for putting this beyond doubt; but we have better reasons than mere likelihood for being of this opinion. Every one acquainted with the Newtonian discoveries in optics, knows that this curious appearance of the coloured rings is the consequence of the action of transparent bodies on the rays of light, by which they are bent aside from their rectilineal course, and that this deflection takes place at a distance from the diaphanous body; a distance which the sagacity of the great philosopher has enabled us to measure. Now, it is known that metals and other opake bodies produce the very same deflections of the rays, bending them toward themselves at one distance, and from them at other distances; in short, attracting or repelling them as the distance varies. Nothing but prepossession can hinder a person from ascribing similar effects to similar causes; and, therefore, thinking it almost certain, that this mutual repulsion is not peculiar to glass, but common to all solid bodies.

To all this we may surely add the celebrated experiment of Mr Huyghens; in which it is evident, that a smooth plate of metal attracts another, even although there be a silk fibre interposed between them. (See Phil. Trans. p. 86.) Is it not highly probable, that at a smaller distance the bodies repel each other? For we observe, that metals, as well as transparent bodies, attract the rays at one distance and repel them at another.

Surely our readers will now grant, that the production of motion by impulsion, as distinguished from the production by action e diffusio, is not to be familiar a phenomenon as was imagined, and that it may even be said to be rare in comparison: for the instances of moderate impulses are numberless. The claim of this mode of explaining difficult phenomena by impulsion, has therefore lost much of its force; and we see much less reason for calling in the aid of invisible fluids, in order to explain the action of gravity, magnetism, and electricity.

But we have still more important information from the optical discovery of Newton. Let the reader turn back again to Optics, Encycl. p. 65, and read the account of the phenomena exhibited by the soap-bubble. The bubble is thinner and thinner as we approach the very uppermost point of it. It also exhibits luminous rings, which vary in their colour, in the same order as in the space between the lenses. These rings come to view in the same manner. First, a coloured spot appears in the summit of the bubble; this becomes a ring, and is succeeded by another spot, as the bubble grows thinner in that part, by the gradual subsiding of the watery film. At last a black spot appears at top, well defined, but of irregular shape, surrounded by a silvery ring. This spot, when viewed very narrowly, is observed to reflect a very minute portion of light, without separating the differently colorific rays of which it consists; but it contains them all, as may be proved by viewing it through a prism. After some little time, the bubble bursts.

Surely we must infer from this, that there is a certain thickness of the transparent plate which renders it unfit for the vivid reflection of light. Does it not legitimately follow from this, that the unreflecting spot between the lenses ceases to entitle us to say, that they are in contact in that place? All that we can conclude from its appearance is, that the distance still between the glasses is too small to fit the place for the vivid reflection of light. This conclusion is indisputable. Were Impulsion, it refused, we are furnished with an incontestible proof by the same bountiful hand. Newton ascribed the colours to the reflection of the plate of air between the glasses, and expected the cessation of them when the air is removed. His friend Mr Boyle had lately invented a commodious air pump. The trial was made, and young Newton found himself mistaken; for the colours still appeared, and he even thought them more brilliant. He then tried the effect of water, expecting that this would diminish their lustre. So it did; and he found that the dimensions of the rings were diminished in the proportion of 4 to 3; namely, the proportion of the refractions of glass and water. By this time Newton had discovered the curious mechanical relation between bodies and the rays of light; and his mind was wholly absorbed by the discovery, and by the revolution he was about to make in the mathematical doctrines of optics. Unfortunately for us, he did not, at that time, attend to the mighty influence which the discovery would have on the whole of mechanical philosophy, and therefore occupied himself only with such phenomena as suited his present purpose. A most important phenomenon passed unnoticed. In repeating Sir Isaac Newton's experiments, we found that the diameters of the rings decreased in the proportion of 4 to 3 only in certain circumstances. When the upper lens was pressed on the other by a heavy metal ring, so as to produce three or four coloured rings, we found, that when water got between them, sometimes no colours whatever appeared; sometimes there was a ring or two, and the diameters were diminished in a much greater proportion than Newton had assigned. We were much puzzled with this discrepancy, and mentioned it to a most respectable and intelligent friend, the late Dr Reid of the university of Glasgow, a mathematician and naturalist of the first rank. He thought it impossible that the glasses separated from each other, lifting up the weight, by attracting the water into the interstice, in the same manner that we observe wood to swell with moisture. We immediately got an apparatus which compressed the glasses by means of four screws; and now we saw Newton's proportion most strictly observed. But in prosecuting the experiment, we found that the introduction of the water always effected a very small spot. This happened after precautions had been taken to prevent all separation of the glasses. As the proportion of 4 to 3 has a relation to refractive power, although we have not been able to deduce it as a necessary consequence, we nevertheless considered it as a sufficient proof, that the distance of the glasses had not changed by introducing the water between them. Therefore we think ourselves well entitled to conclude, that the disappearance of the black spot was not owing to a separation of the glasses, which admits the water into the empty space; and we affirm, that before the entry of the water, there was room for it in the place which reflected no light; that is, that although the glasses were pressed together with a very great force, they were not in contact.

Remark. It deserves remark, that in endeavouring to produce the black spot, when water is between the glasses, we found great and unaccountable anomalies. Sometimes a moderate increase of pressure produced it, and sometimes we were not able to produce it by any pressure.

Several lenses were broken in the trial. We are led to think that the thickness which gives the silvery reflection is much greater than the thousandth part of an inch, and that it is not the same in all glasses. But we were interrupted in these experiments, and indeed in all active pursuits, by bad health, which has never permitted their renewal. The subject is of great importance to the curious mechanician, and we earnestly recommend it to their attention. There is something very remarkable in the abrupt cessation of the coloured reflection. At a certain thickness all colours are reflected, without separation, producing the whiteness of silver. The smallest diminution of it hinders the void reflection of all colours, and then there seems to succeed a thickness which equally reflects a small proportion of all without separation. The finest polish that can be given to glass in the tool of the artist, leaves irregularities which occasion the irregular ragged figure of the spot. It is worth trying, whether smoothing the surfaces (both) by a softening heat will remove this ruggedness. If it does, without destroying the sharp termination, it will prove the abrupt passage from effe to non-effe.

The last remark to be made on this important experiment in optics is, that the distance between the glasses which is unfit for vivid reflection, cannot be determined by means of the other measurable intervals. It may be equal to many of them taken together. The same must be granted with respect to the thickness of the black spot on the top of the soap bubble. We attempted to measure this thickness by letting a drop (of a known weight) of spirit of turpentine spread on the surface of water. As it slowly enlarged in surface, it decreased in thickness, and produced, in regular order, several of the more compounded colours of the Newtonian series. But before it came to the 20th ring from the centre, it became very irregular and spotty.

The inference to be drawn from this combination of the two optical facts is remarkable and important. It is not proved, that we have no authority for affirming that the changes of motion by the collision of bodies is brought about by absolute contact in any instance whatever. The glasses are not in contact where there is vivid reflection; and we have no proof that they are in contact in the black spot, however great the compression may be.

It is hardly necessary now to say, that all attempts therefore to explain gravitation, or magnetism, or electricity, or impulsion any such apparent action at a distance by the impulsion of an unseen fluid, are futile in the greatest degree. Impulsion, by absolute contact, is far from being a familiar phenomenon, that it may justly be questioned whether we have ever observed a single instance of it. The supposition of an invisible impelling fluid is not more gratuitous than it is useless; because we have no proof that a particle of this fluid does or can come into contact with the body which we suppose impelled by it, and therefore it can give no explanation of an action that is apparently effulgent.

The general inference from the whole seems to be, that instead of explaining pressure by impulse, we must not only derive all impulse from pressure, but must also explain all pressure by action from a distance; that is, by constituting properties of matter by which its particles are moved sure, without geometrical contact.

This collection of facts conspires, with many appearances particles of fluid and solid bodies, to prove that even the particles of fluid, or feebly continuous bodies, are not in contact, but are held in their respective situations by the balance of forces which we are accustomed to call attractions and repulsions. The fluidity of water under very strong compressions (which have been known to compress it \( \frac{1}{3} \) of its bulk), is as inconsistent with the supposition of contact as the fluidity of air is. The shrinking of a body in all its dimensions by cold, nay, even the bending of any body, cannot be conceived without allowing that some of its ultimate unalterable atoms change their distances from each other. The phenomena of capillary attraction are also inexplicable, without admitting that particles act on others at a distance from them. The formation of water into drops, the coalescence of oil under water into spherical drops, or into circular spots when on the surface, show the same thing, and are inexplicable by mere adhesion. In short, all the appearances and mutual actions of tangible matter concur in shewing, that the atoms of matter are endowed with inherent forces, which cause them to approach or to avoid each other. The opinion of Boscovich seems to be well founded; namely, that at all sensible distances, the atoms of matter tend toward each other with forces inversely as the squares of the distances, and that, in the nearest approach, they avoid each other with insuperable force; and, in the intermediate distances, they approach or avoid each other with forces varying and alternating by every change of distance. See the article Boscovich, Suppl.

From all that has been said, we learn that physical or sensible contact differs from geometrical contact, in the same manner as physical solidity differs from that of the mathematician. Euclid speaks of cones and cylinders standing on the same base, and between the same parallels. These are not material solids, one of which would press the other out of its place. Physical contact is indicated, immediately and directly, by our sense of touch; that is, by exciting a pressure on our organ of touch when it is brought sufficiently near. It is also indicated by impulsion; which is the immediate effect of the pressure occasioned by a sufficient approximation of the body impelling to the body impelled. The impulsion is the completion of the same process that we described in the example of the magnets; but the extent of space and of time in which it is completed is so small that it escapes our observation, and we imagine it to be by contact and in an instant. We now see that it is similar to all other operations of accelerating or retarding forces, and that no change of velocity is instantaneous; but, as a body, in passing from one point of space to another, passes through the intermediate spaces; so, in changing from one velocity to another, it passes through all the intermediate degrees without the smallest fall.

And, in this way, is the whole doctrine of impulsion avoided within the pale of dynamics, without the admission of any new principle of motion. It is merely the application of the general doctrines of dynamics to cases where every accelerating or retarding force is opposed by another that is equal and contrary. We have found, that the opinion, that there is inherent in a moving body a peculiar force, by which it perseveres in motion, and puts another in motion by shifting into it, is as useless as it is inconsistent with our notions of motion and of moving forces. The impelled body is moved by the insuperable repulsion exerted by all atoms of matter when brought sufficiently near. The retardation of the impelling body does not arise from an inertia, or resisting sluggishness of the body impelled, but because this body also repels anything that is brought sufficiently near to it. We can have no doubt of the existence of such causes of motion. Springs, expansive fluids, cohering fibres, exhibit such active powers, without our being able to give them any other origin than the fiat of the Almighty, or to comprehend, in any manner whatever, how they reside in the material atom. But once we admit their existence and agency, every thing else is deduced in the most simple manner imaginable, without involving us in any thing incomprehensible, or having any consequence that is inconsistent with the appearances. Whereas both of these obstructions to knowledge come in our way, when we suppose any thing analogous to force inherent in a moving body solely because it is in motion. It forces us to use the unmeaning language of force and motion passing out of one body into another; and to speak of force and velocity as things capable of division and actual separation into parts. The force of inertia is one of the bitter fruits of this misconception of things. It is amusing to see how metaphysicians of eminence, such as D'Alembert, endeavour to make its operations tally with acknowledged principles. In his celebrated work on dynamics, the most elaborate of all his performances, he explains how a body, whose mass is 1, moving with the velocity 2, must stop another body whose mass is 2, moving with the velocity 1, in the following manner: He supposes the velocity 2 to consist of two parts, and that, in the instant of collision, one of these parts destroys the motion of one half of the other body, and then the other part destroys the motion of the other half. These are words; but in vain shall we attempt to accompany them by clear conceptions. His distinction between the force of inertia and what he calls the active forces of bodies, such as the force of bodies which strike each other in opposite directions, is equally undefeasible of clear conceptions. Active forces (says he) absorb a part of the motion; but when inertia takes part of the motion from the striking body, this motion passes wholly into the body that is struck, none of it being absorbed or really destroyed. He demonstrates this by the equation \( A \times x = B \times y - t \), which is a mere narration of facts, but no deduction from the nature of inertia, nor even any establishment of that nature by philosophical argument. And in attempting to give still clearer notions (being sensible that some great obscurity still hangs about it), he says, "Inertia therefore, and properly speaking, is the mean of communicating motion from one body to another. Every body resists motion; and it is by resisting that it receives it; and it receives precisely as much as it destroys in the body which acts on it." Surely almost every word of this sentence is doing violence to the common use of language. What can be more incomprehensible than that a body resists motion only when it receives it? Should a man be thought to resist being pushed out of his place when he actually allows another to displace him, and not to resist when he firmly keeps his place? All these difficulties and puzzling questions vanish when we give over speaking of inertia as something distinguishable from the active forces or causes of motion which we find in bodies, and distinguishing by the names of elasticity, cohesion, magnetism, electricity, weight, &c., and which philosophers have clasped under one name, accelerating or retarding force, according as its direction chances to be the same, or the opposite to that of the motion under consideration. To suppose it a peculiar faculty by which a body maintains its condition of motion or rest, is contrary to every conception that we can annex to the words faculty, power, force. It is frivolous in the extreme to say, that snow has the faculty of continuing white or cold; or that it resists being melted because it melts, or because heat must be employed to melt it.

The only argument that we know for giving the name force to the perseverance of matter in its state of motion (or rather for ascribing this perseverance to the exertion of a peculiar faculty), which appears to deserve any attention, is one that we do not recollect the express employment of for this purpose, namely, the composition of a previous motion with the motion which known force would produce in the body at rest. We know, that if a body be moving eastward at the rate of four feet per second, and a force act on it which would impel it from a state of rest at the rate of three feet per second to the south, the body will move at the rate of five feet per second in the direction E. 36° 52' S. We know also, that if a force act on this body at rest, so as to give it a motion eastward at the rate of four feet per second, and if another force act on it at the same instant, so as to give it a motion to the south at the rate of three feet per second, the body will move at the rate of five feet per second in the direction E. 36° 52' S. In this instance, the body previously in motion seems to possess something equivalent to what is allowed to be a moving force. Why therefore refuse it the name? The answer is easy. The term force has been applied, by all parties, to whatever produces a change of motion, and is measured by the change which accompanies its exertion. There is some difference between the parties about the way of estimating this measure; but all agree in making, not the motion, but the change of motion, the basis of the measurement. Now we shewed, at great length, in the article Dynamics, that the change of motion, in every case, is that motion which, when compounded with the former motion, constitutes the new motion. Did we take the new motion itself as the characteristic and measure of the changing force, it would be different in every different previous state of the body, and would neither agree with our general notion of force, nor with the knowledge that we have of the actual pressures and other moving forces that we know. The sole reason why the previous motion is equivalent with a force is, that the only mark or knowledge that we have of a moving force is the motion which it is conceived to produce. The force is equivalent with the previous motion, because we know nothing of it but that motion; and the name that we give it, only marks some external thing to which it has an observed relation. We call it magnetism or electricity, because we observe that a magnet or an electrified body gives occasion to its appearance. We never observe the resistance of inertia, except in cases where we know, from other circumstances, that moving

forces inherent in bodies are really brought into action. Impulse. The inertia of the ball which has been moved by a stroke of another, is inferred from the diminution of that other's motion. But this is occasioned precisely in the same way as the diminution of the motion of the magnet A in the first example; an event which every unprepossessed person attributes to the repulsion of B in the opposite direction, and not to its inertia.

We trust that our readers are not displeased with this detail of the procedure of Nature in the phenomena of impulse. It has been prolix; because we apprehend, that the too synoptical manner in which the laws of collision have always been delivered, leaves the mind in great obscurity concerning the connection of the events. General facts have been taken for philosophical principles and elementary truths; whereas they were deductions from the sum total of our knowledge. They were very proper logical principles for a synthetic discussion; but their previous establishment as general facts was necessary. We have established the two most general facts from which the result of every collision may be deduced with the utmost ease. The first is, that in the instant of greatest compression, the common velocity is $\frac{Aa}{A+B}$; and we have shewn, that this is applicable to the collision of unelastic bodies. The second is, that the change in perfectly elastic bodies is double of the change in unelastic bodies. The conservatio momentumum, and the conservatio virium vivorum, are also general facts; or rather they are the same mentioned with those above, considered in another aspect. They may all be used as the principles of a synthetic treatise of impulse; and they have been so employed. Each has its own advantages.

Mr Maupertuis gives a treatise on the Communication of Motion, that is, of impulse or collision, which has the appearance of being deduced from a new principle, which he calls the Principle of Smallest Action. He supposes, that perfect wisdom will accomplish everything by the smallest expenditure of action; and he chanced to observe, in the equations employed in the common doctrine of impulse, a quantity which is always a minimum. He chooses to consider this as the expression of the action.

His principle or axiom, deduced from the perfect wisdom of God, is thus expressed: "When any change happens in nature, the quantity of action necessary for it is the smallest possible." And then he adds,

"In mechanical changes, the quantity of action is the product of the quantity of matter in the body by the space passed over, and by the velocity of the motion." This is evidently the measure adopted long before by Leibnitz (see Phil. Transf. vol. xliii. p. 423, &c.), and it is equivalent to $mv^2$; because the space multiplied by the velocity is as the square of either. We refer to Dr Jurin's remarks on this passage for proof that this is by no means a just measure of action; and only observe here, that we can form no other notion of velocity than that of a certain space described in a given time. The change produced is not the actual description of a line, but the determination to that motion. It is in this respect alone that the condition of the body is changed; and therefore the product $mv$, and not $mv^2$, is the proper measure of the action. On the authority of this maxim of divine conduct, Maupertuis investigates gates the results which will make this quantity a minimum, and affects that these may be the laws of collision. Luckily this investigation is extremely simple, and very neat and perspicuous; and it gives very easy solutions. For example, the unelastic body A, moving with the velocity \(a\), overtakes the elastic body B, moving with the velocity \(b\). Both move after the collision with the velocity \(x\). This velocity is required.

To determine this, we must make \(A \times x - n^2 + B \times x - b^2\) a minimum; or \(A a^2 - 2 A a x + A x^2 + B x^2 - 2 B b x + B b^2\) is a minimum. Therefore \(-2 A a x + 2 A x^2 + 2 B x^2 - 2 B b x = 0\), or \(2 A a + 2 B b = 2 A x + 2 B x\), and \(x = \frac{A a + B b}{A + B}\); as we have already shown it to be.

The amiable and worthy author grew more fond of his theory, when he saw what he imagined to be its influence extended to an immense variety of the operations of nature. Euler demonstrated, that the quantity called action by Maupertuis was a minimum in the planetary motions, and indeed in all curvilinear motions in free space. But all the while, this principle of least action is a mere whim, and the formula which is so generally found a minimum has no perceptible connection with the quantity of action. In many cases to which Maupertuis has applied it, the conclusions are in direct opposition to any notion that we can form of the economy of action. Nay, it is very disputable whether it does not, on the contrary, express the greatest want of economy; namely, a minimum of effect from a given expenditure of power. In the case of impulsion, this minimum is the mathematical result of the equality and opposition of action and reaction. Maupertuis might have pleased his fancy by saying, that it became the infinite wisdom of God to make every primary atom of matter alike; and this would have answered all his purpose.

There still remains to be considered a very material circumstance in the doctrine of impulsion, which produces certain modifications of the motions that are of mighty practical importance. We have contented ourselves with merely stating the moving force that is brought into action in the points of physical contact; but have not explained how this produces the progressive motion of every particle of the impelled body, and what motion it really does produce in the remote particles. A body, besides the general progressive motion which it receives from the blow, is commonly observed to acquire also a motion of rotation, by which it whirls round an axis. It has not been shown, that when a body has received an impulse by a blow in a particular direction on one point, it will proceed in that direction, or in what direction it will proceed. Experience shows us, that this depends on circumstances not yet considered. The billiard player knows, that by a stroke in one direction he can make his antagonist's ball move in a direction extremely different.

There are questions of great intricacy and difficulty, and would employ volumes to treat them properly. We have already enlarged this article till we fear that we have exhausted the reader's patience, and deviated from the proportion of room justly allowable to impulsion. We must therefore limit our attention to such things only as seem elementary, and indispensably necessary for a useful application of the doctrine of impulsion.

With respect to the direction of the motion produced by impulsion, the very example just now borrowed from billiard playing, shows that it is important, and by no means obvious. We are sorry to say, that we have nothing to offer in solution of this question that will be received by all as demonstration. It is comprehended in the following proposition, which we bring forward merely as a matter of fact.

The direction of the stroke or pressure exerted by two bodies in physical contact, is always perpendicular to the touching surfaces. Of this truth we have a very clear and distinct and pretty example and proof by the billiard table. If two balls A and B (fig. 2.) are laid on the touching table in contact, and A is smartly struck by a third ball surface, C in any direction Cc, so that the line aA, which joins the point of contact a with the centre A, may make an obtuse angle with the line AB, joining the centres of the two balls, the ball B will always fly off in the direction ABF. The pressure on B, which produces the impulsion, is evidently exerted at the point b of contact, and the direction BF is perpendicular to the plane GbH, touching both balls in the point b. The primary stroke is at a, and acts in the direction aA, although C moved in the direction Cc. Had A been alone, it would have gone off in the direction aA produced. But the force acting in the direction aA is equivalent to the two forces dA and dA, of which dA presses the ball on B at b, and produces the motion. In like manner, another ball E, so laid that bB is obtuse, will fly off in the direction ED, which may even be opposite to Cc. These are matters of fact; not indeed precisely so, because billiard balls are not perfectly elastic, retorting their figure with a promptitude equal to that of their compression; and also because there is a little friction, by which the point a of the ball A is dragged a little in the direction of C's motion. This may both give a twist to A, and diminish its pressure on B. The general result, however, is abundantly agreeable to the doctrines now delivered. But we wish to show on what properties of tangible matter this depends; and although we dare not hope for implicit belief, we expect some credit in what we shall offer.

We have evident proofs, that at a distance which is not measurable by its minuteness, and certainly far exceeds the 96th part of an inch, bodies repel each other with very great force. This distance also far exceeds the distance between the particles, if these are discrete. Let \(m\) (fig. 3.) be the distance at which a particle repels another, and let P be a particle situated at a less distance than \(m\) from the surface AC of a solid body. With a radius PA, equal to \(m\), describe a segment of a sphere ABC, and draw PB perpendicular to AC. It is plain, that every particle of matter in the segment ABC repels the particle P, and that it is not affected by any more. Let D be any such particle. It repels P in the direction DP. But there is another particle d similarly situated on the other side of PB. This will repel P with equal force in the direction dP. Therefore the two particles D and d will produce a joint repulsion in the direction BP. The like may be said of every particle and its corresponding one on the other side of PB. Therefore the joint repulsion impulsion, pulsion of all the matter in the segment will have the direction BP. It is plain, that the radius of curvature of every sensible figure may be considered as immensely great in comparison of m n; and therefore the proposition is manifest.

This is a proposition of very great importance to the artist and the engineer, as well as to the philosopher. In all the connections of engines and machines, the mutual action is regulated by this fact. The mutual pressure at the contacts of the teeth of wheels and pinions depend so much on it, that it is easy to make them of such a shape that they shall produce no force whatever that is of any service; and it requires a skilled attention to their forms to obtain the service we want. This will be considered with some care in the article MACHINE.

Having thus discovered the direction of the real impulsion, and that it may be very different from that of the force exerted, we proceed to consider what will be the direction and velocity of the motion, and whether it will be accompanied with any rotation.

Our readers are acquainted with the elementary mechanical property of the centre of gravity. If a body be supported at this point by a force acting vertically upwards, and equal to the united weight of every particle of matter in it, it will not only remain at rest, but will have no tendency to incline to either side; that is, the upward force balances the weight of the whole body, and the mechanical momenta of all the heavy particles balance each other, like the weights in the scales of a steelyard. That this may be the case, we know that if the weight of every particle be multiplied into the horizontal lever by which it hangs (which is a line drawn from the particle perpendicular to a vertical plane passing through the centre of gravity), the sum of all the products on one side must be equal to the sum of all the products on the other side. Therefore, if we suppose the particles all equal, and represent each by unity, the sums of all the perpendiculars themselves must be equal. How is this balancing effected? Every particle tends downwards with a certain force. It must therefore be kept up by a force precisely equal and opposite. This must be propagated to the particle by means of the connecting corporeal forces. The force propagated to any particle is equal and opposite to the force acting on that particle, which it balanced; and if not balanced, it would produce a motion equal and opposite to that produced by the other force. Gravity would cause every particle to descend equally; therefore the force which, by acting on one point, excites those balancing forces on each particle, would cause them to move equally upwards. And since this is true in any attitude of the body, it follows, that a force, acting in any direction through the centre of gravity, will cause all the particles to move in that direction equally; that is, without rotation.

Hence we learn, that when the direction of the stroke given to any body passes through the centre of gravity, the body will move in that direction without any rotation. If the quantity of matter, or number of equal particles in the body, be m, the moving power P will impress on each particle an accelerating force f, equal to the nth part of P. Therefore \( f = \frac{P}{m} \), and \( P = mf \).

An accelerating force is estimated by the velocity v, which it generates by acting uniformly during some impulsive time t, or \( v = ft \), and \( f = \frac{v}{t} \), and \( P = \frac{mv}{t} \), and \( v = \frac{P}{m} t \). The symbol t may be omitted, if we reckon on every force by the velocity which it can produce in a second. Thus may all forces be compared with gravity, by taking 32 feet for the measure of gravity. Then \( mv \) will express the number of pounds which give a pressure equal to the force under consideration. Thus if the force can generate the velocity 48 feet per second in 100 pounds of matter, by acting on it uniformly during a second, its pressure is equal to the weight of 150 pounds.

When a body A, moving with the velocity a, overtakes or meets a body B, moving with the velocity b, and the line perpendicular to their touching surfaces passes through the centres of both in the direction of their motion, all the circumstances of the collision are determined by the rules already laid down. This is called DIRECT IMPULSE; and it is this which admits the application of the simple doctrines of impulsion, deduced, as we have done it, from the action of accelerating forces. All that was said of the changes of motion produced in the magnets obtain here without any farther modification.

We may just be allowed to take notice of a curious observation of Mr Huyghens on the collision of perfectly elastic bodies. Instead of impelling the elastic ball C by the stroke of the elastic ball A, we may cause A to strike an intermediate ball B (also perfectly elastic), which is lying in contact with C. In many cases, the ball B will not fly sensibly from its place, and C alone will fly off. Nay, if a long row of equal billiard balls lie in contact, and one of the extreme balls be hit by another ball in the direction of the row, only the remote ball of the row will fly off. All this is easily seen and understood, by considering them as bodies mutually repelling, and placed at the limits of their mutual action. Or even supposing them elastic balls, at a very small distance from each other: The ball employed to strike the first comes to rest, and the struck ball moves off with its velocity: It strikes the second ball of the row, and is brought to rest: The second strikes the third, and is brought to rest: And this goes on in succession to the last, which is the only one that can fly off. The curious observation of Mr Huyghens is, that a greater velocity will be communicated by a large ball to a small one, if we employ the intermediate of another ball of a size between the two; and that the velocity will be the greatest possible when the intermediate ball is a mean proportional (geometrical) between the two. This is also easily deduced from the similar attention to the action of the accelerating forces, or from the supposition of successive impulses. From this it also follows, that a greater velocity will be produced by the intervention of two, three, or more, mean proportionals.

But the direction of the stroke may not be the same with that of the motion. This is called OBLIQUE IMPULSION. The cases of oblique collisions are extremely different, according to the directions of the motions; and the results are, in many of them, far from being obvious. But we have not room for a particular treatment of them. We shall therefore avail ourselves of some some point C of the line PG, and will also be perpendicular to the same plane. All this has been demonstrated in the article Rotation, no. 94, &c. Complete the parallelogram AFHE. It is plain, that the motion AH is equivalent to AE and AF. By the motion AE, A only slides along the surface of B without pressing it, or causing any tendency to motion in that direction, except perhaps a little arising from friction. It is by the motion AF alone that the impulse is made.

Therefore let \( \phi = V \times \frac{AF}{AH} \); and then \( A \times v \) may be called the effective impulse of the body A in the present circumstances, and \( v \) the effective velocity. This will be diminished by the collision. Let \( x \) be the unknown velocity remaining in A after the collision, or rather in the instant of the greatest compression and common motion of the touching points of A and B, estimated in the direction FP. The effective momentum lost by A must therefore be \( A \times v - x \): but the same must be gained by B, and its centre G must move in the direction GI, parallel to FP, with this momentum; and therefore with the velocity \( \frac{A \times v - x}{B} \). That this may be the case, the point of percussion F must yield with the velocity \( x \), because the bodies are in contact. But because C is the spontaneous axis of conversion, every particle is beginning to describe an arch of a circle round this axis. Therefore F is beginning to move in the direction FG, perpendicular to the momentary radius vector CF. Let FG be a very minute arch, described in a moment of time. Draw \( g \perp \) perpendicular to FP. Then \( f \perp \) is the motion FG reduced to the direction FP, and will express the yielding of B in the direction of the impulse, while G describes a space equal to \( \frac{A \times v - x}{B} \), and A describes a space \( x \). Therefore FG will express \( x \). Let \( P \perp \) be the space described in the same time that FG is described. Draw \( pC \), cutting GR in the point I. GI is the yielding of the body B to the impulse, and must therefore be equal to \( \frac{A \times v - x}{B} \).

The triangles \( Ff \perp \) and CPF are similar; for the angle CFP is the complement of \( f \perp \) to a right angle: It is also the complement of PCF to a right angle. Therefore \( Fg : f \perp = FC : CP \). But \( Fg : P \perp = FC : P \perp \); because the little arches \( Fg, P \perp \) have the same angle at C. Therefore \( P \perp = f \perp = x \). It is plain, that \( CG : CP = GI : P \perp \). Therefore \( CG : CP = \frac{A \times v - x}{B} \), and \( x = \frac{A \times v - x}{B \times CG} \), or \( x = \frac{A \times CP}{B \times CG} \). Wherefore \( x \times B \times CG + x \times A \times CP = v \times A \times CP \), and \( x \times B \times CG + A \times CP = o \times A \times CP \), and \( x = \frac{A \times CP}{B \times CG} \). The velocity remaining in A, estimated in the direction FP.

And \( v \), the velocity with which G will advance, is evident that A will change its direction by the collision: For in the instant of greatest compression, it was react-

Suppl. Vol. I. Part II. ed on by a force \( A \times v - x \) in the direction FA. This must be compounded with \( A \times V_1 \) in the direction AH, in order to obtain the new motion of A; or it may be found by compounding \( x \), which is retained by A, with FH, which has suffered no change by the collision. The bodies will therefore separate, although they be unelastic: If they are elastic, we must double their changes on each. If B was also in motion before the collision, the motion of A must be resolved into two, one of which is equal and parallel to the motion of B: the other must be employed as we have employed the motion AH.

Expressions still more general may be obtained for \( x \) and \( u \); namely, by taking the formulae for the centres of conversion and percussion (Rotation, n° 96, 99):

\[ CG = \frac{\int p r^2}{B \times GP}, \quad CP = \frac{\int p r^2 + B \times CP}{B \times GP} \]

where \( p \) stands for a particle of matter, and \( r \) for its distance from an axis passing through G perpendicular to the plane of the lines GP and PF. In this way

\[ A \cdot \int p r^2 + A \cdot B \cdot GP^2 \]

we obtain \( x = 0 \).

It is plain from this proposition, that the progressive motion of the body depends, not only on the momentum of the impelling body, but also on the place where the other is struck: For even although the original momentum of A be the same, and the obliquity of the stroke, making \( v \) the same, and the body (and consequently \( \int p r^2 \)) also remain the same, we see that \( x \) and \( u \) depend on the ratio of CP to CG. Now C and P are always on opposite sides of G: Consequently, by removing the direction FP of the impulsion farther from G, we diminish CG and increase CP; and therefore increase the value of \( x = \frac{A \cdot CP}{B \cdot CG + A \cdot CP} \); and consequently diminish the value of \( A \times v - x \), to which \( B \times u \) is equal. The greatest momentum of B is produced when the direction of the impulse passes through G, and no rotation is produced. Indeed we are led, by a fort of common sense, to expect this.

This investigation is by no means a piece of mere speculative curiosity. It is the solution of the greatest problem in practical mechanics. It is in this way that we must proceed in computing the actions of the wind and water on the sails and hull of a ship. Were it not that many circumstances concur in determining several of the preparatory steps, it is evident that the task must be almost impracticable. But the pressure and its direction are generally determined by experiment, without the trouble of computation; and we are seldom solicitous about the subsequent motion of the wind or water.

There is another question in impulsion which is of the first practical importance—namely, when the impulse is exerted on the parts of a machine, where the body struck is not at liberty to yield freely to the stroke, but must slide along some solid path, or turn round some axis, or take some other constrained motion. The operations of most engines depend on this. The operation of wedges, axes, and many cutting and piercing instruments, and the penetration of piles, impelled by a rammer, are all ascertained by the same doctrines. But the particular applications can scarcely be elucidated by any classification that occurs to us, the circumstances of the case making such great difference in the result, both in kind and degree. For example, in the simplest case that occurs, the driving of piles, the penetration of the pile depends, in the first place, on the momentum of the rammer. If the mass of the pile be neglected, the penetration through a uniformly resisting substance will be as the square of the velocity of the rammer, (Dynamics, Suppl. n° 95), and its absolute quantity may be determined from a knowledge of the proportion of the weight of the rammer to the resistance of the earth. But when we consider that we have to put in motion the whole matter of the pile, we learn that a great diminution of the effect must take place. We still can compute what this must be, because we have the same momentum, with a velocity diminished in a certain proportion of the sum of the matter in the rammer and pile, to that in the rammer alone.—Another defalcation arises from friction, which continually increases as the pile goes deeper; and a still greater defalcation proceeds from the nature of the pile. If it is a piece of very dry straight grained fir, it is very elastic, and acquires almost a double velocity from the stroke of a rammer of cast iron. If it is moist and soft, especially if it is oak, or other timber of an undulated fibre, it does not acquire so great velocity, and the penetration is very much diminished. It is probable that a pile, headed with moil cork, could not be driven at all. The writer of this article found a remarkable effect of the elasticity in the process of boring limestone. When the boring bit was made entirely of steel, and tempered throughout its whole length to a hard spring temper, the workman bored three inches, in the same time that another bored two inches with a bit made of soft iron; and he would never use any but steel bits, if they could be hindered from chipping by the hammer (which must also be of tempered steel throughout). This has hitherto baffled many attempts. A pretty large round head, like a marlin spike, has succeeded best: but even this cracks after some days use. The improvement is richly worth attention; for the workman is delighted by feeling the hammer rise in his hand after every stroke, and says that the work is not so hard by half. N.B. The stone cutters at Lisbon and Oporto use iron mallets.

The case of impulsion made on part of a machine impellable round an axis has been considered in the article Rotation, Encycl. n° 72; where \( x \) is shown to be \( v \times \frac{A \cdot CP^2}{\int p r^2 + A \cdot CP^2} \). But, in this formula, \( r \) denotes the distance of \( p \) from the point C, and not from G. \( \int p r^2 \) in this formula, is B. CG. CP; whereas, in the formula for a free body, where \( r \) is the distance of a particle from G, \( \int p r^2 \) is B. CG. GP.

In the practical consideration of this question, the reader will do well to consider the whole of that article with attention. Many circumstances occur, which make a proper choice of the point of impulsion, and the direction of the tangent plane, of the greatest consequence to the good performance of the machine; and there there is nothing in which the scientific knowledge of the engineer is of more essential service to him. An engineer of great practice, and a sagacious combining mind, collects his general observations, and stores them up as rules of future practice. But it is seldom that he possesses them with that distinctness and confidence that can enable him to communicate his knowledge to others, or even secure himself against all mistakes; whereas a moderate acquaintance with these elements of real mechanics, may be applied with safety on all occasions, because arithmetical computations, when rightly made, afford the most certain of all results.

There is a circumstance which greatly affects the performance of machines which are actuated by impulses, namely, the yielding and bending of the parts. When the moving power acts by repeated small impulsions, it may sometimes be entirely consumed, without producing any effect whatever at the remote working point of the machine; and the engineer, who founds his constructions on the elementary theories to be had in most treatises of mechanics, will often be miserably disappointed. In the usual theories, even as delivered by writers of eminence, it is asserted, that the smallest impulse will start the greatest weight. But since impulse is only a continued pressure, and requires time for the transmission of its effect through the parts of a yielding solid, it is plain that the motion of the impelling body may be extinguished before it has produced compression enough for exciting the forces which are to raise the remote parts of a heavy body from the ground. What blow with a hammer could start a feather bed? Much oftener may we expect, that a blow, given to one arm of a long lever, will be consumed in bending the whole of its length, so as to bring the remote end into action. Therefore great stiffness, and perfect elasticity, both in the moving parts and in the points of support, are necessary for transmitting the full, or even a considerable part, of the power of the impelling body. Perhaps not the half of the blow given by the wipers of a great forge or tilting mill to the trunk of the hammer is transmitted in the proper instant of time to the hammer-head. The hammer, while it is tossed up by the blow, is quivering as it flies. Should it reach the spring above it in the time of its downward vibration, it will not be returned with such force as if it had hit the spring a moment before or after. A quarter of an inch will produce a great effect in such cases. It is found, that the minute impulses given to the pallets of a clock or watch lose much of their force by the imperfect elasticity of the pendulum or balance: We must therefore make all the parts which transmit the blow to the regulating mass of matter as continuous, hard, elastic, and stiff, as possible. The performance of ruby pallets is very sensibly weakened by putting oil on the face of them, especially in the detached escapements, which act partly by impulse. A wheel of hard tempered steel, working on a dry ruby pallet, excels all others. The intelligent engineer, seeing that, after all his care, much impulsion is unavoidably lost, will avoid employing a first mover which acts in a subtilty manner, and will substitute one of continued pressure when it is in his power. This is one chief cause of the great superiority of overshot water-wheels above the underthrust.

We can now understand how it happens that Galileo, Merfennus, and others, could compare the impulse given by a falling body with the pressure of a weight in the opposite scale of a balance, and can see the reason of the immense differences, yet accompanied by a sort of regularity, in the results of the experiments. Galileo, Merfennus, and Riccioli, found them to be proofs that the forces of moving bodies are as their velocities; because the heights from which the body fell were as the squares of the weights started from the ground. Gravetande found the same thing as long as he held the same opinion; but when he adopted the Leibnitzian measure, he found many faults in the apparatus employed in his former illustrations, and altered it, till he obtained results agreeable to his new creed. But any one who examines with attention all that passes in the bending of the apparatus, and takes into account the mass of matter which must be displaced before the opposite arm rises so far as to detach the spring which gives indication of the magnitude of the stroke, must see that the agreement is purely accidental, and may be procured for any theory we please (see Gravetande's Nat. Phil. translated by Desaguliers, vol. i. p. 241 &c.). The proposition, n° 95, Dynamics, suffices for explaining every thing that can happen in such experiments. And it will shew us, that although the motion of impulsion is produced by pressure alone, yet impulse is incomparable with mere pressure: It is not infinitely greater, but disparate. A weight (which is a pressure) bends a spring to a certain degree, and will derange to a certain degree the fibres of a body on which it presses, before it be balanced. The same weight, falling on this spring from the smallest height, will bend it farther, and may crush or shiver to pieces the body which would have carried it for ever. We shall make some further remarks on this subject, of great practical importance, under the word Percussion.

The method which we have pursued in considering the doctrines of impulsion, differs considerably from that which has generally been followed; but we trust that it will not be found less instructive. Although the reader should not adopt our decided opinion, that we have no proof of pure impulsion ever being observed, and that all the phenomena which go by that name are really the effects of pressures, analogous to gravity, he perceives that our opinion does not lead to any general laws of impulsion that are different from those which are acknowledged by all. We differ only, by exhibiting the internal procedure by which they are unquestionably produced in a vast number of cases, and which takes place in all that we have seen, in some degree. Our method has undoubtedly this advantage, that it requires no principle but one, namely, that accelerating forces are to be estimated by the acceleration which they produce. Even this may be considered, not as a principle, but merely as a definition.—We get rid of all the obscurity and perplexity that result from the introduction of inertia, considered as a power—a power of doing nothing—and we are freed from the unphilosophical fiction (adopted by all the abettors of that doctrine, and even by many others) of conceiving the space, in which motions are performed, and bodies act, to be carried along with the bodies in it.—This furnishes, indeed, in some cases, a familiar way of conceiving the thing, by supposing the experiments to be made in a ship under sail, and by appeal- As no part of mechanical philosophy has been so much debated about as impulsion, it will surely be agreeable to our readers to have a notice of the different treatises which have been published on the subject:

Galilei Opera, T. I. 957. II. 479, &c. Jo. Wallisii Tractatus de Percussione. Oxon. 1669. Chr. Hugenius de Motu Corporum ex Percussione. Op. II. 73. Traité de la Percussion des Corps, par Mariotte, Op. I. r. Hypothesis Physica Nova, qua phenomenon causae ab unico quodam universali motu in nostro globo supposito repetuntur. Auct. G. G. Leibnitio. Moguntiae: 1671.—Leibn. Op. T. II. p. II. 3. Ejusdem Theoria Motus Abstracti. Ibid. 35. Hermannii Phoronomia. Amst. 1716.

END OF THE FIRST VOLUME.

ERRATA.

Page 108. col. 1. Dele lines 24, 25, 26, 27, 28, 29; and in their place read, "which though Lord Auchinleck and his son took the same side, they took it with very different degrees of ardour. The judge saw not the propriety of illuminating his windows, when the cause was finally decided by the House of Peers; and to compel him to illuminate, the advocate got possession of a Chinese gong."

Page 186. col. 1. line 18. For "Henry VI." read "Henry II."

Page 253. col. 1. line 23. For sulphuret read sulphat.

Chemistry Index, page 399. col. 1. line 2. After 464 add, "and Part 3. chap. 2. sect. 11." Page 399. col. 1. line 38. For p. 624, read 359. — Page 400, col. 2. line 18. For p. 624, read p. 359.

Page 451. col. 1. line 34. For "Diderot," read "D'Alembert."

DIRECTIONS FOR PLACING THE PLATES.

| PART I. | PART II. | |---------|----------| | Plate I. | Plate XX. | | II. | 462 | | III. | XXI. | | IV. | XXII. | | V. | XXIII. | | VI. | XIV. | | VII. | XV. | | VIII. | XVI. | | IX. | XVII. | | X. | XVIII. |

Printed by John Brown, Anchor Close.

1799.