1. This name marks that department of physico-mathematical science which contains the abstract doctrine Definition of moving forces; that is, whatever necessarily results from the relations of our ideas of motion, and of the immediate causes of its production and changes.

2. All changes of motion are considered by us as the indications, the characteristics, and the measures of changing causes. This is a physical law of human thought, and therefore a principle to which we may refer, and from which we must derive all our knowledge of those causes. When we appeal to our own thoughts or feelings, we do not find in ourselves any disposition to refer mere existence to any cause, although the beginning of existence certainly produces this reference in an instant. Had we always observed the universe in motion, it does not appear that we should have ascribed it to a cause, till the observation of relative rest, or something leading to it, had enabled us to separate, by abstraction, the notion of matter from that of motion. We might then perceive that rest is not incompatible with matter; and we might even observe, by means of relative motions, that absolute rest might be produced by the concourse of equal and opposite motions. But all this requires reflection and reasoning, whereas we are now speaking of the first suggestions of our minds.

3. We cannot have any notion of motion in abstracto, without considering it as a state or condition of existence which would remain if not changed by some cause. It is from changes alone, therefore, that we infer any agency in nature; and it is in these that we are to find all that we know of their causes.

4. When we look around us, we cannot but observe that the motions of bodies have in most cases, if not always, some relation to the situation, the distance, and the discriminating qualities of other bodies. The motions of the moon have a palpable relation to the earth; the motions of the tides have as evident a relation to the moon; the motions of a piece of iron have a palpable dependence on a magnet. The vicinity of the one seems to be the occasion at least of the motions of the other. The causes of these motions have an evident connection with or dependence on the other body. We are even disposed to imagine that they are inherent in that body, and that it possesses certain qualities which are the causes of those modifications of motion in other bodies. These serve to distinguish some bodies from others, and may therefore be called properties; and since the condition of other bodies so evidently depends on them, these properties express very interesting relations of bodies, and are chiefly attended to in the enumeration of the circumstances which ascertain what we call the nature of any thing. We do not mean to say that these inferences are always just; nay, we know that many of them are ill founded; but they are real, and they serve abundantly for informing us what we may expect from any proposed situation of things. It is enough for us to know, that when a piece of iron is so and so situated in relation to a magnet, it will move in a certain manner.

This mutual relation of bodies is differently considered, according to the interest that we chance to take in the phenomenon. The cause of the approach of the iron to a magnet is generally ascribed to the magnet, which is said to attract the iron, because we commonly employ the magnet in order that these motions may take place. The similar approach of a stone to the earth is ascribed to the stone, and we say that it tends to the earth. In all probability the procedure of nature is the same in both; for they are observed in every instance to be mutual between the related bodies. As iron approaches a magnet, so the magnet approaches the iron. The same thing is observed in the motions of electrified bodies; also in the case of the stone and the earth. Therefore the cause of the motions may be conceived as inherent in either, or in both.

The qualities thus inherent in bodies constituting their mechanical relations, have been called the mechanical actions of matter. But they are more commonly named powers or forces; and the event which indicates their presence is considered as the effect and mark of their agency. The magnet is said to act on the iron, the earth is said to act on the stone; and the iron and the stone are said to act on the magnet and on the earth.

All this is figurative or metaphorical language. All languages have begun with social union, and have improved along with it. The first collections of words expressed the most familiar and the most interesting notions. In the process of social improvement, the number of words did not increase in the same proportion with the notions that became interesting and familiar in their turn: for it often happened that relations of certain ideas so much resembled the relations of certain other ideas, that the word expressing one of them served very well for expressing the other; because the dissimilar circumstances of the two cases prevented all chance of mistake. Thus we are said to surmount a difficulty, without attaching to the word the notion of getting over a steep hill. Languages are thus filled with figurative expressions.

5. Power, Force, and Action, are words which must have appeared in the language of the most simple people; because the notions of personal ability, strength, and exertion, are at once the most familiar and the most interesting that can have a place in the human mind. These terms, when used in their pure, primitive sense, express the notions of the power, force, and action of a sentient, active being. Such a being only is an agent. The exertion of his power or force is (exclusively) action; but the relation of cause and effect so much resembles in its results the relation between this force and the work performed, that the same term may be very intelligibly employed for both. Perhaps the only case of pure unfigurative action is that of the mind on the body. But as this is always with the design of producing some change on external bodies, we think only of them; the instrument or tool is overlooked, and we say that we act on the external body. Our real action therefore is but the first movement in a long train of successive events, and is but the remote cause of the interesting event. The resemblance to such actions is very strong indeed in many cases of mechanical phenomena. A man throws a ball by the motion of his arm. A spring impels a ball in the same manner by unbending. These two events resemble each other in every circumstance but the action of the mind on the corporeal organ; the rest of it is a train of pure mechanism. In general, because the ultimate results of the mutual influence of bodies on each other greatly resemble the ultimate results of our actions on bodies, we have not invented appropriated terms, but have contented ourselves with those already employed for expressing our own actions, the exertions of our own powers or forces. The relation of physical cause and effect is expressed metaphorically in the words which belong properly to the relation of agent and action. This has been attended by the usual consequences of poverty. Dynamics of language, namely, ambiguity, and sometimes mistake, both in our reflections (which are generally carried on by mental discourse), our reasonings, and our conclusions. It is necessary to be on our guard against such mistakes; for they frequently amount to the confounding of things totally different. Many philosophers of great reputation, on no better foundation than this metaphorical language, have confounded the relations of activity and of causation, and even denied that there is any difference; and they have affirmed that there is the same invariable relation between the determinations of the will and the inducements that prompt them, as there is between any physical power and its effect. Others have maintained that the first mover in the mechanical operations, and indeed through the whole train of any complicated event, is a perceptive and intending principle, in the same manner as in our actions. According to these philosophers, a particle of gravitating matter perceives its relation to every other particle in the universe, and determines its own motion according to fixed laws, in exact conformity to its situation. But the language, and even the actions of all men, show that they have a notion of the relation of an agent to the action, easily distinguishable (because all distinguish it) from the relation between the physical cause and its effect. This metaphorical language has affected the doctrines of mechanical philosophy, and has produced a dispute about some of its first principles; the only way to decide this dispute is to avoid most scrupulously all metaphorical language, though at the expense of much circumlocution.

6. When we speak of powers or forces as residing in a body, and the effect as produced by their exertion, the body, considered as possessing the power, is said to act on the other. A magnet is said to act on a piece of iron; a billiard ball in motion is said to act on one that is hit by it; but if we attempt to fix our attention on this action, as distinct both from the agent and the thing acted on, we find no object of contemplation; the exertion or procedure of nature in producing the effect does not come under our view. When we speak of the action as distinct from the agent, we find that it is not the action, properly speaking, but the act, that we speak of. In like manner the action of a mechanical power can be conceived only in the effect produced.

7. A man is not said to act unless he produces some effect. Thought is the act of the thinking principle; motion of the limb is the act of the mind on it. In mechanics also, there is action only in so far as there is mechanical effect produced. I must act violently in order to begin motion on a slide. I must exert force, and this force exerted produces motion. I conceive the production of motion in all cases as the exertion of force; but it requires no exertion to continue the motion along the slide; I am conscious of none; therefore I ought to infer that no force is necessary for the continuation of any motion. The continuation of motion is not the production of any new effect, but the permanency of an effect already produced. We indeed consider motion as the effect of an action; but there would be no effect if the body were not moving. Motion is not the action, but the effect of the action.

8. Mechanical actions have been usually classed under two heads. They are either pressures or impulsions. They are generally considered as of different kinds—the exertions of different powers. Pressure is supposed to differ essentially from impulse.

Instead of attempting to define, or describe, these two kinds of forces and actions, we shall just mention some instances. This will give us all the knowledge of their distinctions that we can acquire.

When a ball lies on a table, and I press it gently on one side, it moves toward the other side of the table. If I follow it with my finger, continuing my pressure, it accelerates Examples continually in its motion. In like manner, when I press of pression on the handle of a common kitchen jack, the fly begins to move. If I continue to urge or press round the handle, the fly accelerates continually, and may be brought into a state of very rapid motion. These motions are the effects of genuine pressure. The ball would be urged along the table in the same manner, and with a motion continually accelerated, by the unbending of a spring. Also, a spring coiled up round the axis of the handle of the jack would, by uncoiling itself, urge round the fly with a motion accelerating in the same way. The more I reflect on the pressure of my finger on the ball, and compare it with the effect of the spring on it, the more clearly do I see the perfect similarity; and I call these influences, exertions, or actions, by one name, pressure, taken from the most familiar instance of them.

Again, the very same motion may be produced in the ball or fly, by pulling the ball or the machine by means of a thread, to which a weight is suspended. As both are motions accelerated in the same manner, I call the influence or action of the thread on the ball or machine by the same name, pressure, and weight is considered as a pressing power. Indeed I feel the same compression from the real pressure of a man on my shoulders that I would feel from a load laid on them. But the weight in our example is acting by the intervention of the thread. By its pressure it is pulling at that part of the thread to which it is fastened; this part is pulling at the next by means of the force of cohesion; and this pulls at a third, and so on, till the most remote pulls at the ball or the machine. Thus may elasticity, weight, cohesion, and other forces, perform the office of a genuine power; and since their result is always a motion beginning from nothing, and accelerating by perceptible degrees to any velocity, this resemblance makes us call them by one familiar name.

But further, I see that if the thread be cut, the weight will fall with an accelerated motion, which will increase to any degree, if the fall be great enough. I ascribe this also to a pressing power acting on the weight. Nay, after a very little refinement, I consider this power as the cause of the body's weight; which word is but a distinguishing name for this particular instance of pressing power. Gravitation is therefore added to the list of pressures; and for similar reasons the attractions and repulsions of magnets or electric bodies may be added to the list; for they produce actual compressions of bodies placed between them, and they produce motions gradually accelerated, precisely as gravitation does. Therefore all these powers may be distinguished by this descriptive name, pressures, which, in strict language, belongs to one of them only.

Several writers, however, subdivide this great class into Gravity, presasions and solicitations. Gravity is a solicitation of attractions, extra, by which a body is urged downward. In like manner the forces of magnetism and electricity, and a vast variety of other attractions and repulsions, are called solicitations. We see little use for this distinction, and the term is too like an affection of mind.

9. Impulsion is exhibited when a ball in motion puts another ball into motion by hitting, or (to speak metaphorically) by striking it. The appearances here are very different. The body that is struck acquires, in the instant of impulse, a sensible quantity of motion, and sometimes a very rapid motion. This motion is neither accelerated nor retarded after the stroke, unless it be affected by some other force. It is also remarked that the rapidity of the motion depends, inter alia, on the previous velocity of the striking body. For instance, if a clay ball, Dynamics moving with any velocity, strike another equal ball which is at rest; the struck ball moves with half the velocity of the other. And it is farther remarkable that the striking body always loses as much motion as the struck body gains. This universal and remarkable fact seems to have given rise to a confused or indistinct notion of a sort of transference of motion from one body to another. The phraseology in general use on this subject expresses this in the most precise terms. The one ball is not said to cause or produce motion in the other, but to communicate motion to it; and the whole phenomenon is called the communication of motion. We call this an indistinct notion; for surely no one will say that he has any clear conception of it. We can form the most distinct notion of the communication of heat, or of the cause of heat; of the communication of saltness, sweetness, and a thousand other things; but we cannot conceive how part of that identical motion which was formerly in A, is now infused into B, being given up by A. It is in our attempt to form this notion that we find that motion is not a thing, not a substance which can exist independently, and is susceptible of actual transference. It appears in this case to be a state, or condition, or mode of existence, of which bodies are susceptible, which is producible, or (to speak without metaphor) causable, in bodies, and which is the effect and characteristic of certain natural qualities, properties, or powers. We are anxious to have our readers impressed with clear and precise notions on this subject, being confident that such, and only such, will carry them through some intricate paths of mechanical and philosophical research.

10. The remarkable circumstance in this phenomenon is, force is the that a rapid motion, which requires for the effecting it distinctive the action of a pressing power, continued for a sensible, character of and frequently a long time, seems to be affected in an instant by impulsion. This has tended much to support the notion of the actual transference of something formerly possessed exclusively by the striking body, inhering in it, but separable, and now transfused into the body stricken. And now room is found for the employment of metaphor, both in thought and language. The striking body affects the body which it thus impels: it therefore possesses the power of impulsion, that is, of communicating motion. It possesses it only while it is in motion. This power, therefore, is the efficient distinguishing cause of its motion, and its only office must be the continuation of this motion. It is therefore called the inherent force, the force inherent in a moving body, vis insita corpori moto. This force is transfused into the body impelled; and therefore the transference is instantaneous, and the impelled body continues its motion till it is changed by some other action. All this is at first sight very plausible; but a scrupulous attention to those feelings which have given rise to this metaphorical conception should have produced very different notions. I am conscious of exertion in order to begin motion on a slide; but if the ice be very smooth, I am conscious of no exertion in order to slide along. My power is felt only while I am conscious of exerting it; therefore I have no primitive feeling or notion of power while I am sliding along. I am certain that no exertion of power is necessary here. Nay, I find that I cannot think of my moving forward without effort otherwise than as a certain mode of my existence. Yet we imagine that the partizans of this opinion did really deduce it in some shape from their feelings. We must continue the exertion of walking in order to walk on; our power of walking must be continually exerted, otherwise we shall stop. But this is a very imperfect, incomplete, and careless observation. Walking is much more than mere continuance in progressive motion. It is a continuously repeated lifting our body up a small height, and allowing it to come down again. This renewed ascent requires repeated exertion.

11. We have other observations of importance yet to make on this force of moving bodies, but this is not the most proper occasion. Meanwhile we must remark, that the instantaneous production of rapid motion by impulse, has induced the first mechanicians of Europe to maintain that the power or force of impulse is unsusceptible of any comparison with a pressing power. They have asserted that impulse is infinitely great when compared with pressure; not recollecting that they held them to be things totally disparate, that have no proportion more than weight and sweetness. But these gentlemen are perpetually enticed away from their creed by the similarity of the ultimate results of pressure and impulse. No person can find any difference between the motion of two balls moving equally swift, in the same direction, one of which is descending by gravity, and the other has derived its motion from a blow. This struggle of the mind to maintain its faith, and yet accommodate its doctrines to what we see, has occasioned some other curious forms of expression. Pressure is considered as an effort to produce motion. When a ball lies on a table, its weight, which they call a power, continually and repeatedly endeavours (mark the metaphorical word and thought) to move the ball downward. But these efforts are ineffectual. They say that this inefficacious power is dead, and call it a vis mortua; but the force of impulsion is called a vis viva, a living force. But this is very whimsical and very inaccurate. If the impelling ball falls perpendicularly on the other lying on the table, it will produce no motion, any more than gravity will; and if the table be annihilated, gravity becomes a vis viva.

We must now add, that in order to prove that impulse Argus is infinitely greater than pressure, these mechanicians turn our attention to many familiar facts which plead strongly in their favour. A carpenter will drive a nail into a board with a very moderate blow of his hammer. This will require a pressure which seems many hundred times greater than the impelling effort of the carpenter. A very moderate blow will shiver into pieces a diamond which would carry the weight of a mountain. Seeing this prodigious superiority in the impulse, how shall they account for the production of motion by means of pressure; for this motion of the hammer might have been acquired by its falling from a height; nay, it is actually acquired by means of the continued pressure of the carpenter's arm. They consider it as the aggregate of an infinity of succeeding pressures in every instant of its continuance; so that the insignificant smallness of each effort is compensated by their inconceivable number.

On the whole, we do not think that there is clear evidence that there are two kinds of mechanical force essentially different in their nature. It is virtually given up by those who say that impulse is infinitely greater than pressure. Nor is there any considerable advantage to be obtained by arranging the phenomenon under those two heads. We may perhaps find some method of explaining satisfactorily the remarkable difference that is really observed in the two modes of producing motion; namely, the gradual production of motion by acknowledged pressure, and the instantaneous production of it by impulse. Indeed we should not have taken up so much of our readers' attention with this subject, had it not been for some inferences that have been made from these premises, which meet us in our very entry on the consideration of first principles, and that are of extensive influence on the whole science of mechanical philosophy, and, indeed, on the whole study of nature. 12. Mechanicians are greatly divided in their opinion about the nature of the sole moving force in nature. Those whom we are now speaking of seem to think that all motion is produced by pressure; for when they consider impulse as equivalent to the aggregate of an infinity of repeated pressures, they undoubtedly suppose any pressure, however insignificant, as a moving force. But there is a party, both numerous and respectable, who maintain that impulsion is the sole cause of motion. We see bodies in motion, say they, and we see them impel others; and we see that this production of motion is regulated by such laws, that there is but one absolute quantity of motion in the universe which remains unalterably the same. It must therefore be transfused in the acts of collision. We also see, with clear evidence, in some cases, that motion can produce pressure. Euler adduces some very whimsical and complicated cases, in which an action precisely similar to pressure may be produced by motion. Thus, two balls connected by a thread may be so struck that they shall move forward, and at the same time wheel round. In this case the connecting thread will be stretched between them. Now, say the philosophers, since we see motion, and see that pressure may be produced by motion, it is preposterous to imagine that it is anything else than a result of certain motions; and it is the business of a philosopher to inquire and discover what motions produce the pressures that we observe.

They then proceed to account for those pressing powers, or solicitations to motion, which we observe in the acceleration of falling bodies, the attractions of magnetism and electricity, and many other phenomena of this kind, where bodies are put in motion by the vicinity of other bodies, or, in the popular language, by the action of other bodies at a distance. To say that a magnet can act on a piece of remote iron, is to say that it can act where it is not, which is as absurd as to say that it can act when it is not. Nihil movetur, says Euler, nisi a contiguo et moto.

The bulk of these philosophers are not very anxious about the way in which these motions are produced, nor do they fall upon such ingenious methods of producing pressure as the one already mentioned, which was adduced by Euler. The piece of iron, say they, is put in motion when brought into the neighbourhood of a magnet, because there is a stream of fluid issuing from one pole of the magnet, which circles round the magnet, and enters at the other pole. This stream impels the iron, and arranges it in certain determined positions, just as a stream of water would arrange the flote grass. In the same manner, there is a stream of fluid continually moving towards the centre of the earth, which impels all bodies in lines perpendicular to the surface; and so on with regard to other like phenomena. These motions are thus reduced to very simple cases by impulsion.

It is unnecessary to refute this doctrine at present: it is enough that it is contrary to all the dictates of common sense. To suppose an agent that we do not see, and for whose existence we have not the smallest argument; with equal propriety we might suppose ministering spirits, or any thing that we please.

13. Other philosophers are so dissatisfied with this notion of the production of pressure, that they, on the other hand, affirm that pressure is the only moving force in nature; not according to the popular notion of pressure, by the mutual contact of solid bodies, but that kind of pressure which has been called solicitation, such as the power of gravity. They affirm that there is no such thing as contact or instantaneous communication of motion by real collision. They say, and they prove it by very convincing facts, that the particles of solid bodies exert very strong repulsions to a small distance; and therefore, when they are brought by motion sufficiently near to another body, they repel it, Dynamics, and are equally repelled by it. Thus is motion produced in the other body, and their own motion is diminished.

And they then show, by a scrupulous consideration of the state of the bodies while the one is advancing and the other retiring, in what manner the two bodies attain a common velocity, so that the quantity of motion before collision remains unchanged, the one body gaining as much as the other loses. They also show cases of such mutual action between bodies, where it is evident that they have never come into contact; and yet the result has been precisely similar to those cases where the motion appeared to be changed in an instant. Therefore they conclude that there is no such thing as instantaneous communication, or transfusion of motion, by contact in collision or impulse. The reason why previous motion of the impelling body is necessary, is not that it may have a vis insita corpori moto, a force inherent in it by its being in motion, but that it may continue to follow the impelled and retiring body, and exert on it a force inherent in itself, whether in motion or at rest. According to these philosophers, therefore, all moving forces are of that kind which has been named solicitation, such as gravity. We shall know it afterwards by the more familiar and descriptive name of accelerating or retarding force.

14. The exertions of mechanical forces are differently Action, termed, according to the reference that we make to the distance, result. If in boxing or wrestling I strike, or endeavour to re-action, throw my antagonist, I am said to act; but if I only parry his blows, or prevent him from throwing me, I am said to resist. This distinction is applied to the exertions of mechanical powers. When one body, A, changes the motion of another, B, we may consider the change in the motion of B either as the indication and measure of A's power of producing motion, or as the indication and measure of A's resistance to the being brought to rest, or having its motion any how changed. The distinction is not in the thing itself, but only in the reference that we are disposed, by other considerations, to make of its effect. They may be distinguished in the following manner: If a change of motion follow when one of the powers ceases to be exerted, that power is conceived as having resisted. The whole language on this subject is metaphorical. Resistance, effort, endeavour, &c. are words which cannot be employed in mechanical discussions without figure, because they all express notions which relate to sentient beings; and the unguarded indulgence of this figurative language has so much affected the imagination of philosophers, that many have almost animated all matter. Perhaps the word re-action, introduced by Newton, is the best term for expressing that mutual force which is perceived in all the operations of nature that we have investigated with success. As the magnet attracts iron, and in so doing is said to act on it; so the iron attracts the magnet, and may be said to re-act on it.

15. With respect to the difficulty that has been objected to the opinion of those who maintain that all the mechanical phenomena are produced by the agency of attractive action or repelling forces; namely, that this supposes the bodies to act on each other at a distance, however Tendency, small those distances may be, which is thought to be absurd, we may observe, that we may ascribe the mutual approaches or recesses to tendencies to or from each other. What we call the attraction of the magnet may be considered as a tendency of the iron to the magnet, somewhat similar to the gravitation of a stone toward the earth. We surely (at least the unlearned) can and do conceive the iron to be affected by the magnet, without thinking of any intermedium. The thing is not therefore inconceivable, which is all that we know about absurdity; and we do not Dynamics know anything about the nature or essence of matter which renders this tendency to the magnet impossible. That we do not see intuitively any reason why the iron should approach the magnet, must be granted; but this is not enough to entitle us to say that such a thing is impossible, or inconsistent with the nature of matter. It appears, therefore, to be very hasty and unwarrantable to suppose the impulse of an invisible fluid, of which we know nothing, and of the existence of which we have no proof. Nay, if it be true that bodies do not come into contact, even when one ball hits another, and drives it before it, this invisible fluid will not solve the difficulty, because the same difficulty occurs in the action of any particle of the fluid on the body. We are obliged to say that the production of motion without any observed contact, is a much more familiar phenomenon than the production of motion by impulsion. More motion has been produced in this way by the gravitation of a small stream of water, running ever since the creation, than by all the impulses in the world twice told. We do not mean by this to say, that the giving to this observed mutual relation between iron and a lodestone the name tendency makes it less absurd than when we say that the lodestone attracts the iron; it only makes it more conceivable. It suggests a very familiar analogy; but both are equally figurative expressions, at least as the word tendency is used at present. In the language of ancient Rome, there was no metaphor when Virgil's hero said, Tendimus in Latium. Tendere versus solen means, in plain Latin, to approach the sun. The safe way of conceiving the whole is to say that the condition of the iron depends on the vicinity of the magnet.

16. When the exertions of a mechanical power are observed to be always directed toward a body, that body is said to attract; but when the other body always moves off from it, it is said to repel. These also are metaphorical expressions. I attract a boat when I pull it toward me by a rope; this is purely Attraction; and it is pure, unfigurative Repulsion, when I push any body from me. The same words are applied to the mechanical phenomena, merely because they resemble the results of real attraction or repulsion. We must be much on our guard to avoid metaphor in our conceptions, and never allow those words to suggest to our mind any opinion about the manner in which the mechanical forces produce their effects. It is plain, that if the opinion of those who maintain the existence and action of the above-mentioned invisible fluid be just, there is nothing like attraction or repulsion in the universe. We must always recur to the simple phenomenon, the motion to or from the attracting or repelling body; for this is all we see, and generally all that we know.

17. We conceive one man to have twice the strength of another man, when we see that he can withstand the united effort of two others. Thus animal force is conceived as a quantity, made up of and measured by its own parts. But we doubt exceedingly whether this be an accurate conception. We have not a distinct notion of one strain added to another, though we have of their being joined or combined. We want words to express the difference of these two notions in our own minds, but we imagine that others perceive the same difference. We conceive clearly the addition of two lines or of two minutes; we can conceive them apart, and perceive their boundaries, common to both, where one ends and the other begins. We cannot conceive thus of two forces combined; yet we cannot say that two equal forces are not double of one of them. We measure them by the effects which they are known to produce. Yet there are not wanting many cases where the action of two men, equally strong, does not produce a double motion.

In like manner, we conceive all mechanical forces as measurable by their effects; and thus they are made the subjects of mathematical discussion. We talk of the proportions of gravity, magnetism, electricity, &c.; nay, we talk of the proportion of gravity to magnetism; yet these, considered in themselves, are disparate, and do not admit of any proportion; but they produce effects, some of which are measurable, and whose assumed measures are susceptible of comparison, being quantities of the same kind. Thus, one of the effects of gravity is the acceleration of motion in a falling body; magnetism will also accelerate the motion of a piece of iron; these two accelerations are comparable. But we cannot compare magnetism with heat, because we do not know any measurable effects of magnetism that are of the same kind with any effects of heat.

When we say that the gravitation of the moon is the 3600th part of the gravitation at the sea-shore, we mean that the fall of a stone in a second is 3600 times greater than the fall of the moon in the same time. But we also mean (and this expresses the proportion of the tendency of gravitation more purely), that if a stone, when hung on a spring steelyard, draw out the rod of the steelyard to the mark 3600, the same stone, taken up to the distance of the moon, will draw it out no farther than the mark 1. We also mean, that if the stone at the sea-shore draw out the rod to any mark, it will require 3600 such stones to draw it out to that mark when the trial is made at the distance of the moon. It is not, therefore, in consequence of any immediate perception of the proportion of the gravitation at the moon to that at the surface of the earth that we make such an assertion; but these motions, which we consider as its effects in these situations, being magnitudes of the same kind, are susceptible of comparison, and have a proportion which can be ascertained by observation. It is these proportions that we contemplate; although we speak of the proportions of the unseen causes, the forces, or endeavours to descend. It will be of material service to the reader to peruse the judicious and acute dissertation on quantity in the works of Dr Reid (vol. ii.), where he will see clearly how force, velocity, density, and many other magnitudes of very frequent occurrence in mechanical philosophy, may be made the subjects of mathematical discussion, by means of some of those proper quantities, measurable by their own parts, which are to be assumed as their measures. Pressures are measurable only by pressures. When we consider them as moving powers, we should be able to measure them by any moving powers, otherwise we cannot compare them; therefore it is not as pressures that we then measure them. This observation is momentous.

One circumstance must be carefully attended to. That those assumed measures may be accurate, they must be invariably connected with the magnitudes which they are employed to measure, and so connected that the degrees of the one must change in the same manner with the degrees of the other. This is evident, and is granted by all. But we must also know this of the measure we employ; we must see this constant and precise relation. How can we know this? We do not perceive force as a separate existence, so as to see its proportions, and to see that these are the same with the proportions of the measures, in the same manner that Euclid sees the proportions of triangles and those of their bases, and that these proportions are the same when the triangles are of equal altitudes. How do we discover that to every magnitude which we call force is invariably attached a corresponding magnitude of acceleration or deflection? Clearly. In fact, the very existence of the force is an inference that we make from the observed acceleration; and the degree of Dynamics. The force is, in like manner, an inference from the observed magnitude of the acceleration. Our measures are therefore necessarily connected with the magnitudes which they measure, and their proportions are the same; because the one is always an inference from the other, both in species and in degree.

18. It is now evident that these disquisitions are susceptible of mathematical accuracy. Having selected our measures, and observed certain mathematical relations of those measures, every inference that we can draw from the mathematical relations of the proportions of those representations is true of the proportions of the motions, and therefore of the proportions of the forces. And thus dynamics becomes a demonstrative science, one of the disciplines accuratae.

19. But moving forces are considered as differing and in kind; that is, in direction. We assign to the force the direction of the observed change of motion, which is not only the indication, but also the characteristic, of the changing force. We call it an accelerating, retarding, deflecting force, according as we observe the motion to be accelerated, retarded, or deflected.

These denominations show us incontestably that we have no knowledge of the forces different from our knowledge of the effects. The denominations are all either descriptive of the effects, as when we call them accelerating, penetrating, protrusive, attractive, or repulsive forces; or they are names of reference to the substances in which the accelerating, protrusive, &c. forces are supposed to be inherent, as when we call them magnetism, electricity, corpuscular, &c.

20. When I struggle with another, and feel, that in order to prevent being thrown, I must exert force, I learn that my antagonist is exerting force. This motion is transferred to matter; and when a moving power which is known to operate produces no motion, we conceive it to be opposed by another equal force, the existence, agency, and intensity of which is detected and measured by these means. The quiescent state of the body is considered as a change on the state of things that would have been exhibited in consequence of the known action of one power, had this other power not acted; and this change is considered as the indication, characteristic, and measure, of another power, detected in this way. Thus forces are recognised not only by the changes of motion which they produce, but also by the changes of motion which they prevent. The cohesion of matter in a string is inferred not only by its giving motion to a ball which I pull toward me by its intervention, but also by its suspending that ball, and hindering it from falling. I know that gravity is acting on the ball, which, however, does not fall. The solidity of a board is equally inferred from its stopping the ball which strikes it, and from the motion of the ball which it drives before it. In this way we learn that the particles of tangible matter cohere by means of moving forces, and that they resist compression with force; and in making this inference, we find that this corpuscular force exerted between the particles is mutual, opposite, and equal; for we must apply force equally to \(a\) or to \(b\), in order to produce a separation or a compression. We learn their equality, by observing that no motion ensues while these mutual forces are known to act on the particles; that is, each is opposed by another force, which is neither inferior nor superior to it.

OF THE LAWS OF MOTION.

Such, then, being our notions of mechanical forces, the causes of the sensible changes of motion, there will result certain consequences from them, which may be called axioms or laws of motion. Some of these may be intuitive, first law offering themselves to the mind as soon as the notions of Motion, which they involve are presented to it. Others may be as necessary results from the relations of these notions, but may not readily offer themselves without the mediation of axioms of the first class. We shall select those which are intuitive, and may be taken for the first principles of all discussions in mechanical philosophy.

FIRST LAW OF MOTION.

Every body continues in a state of rest, or of uniform rectilinear motion, unless affected by some mechanical force.

21. This is a proposition, on the truth of which the whole science of mechanical philosophy ultimately depends. It is therefore to be established on the firmest foundation; and a solicitude on this head is the more justifiable, because the opinions of philosophers have been extremely different, both with respect to the truth of this law, and with respect to the foundation on which it is built. These opinions are, in general, very obscure and unsatisfactory; and, as is natural, they influence the discussions of those by whom they are held through the whole science. Although of contradictory opinions one only can be just, and it may appear sufficient that this one be established and uniformly applied; yet a short exposition, at least, of the rest is necessary, that the greatest part of the writings of the philosophers may be intelligible, and that we may avail ourselves of much valuable information contained in them, by being able to perceive the truth in the midst of their imperfect or erroneous conceptions of it.

22. It is not only the popular opinion that rest is the Does conatural state of body, and that motion is something foreign tinned motion, but it has been seriously maintained by the greatest part of those who are esteemed philosophers. They readily grant that matter will continue at rest unless some moving force act upon it. Nothing seems necessary for matter's remaining where it is, but its continuing to exist. But it is far otherwise, say they, with respect to matter in motion. Here the body is continually changing its relations to other things; therefore the continual agency of a changing cause is necessary (by the fundamental principle of all philosophical discussion), for there is here the continual production of an effect. They say that this metaphysical argument receives complete confirmation (if confirmation of an intuitive truth be necessary) from the most familiar observation. We see that all motions, however violent, terminate in rest; and that the continual exertion of some force is necessary for their continuance.

23. These philosophers therefore assert, that the continual action of the moving cause is essentially necessary for the continuance of the motion; but they differ among themselves in their notions and opinions about this cause. Some maintain that all the motions in the universe are produced and continued by the immediate agency of Deity; others affirm, that in every particle of matter there is inherent a sort of mind, the \(\phi\alpha\sigma\tau\eta\gamma\) of Aristotle, which they call an Elemental Mind, which is the cause of all its motions and changes. An overwhelming reverence for Greek learning has had a great influence in reviving this doctrine of Aristotle. The Greek and Roman languages are affirmed to be more accurate expressions of human thought than the modern languages are. In those ancient languages, the verbs which express motion are employed both in the active and passive voice; whereas we have only the active verb to move, for expressing both the state of motion and the act of putting in motion. "The stone moves down the slope, and moves all the pebbles which lie in its way;" but in the ancient lan- First Law guages, the mere state of motion is always expressed by of Motion, the passive or middle voice. The accurate conception of the speakers is therefore extolled. The state of motion is expressed as it ought to be, as the result of a continual action. Kontrai, movetur, is equivalent to "it is moved." According to these philosophers, everything which moves is mind, and every thing that is moved is body.

The argument is futile, and it is false; for the modern languages are, in general, equally accurate in this instance: "se mouvoir," in French; "fich bewegen," in German; "deigatun," in Slavonic; are all passive or reflected. And the ancients said that "rain falls, water runs, smoke rises," just as we do. The ingenious author of Ancient Meta- physics has taken much pains to give us, at length, the procedures of those elementary minds in producing the ostensible phenomena of local motion; but it seems to be merely an abuse of language, and a very frivolous abuse. This elemental mind is known and characterised only by the effect which we ascribe to its action; that is, by the motions or changes of motions. Uniform and unexpec- ted experience shows us that these are regulated by laws as precise as those of mathematical truth. We consider nothing as more fixed and determined than the common laws of mechanism. There is nothing here that indicates any thing like spontaneity, intention, purpose; none of those marks by which mind was first brought into view; but they are very like the effects which we produce by the exertions of our corporeal forces; and we have ac- cordingly given the name force to the causes of motion. It is surely much more apposite than the name mind, and conveys with much more readiness and perspicuity the very notions that we wish to convey.

24. We now wish to know what reason we have to think that the continual action of some cause is necessary for continuing matter in motion, or for thinking that rest is its natural state. If we pretend to draw any argument from the nature of matter, that matter must be known, as far as is necessary for being the foundation of argument. Its very existence is known only from observation; all our knowledge of it must therefore be derived from the same source.

If we take this way to come at the origin of this opi- nion, we shall find that experience gives us no authority for saying that rest is the natural condition of matter. We cannot say that we have ever seen a body at rest; this is evident to every person who allows the validity of the Newtonian philosophy, and the truth of the Copernican system of the sun and planets; all the parts of this system are in motion. Nay, it appears from many observations, that the sun, with his attending planets, is carried in a certain direction, with a velocity which is very great. We have no unquestionable authority for saying that any one of the stars is absolutely fixed; but we are certain that many of them are in motion. Rest is therefore so rare a condition of body, that we cannot say, from any experience, that it is its natural state.

25. It is easy, however, to see that it is from observa- tion that this opinion has been derived; but the observa- tion has been limited and careless. Our experiments in this sublunar world do indeed always require continued action of some moving force to continue the motion; and if this be not employed, we see the motions slacken every minute, and terminate in rest after no long period. Our first notions of sublunar bodies are indicated by their operation in cases where we have some interest. Perpe- tually seeing our own exertions necessary, we are led to consider matter as something not only naturally quiescent and inert, but sluggish, averse from motion, and prone to rest (we must be pardoned this metaphorical language, because we can find no other term). What is expressed by it, on this occasion, is precisely one of the erroneous or inadequate conceptions that are suggested to our thoughts by reason of the poverty of language. We ani- mate matter in order to give it motion, and then we endow it with a sort of moral character in order to explain the appearance of those motions.

26. But more extended observation has made men gra- dually desert their first opinions, and at last allow that matter has no peculiar aptitude to rest. All the retarda- tions that we observe have been discovered, one after an- other, to have a distinct reference to some external cir- cumstances. The diminution of motion is always observ- ed to be accompanied by the removal of obstacles, as when a ball moves through sand, or water, or air; or it is owing to opposite motions which are destroyed; to roughness of the path, or to friction, &c. We find that the more we can keep these things out of the way, the less are the motions diminished. A pendulum will vibrate but a short while in water; much longer in air; and in the exhaust- ed receiver it will vibrate a whole day. We know that we cannot remove all obstacles; but we are led by such ob- servations to conclude that, if they could be completely removed, our motions would continue for ever. And this conclusion is almost demonstrated by the motions of the heavenly bodies, to which we know of no obstacles, and which we really observe to retain their motions for many thousand years, without the smallest sensible diminution.

27. Another set of philosophers maintain an opinion directly opposite to that of the inactivity of matter, and of assert, that it is essentially active, and continually chan- ging its state. Faint traces of this are to be found in the writings of Plato, Aristotle, and their commentators. Mr. Leibnitz is the person who has treated this question most systematically and fully. He supposes every particle of matter to have a principle of individuality, which he there- fore calls a Monad. This Monad has a sort of perception of its situation in the universe, and of its relation to every other part of this universe. Lastly, he says that the mo- nad acts on the material particle much in the same way that the soul of man acts on his body. It modifies the motion of the material atom (in conformity, however, to unalterable laws), producing all those modifications of motion that we observe. Matter, therefore, or at least particles of matter, are continually active, and continual- ly changing their situation.

It is quite unnecessary to enter on a formal confutation of Mr Leibnitz's system of monads, which differs very lit- tle from the system of elemental minds, and is equally whimsical and frivolous; because it only makes the un- learned reader stare, without giving him any information. Should it even be granted, it would not, any more than the action of animals, invalidate the general proposition which we are endeavouring to establish as the funda- mental law of motion. Those powers of the monads, or of the elemental minds, are the causes of all the changes of motion; but the mere material particle is subject to the law, and requires the exertion of the monad in order to exhibit a change of motion.

28. A third sect of philosophers, at the head of which we may place Sir Isaac Newton, maintain the doctrine enounced in the proposition. But they differ much in re- spect of the foundation on which it is built.

Some assert that this truth flows from the nature of the thing. If a body be at rest, and you assert that it will not remain at rest, it must move in some one direction. If it be in motion in any direction, and with any velocity, and then do not continue its equable, rectilinear motion, it must either be accelerated or retarded; it must turn either to one side or to some other side. The event, whatever it may be, is individual and determinate; but no cause which But considerable objections may be made to this argument of sufficient reason, as it is called. In the immensity and perfect uniformity of space and time, there is no determining cause why the visible universe should exist in the place in which we see it rather than in another, or at this time rather than at another. Nay, the argument seems to beg the question. A cause of determination is required as essentially necessary—a determination may be without a cause, as well as a motion without a cause.

29. Other philosophers, who maintain this doctrine, consider it merely as an experimental truth; and proofs of its universality are innumerable.

When a stone is thrown from the hand, we press it forward while in the hand, and let it go when the hand has acquired the greatest rapidity of motion that we can give it. The stone continues in that state of motion which it acquired gradually along with the hand. We can throw a stone much farther by means of a sling, because, by a very moderate motion of the hand, we can whirl the stone round till it acquire a very great velocity, and then we let go one of the strings, and the stone escapes, by continuing its rapid motion. We see it still more distinctly in shooting an arrow from a bow. The string presses hard on the notch of the arrow, and it yields to this pressure and goes forward. The string alone would go faster forward. It therefore continues to press the arrow forward, and accelerates its motion. This goes on till the bow is as much bent as the string will allow. But the string is now a straight line. It came into this position with an accelerated motion, and it therefore goes a little beyond this position, but with a retarded motion, being checked by the bow. But there is nothing to check the arrow, therefore the arrow quits the string and flies away.

These are simple cases of perseverance in a state of motion, where the procedure of nature is so easily traced that we perceive it almost intuitively. It is no less clear in other phenomena which are more complicated; but it requires a little reflection to trace the process. We have often seen an equestrian showman ride a horse at a gallop, standing on the saddle, and stepping from it to the back of another horse that gallops alongside at the same rate; and he does this seemingly with as much ease as if the horses were standing still. The man has the same velocity with the horse that gallops under him, and keeps this velocity while he steps to the back of the other. If that other were standing still, the man would fly over his head. And if a man should step from the back of a horse that is standing still, to the back of another that gallops past him, he would be left behind. In the same manner, a slack-wire dancer tosses oranges from hand to hand while the wire is in full swing. The orange, swinging along with the hand, retains the velocity; and when in the air, follows the hand, and falls into it when it is in the opposite extremity of its swing. A ball dropped from the mast-head of a ship that is sailing briskly forward, falls at the foot of the mast. It retains the motion which it had while in the hand of the person who dropped it, and follows the mast during the whole of its fall.

We also have familiar instances of the perseverance of a body in a state of rest. When a vessel filled with water is drawn suddenly along the floor, the water dashes over the posterior side of the vessel. It is left behind. In the same manner, when a coach or boat is dragged forward, the persons in it find themselves strike against the hinder part of the carriage or boat. Properly speaking, it is the carriage that strikes on them. In like manner, if we lay a card on the tip of the finger, and a piece of money on the card, we may nick away the card, by hitting it neatly on its edge; but the piece of money will be left behind, lying on the tip of the finger. A ball will go through a wall and fly onward; but the wall is left behind. Buildings are thrown down by earthquakes; sometimes by being tossed from their foundations, but more generally by the ground on which they stand being hastily drawn sidewise from under them, &c.

30. But common experience seems insufficient for establishing this fundamental proposition of mechanical philosophy. We must, on the faith of the Copernican system, insinuate that we never saw a body at rest, or in uniform rectilinear motion; yet this seems absolutely necessary before we can say that we have established this proposition experimentally.

What we imagine, in our experiments, to be putting a body, formerly at rest, into motion, is, in fact, only changing a most rapid motion, not less, and probably much greater, than 90,000 feet per second. Suppose a cannon pointed east, and the bullet discharged at noon day with sixty times greater velocity than we have ever been able to give it; it would appear to set out with this unmeasurable velocity to the eastward, to be gradually retarded by the resistance of the air, and at last brought to rest by hitting the ground. But, by reason of the earth's motion round the sun, the fact is quite the reverse. Immediately before the discharge, the ball was moving to the westward with the velocity of 90,000 feet per second nearly. By the explosion of the powder, and its pressure on the ball, some of this motion is destroyed, and at the muzzle of the gun the ball is moving slower, and the cannon is hurried away from it to the westward. The air, which is also moving to the westward 90,000 feet in a second, gradually communicates motion to the ball, in the same manner as a hurricane would do. At last (the ball dropping all the while) some part of the ground hits the ball, and carries it along with it.

Other observations must therefore be resorted to, in order to obtain an experimental proof of this proposition; and such are to be found. Although we cannot measure the absolute motions of bodies, we can observe and measure accurately their relative motions, which are the differences of their absolute motions. Now, if we can show experimentally that bodies show equal tendencies to resist the augmentation and the diminution of their relative motions, they, ipso facto, show equal tendencies to resist the augmentation or diminution of their absolute motions. Therefore let two bodies, A and B, be put into such a situation that they cannot (by reason of their impenetrability, or the action of their mutual powers) persevere in their relative motions. The change produced on A is the effect and the measure of B's tendency to persevere in its former state; and therefore the proportion of these changes will show the proportion of their tendencies to maintain their former states. Therefore let the following experiment be made at noon.

Let A, apparently moving westward three feet per second, hit the equal body B apparently at rest. Suppose, moreover, that A impels B forward without any diminution of its own velocity. This result would show that B manifests no tendency to maintain its motion unchanged, but that A retains its motion undiminished.

2dly. Suppose that A stops, and that B remains at rest. This would show that A does not resist a diminution of motion, but that B retains its motion unaugmented.

3dly. Suppose that both move westward with the velocity of one foot per second. The change on A is a diminution of velocity, amounting to two feet per second. First Law This is the effect and the measure of B's tendency to maintain its velocity unaltered. The change on B is an augmentation of one foot per second made on its velocity; and this is the measure of A's tendency to maintain its velocity undiminished. This tendency is but half of the former; and this result would show that the resistance to a diminution of velocity is but half of the resistance to augmentation. It is perhaps but one quarter, for the change on B has produced a double change on A.

4thly, Suppose that both move westward at the rate of one and a half feet per second. It is evident that their tendencies to maintain their states unchanged are now equal.

5thly, Suppose \( A = 2B \), and that both move, after the collision, two feet per second; B has received an addition of two feet per second to its former velocity. This is the effect and the measure of A's whole tendency to retain its motion undiminished. Half of this change on B measures the persevering tendency of the half of A; but A, which formerly moved with the apparent or relative velocity three, now moves (by the supposition) with the velocity two, having lost a velocity of one foot per second. Each half of A therefore has lost this velocity, and the whole loss of motion is two. Now this is the measure of B's tendency to maintain its former state unaltered; and this is the same with the measure of A's tendency to maintain its own former state undiminished. The conclusion from such a result would therefore be, that bodies have equal tendencies to maintain their former states of motion without augmentation and without diminution.

What is supposed in the fourth and fifth cases is really the result of all the experiments which have been tried; and this law regulates all the changes of motion which are produced by the mutual actions of bodies in impulsions. This assertion is true without exception or qualification. Therefore it appears that bodies have no preferable tendency to rest, and that no fact can be adduced which should make us suppose that a motion once begun should suffer any diminution without the action of a changing cause.

But we must now observe, that this way of establishing the first law of motion is very imperfect, and altogether unfit for rendering it the fundamental principle of a whole and extensive science. It is subject to all the inaccuracy that is to be found in our best experiments; and it cannot be applied to cases where scrupulous accuracy is wanted, and where no experiment can be made.

Let us therefore examine the proposition by means of principles which contain the foundation of all our knowledge of active nature. These will, we imagine, give a decision of this question that is speedy and accurate; showing the proposition to be an axiom or intuitive consequence of the relations of those ideas which we have of motion, and of the causes of its production and changes.

31. It has been fully demonstrated that the powers or forces of which we speak so much are never the immediate objects of our perception. Their very existence, their kind, and their degree, are instinctive inferences from the motions which we observe and class. It evidently follows from this experimental and universal truth, i.e., that where no change of motion is observed, no such inference is made, that is, no power is supposed to act. But whenever any change of motion is observed, the inference is made; that is, a power or force is supposed to have acted.

In the same form of logical conclusion, we must say that, 2dly, when no change of motion is supposed or thought of, no force is supposed; and that whenever we suppose a change of motion, we in fact, though not in terms, suppose a changing force. And, on the other hand, whenever we suppose the action of a changing force, we suppose the change of motion; for the action of this force, and the change of motion, is one and the same thing. We cannot think of the action without thinking of the indication of that action; that is, the change of motion. In the same manner, when we do not think of a changing force, or suppose that there is no action of a changing force, we in fact, though not in terms, suppose that there is no indication of this changing force; that is, that there is no change.

Whenever, therefore, we suppose that no mechanical force is acting on a body, we in fact suppose that the body continues in its former condition with respect to its motion. If we suppose that nothing accelerates, or retards, or deflects the motion, we suppose that it is not accelerated, nor retarded, nor deflected. Hence follows the proposition in express terms—We suppose that the body continues in its former state of rest or motion, unless we suppose that it is changed by some mechanical force.

Thus it appears that this proposition is not a matter of experience or contingency, depending on the properties which it has pleased the Author of nature to bestow on body; it is to us a necessary truth. The proposition does not so much express anything with regard to body, as it does the operations of our mind when contemplating body. It may perhaps be essential to body to move in some particular direction. It may be essential to body to stop as soon as the moving cause has ceased to act; or it may be essential to body to diminish its motion gradually, and finally come to rest. But this will not invalidate the truth of this proposition. These circumstances in the nature of body, which render those modifications of motion essentially necessary, are the causes of those modifications; and in our study of nature they will be considered by us as changing forces, and will be known and called by that name. And if we should ever see a particle of matter in such a situation that it is affected by those essential properties alone, we shall, from observation of its motion, discover what those essential properties are.

This law turns out at last to be little more than a tautological proposition; but mechanical philosophy, as we have defined it, requires no other sense of it; for even if we should suppose that body, of its own nature, is capable of changing its state, this change must be performed according to some law which characterizes the nature of body; and the knowledge of the law can be had in no other way than by observing the deviations from uniform rectilineal motion. It is therefore indifferent whether those changes are derived from the nature of the thing or from external causes; for, in order to consider the various motions of bodies, we must first consider this nature of matter as a mechanical affection of matter, operating in every instance; and thus we are brought back to the law enounced in this proposition. This becomes more certain when we reflect that the external causes (such as gravity or magnetism), which are acknowledged to operate changes of motion, are equally unknown to us with this essential original property of matter, and are, like it, nothing but inferences from the phenomena.

The above very diffuse discussions may appear superfluous to many readers, and even cumbersome; but we trust that the philosophical reader will excuse our anxiety on this head, when he reflects on the complicated, indistinct, and inaccurate notions commonly had of the subject. We may include Sir Isaac Newton in the number of those who have at least introduced modes of expression which mislead the minds of incautious persons, and suggest inadequate notions, incompatible with the pure doctrine of the proposition. Although in words they disclaim the doctrine that rest is the natural state of body, and that force is necessary for the continuation of its motion,

The Law yet in words they (and most of them in thought) likewise abet that doctrine: for they say that there resides in a moving body a power or force, by which it perseveres in its motion. They call it the vis insita, the inherent force, of a moving body. This is surely giving up the question: for if the motion is supposed to be continued in consequence of a force, that force is supposed to be exerted; and it is supposed that if it were not exerted, the motion would cease; and therefore the proposition must be false. Indeed it is sometimes expressed so as seemingly to ward off this objection. It is said that the body continues in uniform rectilineal motion, unless affected by some external cause. But this way of speaking obliges us, at first setting out in natural philosophy, to assert that gravity, magnetism, electricity, and a thousand other mechanical powers, are external to the matter which they put in motion. This is quite improper. It is the business of philosophy to discover whether they be external or not; and if we assert that they are, we have no principles of argumentation with those who deny it. It is this one thing that has filled the study of nature with all the jargon of ethers, and other invisible, intangible fluids, which has disgraced philosophy, and greatly retarded its progress.

32. We must observe, that the terms vis insita, inherent force, are very improper. There is no dispute among philosophers in calling every thing a force that produces a change of motion, and in inferring the action of such a force whenever we observe a change of motion. It is surely incongruous to give the same name to what has not this quality of producing a change, or to infer (or rather to suppose) the energy of a force when no change of motion is observed. This is one among many instances of the danger of mistake when we indulge in analogical discussions. All our language, at least, on this subject is analogous. I feel that in order to oppose animal force I must exert force. But I must exert force in order to oppose a body in motion; therefore I imagine that the moving body possesses force. A bent spring will drive a body forward by unbending; therefore I say that the spring exerts force. A moving body impels the body which it hits; therefore I say that the impelling body possesses and exerts force. I imagine farther, that it possesses force only by being in motion, or because it is in motion; because I do not find that a quiescent body will put another into motion by touching it. But we shall soon find this to be false in many, if not in all cases, and that the communication of motion depends on the mere vicinity, and not on the motion of the impelling body; yet we ascribe the exertion of the vis insita to the circumstance of the continued motion. We therefore conceive the force as arising from or as consisting in the impelling body's being in motion; and, with a very obscure and indistinct conception of the whole matter, we call it the force by which the body preserves itself in motion. Thus, taking it for granted that a force resides in the body, and being obliged to give it some office, this is the only one that we can think of.

33. But philosophers imagine that they perceive the necessity of the exertion of a force in order to the continuation of a motion. Motion (say they) is a continued action; the body is every instant in a new situation; there is the continual production of an effect, therefore the continual action of a cause.

This, however, is a very inaccurate way of thinking. We have a distinct conception of motion; and we conceive that there is such a thing as a moving cause, which we distinguish from all other causes by the name force. It produces motion. If it does this, it produces the character of motion, which is a continual change of place. Motion is not action, but the effect of an action; and this action is as complete in the instant immediately succeeding the beginning of the motion as it is a minute after.

The subsequent change of place is the continuation of an effect already produced. The immediate effect of the moving force is a determination, by which, if not hindered, the body would go on for ever from place to place. It is in this determination only that the state or condition of the body can differ from a state of rest; for in any instant the body does not describe any space, but has a determination by which it will describe a certain space uniformly in a certain time. Motion is a condition, a state, or mode of existence, and no more requires the continued agency of the moving cause than yellowness or roundness does. It requires some chemical agency to change the yellowness to greenness; and it requires a mechanical cause or a force to change this motion into rest. When we see a moving body stop short in an instant, or be gradually but quickly brought to rest, we never fail to speculate about a cause of this cessation or retardation. The case is no way different in itself although the retardation should be extremely slow. We should always attribute it to a cause. It requires a cause to put a body out of motion, as much as to put it into motion. This cause, if not external, must be found in the body itself; and it must have a self-determining power, and may as well be able to put itself into motion as out of it.

If this reasoning be not admitted, we do not see how any effect can be produced by any cause. Every effect supposes something done; and anything done implies that the thing done may remain till it be undone by some other cause. Without this it would have no existence. If a moving cause did not produce continued motion by its instantaneous action, it could not produce it by any continuance of that action; because in no instant of that action does it produce continued motion.

We must therefore give up the opinion, that there resides in a moving body a force by which it is kept in motion; and we must find some other way of explaining that remarkable difference between a moving body and a body at rest, by which the first causes other bodies to move by hitting them, while the other does not do this by merely touching them. We shall see, with the clearest evidence, that motion is necessary in the impelling body, in order that it may permit the forces inherent in one or both bodies to continue this pressure long enough for producing a sensible or considerable motion. But these moving forces are inherent in bodies, whether they are in motion or at rest.

34. The foregoing observations show us the impropriety of the phrase communication of motion. By this reflection of on the notions that are involved in the general conception of one body being made to move by the impulse of another, we perceive that there is nothing individual transferred from the one body to the other. The determination to motion, indeed, existed only in the impelling body before collision; whereas, afterwards, both bodies are so conditioned or determined. But we can form no notion of the thing transferred. With the same metaphysical impropriety we speak of the communication of joy, of fever.

35. Kepler introduced a term inertia, vis inertiae, into so is vis mechanical philosophy; and it is now in constant use. Inertia writers are very careless and vague in the notions which they affix to these terms. Kepler and Newton seem generally to employ it for expressing the fact, the perseverance of the body in its present state of motion or rest; but they also frequently express by it something like an indifference to motion or rest, manifested by its requiring the same quantity of force to make an augmentation of its motion as to make an equal diminution of it. The popular notion is First Law like that which we have of actual resistance; and it always implies the notion of force exerted by the resisting body. We suppose this to be the exertion of the vis insita, or the inherent force of a body in motion. But we have the same notion of resistance from a body at rest which we set in motion. Now surely it is in direct contradiction to the common use of the word force, when we suppose resistance from a body at rest; yet vis inertiae is a very common expression. Nor is it more absurd (and it is very absurd) to say, that a body maintains its state of rest by the exertion of a vis inertiae, than to say that it maintains its state of motion by the exertion of an inherent force. We should avoid all such metaphorical expressions as resistance, indifference, sluggishness, or proneness to rest (which some express by inertia) because they seldom fail to make us indulge in metaphorical notions, and thus lead us to misconceive the modus operandi, or procedure of nature.

There is no resistance whatever observed in these phenomena, for the force employed always produces its complete effect. When I throw down a man, and find that I have employed no more force than was sufficient to throw down a similar and equal mass of dead matter, I know by this that he has not resisted; but I conclude that he has resisted if I have been obliged to employ much more force. There is therefore no resistance, properly so called, when the exerted force is observed to produce its full effect. To say that there is resistance, is therefore a real misconception of the way in which mechanical forces have operated in the collision of bodies. There is no more resistance in these cases than in any other natural changes of condition. We are guilty, however, of the same impropriety of language in other cases, where the cause of it is more evident. We say that colours in grain resist the action of soap and of the sun, but that Prussian blue does not. We all perceive that in this expression the word resistance is entirely figurative; and we should say that Prussian blue resists soap, if we are right in saying that a body resists any force employed to change its state of motion; for soap must be employed to discharge or change the colour, and it does change it. Force must be employed to change a motion, and it does change it. The impropriety, both of thought and language, is plain in the one case, and it is no less real in the other. Both of the terms inherent force and inertia may be used with safety for abbreviating language, if we be careful to employ them only for expressing either the simple fact of persevering in the former state, or the necessity of employing a certain determinate force in order to change that state, and if we avoid all thought of resistance.

36. From the whole of this discussion we learn that the deviations from uniform motions are the indications of the existence and agency of mechanical forces, and that they are the only indications. The indication is very simple—mere change of place; it can therefore indicate nothing but what is very simple, the something competent to the production of the very motion that we observe; and when two changes of motion are precisely similar, they indicate the same thing. Suppose a mariner's compass on the table, and that by a small tap with my finger I cause the needle to turn off from its quiescent position ten degrees. I can do the same thing by bringing a magnet near it, or by bringing an electrified body near it, or by the unbending of a fine spring pressing it aside, or by a puff of wind, or by several other methods. In all these cases the indication is the same; therefore the thing indicated is the same, namely, a certain intensity and direction of a moving power. How it operates, or in what manner it exists and exerts itself in these instances, outwardly so different, is not under consideration at present. Impulsiveness, intensity, and direction, are all the circumstances of resemblance by which the affections of matter are to be characterized; and it is to the discovery and determination of these alone that our attention is now to be directed. We are directed in this research by the

SECOND LAW OF MOTION.

Every change of motion is proportional to the force impressed, and is made in the direction of that force.

37. This law also may almost be considered as an identical proposition; for it is equivalent to saying that the changing force is to be measured by the change which it produces, and that the direction of this force is the direction of the change. Of this there can be no doubt, when we consider the force in no other sense than that of the cause of motion, paying no attention to the form or manner of its exertion. Thus, when a pellet of tow is shot from a pop-gun by the expansion of the air compressed by the rammer, or where it is shot from a toy pistol by the unbending of the coiled wire, or when it is kicked away by the thumb like a marble—it, in all these cases, it moves off in the same direction, and with the same velocity, we cannot consider or think of the force, or at least of its exertion, as any how different. Nay, when it is driven forward by the instantaneous percussion of a smart stroke, although the manner of producing this effect (if possible) is essentially different from what is conceived in the other cases, we must still think that the propelling force, considered as a propelling force, is one and the same. In short, this law of motion, as thus expressed by Sir Isaac Newton, is equivalent to saying, "that we take the changes of motion as the measures of the changing forces, and the direction of the change for the indication of the direction of the forces:" for no reflecting person can pretend to say that it is a deduction from the acknowledged principle, that effects are proportional to their causes. We do not affirm this law from having observed the proportion of the forces and the proportion of the changes, and that these proportions are the same; and from having observed that this has obtained through the whole extent of our study of nature. This would indeed establish it as a physical law, an universal fact; and it is, in fact, so established. But this does not establish it as a law of motion, according to our definition of that term; as a law of human thought, the result of the relations of our ideas, as an intuitive truth. The injudicious attempts of philosophers to prove it as a matter of observation, have occasioned the only dispute that has arisen in mechanical philosophy. It is well known that a bullet moving with double velocity penetrates four times as far. Many other similar facts corroborate this; and the philosophers observe, that four times the force has been expended to generate this double velocity in the bullet; it requires four times as much powder. In all the examples of this kind it would seem that the ratio of the forces employed has been very accurately ascertained; yet this is the invariable result. Philosophers, therefore, have concluded that moving forces are not proportional to the velocities which they produce, but to the squares of those velocities. It is a strong confirmation to see that the bodies in motion seem to possess forces in this very proportion, and produce effects in this proportion; penetrating four times as deep when the velocity is only twice as great, &c.

But if this be a just estimation, we cannot reconcile it to the concession of the same philosophers, who grant that the velocity is proportional to the force impressed, in the cases where we have no previous observation of the ratio of the forces, and of its equality to the ratio of the velocities. This is the case with gravity, which these philosophers always measure by its accelerating power, or the velocity which it generates in a given time. And this cannot be refused by them; for cases occur where the force can be measured in the most natural manner by the actual pressure which it exerts. Gravity is thus measured by the pressure which a stone exerts on its supports. A weight which at Quito will pull out the rod of a spring steelyard to the mark 312, will pull it to 313 at Spitzbergen; and it is a fact, that a body will fall 313 inches at Spitzbergen in the same time that it falls 312 at Quito. Gravitation is the cause both of the pressure and the fall, and it is a matter of unexcepted observation that they have always the same ratio. The philosophers who have so strenuously maintained the other measure of forces are among the most eminent of those who have examined the motions produced by gravity, magnetism, electricity, &c.; and they never think of measuring those forces any other way than by the velocity. It is in this way that the whole of the celestial phenomena are explained in perfect uniformity with observation, and that the Newtonian philosophy is considered as a demonstrative science.

There must, therefore, be some defect in the principle on which the other measurement of forces is built, or in the method of applying it. Pressure is undoubtedly the immediate and natural measure of force; yet we know that four springs, or a bow four times as strong, give only a double velocity to an arrow.

The truth of our law rests on this only, that we assume the changes of motion as the measure of the changing forces, or at least as the measures of their exertions in producing motion. In fact, they are the measures only of a certain circumstance, in which the actions of very different natural powers may resemble each other, namely, the competency to produce motion. They do not, perhaps, measure their competency to produce heat, or even to bend springs. We can surely consider this apart from all other circumstances, and it is worthy of separate consideration. Let us see what can be, and what ought to be, deduced from this way of treating the subject.

38. The motion of a body may certainly remain unchanged. If the direction and velocity remain the same, we perceive no circumstance in which its condition with respect to motion differs. Its change of place or situation can make no difference; for this is implied in the very circumstance of the bodies being in motion.

But if either the velocity or direction change, then surely its mechanical condition no longer the same; a force has acted on it, either intrinsic or from without, either accelerating, or retarding, or deflecting it. Supposing the direction to remain the same, its difference of condition can consist in nothing but its difference of velocity. This is the only circumstance in which its condition can differ, as it passes through two different points of its rectilineal path. It is this determination by which the body will describe a certain determinate space uniformly in a given time, which defines its condition as a moving body; the changes of this determination are the measures of their own causes, and to those causes we have given the name force. Those causes may reside in other bodies, which may have other properties, characterized and measured by other effects. Pressure may be one of those properties, and may have its own measures; these may or may not have the same proportion with that property which is the cause of a change of velocity; and therefore changes of velocity may not be a measure of pressure. This is a question of fact, and requires observation and experience; but, in the mean time, velocity, and the change of velocity, is the measure of moving force and of changing force. When, therefore, the change of velocity is the same whatever the previous velocity may be, the changing force must be considered as the same. Therefore, finally, if the previous velocity is nothing, and consequently the change on that body is the very velocity or motion that it acquires, we must say, that the force which produces a certain change in the velocity of a moving body is the same with the force which would impart to a body at rest a velocity equal to this change or difference of velocity produced on the body already in motion.

39. This manner of estimating force is in perfect conformity to our most familiar notions on these subjects. We conceive the weight or downward pressure of a body as the cause of its motion downwards; and we conceive it as belonging to the body at all times, and in all places, whether falling, or rising upwards, or describing a parabola, or lying on a table; and, accordingly, we observe, that in every state of motion it receives equal changes of velocity in the same or an equal time, and all in the direction of its pressure.

40. All that we have now said of a change of velocity might be repeated of a change of direction. It is surely possible that the same change of direction may be made on any two motions. Let one of the motions be considered as growing continually slower, and terminating in rest. In every instant of this motion it is possible to make one and the same change on it. The same change may therefore be made at the very instant that the motion is at an end. In this case the change is the very motion which the body acquires from the changing force. Therefore, in this case also, we must say that a change of motion is itself a motion, and that it is the motion which the force would produce in a body that was previously at rest.

41. The result of these observations is evidently this, How ascertain that we must ascertain, in every instance, what is the tained and change of motion, and mark it by characters that are conspicuous and distinguishing; and this mark and measure of change must be a motion. Then we must say, that the changing force is that which would produce this motion in a body previously at rest. We must see how this is manifest as a motion in the difference between the former motion and the new motion; and, on the other hand, we must see how the motion producible in a quiescent body may be so combined with a motion already existing as to exhibit a new motion, in which the agency of the changing force may appear.

Suppose a ship at anchor in a stream, while one man walks forward on the quarter deck at the rate of two miles per hour, another walks from stem to stern at the same rate, a third walks athwart ship, and a fourth stands still. Let the ship be supposed to cut or part her cable, and float down the stream at the rate of three miles per hour. We cannot conceive any difference in the change made on each man's motion in absolute space; but their motions are now exceedingly different from what they were: the first man, whom we may suppose to have been walking westward, is now moving eastward one mile per hour; the second is moving eastward four miles per hour, and the third is moving in an oblique direction, about three points north or south of due east. All have suffered the same change of condition with the man who had been standing still. He has now got a motion eastward three miles per hour. In this instance, we see very well the circumstance of sameness that obtains in the change of these four conditions. It is the motion of the ship which is blended with the other motions. But this circumstance is equally present whenever the same previous motions are changed into the same new motions. We must learn to explicate this; which we shall do by considering the manner in which the motion of the ship is blended with each of the men's motions. 42. This kind of combination has been called the composition of motion; because, in every point of the motion really pursued, the two motions are to be found.

The fundamental theorem on this subject is this: Two uniform motions in the sides of a parallelogram compose an uniform motion in the diagonal.

Suppose that a point A (fig. 1) describes AB uniformly in some given time, while the line AB is carried uniformly along AC in the same time, keeping always parallel to its first position AB. The point A, by the combination of these motions, will describe AD, the diagonal of the parallelogram ABDC, uniformly in the same time.

For it is plain that the velocities in AB and AC are proportional to AB and AC, because they are uniformly described in the same time. When the point has got to E, the middle of AB, the line AB has got into the situation GH, half way between AB and CD, and the point E is in the place e, the middle of GH. Draw EeL parallel to AC. It is plain that the parallelograms ABDc and AEeG are similar; because AE and AG are the halves of AB and AC, and the angle at A is common to both. Therefore, by a proposition in the elements, they are about the same diagonal, and the point e is in the diagonal of AD. In like manner, it may be shown, that when A has described AF, three fourths of AB, the line AB will be in the situation IK; so that AI is three fourths of AC, and the point f, in which A is now found, is in the diagonal AD. It will be the same in whatever point of AB the describing point A be supposed to be found. The line AB will be on a similar point of AC, and the describing point will be in the diagonal AD.

Moreover, the motion in AD is uniform; for Ae is described in the time of describing AE; that is, in half the time of describing AB, or in half the time of describing AD. In like manner, Af is described in three fourths of the time of describing AD, &c. &c.

Lastly, the velocity in the diagonal AD is to the velocity in either of the sides as AD is to that side. This is evident, because they are uniformly described in the same time.

This is justly called a composition of the motions AB and AC, as will appear by considering it in the following manner: Let the lines AB, AC be conceived as two material lines like wires. Let AB move uniformly from the situation AB into the situation CD, while AC moves uniformly into the situation BD. It is plain that their intersection will always be found on AD. The point e, for example, is a point common to both lines. Considered as a point of EL, it is then moving in the direction eH or AB; and considered as a point of GH, it is moving in the direction eL. Both of these motions are therefore blended in the motion of the intersection along AD. We can conceive a small ring at e embracing loosely both of the wires. This material ring will move in the diagonal, and will really partake of both motions.

Thus we see how the motion of the ship is actually blended with the motions of the three men; and the circumstance of sameness which is to be found in the four changes of motion is this motion of the ship, or of the man who was standing still. By composition with each of the three former motions, it produces each of the three new motions. Now, when each of two primitive motions is the same, and each of the new motions is the same, the change is surely the same. If one of the changes has been brought about by the actual composition of motions, we know precisely what that change is; and this informs us what the other is, in whatever way it was produced.

Hence we infer that,

43. When a motion is anyhow changed, the change is in that motion which, when compounded with the former motion, will produce the new motion. Now, because we assume the change as the measure and characteristic of the changing force, we must do so in the present instance; and we must say,

44. That the changing force is that which will produce in a quiescent body the motion which, by composition with the former motion of a body, will produce the new motion.

And, on the other hand,

When the motion of a body is changed by the action of its own force, the new motion is that which is compounded of the former motion, and of the motion which the force would produce in a quiescent body.

When a force changes the direction of a motion, we see that its direction is transverse in some angle BAC; because a diagonal AD always supposes two sides. As we have distinguished any change of direction by the term deflection, we may call the transverse force a deflecting force.

In this way of estimating a change of motion, all the characters of both motions are preserved, and it expresses every circumstance of the change; the mere change of direction, or the angle BAD, is not enough, because the same force will make different angles of deflection, according to the velocity of the former motion, or according to its direction. But in this estimation the full effect of the deflecting force is seen; it is seen as a motion; for when half of the time is elapsed, the body is at e instead of E; when three fourths are elapsed it is at f instead of F; and at the end of the time it is at D instead of B. In short, the body has moved uniformly away from the points at which it would have arrived independent of the change; and this motion has been in the same direction, and at the same rate, as if it had moved from A to C by the changing force alone. Each force has produced its full effect; for when the body is at D, it is as far from AC as if the force AC had not acted on it; and it is as far from AB as it would have been by the action of AC alone.

For all these reasons, therefore, it is evident, that if we are to abide by our measure and character of force as a mere producer of motion, we have selected the proper characteristic and measure of a changing force; and our descriptions, in conformity to this selection, must be agreeable to the phenomena of nature, and retain the accuracy of geometrical procedure; because, on the other hand, the results which we deduce from the supposed influence of those forces are formed in the same mould. It is not even requisite that the real exertions of the natural forces, such as pressure of various kinds, &c., shall follow these rules; for their deviations will be considered as new forces, although they are only indications of the differences of the real forces from our hypothesis. We have obtained the precious advantage of mathematical investigation, by which we can examine the law of exertion which characterizes every force in nature.

45. On these principles we establish the following fundamental elementary proposition, of continual and indispensable use in all mechanical inquiries. If a body or material particle be subjected at the same time to the action of two moving forces, each of which would separately cause it to describe the side of a parallelogram uniformly in a given time, the body will describe the diagonal uniformly in the same time.

For the body, whose motion AB was changed into AD, had got its motion by the action of some force. It was moving along NAB; and when it reached the point A, the force AC acted on it. The primitive motion is the same, or the body is in the same condition in every instant of the primitive motion. It may have acquired this motion when it was in N, or when at O, or any other point of NA. In all these cases, if AC act on it when it is in A, it will always describe AD; therefore it will describe AD when it acquires the primitive motion also in A; that is, if the two forces act on it at one and the same instant. The demonstration may be neatly expressed thus: The change induced by each force on the motion produced by the other is the motion which it would produce in the body if previously at rest. Therefore the motion resulting from joint action is the motion which is compounded of these two motions, or it is a motion in the diagonal of the parallelogram, of which these motions are the sides.

This is called the composition of forces. The forces which produce the motions along the sides of the parallelogram are called the simple forces, or the constituent forces; and the force which would alone produce the motion along the diagonal is called the compound force, the resulting force, the equivalent force.

46. On the other hand, the force which produces a motion along any line whatever may be conceived as resulting from the combined action of two or more forces. We may know or observe it to be so; as when we see a lighter dragged along a canal by two horses, one on each side. Each pulls the boat directly toward himself in the direction of the track-rope; the boat cannot go both ways, and its real motion, whatever it is, results from this combined action. This might be produced by a single force; for example, if the lighter be dragged along the canal by a rope from another lighter which precedes it, being dragged by one horse, aided by the helm of the foremost lighter. Here the real force is not the resulting, or the compound, but the equivalent force.

This view of a motion mechanically produced is called the resolution of forces. The force in the diagonal is said to be resolved into the two forces, having the directions and velocities represented by the sides. This practice is of the most extensive and multifarious use in all mechanical disquisitions. It may frequently be exceedingly difficult to manage the complication of the many real forces which concur in producing a phenomenon; and by substituting others, whose combined effects are equivalent, our investigation may be much expedited. But more of this afterwards.

We must carefully remember, that when the motion AD is once begun, all composition is at an end, and the motion is a simple motion. The two determinations, by one of which the body would describe AB, and by the other of which it would describe AC, no longer co-exist in the body. This was the case only in the instant, in the very act of changing the motion AB into the motion BD; yet is the motion AD equivalent to a motion which is produced by the actual composition of two motions AB and AC; in which case the two motions co-exist in every point of AD.

47. Accordingly this is the way in which the composition of forces is usually illustrated, and thought to be demonstrated. A man is supposed (for instance) to walk uniformly from A to C on a sheet of ice, while the ice is carried uniformly along AB by the stream. The man's real motion is undoubtedly along AD; but this is by no means a demonstration that the instantaneous or short-lived action of two forces would produce that motion; the man must continue to exert force in order to walk; and the ice is dragged along by the stream. Some indeed express this proof in another way, saying; let a body describe AB, while the space in which this motion is performed is carried along AC. The ice may be carried along, and may, by friction or otherwise, drag the man along with it; but a space cannot be removed from one place to another, nor, if it could, would it take the man with it. Should a ship start suddenly forward while a man is walking across the deck, he would be left behind, and fall toward the stern. We must suppose a transverse force, and we must suppose the composition of this force without proof. This is no demonstration.

We apprehend, that the demonstration given above of this fundamental proposition is unexceptionable, when the terms force and deflection are used in the abstract sense which we have affixed to them. The only circumstance in which it can be the subject of discussion is, whether we have selected the proper measure and characteristic of a change of motion. We never met with any objection to it.

48. But some have still maintained that it does not evidently appear from these principles, that the motion to the which results from the joint action of two natural powers, moments whose known and measurable intensities have the same proportions with AB and BC, and which also exert themselves in those directions, will produce a motion having the direction and proportion of AD. They will not, if the velocities produced by these forces are not in the proportion of those intensities, but in the subduplicate ratio of them. Nay, they say that it is not so. If a body be impelled along AC by one spring, and along AB by two springs equally strong, it will not describe the diagonal of a parallelogram, of which the side AB is double the side AC. They add, that an indefinite number of examples can be given where a body does not describe the diagonal of the parallelogram by the joint action of two forces, which separately would cause it to describe the sides. And lastly, they say that at any rate it does not appear evident to the mind, that two incitements to motion, having the directions and the same proportion of intensity with that of the sides of a parallelogram, actually generate a third, which is the immediate cause of the motion in the diagonal. An equivalent force is not the same with a resulting force.

49. Yet we see numberless cases of the composition of incitements to motion, and they seem as determinate and as susceptible of being combined by composition, as the things called moving forces, which are measured by the velocities. We see them actually so combined in a thousand instances, as in the example already given of a lighter dragged by two horses pulling in different directions. Experiment even shows, that this composition follows precisely the same rule as the composition of the forces which are measured by the velocities; for if the point A (fig. I) be pulled by a thread or pressed by a spring in the direction AB, and by another in the direction AC, and if the pressures are proportional to AB and AC, then it will be withheld from moving, if it be pulled or pressed by a third force, acting in the direction Ad, opposite to AD, the pressure being also proportional to AD. This force acting in the direction Ad, would certainly withstand an equal force acting in the direction AD; therefore we must conclude that the two pressures AB and AC really generate a force AD. This uniform agreement shows that the composition is deducible from fixed principles; but it does not appear that it can be held as demonstrated by the arguments employed in the case of motions. A demonstration of the composition of pressures is farther wanted, in order to render mechanics a demonstrative science.

Accordingly, philosophers of the first eminence have turned their attention to this problem. It is by no means easy; being so nearly allied to first principles, that it must be difficult to find axioms of greater simplicity by which it may be proved.

Mechanicians generally contented themselves with the solution given by Aristotle; but this is merely a composition of motions; indeed he does not give it for any thing else, and calls it ενεργεια των συνεργών. The first writer who appears to have considered it as different from the mere composition of motions, was the celebrated Dutch engineer Stevinus, in his work on Sluices; but his solution is obscure. It was sufficient, however, to convince Daniel Bernoulli of the necessity and the difficulty of the problem. He has given the first complete demonstration of it in the first volume of the commentaries of the Imperial Academy of Sciences at St Petersburg. It is extremely ingenious; but it is tedious and intricate, requiring a series of fifteen propositions to demonstrate that two pressures, having the directions and magnitudes of the sides of any parallelogram, compose a third, which has the direction and magnitude of its diagonal. His first proposition is, that two equal pressures, acting at right angles, compose a third, in the direction of the diagonal of a square, and having to either of the other two the proportion of the diagonal of a square to its sides.

M. d'Alembert has greatly simplified and improved this demonstration, by beginning with a case that is self-evident; namely, if three equal forces are inclined to each other in equal angles of 120°, any one of them will balance the combined action of the other two. Surely; for neither of them can prevail. Therefore two equal forces, inclined in an angle of 120°, produce a third, which has the direction and proportion of the diagonal of the rhombus; for this is equal and opposite to one of the three above mentioned. He then demonstrates the same thing of two equal forces inclined in any angle; and by a series of eight propositions more, demonstrates the general theorem. This dissertation is in the Memoirs of the Academy at Paris for 1769. He improves it still farther in a subsequent memoir.

Mr Riccati and Mr Fonsonex, in the Commentaries of the Academy of Turin, have given analytical demonstrations, which are also very ingenious and concise, but require acquaintance with the higher mathematics. There is another very ingenious demonstration in the Journal des Scavans for June 1764, but too obscure for an elementary proposition. It is somewhat simplified by Bellidor in his Ingenieur Francois. Frisius, in his Cosmographia, has given one, which is perhaps the best of all those that are easily comprehended without acquaintance with the higher mathematics; but we imagine that, although no one can doubt of the conclusion, it has not that intuitive evidence for every step of the process that seems necessary.

50. We here offer another, composed by blending together the methods of Bernoulli and D'Alembert; and we imagine that no objection can be made to any step of it. We limit it entirely to pressures, and do not at all consider nor employ the motions which they may be supposed to produce.

(A.) If two equal and opposite pressures or incitements to motion act at once on a material particle, it suffers no change of motion; for if it yields in either direction by their joint action, one of the pressures prevails, and they are not equal.

Equal and opposite pressures are said to balance each other; and such as balance must be esteemed equal and opposite.

(B.) If $a$ and $b$ are two magnitudes of the same kind, proportional to the intensities of two pressures which act in the same direction, then the magnitude $a + b$ will measure the intensity of the pressure, which is equivalent, and may be called equal to the combined effort of the other two; for when we try to form a notion of pressure as a measurable magnitude, distinct from motion or any other effect of it, we find nothing that we can measure it by but another pressure. Nor have we any notion of a double or triple pressure different from a pressure that is equivalent to the joint effort of two or three equal pressures. A pressure $a$ is accounted triple of a pressure $b$, if it balances three pressures, each equal to $b$, acting together. Therefore, in all proportions which can be expressed by numbers, we must acknowledge the legitimacy of this measurement; and it would surely be affectation to omit those which the mathematicians call incommensurable.

In like manner, the magnitude $a - b$ must be acknowledged to measure that pressure which arises from the joint action of two pressures $a$ and $b$ acting in opposite directions, of which $a$ is the greatest.

(C.) Let $ABCD$ and $A'B'C'D'$ (fig. 2) be two rhombuses, which have the common diagonal $AC$. Let the angles $BAB'$, $DAD'$, be bisected by the straight lines $AE$ and $AF$. If there be drawn from the points $E$ and $F$ the lines $EG$, $EH$, $FG$, $FH$, making equal angles on each side of $EA$ and $FA$, and if $Gg$, $Hh$ be drawn, cutting the diagonal $AC$ in $I$ and $L$: then $AI + AL$ will be greater or less than $AQ$, the half of $AC$, according as the angles $GEH$, $gfh$, are greater or less than $GAH$, $gAh$.

Draw $GH$, $gh$, cutting $AE$, $AF$, in $O$ and $o$, and draw $Oo$, cutting $AC$ in $K$.

Because the angles $AEG$ and $EAG$ are respectively equal to $AEH$ and $EAH$, and $AE$ is common to both triangles, the sides $AG$, $GE$, are respectively equal to $AH$, $HE$, and $GH$ is perpendicular to $AE$, and is bisected in $O$; for the same reasons $gh$ is bisected in $o$. Therefore the lines $Gg$, $Oo$, $Hh$, are parallel, and $IL$ is bisected in $K$. Therefore $AI + AL$ is equal to twice $AK$. Moreover, if the angle $GEH$ be greater than $GAH$, $AO$ is greater than $EO$, and $AK$ is greater than $KQ$. Therefore $AI + AL$ is greater than $AQ$; and if the angle $GEH$ be less than $GAH$, $AI + AL$ is less than $AQ$.

(D.) Two equal pressures, acting in the directions $AB$ and $AC$ (fig. 3), at right angles to each other, compose a pressure in the direction $AD$, which bisects the right angle; and its intensity is to the intensity of each of the constituent pressures as the diagonal of a square to one of the sides. It is evident, that the direction of the pressure, generated by their joint action, will bisect the angle formed by their directions; because no reason can be assigned for the direction inclining more to one side than to the other.

In the next place, since a force in the direction AD does, in fact, arise from the joint action of the equal pressures AB and AC, the pressure AB may be conceived as arising from the joint action of two equal forces similarly inclined and proportioned to it. Draw EAF perpendicular to AD. One of these forces must be directed along AD, and the other along AE. In like manner, the pressure AC may arise from the joint action of a pressure in the direction AD, and an equal pressure in the direction AF. It is also plain that the pressures in the directions AE and AF, and the two pressures in the direction AD, must be all equal. And also, any one of them must have the same proportion to AB or to AC, that AB or AC has to the force in the direction AD, arising from their joint action.

Therefore, if it be said that AD does not measure the pressure arising from the joint action of AB and AC, let Ad greater than AD, be its just measure, and make Ad : AB = AB : Ag = AB : Ae. Then Ag and Ae have the same inclination and proportion to AB that AB and AC have to Ad. We determine, in like manner, two forces Af and Ag as constituents of AC.

Now Ad is equivalent to AB and AC, and AB is equivalent to Ae and Ag; and AC is equivalent to Af and Ag. Therefore Ad is equivalent to Ae, Af, Ag, and Ag. But (A) Ae and Af balance each other, or annihilate each other's effect; and there remain only the two forces or pressures Ag, Ag. Therefore (B) their measure is a magnitude equal to twice Ag. But if Ad be greater than the diagonal AD of the square, whose sides are AB and AC; then Ag must be less than AI, the side of the square whose diagonal is AB. But twice Ag is less than AD, and much less than Ad. Therefore the measure of the equivalent of AB and AC cannot be a line Ad, greater than AD. In like manner, it cannot be a line Ad that is less than AD. Therefore it must be equal to AD, and the proposition is demonstrated.

(E.) Cor. Two equal forces, AB, AC, acting at right angles, will be balanced by a force AO, equal and opposite to AD, the diagonal of the square whose sides are AB and AC; for AO would balance AD, which is the equivalent of AB and AC.

(F.) Let AECF (fig. 4), be a rhombus, the acute angle of which, EAF, is half of a right angle. Two equal pressures, which have the directions and measures AE, AF, compose a pressure having the direction and measure AC, which is the diagonal of the rhombus.

It is evident, in the first place, that the compound force has the direction AC, which bisects the angle EAF. If AC be not its just measure, let it be AP less than AC. Let ABCD be a square described on the same diagonal, and make AP : AQ = AE : AO, = AF : Ao. Draw KOG, Kog perpendicular to AE, AF; draw Glg, OHo, EG, EK, Fg, FK, PF, and PE.

The angles CAB and FAE are equal, each being half of a right angle. Also the figures AEPE and AGEK are similar; because AP : AQ = AE : AO. Therefore FA : AP = KA : AE, and EA : AP = GA : AE. Therefore, in the same manner that the forces AE, AF are affirmed to compose AP, the forces AG and AK may compose the force AE, and the forces Ag and AK may compose the force AF. Therefore (B) the force AP is equivalent to the four forces AG, AK, Ag, AK. But (D) AG and Ag are the sides of a square, whose diagonal is equal to twice AI; and the two forces AK, AK are equal to, or are measured by, twice AK. Therefore the four forces AG, AK, Ag, Ak, are equivalent to 2 AI + 2 AK = 4 AH.

But because AP was supposed less than AC, the angle FPE is greater than FAE, and GEK is greater than GAK, AO is greater than OE, and AH is greater than HQ, and 2AH is greater than AQ; and therefore 4AH is greater than AC, and much greater than AP. Therefore AP is not the just measure of the force composed of AE and AF.

In like manner it is shown that AE and AF do not compose a force whose measure is greater than AC. It is therefore equal to AC; and the proposition is demonstrated.

(G.) By the same process it may be demonstrated, that if BAD be half a right angle, and EAF be the fourth of a right angle, two forces AE, AF will compose a force measured by AC. And the process may be repeated for a rhombus, whose acute angle is \( \frac{1}{8} \)th, \( \frac{1}{16} \)th, &c. of a right angle; that is, any portion of a right angle that is produced by continual bisection. Two forces, forming the sides of such a rhombus, compose a force measured by the diagonal.

(H.) Let ABCD, Ab Cd, (fig. 5), be two rhombuses, formed by two consecutive bisections of a right angle. Let AECF be another rhombus, whose sides AE and AF bisect the angles BAB and DAD.

The two forces AE, AF, compose a force AC.

Bisect AE and AF in O and o. Draw the perpendiculars GOH, goh, and the lines Glg, OKo, HLa, and the lines EG, EH, Fg, Fh. It is evident that AGEH and AGFh are rhombuses; because AO = OE, and Ao = oF. It is also plain, that since bAd is half of BAD, the angle GAH is half of bAd. It is therefore formed by a continual bisection of a right angle. Therefore (G) the forces AG, AH, compose a force AE; and Ag, Ah, compose the force AF. Therefore the forces AG, AH, Ag, Ah, acting together, are equivalent to the forces AE, AF, acting together. But AG, Ag compose a force = 2 AI; and the forces AH, Ah compose a force = 2 AL. Therefore the four forces acting together are equivalent to 2 AI + 2 AL, or to 4 AK. But because AO is \( \frac{1}{2} \) AE, and the lines Gg, Oo, Hh, are evidently parallel, 4 AK is equal to 2 AQ, or to AC; and the proposition is demonstrated.

(I.) Cor. Let us now suppose that by continual bisection of a right angle we have obtained a very small angle \( \alpha \) of a rhombus; and let us name the rhombus by the multiple of \( \alpha \), which forms its acute angle.

The proposition (G) is true of \( \alpha, 2\alpha, 4\alpha, \ldots \). The proposition (H) is true of \( 3\alpha \). In like manner, because (G) is true of \( 4\alpha \) and \( 8\alpha \), proposition (H) is true of \( 6\alpha \); and because it is true of \( 4\alpha, 6\alpha, \) and \( 8\alpha \), it is true of \( 5\alpha \) and \( 7\alpha \). And so on continually till we have demonstrated it of every multiple of \( \alpha \) that is less than a right angle.

(K.) Let RAS (fig. 6) be perpendicular to AC, and let ABCD be a rhombus, whose acute angle BAD is some multiple of \( 2\alpha \) that is less than a right angle. Let Ab, Cd be another rhombus, whose sides Ab, Ad bisect the angles RAB, SAD. Then the forces Ab, Ad compose a force AC.

Draw BR, dS parallel to BA, DA. It is evident that ARbB and ASdD are rhombuses, whose acute angles are multiples of \( \alpha \), that are each less than a right angle. Therefore (I) the forces AR and AB compose the force Ab and AS, AD compose Ad; but AR and AS annihilate each other's effect, and there remains only the forces AB, AD. Therefore Ab and Ad are equivalent to AB and AD, which compose the force AC; and the proposition is demonstrated.

(L.) Cor. Thus is the corollary of last proposition extended to every rhombus, whose angle at A is some multiple of \( \alpha \) less than two right angles. And since \( \alpha \) may be taken less than any angle that can be named, the proposition may be considered as demonstrated of every rhombus; and we may say,

(M.) Two equal forces, inclined to each other in any angle, compose a force which is measured by the diagonal of the rhombus, whose sides are the measures of the constituent forces.

(N.) Two forces AB, AC (fig. 7) having the direction and proportion of the sides of a rectangle, compose a force AD, having the direction and proportion of the diagonal.

Draw the other diagonal CB, and draw EAF parallel to it; draw BE, CF parallel to DA.

AEBG is a rhombus; and therefore the forces AE and AG compose the force AB. AFCG is also a rhombus, and the force AC is equivalent to AF and AG. Therefore the forces AB and AC, acting together, are equivalent to the forces AE, AF, AG, and AG acting together, or to AE, AF, and AD acting together; but AE and AF annihilate each other's action, being opposite and equal (for each is equal to the half of BC). Therefore AB and AC acting together are equivalent to AD, or compose the force AD.

(O.) Two forces, which have the direction and proportions of AB, AC (fig. 8) the sides of any parallelogram, compose a force having the direction and proportion of the diagonal AD.

Draw AF perpendicular to BD, and BG and DE perpendicular to AC.

Then ABFG is a rectangle, as is also AFDE; and AG is equal to CE. Therefore (N) AB is equivalent to AF and AG. Therefore AB and AC acting together, are equivalent to AF, AG, and AC acting together; that is, to AF and AE acting together; that is (N) to AD; or the forces AB and AC compose the force AD.

51. Hence arises the most general proposition.

If a material particle be urged at once by two pressures Comp. or inclemencies to motion, whose intensities are proportional to the sides of any parallelogram, and which act in the directions of those sides, it is affected in the same manner as if it were acted on by a single force, whose intensity is measured by the diagonal of the parallelogram, and which acts in its direction: Or, two pressures, having the direction and proportion of the sides of a parallelogram, generate a pressure having the direction and proportion of the diagonal.

52. Thus have we endeavoured to demonstrate from abstract principles the perfect similarity of the composition of pressures, and the composition of forces measured by the motions which they produce. We cannot help being of the opinion that a separate demonstration is indispen- DYNAMICS

What may be fairly deduced from the one case cannot always be applied to the other. No composition of pressures can explain the change produced by a deflecting force on a motion already existing; for the changing pressure is the only one that exists, and there is none to be compounded with it. And, on the other hand, our notions and observations of the composition of motions will not explain the composition of pressures, unless we take it for granted that the pressures are proportional to the velocities; but this is perhaps a gratuitous assumption. At any rate, it is not an intuitive proposition; and we have mentioned some facts where it seems that they do not follow the same proportion. The pressure of four equal springs produces only a double velocity. It would appear, therefore, that there are circumstances which oblige us to say, that the exertion of pressure, as a cause of motion, is not (always at least) proportional to the real measurable pressure. We are therefore anxious to discover in what the difference consists; and in the mean time must allow, that the pressure exerted on a body at rest is different from its exertion in producing motion. We cannot indeed state any immediate comparison between pressure and motion, nor have we any clear conception of the connection between them. It is only by our own sensations of touch that we have any notion of pressure, and it is experience that teaches us that it always accompanies every cause of motion. We can, however, observe the proportions of pressures, and compare them with the proportions of motion. We very often observe them different; and therefore it was indispensably necessary to investigate the laws of combined pressure as we did the laws of combined motion in consequence of pressure. Yet we should err if we hastily asserted that pressures are not proportional to the motions which they produce; all that we are entitled to call in doubt is, whether the pressures in their exertion, while they actually produce motion, or changes of motion, continue to be the same as when they do not produce motion, being withstood or balanced by opposite pressures. Considered as causes of motion, we ought to think that they do not vary while they produce motion, and that the actual pressure, while it produces a double motion, is really double, although it may be quadruple when the body exerting it is made to act on a body that it cannot move. We are confirmed in this opinion by observing, that other facts show us, that even while producing motion, the pressure which we call quadruple, because we have measured it by four equal pressures balancing it, is really quadruple, considered as the cause of motion, and produces a quadruple motion. A bow which requires four times the force to draw it to any given extent, will communicate the same velocity to a bundle of four arrows that a bow four times more easily drawn communicates to one arrow, and will therefore produce a quadruple motion. Yet it will only produce a double velocity in the arrow that acquired a simple velocity from a bow having one fourth of the strength.

These discrepancies should excite the endeavours of mechanicians to investigate the laws observed in the action of pressures producing motion. Had this been done with care and with candour, we should not have had the great difference of opinion which still divides philosophers about the measures of moving forces. But a spirit of party, which had arisen from other causes, gave importance to what was at first only a difference of expression, and made the partizans of Mr Leibnitz avail themselves of the figurative language which has done so much harm in all the departments of philosophy. Notwithstanding all our caution, it is hardly possible to avoid metaphorical conceptions when we employ the language of metaphor. The abettors of the Leibnitzian measure of moving forces, or perhaps, to speak more properly, the abettors of the Leibnitzian measure of that force which is supposed to preserve bodies in their condition of motion, insist that the force which is exerted in producing any change of motion is greater in proportion as the motion changed is greater; and they give a very specious argument for their assertion. They appeal to the exertions which we ourselves make. Here we are conscious of the fact. Then they give similar examples of the action of bodies. A clay ball, moving six feet per second, will make the addition of one foot to the velocity of an equal clay ball that is already moving four feet per second in the same direction. But if this last ball be already moving ten feet per second, we must follow it with a velocity of twelve feet in order to increase its velocity one foot. But, without insisting on the numberless paradoxisms and inconsistencies which this way of conceiving the matter would lead us into, it suffices to observe, that the phenomena give us abundant assurance that there has been the same exertion in both these cases. This acceleration is always accompanied by a compression of the balls; and the compression is the same in both. This compression is a very good measure of the force employed to produce it; and, in the present case, we need not even trouble ourselves with any rule for its measurement: for surely, when the compression is not different, but the same, the force exerted is the same. This is farther confirmed by observing, that it requires the same force to make the same pit, or to give the same motion, to a piece of clay lying on the table of a ship's cabin, whether the ship be sailing two miles or ten miles per hour.

Thus we see that there are strong reasons for believing, that the exertions of pressure in producing motion, or that the pressures actually exerted, are proportional to the changes of motion observed, and that they coincide in this respect with our abstract conceptions of moving forces.

But we have still better arguments. None of the Leibnitzians think of denying the equal exertions of gravity, or of any of those powers which they call solicitations or accelerating forces. They all admit that gravity, or any constant accelerating force, produces equal increments of velocity in equal times, and that a double gravity will produce a double increment in an equal time, and an equal increment in half of the time: and that a quadruple gravity will produce a double velocity in half the time. All these things are granted by them, and their writings are full of reasonings from this principle. Now, from the fact, acknowledged by the Leibnitzians, that the quadruple force of a bow gives a double velocity to the arrow, in every instant of its action, it indisputably follows, that it has acted on it only for half the time of the action of the four times weaker bow, which gives the arrow only half the velocity; and thus has the discrepancy between the effects of pressures and of our abstract moving forces entirely disappeared. For this circumstance of the difference in the time of acting will be found, on strict examination, in all the cases of the change of motion by pressures which we measure by their effects on a body at rest. When this and the appreciable changes of actual pressure, during the time of producing the motion, are taken into consideration, all difference vanishes, and the composition of pressures is in perfect harmony with the composition of motions, or of abstract moving forces. Dynamics is thus made a demonstrative science, and affords the opportunity of investigating, by observation and experiment, the nature of those mechanical powers which reside in bodies, and which appear to us under the form of pressure, inducing us to consider pressure as a cause of motion.

In this, however, we are rather inaccurate. Pressure is one of the sensible effects of that property which is also the cause of motion. It is not the pressure of a piece of lead, but its heaviness, that is the reason that it gives motion to a kitchen jack. Pressure is merely a generic name, borrowed from a similar instance, and given to moving forces, which have the same nature, but different names that serve to mark their connection with certain substances, in which they may be supposed to reside. Natural philosophy is almost entirely employed in examining the nature of these various pressures or accelerative forces; and the general doctrines of dynamics, by ascertaining what is common to them all, enable us to mark with precision what is characteristic of each.

53. We have now advanced very far in this investigation; for we have obtained the criterion by which we learn the direction and the magnitude of every changing force; and, on the other hand, we see how to state what will be the effect of the exertion of any force that is known or suspected to act. All this we learn by the composition of forces; and the greatest part of mechanical disquisition consists in the application of this doctrine. For such reasons it merits minute consideration; and therefore we must point out some general conclusions from the properties of figure, which will greatly facilitate the use of the parallelogram of forces.

54. First, The constituent and the resulting forces, or the simple and compound forces, act in the same plane; for the sides and diagonal of a parallelogram are in one plane.

Fig. 9.

55. Second, The simple and the compound forces are proportional to the sides of any triangle which are parallel to their directions. For if any three lines, ab, bc, cd, be drawn parallel to AB, AC, and AD (fig. 9), they will form a triangle similar to the triangle ABD. For the same reasons they are proportional to the sides of a triangle abcd, which are respectively perpendicular to their directions.

56. Third, Therefore each is proportional to the sine of the opposite angle of this triangle; for the sides of any triangle are proportional to the sines of the opposite angles.

57. Fourth, Each is proportional to the sine of the angle contained by the directions of the other two; for AD is to AB as the sine of the angle ABD to the sine of the angle ADB. Now the sine of ABD is the same with the sine of BAC contained between the directions AB and AC, and the sine of ADB is the same with the sine of CAD; also AB is to AC, or BD, as the sine of ADB (or CAD) to the sine of BAD.

58. We now proceed to the application of this fundamental proposition. And we observe, in the first place, that since AD may be the diagonal of an indefinite number of parallelograms, the motion or the pressure AD may result from the joint action of many pairs of forces. It may be produced by forces which would separately produce the motions AF and AG. This generally gives us the means of discovering the forces which concur in its production. If one of them, AB, is known in direction and intensity, the direction AC, parallel to BD, and the intensity, are discovered. Sometimes we know the directions of both. Then, by drawing the parallelogram or triangle, we learn their proportions. The force which deflects any motion AB into a motion AD, is had by simply drawing a line from the point B (to which the body would have moved from A in the time of really moving from A to D) to the point D. The deflecting force is such as would have caused the body move from B to D in the same time. And, in the same manner, we get the compound motion AD, which arises from any two simple motions AB and AC, by supposing both of the motions to be accomplished in succession. The final place of the body is the same, whether it moves along AD or along AB and BD in succession.

59. This theorem is not limited to the composition of two motions or two forces only; for since the combined action of two forces puts the body into the same state as if their equivalent alone had acted on it, we may suppose this to have been the case, and then the action of a third force will produce a change on this equivalent motion. The resulting motion will be the same as if only this third force and the equivalent of the other two had acted on the body. Thus, in fig. 10, the three forces AB, AC, AE, may act at once on a particle of matter. Complete the parallelogram ABDC; the diagonal AD is the force which is generated by AB and AC. Complete the parallelogram AEFD; the diagonal AF is the force resulting from the combined action of the forces AB, AC, and AE. In like manner, completing the parallelogram AGHF, the diagonal AH is the force resulting from the combined action of AB, AC, AE, and AG, and so on of any number of forces.

Fig. 10.

This resulting force and the resulting motion may be much more expeditiously determined, in any degree of composition, by drawing lines in the proportion and direction of the forces in succession, each from the end of the preceding. Thus, draw AB, BD, DF, FH, and join AH; AH is the resulting force. The demonstration is evident.

60. It is to be noticed here, that in the composition of more than two forces, we are not limited to one plane. The force AD is in the same plane with AB and AC; but AE may be elevated above this plane, and AG may lie below it. AF is in the plane of AD and AE, and AH is in the plane of AF and AG.

Complete the parallelograms ABLE, ACKE, ELFK. It is evident that ABLFKCD is a parallelogram, and that AF is one of its diagonals. Hence we derive a more general theorem of great use.

Three forces having the proportion and direction of the three sides of a parallelogram, compose a force having the proportion and direction of the diagonal.

61. Any number of forces acting together on one particle of matter are balanced by a force that is equal and opposite to their resulting force; for this force would balance their resulting force which is equivalent to them in action. When this is duly considered, we perceive that each force is then in equilibrium with the equivalent of all. the others; for a force can balance only what is equal and opposite to it. It appears very readily by the geometrical construction. If, instead of the circuit A, B, D, F, H, we take B, D, F, H, A, we have BA for the equivalent of the forces AC, AE, AG; but AB is equal and opposite to BA. Therefore the force AB is in equilibrium with the equivalent of all the others.

62. When any number of forces act on one particle of matter, and are in equilibrium, if they be considered as acting in parcels, the equivalents of these parcels are in equilibrium; for let the forces AB, AC, AE, AG, AH, be in equilibrium, and let them be considered in the two parcels AB, AC, and AE, AG, AH; then AD is the equivalent of AB, BD (or AC), and DA is the equivalent of DF, FH, HA (or AH); now AD and DA balance each other. This corollary enables us to simplify many intricate complications of force; it also enables us to draw accurate conclusions from very imperfect observations. In most of our practical discussions we know, or at least we attend to, a part only of the forces which are acting on a material particle; and in such cases we reason as if we saw the whole; yet is our mathematical reasoning good with respect to the equivalent of all the parcels which we are contemplating; and the equivalents of the smaller parcels of which it consists; and the neglected force, or parcel of forces, induces no error on our conclusions.

63. In the spontaneous phenomena of nature, the investigation and discovery of our ultimate object of search is frequently very difficult, on account of the multiplicity of directions and intensities of the operating forces or motions. We may generally facilitate the process, by substituting equivalent forces or motions acting in convenient directions. It is in this way that the navigator computes the ship's place with very little trouble, by substituting equivalent motions in the meridional and equatorial directions for the real oblique courses of the ship. Instead of setting down ten miles on a course, S. 86° 52' W. he supposes that the ship has sailed eight miles due south, and six miles due west, which brings her near to the same place. Then, instead of fourteen miles south-west, he sets down ten miles south and ten miles west; and he proceeds in the same way for every other course and distance. He does this expeditiously by means of a traverse table, in which are ready calculated the meridional and equatorial sides of right-angled triangles, corresponding to every course and distance. Having done this for the course of a whole day; he adds all the southings into one sum, and all the westings into another; he considers these as forming the sides of a right-angled triangle; he looks for them, paired together, in his traverse table, and then notices what angle and what distance corresponds to this pair. This gives him the position and magnitude of the straight line joining the beginning and end of his day's work.

The miner proceeds in the same way when he takes the plan of subterraneous workings, measuring, as he goes along, and noticing the bearing of each line by the compass, and setting down, from his traverse table, the northing or southing, and the easting or westing, for each oblique line; but there is another circumstance which he must attend to, namely, the slope of the various drifts, galleries, and other workings. This he does by noting the rise or the dip of each sloping line. He adds all these into two sums, and taking the risings from the dips, he obtains the whole dip. Thus he learns how far the workings proceed to the north, how far to the east, and how far to the dip.

The reflecting reader will perceive that the line joining the two extremities of this progression will form the diagonal of a rectangular parallelopiped; one of whose sides lies north and south, the other lies east and west, and the third is right up and down.

The mechanician proceeds in the very same way in the investigation of the very complicated phenomena which frequently engage his attention. He considers every motion as compounded of three motions in some convenient directions, at right angles to each other. He also considers every force as resulting from the joint action of three forces, at right angles to each other, and takes the sum or difference of these in the same or opposite directions. From this process he obtains the three sides of a parallelopiped, and from these computes the position and magnitude of the diagonal. This is the motion or force resulting from the composition of all the partial ones.

This procedure is called the estimation or reduction of forces motions and forces.

Fig. 11.

64. A motion or force AB (fig. 11) is said to be estimated in the direction EF, or to be reduced to this direction when it is conceived as compounded of the motions or forces AC, AD, one of which, AC, is parallel to EF, and the other, AD, is perpendicular to it. This expression is abundantly significant; for it is plain that the motion AD neither promotes nor hinders the progress along EF, and that AC expresses the whole progress in this direction.

Fig. 12.

65. In like manner, a force AB (fig. 12) is said to be estimated in, or reduced to, a given plane EFG when it is conceived as resulting from the joint action of two forces AC, AD, one of which is parallel to a line ab drawn in that plane, and the other AD is perpendicular to it. The position of the line ab is determined by letting fall Bb perpendicular to the plane, and drawing bP to the point P, in which BA meets the plane; then Aa being drawn parallel to Bb, will cut off ba, which is the reduction of the motion AB to the plane. Drawing AC parallel to ab, and completing the parallelogram ACBD, it is evident that the motion AB is equivalent to AD and AC, which is parallel to ab, and the three forces AB, AC, AD, are, as they should be, in one plane perpendicular to the plane EG.

66. If three forces AB, AC, AD (fig. 13) are in equilibrium, Equilibria are reduced to any one direction dA, or to one plane of forces so estimated or reduced.

First, let them be reduced to one direction dl by drawing the perpendiculars Ba, Ce, Dd; make AL equal to AD, and join BL, CL, and draw the perpendiculars Lz, Ce; then, because the forces AB, AC, AD, are in equilibrium, ABLC must be a parallelogram, and AL is the force equivalent to AB and AC combined; then, because the lines Dd, Bb, Ce, Lz are parallel, dA is equal to AL, and Ab to Co, or to cL; therefore Al is equal to the sum of Ab and Ac, which are the reductions of AB and AC; Secondly, let them be reduced to one plane, EFGH, and let \(a_3\), \(a_2\), \(a_1\) be the reduced forces. The lines \(D_3\), \(A_3\), \(B_3\), \(C_3\), \(L_3\) are all parallel, being perpendicular to the plane; therefore the planes \(AB\), \(BC\), and \(CL\), \(L_3\) are parallel, and \(a_3\), \(a_2\), \(a_1\) are parallel. For similar reasons, \(b_3\), \(b_2\), \(b_1\) are parallel; therefore \(a_3b_3\) is a parallelogram.

Also, because the lines \(D_3\), \(A_3\), \(L_3\) are parallel, and \(DA\) is equal to \(AL\); therefore \(a_3\) is equal to \(a_1\). But because \(a_3b_3\) is a parallelogram, the forces \(a_3\), \(a_2\), \(a_1\) are equivalent to \(a_3\); and \(a_3\) is equal and opposite to \(a_3\), and will balance it; and therefore will balance \(a_3\) and \(a_2\), which are the reductions of \(AB\) and \(AC\) to the plane EFGH, while \(a_3\) is the reduction of \(AD\); therefore the proposition is demonstrated.

The most usual and the most useful mode of reduction is to estimate all forces in the directions of three lines drawn from one point, at right angles to each other, like the three plane angles of a rectangular chest, forming the length, the breadth, and the depth of the chest. These are commonly called the three co-ordinates. The resulting force will be the diagonal of this parallelopped. This process occurs in all disquisitions in which the mutual action of solids and fluids is considered, and when the oscillation or rotation of detached free bodies is the subject of discussion.

67. The only other general theorem that remains to be deduced from this law of motion is, that if a number of bodies are moving in any manner whatever, and an equal force act on every particle of matter in the same or parallel directions, their relative motions will suffer no change; for the motion of any body \(A\) (fig. 14), relative to another body \(B\), which is also in motion, is compounded of the real motion of \(A\), and the opposite to the real motion of \(B\); for let \(A\) move uniformly from \(A\) to \(C\), while \(B\) describes \(BD\) uniformly; draw \(AB\), also draw \(AE\) equal and parallel to \(BD\); join \(EC\), \(DC\), \(ED\). The motion of \(A\), relative to \(B\), consists in its change of position and distance.

Had \(A\) described \(AE\), while \(B\) described \(BD\), there would have been no change of relative place or distance; but \(A\) is now at \(C\), and \(DC\) is its new direction and distance. The relative or apparent motion of \(A\) therefore is \(EC\). Complete the parallelogram \(ACFE\); it is plain that the motion \(EC\) is compounded of \(EF\), which is equal and parallel to \(AC\), the real motion of \(A\), and of \(EA\), the equal and opposite to \(BD\), the real motion of \(B\).

Now let the motions of \(A\) and \(B\) sustain the same change; let the equal and parallel motions \(AG\), \(BH\), be compounded with the motions \(AC\) and \(BD\); or let forces act at once on \(A\) and \(B\), in the parallel directions \(AG\), \(BH\), and with equal intensities; in either supposition, the resulting motions will be \(Ac\), \(Bd\), the diagonals of the parallelograms \(AGcG\) and \(BHDd\). Construct the figure as before, and we see that the relative motion is now \(ec\), and that it is the same with \(EC\) both in respect of magnitude and position.

Here we still see the constant analogy between the composition of motions and the composition of forces. In the first case, the relative motions of things are not changed, whatever common motion be compounded with them all; or, as it is usually but inaccurately expressed, although the space in which they move be carried along with any motion whatever. In the second case, the relative motions and actions are not changed by any external force, however great, when equally exerted on every particle in parallel directions.

Thus it is that the evolutions of a fleet in a uniform current are the same, and produced by the same means, as in still water. Thus it is that we walk about on the surface of this globe in the same manner as if it neither revolved round the sun, nor turned round its axis. Thus it is that the same strength of a bow will communicate a certain velocity to an arrow, whether it is shot east, or west, or north, or south. Thus it is that the mutual actions of sublunary bodies are the same, in whatever directions they are exerted, and notwithstanding the very great changes in their velocities by reason of the earth's rotation and orbital revolution. The real velocity of a body on the earth's equator is about 1000 feet per second greater at midnight than at mid-day. For at midnight the motion of rotation nearly conspires with the orbital motion, and at mid-day it nearly opposes it. The difference between the velocities at the beginning of January and the beginning of July is vastly greater. And at other times of the day, and other seasons of the year, both motions of the earth are transversely compounded with the easterly or westerly motion of an arrow or cannon bullet; yet we can observe no change in the effects of the mutual actions of bodies.

68. This is an important observation, because it proves that forces are to be measured by no other scale than by the motions which they produce. We have had repeated occasions to mention the very different estimation of moving forces by Mr Leibnitz; and have shown how, by a close consideration of what passes when motion is produced by measurable pressures, proves that the forces really exerted are as the velocities produced. But the most copious proof is had from the present observation, that, in fact, the mutual actions of bodies depend on their relative motions alone.

69. The Leibnitzian measure of moving force is altogether incompatible with the universal fact now mentioned,

viz. that the relative motions of bodies, resulting from their mutual actions, are not affected by any common motion, or the action of any equal and parallel force on both bodies; for this universal fact imports, that when two bodies are moving with equal velocities in the same direction, a force applied to one of them, so as to increase its velocity, gives it the same motion relative to the other, as if both bodies had been at rest. Here it is plain that the space described by the body in consequence of the primitive force, and of the force now added, is the sum of the spaces which each of them would generate in a body at rest. Therefore the forces are proportional to the velocities or changes of motion which they produce, and not to the squares of those velocities. This measure of forces, or the position that a force makes the same change on any velocity whatever, and the independence of the relative motions on any motion that is the same on all the bodies of a system, are counterparts of each other. Since this independence is a matter of observation in all terrestrial bodies, we are entitled to say that the powers which the Author of nature has imparted to natural bodies are no way different from what are competent to matter once called into existence. And it also follows from this, that we must always remain ignorant of the absolute motions of bodies. The fact that it has required the unremitting study of ages to discover even the relative motions of our solar system, is an argument to prove that the influence of this mechanical principle extends far beyond the limits of this sublunary world; nor has any phenomenon yet been exhibited which should lead us to imagine that it is not universal.

When we have made use of these arguments with some zealous partizans of Mr Leibnitz's doctrine, they have answered, that if indeed this independence of the relative motions of terrestrial bodies were observed to obtain exactly, it would be a conclusive argument. But the motion with which all is carried along is so great in comparison with the motions which we can produce in our experiments, that the small additions or diminutions that we can make to the velocity of this common motion must observe very nearly the proportions of the additions or diminutions of their squares. The differences of the squares of 2, 3, and 4, are very unequal; but the differences of the squares of 9, 10, 11, are much nearer to the ratio of equality; and the differences of the squares of 1000001, 1000002, 1000003, do not sensibly deviate from this ratio. But it is not fact that we cannot produce motions which have a very sensible proportion to the common motion. The motion of a cannon ball discharged with one third of its weight of powder, is nearly equal to that of the rotation of the earth's equator. When, therefore, we discharge the ball eastward, we double its motion; when to the westward, we destroy it. Therefore, according to Leibnitz, the action in the first case is three times the action in the second. In the first case it changes the square of the velocity (which we may call 1) from 1 to \( \frac{1}{4} \); and, in the second, it changes it from 1 to 0. But, say the Leibnitzians, the velocity of rotation is but \( \frac{1}{365} \) of the orbital velocity of the earth, and our observations of the velocities of cannon bullets are not sufficiently exact to ensure us against an error of \( \frac{1}{365} \). But the latter observations on the peculiar motions of the fixed stars concur in showing, that the sun, with his attending planets, are carried along with a very great motion, which, in all probability, has a sensible ratio to the orbital motion of the earth. This must make a prodigious change on the earth's absolute motion, according as her orbital motion conspires with, opposes, or crosses, this other motion; the earth may even be at absolute rest in some points of its orbit. Thus will the composition with the motions produced in our experiments be so varied that cases must occur when the difference of the results of the two measures of force will be very sensible.

But, further, they have not attended to the agreement of our experiments when the discharges of cannon are made in a direction transverse to that of the common motion. Here the immensity of the common motion, and the minuteness of our experimental velocities, can have no effect in diminishing the difference of the results of the two doctrines. This will appear distinctly to every reader who is much conversant in disquisitions of this kind; and it is in these more moderate motions that the complete independence of the relative motions on the common motions most accurately appears. Pendulum clocks and watches have been often executed which do not deviate from perfect equability of motion one part in 86,400. This could not be obtained in all directions of the oscillations, if the forces deviated from the ratio of the velocities one part in 86,400.

On the whole, we may consider it as established on the Perfect surest foundation, that the action of those powers of natural bodies which we call pressures, such as the force of the abstractive springs, the exertions of animals, the cohesion of bodies, as well as the action of those other incitements to motion which we call attractions and repulsions, such as gravitation, all our ac- magnetism, and electricity, is proportional to the change curate ob- of velocity produced by it. And we must observe here, that this is not a mere mode of conception, the result of the laws of human thought, which cannot conceive a natural power as the cause of motion otherwise than by its producing motion, and which cannot conceive any degree of moving power different from the degree of the motion. This is the abstract doctrine, and is true whether the pressures are proportional to the velocities or to the squares of the velocities.

But we see further, that whatever is the pressure of a spring (for example) on a quiescent body, yet the pressure actually exerted in producing a double velocity is only double, and not quadruple, as our first imperfect observations make us imagine.

70. Sir Isaac Newton has added another proposition to Newton's number of laws of motion; namely, that every action third law is accompanied by an equal and contrary re-action. But in motion is founded affirming this to be a law of nature, he only means that it on experi- is a universal fact; and he makes this affirmation on the case alone, authority of what he conceives to be a law of human and is not thought; namely, that those qualities which we find in a necessary all bodies on which we can make experiments and ob- truth. tractions, are to be considered as universal qualities of body. But we have limited the term law of motion to those consequences that necessarily flow from our notions of motion, of the causes of its production and changes. Now this third Newtonian proposition is not such a result. A magnet is said to act on a piece of iron, when, and only when, the vicinity of the magnet is observed to be accom- panied by certain motions of the iron. But it by no means follows from this observation, that the presence of the iron shall be accompanied by any motion, or any change of state whatever of the magnet, or any appearance that can suggest the notion that the iron acts on the magnet. When this was observed, it was accounted a discovery. Newton discovered that the sun acts on the planets, and that the earth acts on the moon; and Kepler discovered that the moon reacts on the earth. Newton had observed that the iron reacts on the magnet; that the actions of electrified bodies were mutual; and that every action of sublunary bodies was, in fact, accompanied by an equal and contrary re-action. On the authority of his rule of philosophizing, he affirmed that the planets react on the sun, and that the sun is not at rest, but is continually agi- tated by a small motion round the general centre of gravitation. He pointed out several consequences of this reaction. Astronomers examined the celestial motions more narrowly, and found that those consequences do really obtain, and disturb all the planetary motions. It is now found that this reciprocity of action obtains throughout the solar system with the utmost precision, and that the third Newtonian proposition is really a law of nature, although it is not a law of human thought. It is a discovery. The contrary involves no absurdity or contradiction. It would indeed be contrary to experience; but things might have been otherwise. It is conceivable and possible that a ball A shall strike another equal ball B, and carry it along with it, without any diminution of its velocity. The fact that the velocity of A is reduced to one half, is the indication of a force residing in B, which force changes the motion of A; and the intensity of this force is learned from the change which it produces. This is found to be equal to the change produced by A on B. And thus the re-action of B is discovered to be equal to the action of A.

It is highly probable that this universality and equality of re-action to action is the consequence of some general principle, which we may in time discover; meanwhile we are entitled to suppose it universal, and to reason from this topic in our disquisitions about the actions of bodies on each other.

Although the celebrated philosophers of Europe have at last agreed in the reception of the two propositions so largely discussed by us as the laws of motion, they have differed exceedingly in their opinion about their origin and validity. Some asserted that they are entirely matters of experience, while others affirmed them to be necessary truths. The Royal Academy of Berlin made this question the subject of their prize dissertation in the year 1744. Mr Maupertuis, president of the academy, published a dissertation, in which he endeavoured to prove that they are necessary truths, only because they are such as make the quantity of action the least possible, an economy which is worthy of infinite wisdom, and therefore certainly directs the choice of the Author of nature. On this account alone are they necessary truths.

But this is not the way to consider a question of this kind. We know too little about infinite wisdom to be able to say, with Messrs Leibnitz and Maupertuis, that the Deity should or should not impress on bodies laws different from those which are essential to matter; and we are not to inquire whether God could or could not do this. We know from our own experience, that matter, when subjected to the action of intelligence, may be moved in a way extremely different from what it would follow if left to itself; and that its motions may either be regulated by fixed but contingent law, or may be without any constancy whatever, and vary in every instance. When we suppose the existence of matter and motion, a variety of truths are involved in the supposition, in the same manner as all the theorems in the third book of Euclid's Elements are involved in the conception of a circle and a straight line. Our first employment should be to evolve those truths. We can do this in no way but by first noticing the relations of the ideas that we have of the different objects of contemplation, and then following the laws of human thought in our judgments concerning those relations. This process of the mind is expressed in the train of a geometrical demonstration. The different parts or argumentations of this train are not the causes of our conclusions, but the means by which we form our judgment; not the reasons of the truth of our ultimate conclusion, but the steps by which we arrive at the knowledge of it. The young geometer generally thinks otherwise. But that this is the matter of fact, is plain from this, that more than one demonstration, and often very different, can be given of the same theorem. We must proceed in the same manner in the present question; and the first general truths which we find involved in the notions of matter, motion, and force, must be received as necessary truths. The steps by which we arrive at the discovery are the laws of human thought; and the expression of the discovery, involving both the truth itself, and the manner of conceiving it, is a necessary law of motion. There may be other facts, perhaps as general as any of those necessary laws, but which do not necessarily result from the relations of our notions of motion and of force. These are discovered by observation only, and they serve to characterize the forces which nature presents to our view. These facts are contingent laws of motion.

We apprehend that this method has been followed in treating this article. The first proposition, termed a law of motion, is only a more convenient way of expressing our contemplation of a motion in body as an effect of the general cause which we term force. The second proposition does nothing but express more distinctly the relation between this cause and its effect; it expresses what we mean by the magnitude and the kind of the cause. The proposition, stating the composition of forces, is but another form of the same law, better suited to the ordinary procedure in geometrical disquisitions.

These propositions might have completed the doctrines of dynamics; but it appears that, in order to the production of a material universe which should accomplish the purposes of the Creator, it was necessary that there be certain characteristic differences between the forces inherent in the various collections of matter which compose this universe. The facts or physical laws (for the above-mentioned laws are metaphysical) of motion may be different from those which would have been observed had matter been left entirely to itself. This difference may have introduced other laws of motion as necessarily resulting from the nature of the forces. We have occasionally mentioned some instances where this appears to obtain, but gave good reasons for affirming that a due examination of all circumstances which may be observed in the production or variation of motion by those forces, has demonstrated, that there are no such deviations from the two laws of motion already determined, but that all the mechanical powers of bodies, when considered merely as causes of motion, act agreeably to the same laws. Careful examination was, however, said to be necessary.

This examination must consist in distinctly noticing the circumstances that occur in the production of motion by any force whatever. It is by no means enough to state simply the intensity of the force and the direction of its exertion. If a force continue to act, it continues to vary the motion already produced. Should the force change its intensity or direction while it is acting, these circumstances must induce still farther changes in the motion; and it is not till all action has ceased that the motion is brought to its ostensible state, in which it is the object of our attention and our future discussions. Instances of the effects of such continued and such varied actions are to be seen in most of the phenomena of nature or art. The communication of motion by impulse is perhaps the only instance (very frequent indeed) that can be produced where this is not necessary; nay, we shall perhaps find reason to conclude that this instance is not an exception, and that even the communication of motion from one billiard hall to another is brought about by an action continued for some time, and greatly varied during that time. Much preparation is therefore necessary before we can apply the OF ACCELERATED AND RETARDED MOTIONS.

71. All men can perceive that a stone dropped from the hand, or sliding down an uniform slope, has its motion continually accelerated, and that the motion of an arrow rising perpendicularly through the air is continually retarded; and they feel no difficulty in conceiving these changes of motion as the effects of the continual operation of their weight or heaviness. The falling stone is in a different condition in respect of motion in the beginning and end of its fall. In what respect do these states of the body differ? Only in respect to what we call its velocity. This is an affection of motion; it is an expression of the relation between the two notions or ideas which concur to form the idea of motion, namely, the space and the time. These are all the circumstances that we observe in a motion. Time elapses, and during its currency a space is described. The term velocity expresses the magnitude of the space, which corresponds to some unit of time. Thus the rate of a ship's motion is determined when we say that it is nine miles in an hour, or nine miles per hour. We sometimes say (but awkwardly) "the motion is at the rate or with the velocity of a mile in three days." It is most conveniently expressed by a number of some given units of length, which completely make up the line described during this unit of time. But the mechanicians express it in a way more general by a fraction, of which the numerator is a number of inches, feet, yards, fathoms, or miles, and the denominator is the number of seconds, minutes, or hours employed in moving along this line. This is a very proper expression; for when we speak of any velocity, and continue to reason from it, we conceive ourselves to speak of something that remains the same, in the different occasions of using the term. Now if the velocity be constant, it is indifferent how long the line may be, because the time of its description will be lengthened in the same proportion. Thus if 48 feet be described in 12 seconds, 36 feet will be described in 9 seconds, 16 feet will be described in 4 seconds, &c. Now $\frac{4}{3}$, $\frac{3}{2}$, and $\frac{2}{1}$, are fractions of equal value, being equal to $\frac{4}{3}$, or 4, that is, to the velocity of 4 feet per second. The value of this fraction, or the quotient of the number of the units of length, divided by the number of units of time, is the number of those units of length described uniformly in one unit of time.

But how shall we determine the velocity in any instant or in any point of a motion that is continually changing? Suppose that a body has fallen 144 feet, and that we would ascertain its velocity in that point of its fall, or the velocity which it has in passing through that point. In the next second the body falls 112 feet farther. This cannot be the measure of the velocity at the beginning of the fourth or the end of the third second. It is too great. The fall during the preceding second was 80 feet. This is too small. The mean of these two, or $\frac{80 + 112}{2} = \frac{192}{2} = 96$, Of Accelerated and Retarded Motions.

Let ACG (fig. 15) be a line described by a body with a motion any how continually but gradually varied; and let mental requisites in any point C to the velocity in any other point F.

AXIOM.—If A be to B in a ratio that is greater than any ratio less than that of C to D, then A is to B as C to D.

Take the straight line acg to represent the time of the body's motion along ACG, so that the points a, c, f, g, may represent the instants of time in which the body passes through the points A, C, F, G; and the portions ac, cf, fg, of the line acg, may represent the times employed in describing the portions AC, CF, FG; and therefore ac is to af as the time of describing AC to the time of describing AF. Moreover, let $hkn$ be a line so related to the straight line $acfg$, by the perpendicular ordinates $ah, ck, fn, go$, that the areas $ackh, afnh, agoh$, may be proportional to the portions $AC, AF, AG$, of the line described by the moving body; and let this relation be true with respect to every point $B, D, E, \&c.$ and the corresponding points $b, d, e, \&c.$

Then it is affirmed that the velocity in the point $C$ is to the velocity in the point $F$ as $ck$ is to $fn$.

Let the equal lines $be, cd, ef, fg$, represent equal moments of time, and let $B, D, E, G$, be the points through which the body is passing at the instants $b, d, e, g$. Then the areas $bibe, cldl, emgf, fnog$, will represent, and be proportional to, the spaces $BC, CD, EF, FG$, which are described during the moments $be, cd, ef, fg$.

Draw $fp$ parallel to $ag$, so as to make the rectangle $bipe$ equal to the trapezium $bibe$; and draw the lines $qe, nr, ss$, in the same manner, so that each rectangle may be equal to its corresponding trapezium.

If the motions had been uniform during the moments $be$ and $fg$, that is, if the spaces $BC$ and $FG$ had been uniformly described, then the velocity in the point $C$ would have been to the velocity in the point $F$ as $cp$ to $fs$; for since the rectangles $bipe$ and $fsgy$ are respectively equal to the trapeziums $bibe$ and $fnog$, and since $bibe$ is to $fnog$ as $BC$ is to $FG$, the rectangle $bipe$ is to the rectangle $fsgy$ as $BC$ to $FG$. But because these two rectangles have equal altitudes $be$ and $fg$, they are to each other in the proportion of their bases $cp$ and $gs$, or $cp$ to $fs$. Therefore $BC$ is to $FG$ as $cp$ to $fs$. But if $BC$ and $FG$ are uniformly described in equal times, they are proportional to the velocities of those uniform motions. Therefore $cp$ is to $fs$ as the velocity with which $BC$ is uniformly described to the velocity with which $FG$ is uniformly described in an equal time.

But the motion expressed by the figure is not uniform, because the line $hlo$ recedes from the axis $ag$, and the areas, cut off by the parallel ordinates, increase in a greater proportion than the corresponding parts of the axis; that is, the spaces increase faster than the times; for the moments $be, cd, ef, fg$, being all equal, it is evident that the corresponding slips of the area continually augment. The motion is swifter at the instant $e$ than at the instant $b$, and the velocity at the instant $e$ is greater than that with which the space $BC$ would be uniformly described in the same time. For the same reason, the velocity at the instant $f$ is less than that with which the space $FG$ would be uniformly described in the same time. Therefore the velocity at the instant $e$ is to the velocity at the instant $f$ in a greater ratio than that of $cp$ to $fs$. In the very same manner, it will appear, by comparing the motion during the moment $cd$ with the motion during the moment $ef$, that the velocity at the instant $e$ is to the velocity at the instant $f$ in a less ratio than that of $cq$ to $fr$.

Therefore the velocity in the point $C$ is to the velocity in the point $F$ in a greater ratio than that of $cp$ to $fs$, but in a less ratio than that of $cq$ to $fr$.

But by continually diminishing the equal moments $be, cd, ef, fg$, it is evident that $cp$ and $cq$ continually approach to equality with $ck$, and $fr$ and $fs$ continually approach to equality with $fn$; that when $cp$ is less than $ck$, $fs$ is greater than $fn$, and when $cq$ is greater than $ck$, $fr$ is less than $fn$.

Therefore the velocity in the point $C$ is to the velocity in the point $F$ in a ratio that is greater than the ratio of any line less than $ck$ to any line greater than $fn$, but which is less than the ratio of any line greater than $ck$ to any line less than $fn$. Therefore the ratio of the velocity in $C$ to the velocity in $F$ is greater than any ratio that is less than that of $ck$ to $fn$; but it is less than any ratio that is greater than that of $ck$ to $fn$. Therefore the velocity in the point $C$ is to the velocity in the point $F$ as $ck$ is to $fn$.

This important theorem may be expressed in more general terms as follows:

If the abscissa $ag$ of a line $hko$ represent the time of any motion, and if the areas bounded by parallel ordinates be proportional to the spaces described, the ordinates are proportional to the velocities.

Remark.—The propriety or aptitude of expressing the time by the portions of the axis $ag$ will perhaps appear more clearly in the following manner:

Let $ag$ be any straight line, and let $hko$ be another line, straight or curved. Let the straight line $alz$, perpendicular to $ag$, be carried uniformly down along this line, keeping always perpendicular to it, and therefore always parallel to its first position $ahz$. In its various situations $elz, emz, \&c.$ it will cut off areas $ackh, acmh, \&c.$ bounded by the axis, by the ordinates $ah$ and $ck$, or by the ordinates $ah$ and $em$, &c., and by the line $hko$. By this motion the moveable ordinate is said, in the language of modern geometry, to generate the areas $ackh, acmh, \&c.$ At the same time, let a point $A$ move along the line $ACG$, setting out from $A$ at the instant when the line $az$ sets out from $a$, and let the motion of the point $A$ be so regulated that the spaces $AB, AC, AD, \&c.$ generated by this motion may increase at the same rate with the areas $ackh, acmh, acil, \&c.$ such that we shall have $AB$ to $AC$ as $ackh$ to $acmh$, &c. It is plain that the motion along $AG$ is the same with that described in the enunciation of the proposition; for because the motion of the ordinate $az$ along the axis $ag$ is supposed to be uniform, the spaces $ab, ac, ad, \&c.$ are proportional to the times in which they are described, and may therefore be taken to measure or to represent those times.

73. Cor. 1. In a motion continually varied, the velocities in the different points of the path are to each other in the limiting or ultimate ratio of the spaces described in equal times, those times being supposed to diminish continually; for it is evident that if the equal moments $be, cd, ef, fg$, are supposed to diminish continually till the instants $b$ and $d$ coalesce with $e$, and the instants $e$ and $g$ coalesce with $f$, then the ratio of $ck$ to $fn$ is the limit of the continually increasing ratio of $cp$ to $fs$, or of the continually diminishing ratio of $cq$ to $fr$. Sir Isaac Newton calls this the ultimate ratio of $cp$ to $fs$, or of $cq$ to $fr$. Now the ratio of $cp$ to $fs$, by construction, the same with the ratio of the rectangle $bipe$ to the rectangle $fsgy$, and the ratio of $cq$ to $fr$ is the same with the ratio of the rectangle $eyfd$ to the rectangle $eygf$. But the ratio of the rectangle $bipe$ is the same with the ratio of the space $bibe$ to the space $fnog$; that is (by hypothesis) the same with the ratio of the space $BC$ to the space $FG$; and the ratio of the rectangles $eyfd$ and $eygf$ is the same with that of the spaces $CD$ and $EF$. Therefore the ratio of the velocity at $C$ to the velocity at $F$ is the same with the ultimate ratio of the small increments $BC, FG, or CD, EF$ of the spaces generated in very small and equal times.

It is also evident, that because the ratio of $ck$ to $fn$ is the limit both of the ratio of $cp$ to $fs$ and of the ratio of $cq$ to $fr$, these ultimate ratios are the same, and that we may say that the velocity in $C$ is to the velocity in $F$ in the ultimate ratio of $BC$ to $EF$, or in the ultimate ratio of $CD$ to $FG$.

We also can easily perceive that the ratio of the area $bibe$ to the area $emgf$ approaches more near to the ratio of $ck$ to $fn$ as we take the moments $be$ and $ef$ smaller. Therefore, in many cases of practice, where it may be easy to measure the spaces described in the different small moments of the motion, but difficult to ascertain their ultimate ratio, so as to obtain accurate measures of the proportions of the velocities, we may reduce the errors of measurement to something very insignificant, by taking these moments extremely small; and we shall diminish the error still more by taking the proportion of the half sum of BC and CD to the half sum of EF and FG for the proportion of the velocities in C and F.

It often happens that we have it not in our power to compare the spaces described in small moments which are precisely equal. Still we can find the exact proportion of the velocities, if we can ascertain the ultimate ratio of the increments of the spaces, and the ultimate ratio of the moments of time in which these increments are described; for it is plain, by considering the gradual approach of the points p and r to the points k and n, that the ratio of ck to fn is still the ultimate ratio of the bases of rectangles equal to the mixtineal areas, whether the altitudes (representing the moments) are equal or not. Now the bases of two rectangles are in the proportion of the rectangles directly, and of their altitudes inversely. But the ultimate ratio of the altitudes is the ultimate ratio of the moments, and the ultimate ratio of the rectangles is the ultimate ratio of the spaces described in those unequal moments. Therefore, in such cases, we have,

74. Cor. 2. The velocities are in the ratio compounded of the direct ultimate ratio of the momentary increments of the spaces, and the inverse ultimate ratio of the increments (or moments) of the times in which these increments of the spaces are made.

If s, v, and t, are taken to represent the magnitudes of the spaces, velocities, and times, and if ds, dt, and dv, are taken always in the limiting or ultimate ratio of their momentary increments,1 we shall have v always in the proportion of ds directly, and of dt inversely. We express this by the proportional equation \( \frac{v}{s} = \frac{ds}{dt} \), which is equivalent to the analogy \( V : v = \frac{dS}{dT} : \frac{ds}{dt} \) or \( V : v = dSdt : dadtT \).

75. N. B. Here observe that this is not the only way of stating the relation of space and time—the abscissa may be made the time, and the ordinate the space: then

\[ \text{the velocity } = \frac{dy}{dx}. \]

The converse of this proposition may be thus expressed.

76. If the axis ag of the line which represent the time of a varied motion along the line AG, and if the ordinates ah, bh, ch, &c., be as the velocities in the instants a, b, c, or in the points A, B, C; then the areas abih, acih, adih, &c. are proportional to the spaces AB, AC, AD, &c.

This may be demonstrated in the same way with the former; but the indirect demonstration is more brief, and equally strict.

If the spaces AC, AF, &c. are not proportional to the areas ackh, afkh, &c. they are proportional to some other areas ackh', afkh', &c. which are bounded by the same ordinates, and by another line Rk'. But because the areas ackh', afkh', &c. are always proportional to the spaces AC, AF, &c. described on the line AG, the velocity in the point C is to the velocity in the point F as the ordinate ck is to the ordinate fn'; but, by hypothesis, the velocity in C is to the velocity in F as ck to fn, and fn' is equal to fn, which is absurd. Therefore the spaces AC, AF, are not proportional to any other areas, &c.

77. Cor. The ultimate ratio of the momentary incre-Of Accelerations of the spaces is compounded of the ratio of the veloci-ties, and the ultimate ratio of the increments of the times; Retarded Motions. for when the moments bc, ef, are equal, it is evident that the ultimate ratio of the rectangles bept, efpu is the same with the ultimate ratio of the increments of the spaces. But the ultimate ratio of these rectangles is the same with that of their bases ep and fr; that is, the ratio of ck to fn, that is, the ratio of the velocities. And when the moments are unequal, the ratio of the rectangles is compounded of the ratio of their bases and the ratio of their altitudes; that is, compounded of the ratio of the velocities and the ultimate ratios of the moments of time.

We have therefore \( dS : ds = VdT : edt \) and \( ds = edt \).

It most commonly happens that we can only observe the accumulated results of varied motions, and in them we only observe a space passed over, and a certain portion of time that has elapsed during the motion; but being able to distinguish the portions of the whole space which are described in known portions of the whole time, and having made such observations in several parts of the motion, we discover the general law that the motion affects, and we affirm this law to hold universally, even though we have not observed it in every point. We do this with a degree of probability and confidence proportioned to the frequency of our observation. It is not till we have done this that we can make use of the first of these two propositions, which enables us to ascertain the velocity of the motion in its different moments. Thus, if we observe that a stone in falling descends one foot in the quarter of a second, 16 feet in a second, 64 feet in two seconds, and 144 feet in three seconds, the general law immediately observed is, "that the spaces described are as the squares of the times;" for 1 is to 16 as the square of \( \frac{1}{4} \) to the square of 1. Again, 16 is to 64 as \( 1^2 \) to \( 2^2 \); and 16 is to 144 as \( 1^2 \) to \( 3^2 \). Hence we infer, with great probability, that the stone would fall 36 feet in a second and a half; for 16 is to 36 as \( 1^2 \) to \( 1\frac{3}{2} \); and we conclude in the same way for all other parts of the motion.

78. This immediate observation of the analogy between the spaces and the squares of the times suggests an easy determination of the velocity in this particular kind of geometrical motion, and it merits particular notice, being very often referred to. We can take ag to represent the time, and then, because the areas which are to represent the spaces described must be proportioned to the squares of the portions of ag, we perceive that the line which comes in place of hko must be a straight line drawn from a; for example, the straight line aoy; for this is the only boundary which will give areas abo, acx, ado, &c. proportional to ab², ac², ad², &c. And we perceive that any straight line drawn from a will have this property.

Having thus got our representations of the times and the spaces, we say, on the authority of our theorem, that the velocity at the instant b is to the velocity at the instant d as bS to dS, &c. And now we begin to make inferences purely geometrical, and express our discovery of the velocities in a very general and simple manner. We remark, that bS is to dS as ab is to ad; and we make the same affirmation concerning the magnitudes represented by these lines. We say that the velocity at the instant b is to the velocity at the instant d as the time ab is to the time ad. We say, in terms still more general, that the velocities are proportional to the times from the beginning of the motion. We moreover perceive, that the spaces are

---

1 In this article, as originally printed, the symbols \( s, t, \) and \( v \) were employed to represent the things here denoted by \( ds, dt, \) and \( dv; \) a like change in the notation of the fluxions of the quantities will be made throughout the article. Of Accelerating and Retarding Forces.

We can further infer, from the properties of the triangle, that the momentary increments of the spaces are proportional to the momentary increments of the squares of the times, or of the squares of the velocities.

We also observe, that not only the whole acquired velocities are proportional to the whole elapsed times, but that the increments of the velocities are proportional to the times in which they are acquired; for \( \frac{dx}{dt} \) is to \( \frac{dy}{dt} \) as \( \frac{dt}{dt} \), &c. Equal increments of velocity are therefore acquired in equal times. Therefore such a motion may, in great propriety of language, be denominated a uniformly accelerated motion; that is, a motion in which we observe the spaces proportioned to the squares of the times, is a motion uniformly accelerated; and spaces in the duplicate ratio of the times form the ostensible characteristic of an uniformly accelerated motion.

Lastly, if we draw \( \overline{ab} \) parallel to the axis \( ab \), we perceive that the rectangle \( acb \) is double of the triangle \( ace \). Now, because \( ae \) represents the time of the motion, and \( ec \) represents the acquired velocity, the rectangle \( ace \) will represent the space which would be uniformly described with the velocity \( ec \) during the time \( ae \). But the triangle \( ace \) represents the space really described with the uniformly accelerated motion during the same time. Hence we infer, that the space that is described in any time, with a motion increasing uniformly from nothing, is one half of the space which would be uniformly described during the same time with the final velocity.

These are but a part of the inferences which we may draw from the geometrical properties of those representations which we had selected of the different measurable affections of motion. We may affirm, with respect to the motions themselves, all the inferences which relate to magnitude and proportion, and thus improve our knowledge of the motions.

We took the opportunity of this very simple and perspicuous example, to give our readers a just conception of the mathematical method of prosecuting mechanical knowledge, and to make them sensible of the unquestionable authority for every theorem deduced in this manner.

One of the most important is, to discover the accumulated result of a motion of which we only observe the momentary increments. This is to be done by finding the area, or portions of the area, of the mixtilinear space \( agob \); and it is evidently analogous to the inverse method of fluxions, or the integral calculus.

In most cases, we must avail ourselves of the corollary \( ds = vdt \), and we obtain the solution of our question only in the cases where our knowledge of the quantities \( ds, dt \), and \( v \) (considered as geometrical magnitudes, that is, as lines and surfaces), enables us to discover \( s \) and \( t \).

OF ACCELERATING AND RETARDING FORCES.

Having thus discovered the proportions of the velocities in motions varying in any manner whatever, we can observe the variations which happen in them. These variations are the effects, and the only marks and measures, of the changing forces. They are the characteristics of their kinds (considered merely as moving forces); that is, the indications of the directions in which they act; for this is the only difference in kind of which they are susceptible in this general point of view. If they increase the velocity, their direction must be conceived as the same with that of the previous motion; because the result of the action of a force is equivalent to the composition of the motion which that force would produce in a quiescent body with the motion already existing; and an increase of velocity is equivalent to the composition of a motion in the same direction.

Having no other mark of the force but the acceleration, we have no other name for it in the abstract doctrines of dynamics, and we call it an accelerating force. Had it retarded the motion, we should have called it a retarding force.

In like manner we have no measure of the magnitude or intensity of an accelerating force, but the acceleration which it produces. In order therefore to investigate the powers which produce all the changes of motion, we must endeavour to obtain measures of the acceleration.

A continual increase of velocity is the effect of the continued action of accelerating forces. If equal increments of velocity are produced in every succeeding equal moment of time, we cannot conceive that there is any change in the accelerating force. Therefore a uniformly accelerated motion is the mark of the unvaried action of an accelerating force, that is, of the continued action of a constant force; of a force whose intensity is always the same. When therefore we observe a body describe spaces proportional to the squares of the times, we must infer that it is urged forward by a force whose intensity does not change; and, on the other hand, a constant force must produce a uniformly accelerated motion by its continued action. And if any previous circumstances assure us of this continued action of an invariable force, we may make all the inferences which were mentioned under the article of uniformly accelerated motion.

That force must surely be accounted double which produces a double increment of velocity in the same time in action by its uniform action; we can form no other estimation of raising its magnitude. And in general, accelerating forces must be accounted proportional to the increments of velocity which they produce, by acting uniformly during the same or equal times.

Supposing them to act on a body at rest. Then the velocity produced is itself the increment; and we must say that accelerating forces are proportional to the velocities which they generate in a body in equal times. And because we found (No. 79), that the space described with a uniformly accelerated motion is half the space which would be uniformly described in the same time with the final velocity, which space is the direct measure of this velocity, and because halves have the same proportion with the wholes—we may say that accelerating forces are proportional to the spaces through which they impel a body from rest in equal times by their uniform action.

This is an important remark; because it gives us an easy measure of the force, without the trouble of first computing the velocities. It also gives us the only distinct notion that we have of the measurement of forces by the motions which they produce. When speaking of the composition of forces, we distinguished and denominated them by the sides and diagonal of a parallelogram. These lines must be conceived as proportional to the spaces through which the forces urge the body uniformly during the small and insensible time of their action, which time is supposed to be the same for both forces; for the sides of the parallelogram are supposed to be separately described in equal times, and therefore to be proportional to the velocities generated by the constituent forces. If indeed the forces do not act uniformly, nor similarly, nor during equal times, we cannot say (without farther investigation) what is the proportion of the intensity of the forces, nor can we infer the composition of their action. We must at least suppose, that in every instant of this very small time of their joint action, their direction remains unchanged, and that their intensities are in the We shall see by and by, that with these conditions the sides of the parallelogram are still proportional to the velocities generated. In the mean time we may take the spaces through which a body is uniformly impelled from rest (that is, with a uniformly accelerated motion) as the measures of the forces; yet these spaces are but the halves of the measures of the velocities. Then, if a body be moving with the velocity of thirty-two feet per second, and an accelerating force acts on it during a second, and if this force be such that it would impel the body (from a state of rest) sixteen feet, it will add to the body a velocity of thirty-two feet per second. Accordingly, this is the effect of gravity—the weight of a pound of lead may be considered as a force which does not vary in its intensity. We know that it will cause the lead to fall sixteen feet in a second; but if the body has already fallen sixteen feet, we know that it is then moving with the velocity of thirty-two feet per second. And the fact is, that it will fall forty-eight feet farther in the next second, and will have acquired the velocity of sixty-four feet per second. It has therefore received an augmentation of thirty-two feet of velocity by the action of gravity during the second; and gravity is in fact a constant force, causing equal increments of velocity in equal times, however great the velocities may be. It does not act like a stream of fluid, whose impulse or action diminishes as the solid body withdraws from it by yielding.

But supposing that we have not compared the increments of velocity uniformly acquired during equal times, in what manner shall we measure the accelerating forces? In such a case that force must be accounted double which generates the same velocity, by acting uniformly during half the time; for when the force is supposed invariable, the changes of velocity which it produces are proportional to the times of its action; therefore if it produces an equal velocity in half the time, it will produce a double velocity in an equal time, and is therefore a double force. The same may be said of every proportion of time in which an equal change of velocity is produced by the uniform action of an accelerating force. The force must be accounted greater in the same proportion that the time required for the production of a given velocity in a body is less. Hence we infer that accelerating forces are inversely proportional to the times in which a given change of velocity is produced by their uniform action.

82. By combining these two propositions, we establish this general theorem:

Accelerating forces are proportional to the changes of velocity which they produce in a body by their uniform action directly, and to the times in which these changes are produced inversely.

If, therefore, A and a are the forces, V' and v' the changes of velocity, and T' and t' the portions of time in which they are uniformly produced, we have

\[ A : a = V' : v' \]

and \( a = \frac{v'}{t'} \).

The formula \( a = \frac{v'}{t'} \) is not restricted to any particular magnitude of v' and t'. It is true, therefore, when the portion of time is diminished without end; for since the action is supposed uniform, the increment of velocity is lessened in the same proportion, and the value of the fraction \( \frac{v'}{t'} \) remains the same. The characters or symbols v' and t' are commonly used to express finite portions of v and t. The symbols v and t are used by Newton to express the same things taken in the ultimate or limiting ratio. (These have been changed to dv and dt throughout this article.) They are usually considered as indefinitely small portions of v and t. We shall abide by the formula \( a = \frac{dv}{dt} \).

83. It must always be kept in mind, that v and t are abstract numbers; and that v refers to some unit of space, such as a foot, an inch, a yard; and that t refers to some number of time, such as an hour, a minute, a second; and especially that a is the number of the same units of space, which will be uniformly described in one unit of the time with the velocity generated, by the force acting uniformly during that unit. It is twice the space actually described by the body during that unit when impelled from rest by the accelerating force. It is necessary to keep hold of these clear ideas of the quantities expressed by the symbols.

On the other hand, when the measure of the accelerating force is previously known, we employ the theorem of a change at \( v' \); that is, the addition made to the velocity during the whole or any part of the time of the action of the force is obtained by multiplying the acceleration of one unit of time by the number of such units contained in v'.

These are evidently leading theorems in dynamics, because all the mechanical powers of nature come under the measures of accelerating and retarding forces. It is the greatest collection of these in any subject, and the manner in which they accompany or are inherent in it, which determine the mechanical character of that subject; and of mechanics therefore the phenomena by which they are brought into view are the characteristic phenomena. Nay, it may even be questioned whether the phenomena bring anything more into view. This force, of which we speak so familiarly, is no object of distinct contemplation; it is merely a something that is proportional to \( \frac{dv}{dt} \). And when we observe that the \( \frac{dV}{dT} \) found in the motions that result from the vicinity of a body A, is double of the \( \frac{dv}{dt} \) which results from the vicinity of another body B, we say that a force resides in A, and that it is double of the force residing in B. The accelerations are the things immediately and truly expressed by these symbols. And the whole science of dynamics may be completely taught without once employing the word force, or the conception which we imagine that we form of it. It is of no use till we come to study the mechanical history of bodies. Then, indeed, we must have some way of expressing the fact, that an acceleration \( \frac{32}{1^{\circ}} \) feet is observed in every thing on the surface of this globe; and that an acceleration \( \frac{418}{1^{\circ}} \) feet is observed over all the surface of the sun.

These facts are characteristic of this earth and of the sun; and we express them shortly by saying that such and such forces reside in the earth and in the sun. It will preserve us from many mistakes and puzzling doubts, if we absolutely adhere to this meaning of the term force; and this will carry mathematical evidence through the whole of our investigations.

84. As velocity is not an immediate object of contemplation, and all that we observe of motion is a space and measure of a time, it may be proper to give an expression of this men-accelerating force which involves no other idea. Supposing the body to have been previously at rest, we Of Accelera- rating and have \( \frac{a}{t} = \frac{v}{t} \). Multiply both parts of the fraction by \( t \) which does not change its value, and we have \( \frac{a}{t} = \frac{vt}{t^2} \).

But \( vt = s \); and therefore \( \frac{a}{t} = \frac{s}{t^2} \).

The formula \( a = \frac{s}{t^2} \) is equivalent to the proportion \( \frac{t^2}{1} = s : a \); and \( a \) would then be the space through which the accelerating force would impel the body in one unit of the time \( t \). But this is only half of the measure of the velocity which the accelerating force generates during that unit of time. For this reason we did not express the accelerating force by an ordinary equation, but used the symbol \( \frac{d^2s}{dt^2} \). In this case, therefore, of uniform action, we may express the accelerating force by \( a = \frac{2s}{t^2} \).

85. The following theorem is of still more extensive use in all dynamical disquisitions.

Accelerating forces are proportional to the momentary increments of the squares of the velocities directly, and as the spaces along which they are uniformly acquired inversely.

Let \( A'B, A'C, \) and \( AD \) (fig. 16) be three lines, described in the same or equal times by the uniform action of accelerating forces; the motions along these lines will be uniformly accelerated, and the lines themselves will be proportional to the forces, and may be employed as their measures. On the greatest of them, \( AD \), describe the semicircle \( ABCD \), and apply the other two lines \( A'B, A'C \) as chords \( AB, AC \). Draw \( EB, FC \) perpendicular to \( AD \). Take any small portions \( Bb, Cc \) of \( AB \) and \( AC \), and draw \( be, cf \) perpendicular to \( AD \), and \( Ea \) and \( Fa \) parallel to \( AB \) and \( AC \).

Then, because the triangles \( DAB \) and \( BAE \) are similar, we have \( AD : AE = AD^2 : AB^2 \). And because \( AD \) is to \( AB \) as the velocity generated at \( D \) is to the velocity generated at \( B \) (the times being equal), we have \( AD \) to \( AE \) as the square of the velocity at \( D \) to the square of the velocity at \( B \); which we may express thus:

\[ AD : AE = V^2, D : V^2, B. \]

For the same reasons we have also

\[ AD : AF = V^2, D : V^2, C. \]

Therefore

\[ AE : AF = V^2, B : V^2, C. \]

But because in any uniformly accelerated motion, the spaces are as the squares of the acquired velocities, we have also

\[ AE : Ae = V^2, B : V^2b, \text{ and } AF : Af = V^2, C : V^2c. \]

Therefore \( Ee \) is to \( Ef \) as the increment of the square of the velocity acquired in the motion along \( Bb \) to the increment of the square of the velocity acquired along \( Cc \).

But by similarity of the triangles \( ABD \) and \( Ech \), we have

\[ AB : AD = Ee : Ea; \text{ and in like manner, } AD : AC = Fk : If. \text{ Therefore } AB : AC = Ee \times Fk : If \times Ea. \]

Now \( AB \) and \( AC \) are proportional to the forces which accelerate the body along the lines \( A'B \) and \( A'C \); \( Ee \) and \( If \) are proportional to the increments of the squares of the velocities acquired in the motions along the portions \( Bb \) and \( Cc \); and \( Eh \) and \( Fk \) are equal to those portions respectively. The ratio of \( AB \) to \( AC \) is compounded of the direct ratio of \( Ee \) to \( If \), and the inverse ratio of \( Ea \) to \( Fk \). The proposition is therefore demonstrated.

The proportion may be expressed thus:

\[ AB : AC = \frac{Ee}{Eh} : \frac{Ff}{If}, \text{ and may be expressed by the proportional equation } AB = \frac{Ee}{Eh} \text{ or symbolically, } a = \frac{d(v^2)}{ds}. \]

Remark.—Because the motion along any of these three lines is uniformly accelerated, the relation between one line, spaces, times, and velocities, may be represented by means of the triangle \( ABC \) (fig. 17); where \( AB \) represents the time, \( BC \) the velocity, and \( ABC \) the space. If \( BC \) be taken equal to \( AB \), the triangle is half of the square \( ABCF \) of the velocity \( BC \); and the triangle \( ADE \) is half of the square \( ADEG \) of the velocity \( DE \). Let \( Dd \) and \( Bb \) be two moments of time, equal or unequal. Then \( DdeE \) and \( BbcC \) are half the increments of the squares of the velocities \( DE \) and \( BC \), acquired during the moments \( Dd \) and \( Bb \). It was demonstrated that the ratio of the area \( DdeE \) to the area \( BbcC \) is compounded of the ratio of \( DE \) to \( BC \), and the ultimate ratio of \( Dd \) to \( Bb \). But \( Dd \) and \( Bb \) are respectively equal to \( se \) and \( xe \). Therefore \( DdeE \) is to \( BbcC \), in the ratio compounded of the ratio of \( DE \) to \( BC \), and the ultimate ratio of \( se \) to \( xe \). If we represent \( DE \) and \( BC \) by \( V \) and \( v \), then \( se \) and \( xe \) must be represented by \( V' \) and \( v' \), the increments of \( V \) and \( v \); and then the compound ratio will be the ratio of \( VV' \) to \( vv' \); and if we take the ultimate ratio of the moments, and consequently the ultimate ratio of the increments of the velocities, we have the ratio of \( VdV \) to \( vde \). If therefore \( V^2 \) and \( v^2 \) represent the squares of the velocities, \( VdV \) and \( vde \) will represent, not the increments of those squares, but half the increments of them.

We may now represent this proposition concerning accelerating forces by the proportional equation \( a = \frac{d^2v}{dt^2} \); and we must consider this as equivalent with \( a = \frac{V^2 - v^2}{2(S-t)} \); keeping always in mind that \( a, V, \) and \( v, \) relate to the same units of time and space, and that \( a \) is that number of units of the scale on which \( S \) and \( s \) are measured, which is run over in one unit of time. This will be more clearly conceived by taking an example. Let us ascertain the accelerative power of gravity, supposing it to act uniformly on a body. Let the spaces be measured in feet and the time in seconds. It is a matter of observation, that when a body has fallen 64 feet, it has acquired a velocity of 64 feet per second; and that when it has fallen 144 feet, it has acquired the velocity of 96 feet per second. We want to determine what velocity gravity communicated to it by acting on it during one second. We have $V^2 = 9216$, and $v^2 = 4096$; and therefore $V^2 - v^2 = 5120$. $S = 144$, and $s = 64$, and $S - s = 80$, and $2(S - s) = 160$. Now $a = \frac{5120}{160} = 32$.

Therefore gravity has generated the velocity 32 feet per second by acting uniformly during one second.

86. The augmentation of the square of the velocity is proportional to the force and to the space jointly. For, because $a = \frac{vdr}{ds}$, we have $ads = vdr$.

Thus we learn, that a given force, acting uniformly on a body along a given space, produces the same increment of the square of the velocity, whatever the previous velocity may have been. Also, in the same manner as we formerly found that the augmentation of the velocity was proportioned to the time during which the force has acted, so the augmentation of the square of the velocity is proportional to the space along which it has acted.

It is pretty plain, that all that we have said of the uniform action of an accelerating force may be affirmed of a retarding force, taking a diminution or decrement of velocity in place of an increment. A uniformly retarded motion is that in which the decrements of velocity in equal terms are equal, and the whole decrements are proportional to the whole times of action. Such a motion is the indication of a constant or invariable force acting in a direction opposite to that of the motion. We conceive this to be the case, when an arrow is shot perpendicularly upwards; its weight is conceived as a force continually pressing it perpendicularly downwards.

In such motions, however great the initial velocity may be, the body will come to rest; because a certain determined velocity will be taken from the body in each equal successive moment, and some multiple of this will exceed the initial velocity. Therefore the velocity will be extinguished before the end of a time that is the same multiple of the time in which the velocity was diminished by the quantity above mentioned. It is no less evident that the time in which any velocity will be extinguished by an opposing or retarding force, is equal to the time in which the same force would generate this velocity in the body previously at rest. Therefore,

87. (1.) The times in which different initial velocities will be extinguished by the same opposing force are proportional to the initial velocities.

88. (2.) The distances to which the body will go till the extinction of its velocity, are as the squares of the initial velocities.

(3.) They are also as the squares of the times elapsed.

89. (4.) The distance to which a body, projected with any velocity, will go till its motion be extinguished by the uniform action of a retarding force, is one half of the space which it would describe uniformly during the same time with the initial velocity.

It very rarely happens that the force which accelerates the body acts uniformly, or with an unvaried intensity. The attraction of a magnet, for example, increases as the iron approaches it. The pressure of a spring diminishes as it unbends. The impulse of a stream of water or wind diminishes as the impelled surface retires from it by yield-

ing. Therefore the effects of accelerating forces are very imperfectly explained, till we have shown what motions result from any given variation of force, and how to discover the variation of force from the observed motion. This last question is perhaps the most important in the study of mechanical nature. It is only thus that we learn what is usually called the nature of a mechanical force. This chiefly consists in the relation subsisting between the intensity of the force and the distance of the substance in which it resides. Thus the nature of that power which produces all the planetary motions, is considered as ascertained, when we have demonstrated that its pressure or intensity is inversely as the square of the distance from the body in which it is supposed to reside.

Acceleration expresses some relation of the velocity and time. This relation may be geometrically expressed in a variety of ways. In fig. 18 the uniform acceleration, or the unvaried relation between the velocity and the time, is very aptly expressed by the constant ratio of the ordinates and abscissas of the triangle agy. The ratio of $da$ to $ad$ is the same with that of $es$ to $ae$, or that of $f\beta$ to $\alpha f$, &c.; or the ratio of the increment of velocity $\pi x$ to the increment of the time $\beta x$ or $be$, or that of $i\beta$ to $u$, &c.

This ratio $\pi x : \beta x$ is equivalent to the symbol $\frac{dv}{dt}$.

But when the spaces described in a varied motion are represented by the areas bounded by a curve line hko, we no longer have that constant ratio of the increments of the ordinates and abscissas.

90. Therefore, in order to obtain measures of the accelerating forces, or at least of their proportions, let the abscissas in scissa agy (fig. 18) of the line hko again represent the time such cases of a motion. But let the areas bounded by parallel ordi-

nates now represent the velocities; that is, let the whole areas increase during the time ag at the same rate with the most extreme velocities of the motion along the line AG. In this case extensive the ordinates bi, ck, dl, &c. will be as the accelerations at use.

The instants b, c, d, &c. or in the points B, C, D, &c.

This is demonstrated in the same way as the former proposition (No. 72). If the accelerating force be supposed constant during any two equal moments be and fg, the rectangles bept and fjsx would express the increments of velocity uniformly acquired in equal times, and their bases ep and fs would have the ratio of the accelerations, or of the accelerating forces. But as the velocities expressed by the figure increase faster than the times during every moment, the force at the instant e is to the force at the instant f in a greater ratio than that of ep to fs; but, for similar reasons, it is in a less ratio than that of ep to fr; and therefore (as in the other proposition) the force at the instant e is to the force at the instant f as ck to fn.

91. Cor. Because ep is to fs in the ratio compounded of Of Accel—The direct ratio of the rectangle opnb to the rectangle fayg, rating and the inverse ratio of the altitude be to the altitude fy; Retarding Forces.

and because these rectangles are proportional to the increments of velocity, and the ultimate ratio of the altitudes is the ultimate ratio of the moments or increments of the time, we must say that the accelerating forces (that is, their intensities or pressures producing acceleration) are directly as the increments of velocity, and inversely as the increments of the times; which proposition may be expressed in regard to two accelerations A and a, by this analogy:

\[ A : a = \frac{dV}{dT} : \frac{dv}{dt}. \]

Or by the proportional equation \( a = \frac{dv}{dt} \). Also \( adt = dv \),

and \( \int adt = v \). And thus do these theorems extend even to the cases where there cannot be observed an immediate measure either of velocity or of acceleration, because neither the space nor the velocity increases uniformly.

The theorem \( a = \frac{dv}{dt} \) is employed when we would discover the variation in the intensity of some natural power. We observe the motion, and represent it by a figure analogous to fig. 18, where the abscissa represents the times, and the area is made to increase at the same rate with the spaces described. Then the ordinates will represent the velocities, or have the proportion of the velocities. Then we may draw a second curve on the other side of the same abscissa, such that the areas of this last curve shall be proportional to the ordinates of the first. The ordinates of this last curve are proportional to the accelerating forces.

92. On the other hand, when we know from other circumstances that a force, varying according to some known law, acts on a body, we can determine its motion. The intensity of the force in every instant being known, we can draw a line so related to another line representing the time, that the ordinates shall be proportional to the forces. The areas will be proportional to the velocities. We can draw another curve to the same abscissa, such that the ordinates of this shall be proportional to the areas of the other, that is, to the velocities of the motion. The areas of this second curve will be proportional to the spaces described.

93. We must now observe, that all that has been said concerning the effects of accelerating forces continually varying, relates to changes of motion, independent of what the absolute motions may be. The areas of the line whose ordinates represent the velocities do not necessarily represent the spaces described, but the change made on the spaces described in the same time; not the motions, but the changes of motion. If, indeed, the body be supposed to be at rest when the forces begin to act, these areas represent the very spaces that are passed over, and the ordinates are the very velocities. In every case, however, the accelerations are the real increments of the velocities.

This circumstance gives a great extension to our theorems, and enables us to ascertain the disturbances of any species of regular motion, apart from the motions themselves, and thus avoid a complication which would frequently be inextricable in any other way. And this process, which is merely mathematical, is perfectly conformable to mechanical principles. It is in fact an application of the doctrine of the composition of motion; a doctrine rigidly demonstrated when we measure a mechanical force by the change of motion which it produces. Acceleration is the continual composition of a new motion with the motion already produced.

We may learn from this investigation of the value of an accelerating force, that no finite change of velocity is effected in an instant by the action of an accelerating force. When the fig. 18 is used for the scale of accelerations, and they are represented by the ordinates of the line hko, each increment of velocity is represented by an area, that is, by a slip of the whole area; which slip must have some ratio to the altitude, or must occupy some portion of the abscissa which represents time. Some portion of time, however small it may be, must elapse before any measurable addition can be made to the velocity. The velocity must change continuously. As no motion can be conceived as instantaneous, because this would be to conceive that in one instant the moving particle is in every point of its momentary path,—so no velocity can change, by a finite quantity, in one instant, because this would be to conceive that in that instant the particle had all the intervening velocities. The instant of change is at once the last instant of the preceding velocity, and the first of the succeeding; and therefore must belong to both. This cannot be conceived, or is absurd. As a body, in passing from one part of space to another, must pass in succession through all the intermediate places; so, in passing from one velocity to another, it must in succession have all the intermediate velocities. It must be continuously accelerated—we must not say gradually—however small the steps.

But to return from this digression:

94. The most frequent cases which come under examination do not show us the relation between the forces and velocities, but the relation between the forces and spaces, namely, Thus, when a piece of iron is in the neighbourhood of a magnet, or a planet is considered in the neighbourhood of the sun, a force is acting on it in every point of its path, and we have discovered that the intensity of this force more or less varies in a certain proportion. Thus, a spring varies in its pressure as it unbends; gunpowder presses less violently as it expands, &c. &c.

Our knowledge is generally confined to some such effect as this. We know, that while a body is moving along a line, ADE (fig 19), it is urged forward by a force, of which the intensity varies in the proportion of the ordinates BF, CG, DH, EI, &c. of the line FGHL.

To investigate the motion or change of motion produced by the action of this force, let CD be supposed a very small portion of the space s, which we may express by \( x' \). Draw GK perpendicular to DH. Then, if we suppose that the

---

1 See Barrow's Lect. Geometr. passim.

If the accelerating force acts with the unvaried intensity CG through the whole space CD, the rectangle CDKG will express half of the increment of the square of the velocity (No. 83). We may suppose that the force acts uniformly along the adjoining small space Dr with the intensity DH. The rectangle DHer will in like manner express another half increment of the square of the velocity. And in like manner we may obtain a succession of such increments. The aggregate or sum of them all will be half the difference between the square of the velocity at B and the square of the velocity at E.

If we employ \( f \) to express the indetermined or variable intensity of the accelerating force, and \( v \) to express the variable velocity, and \( v' \) its increment uniformly acquired, then the rectangle CDKG will be expressed by \( f \cdot v' \). We have seen that this is equal to \( v^2 \). Therefore, in every case where we can tell the aggregate of all the quantities \( f \cdot v' \), it is plain that we will obtain half the difference between the squares of the velocities in B and E, on the supposition that the intensity of the force was constant along each little space, and varied by steps. Then, by increasing the number and diminishing the magnitude of those little portions of the space without end, it is evident that we terminate in the expression of the real state of the case, i.e., of a force varying continually; and that in this case the aggregate of these rectangles occupies the whole area AEIF, and is equivalent to the fluent of \( f \cdot v' \), or to the symbol \( \int f \cdot v' \), used to express this fluent, which is conceived an aggregate of small rectangles \( f \cdot v' \). And we see that this area expresses half of the augmentation of the square of the velocity. Therefore,

95. If the abscissa AE (fig. 19) of a line FGI is the path along which a body is urged by an accelerating force, and if the ordinates BE, CG, DH, &c. are proportional to the forces acting in the points B, C, D, &c. the intercepted areas BCGF, BEIF, &c. are proportional to the augmentations of the square of the velocity.

Observe that the areas BCGF and DEIH are also proportional to the augmentations made on the squares of the velocities in B and in D.

Observe also, that it is indifferent what may have been the original velocity. The action of the forces represented by the ordinates make always the same addition to its square; and this addition is half the square of the velocity which those forces would generate in the body by impelling it from rest in the point A.

Lastly, on this head, observe, that we can state what constant or variable force will make the same augmentation of the square of the velocity by impelling the body uniformly along the same space BE, or along what space a given force must impel the body in order to produce the same increase of the square of its velocity. In the first case, we have only to make a rectangle BENP, equal to the area BEIF, and then BP is the intensity of the constant force wanted. In the second case, in which the force EO is given, we must make the rectangle AaOE equal to the area BEIF, and AE is the space required.

97. The converse of this proposition, viz., if the areas are as the increments of the square of the velocity, the ordinates are as the forces, is easily demonstrated in the same way; for if the elementary areas CDKG and EIMe represent increments of the squares of the velocity, the accelerating forces are in the ratio compounded of the direct ratio of these rectangles and the inverse ratio of their altitudes, because these altitudes are the increments of the space (No. 85). Now the base CG of the rectangle CDKG is to the base EI of the rectangle EIMe in the same compounded ratio; therefore the force in C is to the force in E as CG to EI.

98. The line hlo (fig. 18) was called by Dr Barrow (who first introduced this extensive employment of motion into geometry) the scale of velocities; and the line FHI (fig. 19) was named by him the scale of accelerations. Hermann, in his Phoronomia, calls it the scale of city accelerations. We shall retain this name, and we may call hlo of acceleration fig. 18 the scale of accelerations, when the areas represent time, &c. The velocities. Sir Isaac Newton added another scale of very great use, viz., a scale of times. It is constructed as follows.

99. Let ABE (fig. 19) be the line along which a body is accelerated, and let FHI be the scale of forces, that is, having its ordinates FB, HD, IE, &c. proportional to the forces acting at B, D, E, F, &c.; let fgi be another line so related to ABE, that Cg is to Ei in the inverse subduplicate ratio of the area BFGC to the area BFIE; or, to express it more generally, let the squares of the ordinates to the line fgi be inversely as the areas of the line FHI intercepted between these ordinates and the first ordinate drawn through B; then the times of the bodies moving from a state of rest in B are as the intercepted areas of the curve fgi.

For let CD and Ee be two very small portions of the space described in equal times. They will be ultimately as the velocities in C and E. The area FBCG is to the area FBEI as the square of Ei to the square of Cg (by construction); but the area FBCG is to FBEI as the square of the velocity at C to the square of the velocity at E (by the proposition); therefore the square of the velocity at C is to the square of the velocity at E as the square of Ei to the square of Cg; therefore Ei is to Cg as the velocity at C to the velocity at E, that is, as CD to Ee; but since Ei : Cg = CD : Ee, we have Ei × Ee = Cg × CD, and the elementary rectangles CgkD and Eime are equal, and may represent the equal moments of time in which CD and Ee were described. Thus the areas of the line fgi will represent or express the times of describing the corresponding portions of the abscissa.

We may express the nature of this scale more briefly thus. Let BE be the space described with any varied motion, and fgi a curve, such that its ordinates are inversely as the velocities in the different points of the abscissa, then the area will be as the times of describing the corresponding portions of the abscissa.

100. In all the cases where our mathematical knowledge enables us to assign the values of the ordinates of the figure 19, we can obtain the law of action of the forces, or the nature of the force; and where we can assign the value of the areas from our knowledge of the proportions of the ordinates or forces, we can ascertain the velocities of the motion. We shall give an example or two, which will show the way in which we avail ourselves of the geometrical properties of figure in order to ascertain the effects of mechanical forces.

(1.) In fig. 19, let the accelerating force which impels the body along the line AB be constant, and let the body be previously at rest in B; the line which bounds the ordinates that represent the forces must be some line φHN parallel to AB. The area BDHp is to the area BENP as the square of the velocity at D to the square of the velocity at E. These areas, having equal bases DH and EN, are as their altitudes BD and BE; that is, the spaces described are as the squares of the acquired velocities. And we see that this characteristic mark of uniformly accelerated motion is included in this general proposition.

Newton's Principia, I. 39. 101. (2.) Let us suppose that the body is impelled from A (fig. 20), towards the point C, by a force proportional to its distance from that point. This force may be represented by the ordinates DA, EB, eb, &c. to the straight line DC. We may take any magnitude of these ordinates; that is, the line DC may make any angle with AC. It will simplify the investigation if we make the first force AD = AC. About C describe the circle AHa, cutting the ordinate EB in F; let ob be another ordinate, cutting the circle in f very near to F; draw CH perpendicular to AC, and make the arch Hb = fF, and draw he parallel to HC; join FC and DH, and draw Fg perpendicular to fb. Let IML be another ordinate.

The area DABE is to the area DAKL as the square of the velocity at B to the square of the velocity at K. But DABE is the excess of the triangle ADC above the triangle EBC, or it is half of the excess of the square of CA or CF above the square of CB, that is, half the square of BF. In like manner, the area DAKL is equal to half the square of KM; but halves have the same ratio as the integers; therefore the square of BF is to the square of KM as the square of the velocity at B to the square of the velocity at K; therefore the velocity at B is to the velocity at K as BF is to KM. The velocities are proportional to the sines of the arches of the quadrant AFH described on AC.

Cor. 1. The final velocity with which the body arrives at C is to the velocity in any other point B as radius to the sine of the arch AF.

Cor. 2. The final velocity is to the velocity which the body would acquire by the uniform action of the initial force at A as 1 to $\sqrt{2}$; for the rectangle DACH expresses the square of the velocity acquired by the uniform action of the force DA; and this is double of the triangle DAC; therefore the squares of these velocities are as 1 and 2, and the velocities are as $\sqrt{1}$ and $\sqrt{2}$, or as 1 to $\sqrt{2}$.

102. Cor. 3. The time of describing AB is to the time of describing AC as the arch AF to the quadrant AFH.

For when the arch Ef is diminished continually, it is plain that the triangle fF is ultimately similar to CFB, by reason of the equal angles Cib (or CFB) and fFi, and the right angles CBF and fFi; therefore the triangles fFf and CBF are also similar. Moreover, Bb is equal to Fg; Ef is equal to AH, which is ultimately equal to eC; therefore, since the triangles fFf and CFB are similar, we have Fg : Ff = FB : CF = FB : HC; therefore Bb is to eC as FB to HC, that is, as the velocity at B to the velocity at C; therefore Bb and eC are described in equal moments when indefinitely small; therefore equal portions Ff, AH, of the quadrant correspond to equal moments of the accelerated motion along the radius AC; and the ring arches AF, FM, MH, &c. are proportional to the times of describing AB, BK, KH, &c.

Cor. 4. The time of describing AC with the unequally accelerated motion, is to the time of describing it uniformly with the final velocity as the quadrantal arch is to the radius of a circle; for if a point move in the quadrantal arch so as to be in F, f, M, H, &c. when the body is in B, b, K, C, it will be moving uniformly, because the arches are proportional to the times of describing those portions of AC; and it will be moving with the velocity with which the body arrives at C, because the arch AH is ultimately = Cc. Now if two bodies move uniformly with this velocity, one in the arch AFH, and the other in the radius AC, the times will be proportional to the spaces uniformly described; but the time of describing AFH is equal to the time of the accelerated motion along AC, therefore the proposition is manifest.

103. Cor. 5. If the body proceed in the line Ca, and be retarded in the same manner that it was accelerated along AC, the time of describing AC uniformly with the velocity which it acquires in C is to the time of describing ACa with the varied motion, as the diameter of a circle to the circumference; for because the momentary retardations at K', B, &c. are equal to the accelerations at K and B, &c. the time of describing ACa is the same with that of describing AHu uniformly with the greatest velocity; that is, to the time of describing AC uniformly as AHu to AC, or as the circumference of a circle to the diameter; therefore, &c. N.B. In this case of retarding forces it is convenient to represent them by ordinates KL', BE, eD, lying on the other side of the axis ACa; and to consider the areas bounded by these ordinates as subtractive from the others. Thus the square of the velocity at K' is expressed by the whole area DACKLC, the part CK'L' being negative in respect of the point DAC. This observation is general.

Cor. 6. The time of moving along KC, the half of AC, by the uniform action of the force at A, is to that of describing ACa by the varied action of the force directed to C, and proportional to the distance from it, as the diameter of a circle to the circumference; for when the body is uniformly impelled along KC by the constant force IK, the square of the velocity acquired at C is represented by half the rectangle IKCH, and therefore it is equal to the velocity which the variable force generates by impelling it along AC (by the way, an important observation). The body will describe AC uniformly with this velocity in the same time that it is uniformly accelerated along KC. Therefore by Cor. 5 the proposition is manifest.

Cor. 7. If two bodies describe AC and KC by the action of forces which are everywhere proportional to the distances from C, their final velocities will be proportional to the distances run over, and the times will be equal.

For the squares of the final velocities are proportional to the triangle ADC, LKC, that is, to AC, KC, and therefore the velocities are as AC, KC. The times of describing AC and KC uniformly, with velocities proportional to AC and KC, must be equal; and these times are in the same ratio (viz. that of radius to $\frac{1}{2}$ of the circumference) to the times of describing AC and KC with the accelerated motion. Therefore, &c.

Thus, by availing ourselves of the properties of the circle, we have discovered all the properties or characters of a motion produced by a force always directed to a fixed point, and proportional to the distance from it. Some of these are remarkable, such as the last corollary; and they are all important, for there are innumerable cases where this law of action obtains in nature. It is nearly the law

Of elastic bodies, and of all elastic bodies, when their change of figure during their mutual action is moderate; and it has been by the help of this proposition first demonstrated in a particular case by Lord Brouncker and Mr. Huygens, that we have been able to obtain precise measures of time, and consequently of actual motions, and consequently of any of the mechanical powers of nature. It is for this reason, as well as for the easy and perspicuous employment of the mathematical method of proceeding, that we have selected it.

Instead of giving any more particular cases, we may observe in general, that if the intensity of the force be proportional to any power whose index is \( n - 1 \) of the distance, and if \( a \) be the distance from the fixed point at which the body begins to be accelerated, and \( x \) its distance from the point in any part of the motion, the velocity will be \( \sqrt{a^n - x^n} \). This is very plain, because the increment CGHD of the area of fig. 19, which is also the increment of the square of the velocity, is \( \frac{d}{dx} \left( a^n - x^n \right) dx \), and the area is \( \frac{a^n - x^n}{n} \); and the whole area, corresponding to the distance \( a \), is \( a^n \). Therefore the portion of the area lying beyond the distance \( x \) is \( a^n - x^n \). This is as the square of the velocity, and therefore the velocity is as the square root \( \sqrt{a^n - x^n} \) of this quantity.

This proposition, \( f = \frac{v}{x} \), or \( f = \frac{v}{x} \), is the 39th of the first book of Newton's Principia, and is perhaps the most important in the whole doctrine of dynamics, whether employed for the investigation of forces or for the explanation of motions. It furnishes the most immediate data for both purposes, but more especially for the last. By its help Sir Isaac Newton was able to point out the numerous disturbances of the planetary motions, and to separate them from each other; thus unravelling, as it were, that most intricate motion in which all are blended together. He has given a most wonderful specimen of its application in his Lunar Theory.

We now are able to explain all the puzzling facts which were adduced by Leibnitz and his partisans in support of their measure of the forces of bodies in motion. We see why four springs, equally bent, communicate but a double velocity, and nine springs but a triple velocity; why a bullet moving twice as fast will penetrate an earthen rampart to a quadruple depth, &c., &c.

This theorem also gives a most perspicuous explanation of the famous doctrine called conservatio virium vivarum. When perfectly elastic bodies act on each other, it is found that the sum of the masses multiplied by the squares of the velocities is always the same. This has been substituted, with great encomiums, by the German philosophers, in place of Descartes's principle, that the quantity of motion in the universe, estimated in one direction, remains always the same. They are obliged, however, to acknowledge, that in the actions of perfectly hard bodies, there is always a loss of vis viva, and therefore have denied the existence of such bodies. But there is the same loss in the mutual actions of all soft or ductile, or even imperfectly elastic, bodies; and they are miserably puzzled how to explain the fact; but both the conservatio and the amissio are necessary consequences of this theorem.

In the collision of elastic bodies, the whole change of motion is produced during the short time that the bodies are compressed, and while they regain their figure. When this is completed, the bodies are at the same distance from each other as when the mutual action began. Therefore the preceding body has been accelerated, and the following body has been retarded along equal spaces; and in every point of this space the accelerating and the retarding force has been equal. Consequently the same area of fig. 20 expresses the change made on the square of acceleration of both bodies. Therefore, if \( V \) and \( U \) are the velocities before collision, and \( v \) and \( u \) the velocities after collision, of the two bodies A and B, we must have

\[ A \times V^2 - v^2 = B \times u^2 - U^2, \]

and therefore

\[ A \times V^2 + B \times U^2 = A \times v^2 + B \times u^2. \]

But in the other class of bodies, which do not completely regain their figure, but remain compressed, they are nearer to each other when their mutual action is ended than when it began. The foremost body has been accelerated along a shorter space than that along which the other has been retarded. The mutual forces have in every instant been equal and opposite. Therefore the area which expresses the diminution of the square of the velocity must exceed the area expressing the augmentation by a quantity that is always the same when the permanent compression is the same; that is, when the relative motion is the same. \( A \times V^2 - v^2 \) must exceed \( B \times u^2 - U^2 \), and \( A \times V^2 + B \times U^2 \) must exceed \( A \times v^2 + B \times u^2 \).

This same theorem is of the most extensive use in all practical questions in mechanic arts; and without it mechanics can go no farther than the mere statement of equilibrium.

Hermann, professor of mathematics at Pavia, one of the ornaments of the mathematical class of philosophers, has given a pretty demonstration of this valuable proposition, curious in the Acta Eruditorum Lipsiae for 1709; and says, that having searched the writings of the mathematicians with great care, he found himself warranted to say, that Newton was the undoubted author, and boasts of his own as the first synthetical demonstration. The purpose of this assertion was not very apparent at the time; but long after, in 1746, when Hermann's papers, preserved in the town-house of Pavia, were examined, in order to determine a dispute between Maupertuis and Koenig about the claim to the discovery of the principle of least action, letters of Leibnitz's were found, requesting Hermann to search for any traces of this proposition in the writings of the mathematicians of Europe. Leibnitz was by this time the envious detractor from Newton's reputation, and could not but perceive that all his contorted arguments for his doctrine received a clear explanation by means of this proposition, in perfect conformity to the usual measure of moving forces. Newton had discovered this theorem long before the publication of the Principia, and even before the discovery of the chief proposition of that book in 1666; for in his Optical Lectures, the materials of which were in his possession in 1664, he makes frequent use of a proposition founded on this. (See No. 42.) We may here remark, that Hermann's demonstration is in every step the same with Dr Barrow's demonstration of it as a theorem merely geometrical, without speaking of moving forces (see Lect. Geometr. xi. p. 85, edit. 16), but giving it as an instance of the transformation of curves, which he calls scales of velocity, of time, of acceleration, &c. It is very true that Barrow in these mathematical lectures approached very near to both of Newton's discoveries, the fluxionary geometry, and the principles of dynamics; and the junto on the Continent, who were his continual detractors, charge him with impudent plagiarism from Dr Barrow, and even say that he has added nothing to the discoveries of his teacher. But surely Dr Barrow was the best judge of this matter; and, so far from representing the use which Newton has made of what he had taught him, he was charmed with the genius of the juvenis spectatissimus his scholar, and of his own accord gave him his professorial chair, and ever after lived in the utmost harmony and friendship with him. Nay, it would even appear Of Accele from some expressions in those very lectures, that Dr Bar- ratingand row owed to young Newton the first thought of making Retarding such extensive use of motion in geometry. We recommend this work of Barrow's to the serious perusal of our read- ers, who wish to acquire clear notions of the science of motion, and an elegant taste in their mechanical disquisi- tions. After all the cultivation of this science by the commentators and followers of Newton, after the Phoro- nomia of Hermann, the Mechanica of Euler, the Dyna- mique of D'Alembert, and the Mechanique Analytique of De la Grange, which are undoubtedly works of transcen- dent merit and utility, the Principia of Newton will still remain the most pleasing, perspicuous, and elegant speci- men of the application of mathematics to the science of universal mechanics, or what we call Dynamics.

The two fundamental theorems \( \frac{d}{dt} = dv \) and \( \frac{ds}{dt} = v \), enable us to solve every question of motion accelerated or retarded by the action of the mechanical powers of na- ture. But the employment of them may be greatly ex- pedited and simplified by noticing two or three general cases which occur very frequently.

104. These may be called similar instants of time, and similar points of space which divide given portions of time, and of space in the same ratio. Thus the middle is a si- milar instant of an hour or of a day, and is the similarly situated point of a foot or of a yard. The beginning of the 21st minute, and of the 9th hour, are similar instants of an hour and of a day. The beginning of the fifth inch, and of the second foot, are similar points of a foot and of a yard.

105. Forces may be said to act similarly when their in- tensities in similar instants of time, or in similar points of space, are in a constant ratio. Thus in fig. 20, when one body is impelled towards C from A, and another from K, each with a force proportional to the distance of every point of its motion from C, these forces may be said to act similarly along the spaces AC and KC, or during the times represented by the quadrant arches AFH, KNO. The following propositions on similar actions will be found very useful on many occasions; but we must premise a geometrical lemma.

Let the times be represented by the straight lines ABC (fig. 22), and abc, and the forces by the ordinates AD, BE, CF, and ad, be, cf. Then if B and b are similar in- stants (suppose the middles) of the whole times, we have BE : be = AD : ad. Therefore, by the lemma, the area ACFD is to acfd as AC × AD to ac × ad. But these areas are proportional to the velocities (No. 72), and the proposition is demonstrated. For the same reason, the change of velocity during the time AB is to the change during ab as AB × AD to ab × ad.

Cor. 1. If the times and forces are reciprocally propor- tional, the changes of velocity are equal; and if the forces are inversely as the times, the changes of velocity are equal.

108. If two particles be similarly urged along given spaces, the changes made on the squares of the velocities are as the forces and spaces jointly.

For if AC (fig. 22) and ac are the spaces along which the particles are impelled, and the forces are as the ordi- nates AD and ad, the areas ACFD and acfd are as the changes on the squares of the velocities. But these areas are as AC × AD, and ac × ad. Therefore, &c.

Cor. 2. If the spaces are inversely as the forces, the changes of the squares of the velocities are equal; and if these are equal, the spaces are inversely as the forces.

Cor. 3. If the spaces along which the particles have been impelled from a previous state of rest, are directly as the forces, the velocities are also as the forces. For, because the changes of the squares of the velocities are as the spaces and forces jointly, they are in this case as the squares of the forces or of the spaces; but the changes of the squares of the velocities are in this case the whole squares of the velocities; therefore the squares of the ve- locities are as the squares of the forces, and the velocities are as the forces. N.B. This includes the motions re- presented in fig. 20.

109. If two particles be similarly impelled along given spaces, from a state of rest, the squares of the times are pro- portional to the spaces directly, and to the forces inversely.

Let ABC (fig. 22), abc, be the spaces described, and AD, ad, the accelerating forces at A and a. Let V, B express the velocity at B, and v, b the velocity at b.

Let GHK and ghk be curves whose ordinates are in- versely as the velocities at the corresponding points of the abscissa. These curves are therefore exponents of the times (No. 99). Then, because the forces act simi- larly, we have, by the last theorem, AC × AD : ac × ad = V : v, B : b, = hb : HB². Therefore HB

106. If there be two lines EFGH (fig. 21), efgh, so re- lated to their abscisses AD, ad, that the ordinates BF, bg, drawn from similar points B and b of the abscisses, are in the constant ratio of AE to ae; then the area ADHE is to the area adhe as the rectangle of AD × AE to the rect- angle ad × ae.

For, let each abscissa be divided into the same number of equal and very small parts, of which let CD and cd be one in each. Inscribe the rectangles CGID, cgid. Then because the number of parts in each axis is the same, the lengths of the portions CD and cd will be proportional to the whole abscissas AD and ad. And because C and c are similar points, CG is to cg as AE is to ae. Therefore CD × CG : cd × cg = AD × AE : ad × ae. This is true of each pair of corresponding rectangles, and therefore it is true of their sums. But when the number of these rect- angles is increased, and their breadth diminished without end, it is evident that the ultimate ratio of the sum of all the rectangles, such as CDHG, to the sum of all the rec- tangles edhg, is the same with that of the area ADHE to the area adhe, and the proposition is manifest. Call this the ratio of \( m \) to \( n \). But since the ordinates of the lines GHK, ghh, are inversely as the velocities, the areas are as the times (No. 99); and since these ordinates are in the constant ratio of \( m \) to \( n \), the areas are in the ratio of \( AC \times m \) to \( ac \times n \). Therefore (calling the times of the motions \( T \) and \( t \)), we have

\[ T : t = m \times AC : n \times ac ; \quad \text{and therefore} \]

\[ T^2 : t^2 = m^2 \times AC^2 : n^2 \times ac^2. \quad \text{But} \]

\[ m^2 : n^2 = ac \times ad : AC \times AD. \quad \text{Therefore} \]

\[ T^2 : t^2 = ac \times ad \times AC^2 : AC \times AD \times ac^2, \]

or \( T^2 : t^2 = ad \times AC : AD \times ac. \)

Or \( T^2 : t^2 = AC : AD \times ac : ad. \)

The attentive reader will observe that these three propositions give a great extension to the theorems which were formerly deduced from the nature of uniformly accelerated motion, or of uniform action of the forces, and were afterwards demonstrated to obtain in the momentary action of forces any how variable.

The first of the three propositions, \( V : v = F \times T : f \times t \), is the extension of the theorem \( \int dt = dv \). The second, \( V^2 : v^2 = F \times S : f \times s \), is the extension of the theorem \( \int ds = ede \). And the third, \( T^2 : t^2 = S : s \), is the extension of \( \int ds = \frac{ds}{dt} \), or of \( \int dt = ds \). These theorems hold true of all similar actions; and, only for this reason, are true of uniformly accelerated motions, or uniform actions.

There remains one thing more to be said concerning the action of accelerating forces. Their magnitude is ascertained by their effect. Therefore that is to be considered as a double force which produces a double quantity of motion. Therefore when a body A contains twice the number of equal atoms of matter, and acquires the same velocity from the action of the force \( F \), that another body \( a \), containing half the number of atoms, acquires from the action of a force \( f \), we conceive \( F \) to be double of \( f \). That this is a legitimate inference appears clearly from this, that we conceive the sensible weight of a body, or that pressure which it exerts on its supports, as the aggregate of the equal pressure of every atom accumulated perhaps on one point; as when the body hangs by a thread, and, by its intervention, pulls at some machine. Without inquiring in what manner or by what intervention this accumulation of pressure is brought about, we see clearly that it results from the equal accelerating force of gravity acting immediately on each atom. When this weight is thus employed to move another body by the intervention of the thread, which is attached to one point perhaps of that body, it puts the whole into motion, generating a certain velocity \( v \) in every atom, by acting uniformly during the time \( t \). We conceive each atom to have sustained the action of an equal accelerating force whose measure is \( \frac{v}{t} \). Without considering how this force is exerted on each atom, or by what it is immediately exerted, or how it is diffused through the body from the point to which the weight of the other body is applied by means of the thread; we still consider it as the aggregate of the action of gravity on each atom of that other body. Moreover, attending only to the motion produced by it, and perhaps not knowing the weight of the impelling body, we measure it, as a moving force, by considering it as the aggregate of the forces propagated to each atom of the impelled body, and measured by \( \frac{v}{t} \). If we know that the impelled body contains the number \( m \) of atoms, the aggregate of forces is \( m \frac{v}{t} \) or \( \frac{mv}{t} \).

But since we measure forces by the quantity of motion which they produce, we must conceive, that when the same force is applied to a body which consists of \( n \) particles, and produces the velocity \( v \), by acting uniformly during the same time \( t \), the force \( \frac{n}{t} \) is equal to the force \( \frac{mv}{t} \).

110. Sir Isaac Newton found it absolutely necessary, moving in the disquisitions of natural philosophy, to keep this force, circumstance of acceleration clear of all notions of quantitative entity of matter, or other considerations, and to contemplate the affections of motion only. He therefore considered \( \frac{v}{t} \) as the true original measure of accelerating force, and \( \frac{mv}{t} \) as an aggregate. He therefore calls the aggregate a vis motrix, a moving force, measured by the quantity of motion that it generates. And he confines the term accelerating force to the quantity \( \frac{v}{t} \) measured by the acceleration or velocity only. It would be convenient, therefore, also to confine the symbol \( f \) to \( \frac{v}{t} \), and to retain the symbol \( a \) for expressing the accelerating force \( \frac{v}{t} \).

This appellation of motive force is perfectly just and simple; for we may conceive it as the same with the accelerating force which produces the velocity \( m \) times \( v \) in one particle, by acting on it uniformly during the time \( t \). This motion of one particle having the velocity \( mv \), is the same with that of \( m \) particles having each the velocity \( v \).

If therefore a motive force \( f \) act on a body consisting of \( m \) particles, the accelerating force \( a \) is \( \frac{f}{m} \frac{v}{t} \).

Therefore the three last propositions concerning the similar, the uniform, or the momentary actions of moving forces, when expressed in the most general terms, are,

\[ v^2 = \frac{f}{mt}, \]

\[ v^2 = \frac{f}{m}, \quad \text{or} \quad ede = \frac{fds}{m}, \]

\[ t^2 = \frac{mS}{f}. \]

OF DEFLECTING FORCES.

111. When we observe the direction of a body to Deflecting change, we unavoidably infer the agency of a force which acts in a direction that does not coincide with that of the body's motion; and we may distinguish this circumstance by calling it a deflecting force. We have already shown how to estimate and measure this deflecting force, by considering it as competent to the production of that motion which, when compounded with the former motion, will produce the new motion (No. 44). Now, as all changes of motion are really compositions of motions or forces, it Deflecting forces when we show this composition.

We may almost venture to say *a priori*, that all deflections must be continual, or exhibit curvilinear motions; for as no finite velocity, or change of velocity, can be produced in an instant by the action of an accelerating force, no polygonal or angular deflection can be produced; because this is the composition of a finite velocity produced in an instant. Defective motions are all produced by the composition of the former motion, having a finite velocity, with a transverse motion continually accelerated from a state of rest. Of this we can form a very distinct notion, by taking the simplest case of such accelerated motion, namely, a uniformly accelerated motion.

Fig. 23.

Example. Determination of the path.

112. Let a body be moving in the direction AC (fig. 23) with any constant velocity, and when it comes to A, let it be exposed to the action of an accelerating force, acting uniformly in any other direction AE. This alone would cause the body to describe AE with a uniformly accelerated motion, so that the spaces AD, AE, would be as the squares of the times in which they are described. Therefore, if AB be the space which it would have described uniformly in the time that it describes AD by the action of the accelerating force, and AC the space which it would have described uniformly while it describes AE by the action of the accelerating force, nothing more is wanted for ascertaining the real motion of the body but to compound the uniform motion in the direction AC with the uniformly accelerated motion in the direction AE. AD is to AE as the square of the time of describing AD to the square of the time of describing AE; that is, as the square of the time of describing AB to the square of the time of describing AC; that is, as \(AB^2 : AC^2\) (by reason of the uniform motion in AC). This composition is performed by taking the simultaneous points B, D, and the simultaneous points C, E, and completing the parallelograms ABFD, ACGE. The body will be found in the points F and G in the instants in which it would have been found at B and C by the uniform motion, or in D and E by the accelerated motion. In the same manner may be found as many points of the real path as we please. It is plain that these points will be in a line AFG, so related to AE that \(AD : AE = DF^2 : EG^2\); or so related to the original motion AC, that \(AB^2 : AC^2 = BF : CG\), &c. This line is therefore a parabola, of which AE is a diameter, DF and EG are ordinates, and which touches AC ordinate in A.

Having thus ascertained the path of the body, we can also ascertain the motion in that path; that is, the velocity in any point of it. We know that the velocity in the point G is to the velocity of the uniform motion in the direction AC as the tangent TG is to the ordinate EG; because this is the ultimate ratio of the momentary increment of the arch AFG to the momentary increment of the ordinate EG. Thus is the velocity in every point of the curve determined. We have taken it for granted that the line of projection touches the path, and that the direction in every point is that of the tangent. To suppose that the curve, in any portion of it, coincides with the tangent, is to suppose that the body is not deflected, that is, is not acted on by a transverse accelerating force; and to suppose that the tangent makes a finite angle with any part of the path, is to suppose that the deflection is not continual, but by starts, both of which are contrary to the conditions of the case. No straight line can be drawn between the direction of the body and the succeeding portion of the path, otherwise we must again suppose that the deflection is subsultory, and the motion angular.

113. But while the investigation is so easy when the direction and intensity of the deflecting force in every point of the curve are known, the investigation of the deflecting force from the observed motion is by no means easy. The observed curvilinear motion always arises from a composition of a uniform motion in the tangent with some transverse motion; but the same curvilinear motion may be produced by compounding the uniform motion in the tangent with an infinity of transverse motions, and the law of action will be different in these transverse motions according as their directions differ. We must learn not only the intensity of the deflecting force, and the law of its variation, but also its direction in every point of the curve. It is not easy to find general rules for discovering the direction of the transverse force; most commonly this is indicated by extrinsic circumstances. The deflecting force is frequently observed to reside in or to accompany some other body. It may be presumed, therefore, that it acts in the direction of the line drawn to or from that body; yet even this is uncertain. The most general rule for this investigation is to observe the place of the body at several intervals of time before and after its passing through the point of the curve, where we are interested to find its precise direction. We then draw lines joining those places of the tangent where the body would have been by the uniform motion only. We shall perhaps observe these lines of junction keep in parallel positions; we may be assured that the direction of the transverse force is the same with that of any of these lines. This is the case in the example just now given of a parabolic motion; but when these lines change position they will change it gradually, and their position in the point of contact is that to which their positions on both sides of it gradually approximate.

But all this is destitute of the precision requisite in philosophical discussion. We are indebted to Sir Isaac Newton for a theorem which ascertains the direction of the transverse force with all exactness, in the cases in which we most of all wish to attain mathematical accuracy, and which not only opened the access to those discoveries which have immortalized his name, but also pointed out to him the path he was to follow, and even marked his first steps. It therefore merits a very particular treatment. 114. If a body describes a curve line ABCDEF (fig. 24) lying in one plane, and if there be a point S so situated in this plane that the line joining it with the body describes areas ASB, ASC, ASD, &c. proportional to the times in which the body describes the arches AB, AC, AD, &c., the force which deflects the body from rectilinear motion is continually directed to the fixed point S.

Let us first suppose that the body describes the polygon ABCDEF, &c. formed of the chords AB, BC, CD, DE, EF, &c. of this curve, and, for greater simplicity of argument, let us consider areas described in equal successive times; that is, let us suppose that the triangles ASB, BSC, CSD, &c. are equal, and described in equal times. Make BC = AB, and draw eS.

Had the motion AB suffered no change in the point B, the body would have described Be in the equal moment succeeding the first; but it describes BC. The body has therefore been deflected by an external force, and BC is the diagonal of a parallelogram (No. 45, 46), of which Be is one side, and eC is another. The deflecting force will be discovered, both in respect of direction and intensity, by completing the parallelogram BeCb. Bb is the space which the deflecting force would have caused the body to describe in the time that it describes Be or BC. Because Be is equal to BA, the triangles BSc, BSA are equal. But, by the nature of the motion, BSA is equal to BSC; therefore the triangles BSC and BSc are equal. They are also on the same base BS; therefore they lie between the same parallels, and Cc is parallel to SB. But eC is parallel to Bb; therefore Bb coincides with BS, and the deflecting force at B is directed toward S. By the same argument, the deflecting force at the angles D, E, F, &c. is directed to S.

Now, let the sides of the polygon be diminished, and their number increased without end. The demonstration remains the same, and continues, when the polygon finally coalesces with the curve, and the deflection is continual.

When areas are described proportional to the times, equal areas are described in equal times; and therefore the deflection is always directed to S. Q. E. D.

The point S may, with equal propriety of language, be called the centre of deflection, or the centre of forces; and forces which are thus continually directed to one fixed point may be distinguished from other deflecting forces by the name central forces.

The line joining the centre of forces with the body, and Of Deflecting Forces, which may be conceived as a stiff line carrying the body round, is usually named the radius vector.

115. The converse of this proposition, viz. that if the deflecting forces be always directed to S, the motion is Central performed in one plane, in which S is situated, and areas forces pro-are described proportional to the times, is easily demonstrated by reversing the steps of this demonstration. The proportion- motion will be in the plane of the lines SB and Be, be-cause the cause the diagonal BC of the parallelogram of forces is in the plane of the sides. Areas are described proportional to the times; for Ce being parallel to SB, the triangles SCB and SeB are equal; and therefore SCB and SAB are equal, &c. &c.

116. Cor. 1. When a body describes areas round S pro-portionally to the times, or when it is continually deflected inversely toward S, or acted on by a transverse force directed to S, as the per-the velocities in the different points A and E of the curve perpendicular are inversely proportional to the perpendiculars Sr and St, drawn from the centre of forces to the tangents in those points; that is, to the perpendiculars from the centre on the momentary directions of the motion: for since the triangles ASB, ESF are equal, their bases AB, EF are inversely as their altitudes Sr, St; but these bases being described in equal times, are as the velocities, and they ultimately coincide with the tangents at A and E.

117. Cor. 2. If Bz and Fs be drawn perpendicular to SA and SE, we have SA × Bz = SE × Fs, and SA : SE = Fs : Bz. For SA × Bz is double of the triangle BSA, and SE × Fs is double of the equal triangle SFE.

118. Cor. 3. The angular velocity round S, that is, the Angular magnitude of the angle described in equal times by the velocity is radius vector, is inversely proportional to the square of inversely the distance from S. For when the arches AB, EF are as the diminished continually, the perpendiculars Bz and Fs will the dis-ultimately coincide with arches described round S with tance from the radii SB and SF. Now the magnitude of an angle is the centre proportional to the length of the arch which measures it of forces directly, and to the radius of the arch inversely. In any circle, an arch of two inches long measures twice as many degrees as an arch one inch long; and an arch one inch long contains twice as many degrees of a circle whose radius is twice as short. Therefore, ultimately, the angle ASB is to the angle ESF as Bz to Fs, and as SF to SB jointly; that is, as Bz × SF to Fs × SB. But Bz : Fs = SE : SA (Cor. 2); therefore ASB : ESF = SE × SF : SB × SA, = ultimately SE² : SB².

This corollary gives us an ostensible mark, in many very important cases, of the action of a deflecting force being always directed to a fixed point. We are often able to measure the angular motion when we cannot measure the real velocities.

Having thus discovered the chief circumstances which Intimate enable us to ascertain the direction of the deflecting force, connection we proceed to investigate the quantity of this deflective determination in the different points of a curvilinear mo-tion. This is a more difficult task. The momentary ef-fect of the deflecting force is a small deviation from the tangent; and this deviation is made with an accelerated motion. The law of this acceleration regulates the cur-vature of the path, and is to be determined by it. We may be allowed to observe by the way, that it appears clearly, from the form in which Newton has presented all his dynamical theorems, that we are indebted to these problems for the immense improvement which he has made in geometry by his invention of fluxions. The pur-poses he had in view suggested to his penetrating mind the means for attaining them; and the connection be-tween dynamics and geometry is so intimate, that the same theorems are in a manner common to both. This is Of Deflect—particularly the case in all that relates to curvature. Or shall we say that the geometry of Dr Barrow suggested the dynamical theorems to Newton? We have seen how the curvature of a parabola is produced by a force acting uniformly. The momentary action of all finite forces may be considered as uniform, and therefore the curvature will be that of some portion of some parabola; but it will be difficult to determine the precise degree without some farther help. We are best acquainted with the properties of the circle, and will have the clearest notions of the curvature of other curves by comparing them with circles.

Measure of curvature. The curvature of a circular arch of given length is so much greater as its radius is shorter, for it will contain so many more degrees in the same length; and therefore the change of direction of its extremities is so much greater. Curvatures may always be measured by the length of the arch directly, and the radius inversely.

Fig. 25.

Evolution and involution of curves. Suppose a thread made fast at one end of a material curve ABCD (fig. 25), and applied to it in its whole length. Taking hold of its extremity D, unfold it gradually from the curve DCBA; the extremity D will describe another curved Deba. This geometrical operation is called the evolution of curves, and Deba is called the evolute of DCBA, which is called the involute of Deba. Perhaps this denomination has been given from the genesis of the area or surface contained by the two lines, which is folded up and unfolded somewhat like a fan. When the describing point is in b, the thread bb is undoubtedly the momentary radius of a circle ebf, whose centre is B, the point of the involute which it is just going to quit. The momentary motion of b is the same, whether it is describing an arch of the evolute passing through b, or an arch of a circle round the centre B. The same line bb, perpendicular to the thread bb, touches the circle ebf and the curve Deba in the point b. This circle ebf must lie within the curve Deba on the side of bb toward a, because on this side the momentary radius is continually increasing. For similar reasons, the circle ebf lies without the curve on the other side of bb. Therefore the circle ebf both touches and cuts the curve Deba in the point b. Moreover, because every portion of the curve between b and D is described with radii that are shorter than bb, it must be more incurvated than any portion of the circle ebf. For similar reasons, every portion of the curve between b and a must be less incurvated than this circle; therefore the circle has that precise degree of curvature that belongs to the curve in the point b; it is therefore called the equicurve circle, or the circle of curvature, and B is called the centre, and bb the radius of curvature. It is easy to perceive that no circle can be described which shall touch the curve in b, and come between it and the circle ebf; for its centre must be in some point i of the radius bb. If ib be less than bb, it must fall within the curve on both sides of b, and if ib is greater than bb, the circle must fall without the curve on both sides of bb. The circle ebf lies closer to the curve, has closer contact with it than any other, and has therefore got the whimsical name of osculating circle; and this sort of contact was called osculation.

This view of the genesis of curve lines is of particular use in dynamical discussions. It exhibits to the eye the perfect sameness of the momentary motion, and therefore of the momentary deflection, in the curve and in the equicurve circle, and leaves the mind without a doubt but that the forces which produce the one will produce the other. A great variety of curves may be described in this way. If perpendiculars be drawn to the curve Deba in every point, they will intersect each other, each its immediate neighbour, in the circumference of the curve DBA; and geometry teaches us how to find the curve DBA which shall produce the curve Deba by evolution. (See Fluxions.)

It is a matter worthy of remark, that the path of a body that is deflected from rectilineal motion by a finite force, varying according to any law whatever, may always be described by evolution. This includes almost every case of the action of deflecting forces; none being excepted but when, by the opposite action of different forces, the body is in equilibrium in one single point of its path.

Our task is now brought within a very narrow compass, namely, to measure the deflection in the arch of a circle.

Fig. 26.

Had the motion represented in fig. 26 been polygonal, it is plain that the deflecting force in the point B is to that in the point E as the diagonal Bb of the parallelogram ABCb to the diagonal Eb of the parallelogram DEFb; therefore let ABCZY be a circle passing through the points A, B, C, and let the radius vector BS cut the circumference in Z; draw AZ, CZ, and the diagonal AC, which necessarily bisects and is bisected by the diagonal Bb. The triangles bBC and CBZ are similar; for the angle CbB is equal to the alternate angle ABb or ABZ, which is equal to the angle ACZ, standing on the same chord AZ. And the angle CBb, or CBZ, is equal to CAZ, standing on the same chord CZ; therefore the remaining angle bCB is equal to the remaining angle... Now let the points A and C continually approach, and ultimately coalesce with B; it is evident that the circle ABCZY is ultimately the equicurve or coinciding circle at the point B, and that AS ultimately coalesces with and is equal to BS, and that AC × BC is ultimately \(2BC^2\); therefore ultimately \(Bb : Ei = \frac{2BC^2}{BZ} : \frac{2EF^2}{Ez}\), or

\[= \frac{BC^2}{BZ} : \frac{EF^2}{Ez}.\]

Now BC and EF being described in equal times, are as the velocities: \(Bb\) and \(Ei\) are the measures of the velocities which the deflective forces at B and E would generate in the time that the body describes BC or EF, and are therefore the measures of those forces. They are as the squares of the velocities directly, and inversely as those chords of the equicurve circles which have the directions of the deflection.

Observe that \(Bb\) or \(Ei\) is the third proportional to half of the chord and the arch described; for \(Bb : BC = BC : \frac{BZ}{2}\).

It is evident, that as the arches AB, BC, continually diminish, AC is ultimately parallel to the tangent Br, and BO is equal to the actual deflection from the tangent.

The triangles BOC and AOZ are similar, and \(BO = \frac{OC}{OZ}\),

or ultimately \(= \frac{BC^2}{BZ}\). We may measure the forces by the actual deflections, because they are the halves of the measures of the generated velocities; and we may say that

120. The actual momentary deflection from the tangent is a third proportional to the deflective chord of the equicurve circle and the arch described during the moment.

Either of these measures may be taken, but we must take care not to confound them. The first is the most proper, because the change produced on the body (which is the immediate effect and measure of the force) is the determination left inherent in it to move with a certain velocity. This is the measure also which we obtain by means of the differential or fluxionary calculus; but the other measure must be obtained when our immediate object is to mark the actual path of the body. The steps of this demonstration, which is Newton's, naturally terminate in the equicurve circle, and give at once the immediate measure of the deflecting force. At the same time the reader must perceive that this measure does not depend on the force being always directed to one centre; it is enough that the two sides of the polygon, in immediate succession, are described in equal times. This is necessary in order that \(ABC\) may be a parallelogram, and that the diagonals AC and Bb may mutually bisect each other.

Thus have we obtained a measure of deflecting force, and, in the most important cases, a method of discovering its direction. It only remains to point out the relation between the intensity of the force, the curvature of the path, and the velocity of the motion. These three circumstances have a necessary connection; for we see that the intensity is expressed by certain values of the other two in the formula \(f = \frac{\text{Arch}^2}{\frac{1}{2} \text{Chord}}\), or \(f = \frac{2BC^2}{BZ}\). The deflective velocity \(Bb\) is acquired in the time that the body describes BC; therefore the deflective velocity is to the velocity in the curve as \(Bb\) to BC. The velocity \(Bb\) is acquired by an accelerated motion along BO; for while, by progressive motion, the body describes BC, it deflects from the tangent through a space equal to the half of Bb, because the momentary action of the deflecting force may be considered as uniform. The progressive velocity BC may be generated by the same force uniformly acting through a space greater than BC; call this space \(x\). The spaces along which a body must be uniformly impelled in order to acquire different velocities are as the squares of those velocities; therefore \(Bb : BC = Bo : x\); but \(Bb : BC = BC : \frac{1}{2} BZ\); therefore \(Bb : BC^2 = Bo : \frac{1}{2} BZ\), and \(Bb : \frac{1}{2} BZ = Bo : x\); and \(Bb : Bo = \frac{1}{2} BZ : x\); but Bo is \(\frac{1}{2}\) of Bb, therefore \(x = \frac{1}{4}\) of BZ; that is,

121. The velocity in any point of a curvilinear path is that which the deflecting forces in that point would generate in the body by impelling it uniformly along one fourth part of the deflective chord of the equicurve circle. If the velocity increase, the chord of the equicurve circle must increase; that is, the path becomes less incurvated. If the force be increased, the curvature will also increase, for the chord of curvature will be less.

There is another general observation to be made on the velocity of a curvilinear motion, which greatly assists us in our investigation.

122. If a body describes a curve by the action of a force always directed to a fixed point, and varying according to some orbit any proportion whatever of the distances from that point, and if another body, acted on by the same centripetal force, move with direct toward the centre in a straight line, and if in any one case of the centre, equal distances from the centre of force the two bodies have equal velocities, they will have equal velocities in every other case of equal distances from the centre.

Let one body be impelled from A (fig. 27) toward C along the straight line AVDEC, and let another be deflected along the curve line VIKk. About the centre C describe concentric arches ID, KE, very near to each other, and cutting the curve in I and K, and the line AC in D and E; draw IC, cutting KE in N, and draw NT perpendicular to the arch IK of the curve, and complete the parallelogram ITNO. Let the bodies be supposed to have equal velocities at I and at D.

Then, because the centripetal forces are supposed to be the same for both bodies when they are at equal distances, the accelerating forces at D and I may be represent- Of Deflected by the equal lines DE and IN; but the force IN is not wholly employed in accelerating the body along the arch IK, but, acting transversely, it is partly employed in incurring the path. It is equivalent to the two forces IO and IT, of which only IT accelerates the body. Now IKN is a right-angled triangle, as is also the triangle INT; and they are similar; therefore IN : IT = IK : IN, or DE : IT = IK : DE; that is, the force which accelerates the body along DE is to the force which accelerates the body along IK as the space IK is to the space DE; therefore (No. 86) the increment of the square of the velocity acquired along DE is equal to the increment of the square of the velocity acquired along IK. But the velocities at D and I were equal, and consequently their squares were equal; and those having received equal increments, therefore the squares of the velocities at E and K are equal, and the velocities themselves are equal. And since this is the case in all the corresponding points of the line AC and the curve VIK, the velocities at all equal distances from C will be equal.

It is evident that the conclusion will be the same, if the bodies, instead of being accelerated by approaching the centre in the straight line AC, and in the curve VIK, are moving in the opposite directions from E to A, or from I to V, and are therefore retarded by the centripetal force.

123. Cor. Hence it follows, that if a body be projected from any point, such as V, of the curve, in a line tending straight from the centre, with the velocity which it had in that point of the curve, it would go to a distance VA, such, that if it were impelled along AV by the centripetal force, it would acquire its former velocity in the point V; also in any point between V and A it will have the same velocity in its recess from the centre that it has there in its approach to the centre.

The line BLFG, whose ordinates are as the intensities of the centripetal force in A, V, D, E, or in A, V, I, K, may be called the scale or exponent of force; the areas bounded by the ordinates AB, VL, DF, EG, &c. drawn from any two points of the axis, are as the squares of the velocity acquired by acceleration along the intercepted part of the axis, or in any curvilinear path, while the body approaches the centre, or which are lost while the body retires from it. When we can compute these areas we obtain the velocities (see No. 102).

We are now in a condition to solve the chief problem in the science of dynamics, to which the whole of it is in a great measure subservient. The problem is this:

124. Let a body be projected with a known velocity from a given point and in a given direction, and let it be under the influence of a mechanical force, whose direction, intensity, and variation, are all known; it is required to determine its path, and its motion in this path, for any given time.

This problem is susceptible of three distinct classes of conditions, which require different investigation.

1. The force may act in one constant direction; that is, in parallel lines.

2. The force may be always directed to a fixed point.

3. It may be directed to a point which is continually changing its place.

1. When the force acts in parallel lines, the problem is solved by compounding the rectilineal accelerated motion which the force would produce in its own direction with the uniform motion which the projection alone would have produced. The motion must be curvilinear when the accelerating force is transverse, in any degree whatever, to the projectile motion; and the curvilinear path must be concave on that side to which the deflecting force tends, for the force is supposed to act incessantly. The place of the body will be had for any time, by finding where the body would have been at the end of that time by each force acting alone, and by completing the parallelogram. Thus, suppose a body projected along AB (fig. 23), while it is continually acted on by a force whose direction is AD. Let D and B be the places where the body would be at the end of a given time. Then the body will, at the end of that time, be in F, the opposite angle of the parallelogram ABED. But it has not described the diagonal AF, because its motion has been curvilinear, as we shall find by determining its place at other instants of this time.

The velocity in any point F is found by first determining the velocity at D, and making DF to DF as the velocity at D to the velocity at B (that is, the velocity of projection, because the motion along AB is uniform). Then draw dF. Then AB is to dF as the constant velocity of projection to the velocity at F. We have seen already (No. 112–119) that dF is a tangent to the curve in F. Hence we may determine the velocity at F in another way. Having determined the form of the path in the way already described, by finding its different points, draw the tangent Fd, cutting the line DA in d. Then the velocity at A is to that at F as AB to dF. Hence also we see that the velocities in every point of the curve are proportional to the portion of the tangents at those points which are intercepted between any two lines parallel to AD.

Either of these methods for ascertaining the velocity, in this case of parallel deflections, will in general be easier than the general method in No. 121, by the equicurve circle.

It was thus that Galileo discovered the parabolic motion of heavy bodies.

2. We must consider the motions of bodies affected by inverse centripetal or centrifugal forces, always tending to one fixed point. This is the celebrated inverse problem of centripetal forces, and is the forty-second proposition of the first book of Newton's Principia. We shall give the solution after the manner of its illustrious author, because it is elementary in the purest sense of the word, keeping in view the two leading circumstances, and these only, namely, the motion of approach and recess from the centre, and the motion of revolution. By this judicious process, it becomes a pattern by which more refined, and, in some respects, better solutions should be modelled. At the same time we shall supply some steps of the investigation which his elegant conciseness has made him omit.

125. Let a body which tends to C (fig. 23) with a force proportional to the ordinates of the exponent BLFG, having the axis CA, be projected from V in the direction VQ, with the velocity which the centripetal force would generate in it by accelerating it along AV. It is required to determine the path or orbit VIKI of the body, and its place I in this orbit, at the end of the assigned time T.

Suppose the thing done, and that I is the place of the body. About the centre C, with the distances CV and CI, describe the circles YV and ID. Draw CIX to the circumference, and draw the ordinate DF of the exponent of forces, producing it toward x, and produce the ordinate VL toward a. Let vt be the distance to which the body would go along the tangent VQ in the time T, and join tC. Let this be supposed done for every point of the curve. Let axk and axy be two curves so related to the curve VIK, that the ordinate DF cuts off an area VoID equal to the orbital sector VCI, and an area VaxD equal to the circular sector VCX.

Then, because the velocity of projection is given, the

If deflecting distance \( Vt \) is known, and the area of the triangle \( Vct \) is given, then this is equal to the area \( VCI \), by the laws of central forces (No. 115). Therefore the area \( VaID \) is given. Also, because the area \( VCI \) increases in the proportion of the time, the area \( VaID \) increases at the same rate.

Fig 28.

Therefore, having these subsidiary curves \( aik \), \( axy \), the problem is solved as follows:

Draw an ordinate \( Di \), cutting off an area \( VaID \) proportional to the time, and describe a circle \( DIR \). Then draw a line \( CX \), cutting off a sector \( VCX \), equal to the area \( VaID \) cut off by the ordinate \( Dix \). This line will cut the circle \( DR \) in the point \( I \), which is the point of the orbit that was demanded.

But the chief difficulty of the problem consists in the description of the two subsidiary curves \( aik \) and \( axy \), into which the lines \( VIK \) and \( VXY \) are transformed. We attain this construction by resolving the motion in the arch of the orbit into two motions, one of which is in the direction of the transverse force, or of the radius vector, and the other is in the direction of revolution, or perpendicular to the radius.

Let \( VK \) and \( IK \) be two very small arches described in equal moments, and therefore ultimately in the ratio of the velocities in \( V \) and \( I \) (No. 73). Describe the circle \( KE \), cutting \( IC \) in \( N \). Draw \( KC \) and \( AC \), and \( kn \) perpendicular to \( VC \).

The element \( ICK \) of the orbit is \( \frac{IC \times KN}{2} \), or to \( \frac{1}{2} IC \times KN \). This is equal to the element \( DiE \) of the area \( VaID \), or to \( Di \times DE \), or to \( Di \times IN \). Therefore \( IN : KN = \frac{1}{2} IC : Di \), or \( 2IN : KN = IC : Di \), and \( Di = \frac{IC \times KN}{2IN} \).

Now let \( Afg \) be the exponent of the velocities, that is (No. 86), let \( VF \) be to \( DF^2 \) as \( ABLV \) to \( ABFD \), or \( VF : DF = \sqrt{ABL} : \sqrt{ABFD} \). Make \( Ve \) and \( li \) in the tangents respectively equal to \( VI \) and \( DF \). Draw \( ve \) and \( io \) perpendicular to \( VC \) and \( IC \), and \( em \) perpendicular to \( LV \) produced. Let \( mrs \) be an equilateral hyperbola, having \( VC \), \( ZC \), for its asymptotes, and cutting \( FD \) produced in \( r \). Then the ordinates \( Vm \), \( Dr \), are inversely proportional to \( CV \), \( CD \), or \( Vm : Dr = CD : CV = CI : CV \). But because the momentary sectors \( VCK \) and \( ICK \) are equal, \( kn : KN = CI : CV \). Therefore,

\[ Vm : Dr = kn : KN; \] but \( Vv : Vm = Vk : kn, \) and \( ii \) (or \( Df \)) : \( Vv = IK : Vk, \) therefore \( Di : Dr = IK : KN; \) but \( ii : IO = IK : KN, \) by sim. triang. therefore \( Dr = iO, \) and \( iO : Vm = VC : CI. \)

Also, by similarity of triangles, \( IO : iO = IN : KN, \) and \( 2IO : iO = 2IN : KN. \)

Now it was shown, that in order that the space \( DiE \) may be equal to the space \( ICK, \) we must have

\[ 2IN : KN = IC : Di, \] or \( 2IO : iO = IC : Di; \) but \( iO : Vm = VC : IC, \) therefore \( 2IO : Vm = VC : Di, \) and \( Di = \frac{VC \times Vm}{2IO}. \)

Having obtained \( Di, \) we easily get \( Dx; \) for the ultimate ratio of \( ICK \) to \( XCY \) is that of \( IC^2 \) to \( VC^2. \) Therefore make

\[ IC^2 : VC^2 = Di : Dx. \]

Thus are the points of the two subsidiary curves \( aik, \) \( axy, \) determined.

The rectangle \( VC \times Vm \) is a constant magnitude, and is given because \( VC \) is given, and \( Vm \) is the given velocity \( VI, \) diminished in the ratio of radius to the sine of the given angle \( CVQ. \)

But the line \( 2IO \) is of variable magnitude; but it is also given by means of known quantities. \( IO^2 \) is \( = I^2 - iO^2, \) \( = DJ^2 - DR^2, \) and \( IO = \sqrt{DJ^2 - DR^2}. \) Moreover, \( DJ^2 = ABFD, \) and \( DR^2 = \frac{VC^2 \times Vm^2}{IC^2}. \) Therefore \( 2IO = 2\sqrt{ABFD - \frac{VC^2 \times Vm^2}{IC^2}}, \) expressed in known quantities, because \( ABFD \) is known from the nature of the centripetal force.

Let the indeterminate distance \( CI \) or \( CD \) be \( x, \) and let the ordinate \( DF, \) expressing the force, be \( y. \) Let \( VC \) be \( a, \) and \( Vm \) be \( c, \) and let \( ab \) be a rectangle equal to the whole area of the exponent of force lying between the ordinate \( AB \) and the ordinate \( CZ, \) so that \( ba - \int ydx \) may represent the indeterminate area \( ABFD. \)

We have \( Di = \frac{ac}{2\sqrt{ab - \int ydx}} \) and \( Dx = \frac{ac}{2x^2\sqrt{ab - \int ydx}}. \)

Remark.—We have hitherto supposed that the velocity of projection is acquired by acceleration along \( AV. \) But this was merely for greater simplicity of argument, and that the final values of \( Di \) and \( Dx \) might be easier conceived. In whatever way the velocity is acquired, it will still be true, that when in any point \( V \) we make \( VI \) to \( Vm \) as the momentary increment \( VK \) of the arch is to the perpendicular \( kn \) on the radius vector, we shall have in every other point, such as \( I, \) the line \( DF \) to the line \( Dr \) as the increment \( IK \) of the arch to \( KN. \) And in the final equation \( DF \) will still be expressed by \( \sqrt{ab - \int ydx}. \)

126. Cor. 1. The angle which the path of the projectile makes with the radius vector is determined by this solution; for \( ii \) is to \( iO \) as radius to the sine of this angle; which sine is therefore \( \frac{ac}{x\sqrt{ab - \int ydx}}. \)

127. Cor. 2. When the magnitude \( \frac{ac}{x} \) is equal to Apsidal determinants \( \sqrt{ab - \int ydx}, \) the path is perpendicular to the radius vec-ed; OF Deflector, and the body is at one of the apsides of its orbit, and ing Forces begins to recede from the centre after having approached to it, or begins to approach after having receded.

128. Cor. 3. The curvature of the orbit VIK is also determined in every point; for the curvature of any line is inversely as the radius of the equicurve circle, and this is to the chord which passes through C as radius to the sine of the angle CII. Because the velocity in any point I = \( \sqrt{ABFD} \), and is equal to what the centripetal force at I would produce, by impelling the body along one fourth of the defective chord of the equicurve circle, we have this

\[ \text{chord} = 4 \frac{ABFD}{DF}. \]

Or we obtain it by taking a third proportional to the momentary deflection and the momentary arch of the curve, or by other processes of the higher geometry, all proceeding on the quantities furnished in this investigation.

129. Such is the solution of this celebrated problem given by Sir Isaac Newton, who may justly be called the inventor of the science of which it is the chief result, as well as of the geometry by help of which it is prosecuted. For we cannot give this glory to Galileo; for his simple problem of the motion of bodies affected by uniform and parallel gravity, however just and elegant his solution may be, was peculiar; and the same must be said of Mr Huygens's doctrine of centrifugal forces. Besides, these theorems had been investigated by Newton several years before, sua mathe faciem preferente, as corollaries which he could not pass unnoticed, from his general method. This is proved by letters from Huygens. Newton's investigation is extremely but elegantly concise, and is one of the best exertions of his sagacious mind.

130. Whether we consider this problem as a piece of mere mathematical speculation, or attend to its consequences, which include the whole of the celestial motions in all their extent and complication, we must allow it to be highly interesting, and likely to have engaged much attention in the period of ardent inquiry which closed the last century. Accordingly, it was no sooner known, by the publication of the Mathematical Principles of Natural Philosophy in 1686, than it occupied the talents of the most eminent mathematicians; and many solutions were published, some of which differ considerably from Newton's, and some are more expeditions, and better fitted for computation. Of these, the most remarkable for originality and ingenuity are those of De Moivre, Hermann, Keill, and Stewart. The last differs most from the methods pursued by others. McLaurin's propositions on this subject, and in that part of his fluxions which treats of curvature, are highly valuable, classing the chief affections of curvilinear motions geometrically, as they are suggested by the fluxionary method; and then showing, in a very instructive manner, the connection between these mathematical affections of motion and the powers of nature which produce them. This part of his excellent work is a fine example of the real nature of all inquiries in dynamics, showing that it differs from geometry little more than in the language, in which the word force is substituted for acceleration, retardation, or deflection. We recommend the careful perusal of these propositions to all who wish to have clear conceptions of the subject. Dr John Keill and Dr Horsley (bishop of Rochester) have given particular treatises on the motions of bodies deflected by centripetal forces inversely proportional to the cubes of the distances; induced by the singular motions which result from this law of action, and the multitude of beautiful propositions which they suggest to the mathematician. Newton, indeed, first perceived both of these peculiarities, and has begun this branch of the general problem. He first demonstrated the description of the logarithmic and hyperbolic spirals, and indicated a variety of curious recurring elliptical spirals, which would be described by means of this force, showing that they are all susceptible of accurate quadrature. Several of those authors affect to consider their solutions as more perfect than Newton's, and as more immediately indicating the remarkable properties of such motions; and also affect to have deduced them from different and original principles. But we cannot help saying that their claims to superiority are very ill founded; there is not a principle made use of in their solutions which was not pointed out by Newton, and employed by him. The appearance of originality arises from their having taken a more particular concern in some general property of curvilinear motions, such as the curvature, the centrifugal force, &c., and the making that the leading step of their process. But Newton's is still the best, because it is strictly elementary, aiming at the two leading circumstances, the motion to or from the centre, and the motion of revolution round that centre. To these two purposes he adapted his two subsidiary curves.

131. Is it not surprising, that, twenty-five years after the publication of Newton's Principia, a mathematician on the Continent should publish a solution in the Memoirs of the French Academy, and boast that he had given the first demonstration of it? Yet John Bernoulli did this in 1710! Is it not more remarkable that this should be precisely the solution given by Newton, beginning from the same theorem, the 40th I. Prin. following Newton in every step, and using the same subsidiary lines? Yet so it is. Bernoulli actually reduces the whole to two functions, namely,

\[ \sqrt{\frac{ab - \int \phi dx}{a^2 c}} \quad \text{and} \quad \sqrt{\frac{abc - x \int \phi dx - a^2 c}{a^2 c}}, \]

which last is plainly the same with Newton's

\[ Q \times CX^2, \]

because Newton's \( \frac{Q}{A} \) is the same with \( \frac{ac}{x} \), and Newton's \( A^2 \sqrt{ABFD - Z} \) is the same with

\[ x \sqrt{\frac{ab - \int \phi dx}{a^2 c}}, \]

which Bernoulli has changed into

\[ \sqrt{\frac{abc - x \int \phi dx - a^2 c}{a^2 c}}. \]

Bernoulli's chief boast in this dissertation is, that now philosophers may be assured that the planets will always describe conic sections; a truth of which they had not as yet received any proof; because, says he, Newton's argument for it in the corollary of the 13th proposition is inconclusive, and because he had not been able to accommodate his demonstration of the 41st and 42d propositions to the particular case of the planetary gravitation. Newton's demonstration in the corollary of the 13th proposition is just founded on the principle on which the very demonstration of the 42d, adopted by Bernoulli, proceeds, and without which that demonstration is of no force; namely, that a body, in given circumstances of situation, velocity, direction, and centripetal force, can describe no other figure than what it really describes. Newton did not accommodate the demonstration of the 42d proposition to the planetary motions, because he had already demonstrated the nature of their orbits; but mentions the case of a force proportional to the reciprocal of the cubes of their distance, not as a deduction from the 42d, but because it was not a deduction from it, and admitted a very singular and beautiful investigation by methods totally essentially different. It cannot be expected that, in the narrow limits prescribed to a work like ours, we can proceed to consider the various departments of this celebrated problem. We are only giving the outlines of the general doctrines of dynamics; and we have bestowed more time on those which are purely elementary than some readers may think they deserve. We were anxious to give just conceptions of the fundamental principles of dynamics; because we know that nothing else can entitle it to the name of a demonstrative science, and because we see much indistinctness and uncertainty, and a general vagueness or want of precision, in several elementary works which are put into the hands of persons entering on the study. This leads to errors of more consequence than a person is apt to think; because they affect our leading thoughts of mechanism itself, and our notions of the intimate nature of the visible universe.

But we must conclude the article with this great problem. Many very general doctrines of dynamics remain untouched; all, namely, that relate to the rotative motion of rigid bodies, and all that relate to the mutual action of bodies on each other in the way of impulse.

Notwithstanding these great omissions, we must observe that no new principle remains to be considered. We have given all that are necessary; and there is no question that occurs in the cases omitted which cannot be completely answered by means of the propositions already established. We have taught how to discover the existence and agency of a mechanical force, to measure and characterize it, and then to state what will be its various effects, according to the circumstances of the case.

Proceeding by these principles, men have discovered an universal fact, that every action of one body on another is accompanied by an equal re-action of that other on the first in the opposite direction; that is, to express it in the language of dynamics, "all the phenomena which make us infer that the body A possesses a force by which it changes the motion of the body B, show, at the same time, that B possesses a force by which it makes an equal and opposite alteration in the motion of A." This, however, is not a doctrine of abstract dynamics; it does not flow from our idea of force; therefore it was not included in our list of the laws of motion. It is a part of the mechanical history of nature, just as the law of universal gravitation is; and it might be called the law of universal re-action. Sir Isaac Newton has, in our humble apprehension, deviated from his accustomed logical accuracy, when he admits, as a third axiom or law of motion, that re-action is always equal and contrary to action. It is a physical law, in so far as it is observed to obtain through the whole extent of the solar system. But Newton himself did not, in the subsequent part of his noble work, treat it as a logical axiom; that is, as a law of human thought with respect to motion; for he labours with much solicitude, and with equal sagacity, to prove, by fact and observation, that it really obtains through the whole extent of the solar system; and it is in this discovery that his chief claim to unequalled penetration and discernment appears.

Availing ourselves of this fact, we, with very little trouble, state all the laws of impulsion. The body A, for example, moving to the westward at the rate of eight feet per minute, overtakes the double body B, moving at the rate of four feet per minute. What must be the consequence of their impenetrability, and of the equality and contrariety of action and re-action? Their motions must be such that both sustain equal and opposite changes. They must give, in some way or other, this indication of possessing equal and opposite forces. This will be the case if, when the changes are completed, A and B move on in contact at the rate of four feet per minute; for here A has produced in each half of B a change of motion Conulsion two, and therefore a totality of change equal to four. This is the effect, the mark, the measure of the impulsive force of A; for it is the whole impulsion. B has produced in A a change of motion four, equal to the former, and in the opposite direction. This is the effect, mark, and measure of the repulsive force of A; for it is the whole repulsion. And this is all that we observe in the collision of two lumps of clay; and the observation is one of the facts on which the reality of the physical law of equal action and re-action is founded; and we can make no further inference from this fact.

But the event might have been very different. A and B may be two magnets floating on corks on water, with their north poles fronting each other. We know, by other means, that they really possess forces by which they equally repel each other. The dynamical principles already established tell us also what must happen in this case. That both conditions of equal re-action and sensible repulsion may be fulfilled, A must come to rest, and B must move forward at the rate of four feet per minute. The same thing must happen in the meeting of perfectly elastic bodies, such as billiard balls. If elasticities are known to be imperfect in any degree, our dynamical principles will still state the effect of their collision in conformity to the law of equal reaction.

In like manner, all the motions of rotation are explained or predicted by means of the same principles of dynamics applied to the force of cohesion. This is considered as a moving force, because, when the attraction of a magnet acts on a bit of iron attached to one end of a long lath floating on water, the whole lath is moved, although the magnet does not act on it at all; some other force acts on it; it is its cohesion, which is therefore a moving force, and the subject of dynamical discussion.

And thus it appears that these subjects do not come necessarily, nor perhaps with scientific propriety, under the category of dynamics, but are parts of the mechanical history of nature. Yet, did a work like ours give room in this place, the study of mechanical nature might be considerably improved, by giving a system of such general doctrines as involve no other notions but those of force and its measures, and the hypothesis of equal re-action. Some very general, nay universal, consequences of this combination might be established, which would greatly assist the mechanician in the solution of difficult and complicated problems. Such is the proposition, that the mutual actions of bodies depend on their relative motions only, and require no knowledge of their real motions. This principle simplifies in a wonderful manner the most difficult and the most frequent cases of action which nature presents to our view; but at the same time gives a severe blow to human vanity, by forcing us to acknowledge that we know nothing of the real motion of any thing in the universe, and never shall know any thing of it, till our intellectual constitution, or our opportunities of observation, are completely changed.

M. d'Alembert had made this principle still more serviceable for extricating ourselves from the immense complication of actions that occurs in all the spontaneous phenomena of nature, by presenting it to us in a different form, which more distinctly expresses what may be called the elements of the actions of bodies on each other. His proposition is as follows (see his Dynamique, page 73):

"In whatever manner a number of bodies change their D'Alembert's motions, if we suppose that the motion which each body bont's goes would have in the following moment, if it were perfectly neutral free, is decomposed into two others, one of which is the ciple of motion which it really takes in consequence of their mutual actions, the other will be such, that if each body were..." impressed by this force alone (that is, by the force which would produce this motion), the whole system of bodies would be in equilibrio."

This is almost self-evident; for if these second constituent forces be not such as would put the system in equilibrio, the other constituent motions could not be those which the bodies really take by the mutual action, but would be changed by the first.

For example, let there be three bodies P, Q, R, and let the forces A, B, C act on them, such as would give them the velocities p, q, r, in any directions whatever, producing the momenta, or quantities of motion, \(P \times p\), \(Q \times q\), \(R \times r\), which we may call A, B, C, because they are the proper measures of the moving force. Let us moreover suppose, that, by striking each other, or by being any how connected with each other, they cannot take these motions, A, B, and C, but really take the motions a, b, and c. It is plain that we may conceive the motion A impressed on the body P, to be composed of the motion a, which it really takes, and of another motion α. In like manner, B may be resolved into b, which it takes, and another β; and C into c and x. The motions will be the same, whether we act on P with the force A, or with the two forces a and α; whether we act on Q with the force B, or with b and β; and on R with the force C, or with c and x. Now, by the supposition, the bodies actually take the motions a, b, and c; therefore the motions α, β, and x, must be such as will not derange the motions a, b, and c; that is to say, that if the bodies had only the motions a, b, and x, impressed on them, they would destroy each other, and the system would remain at rest.

M. d'Alembert has applied this proposition with great address and success to the very difficult questions that occur in the motions and actions of fluids, and many other most difficult problems, such as the precession of the equinoxes, &c. The cause of its utility is, that in most cases it is not difficult to find what forces will put a system in equilibrio; and, combining these with the known extraneous forces whose effects we are interested to discover, we obtain the motions which really follow the mutual action of the bodies.

This is not, properly speaking, a principle; it is a form in which a general fact may be conceived. In the same way the celebrated mathematician De la Grange observed, that a system of bodies, acting on each other in any way, is in equilibrio, if there be impressed on its parts forces in the inverse proportion of the velocities which each body takes in consequence of their action or connection; and he expresses this universal fact by a very simple formula; and, calling this also a principle, he solves every question with ease and neatness, by reducing it to the investigation of those velocities. In this way he has written a complete system of dynamics, to which he gives the title of Mécanique Analytique, full of the most ingenious and elegant solutions of very interesting and difficult problems; and all this without drawing a line or figure, but accomplishing the whole by algebraic operations.

But this is not teaching mechanical philosophy; it is merely employing the reader in algebraic operations, each of which he perfectly understands in its quality of an algebraic or arithmetical operation, and where he may have the fullest conviction of the justness of his procedure. But all this may be (and, in the hands of an expert algebraist, it generally is), without any notions, distinct or indistinct, of the things or the process of reasoning that are represented by the symbols used of.

---