in virtue of their position their velocities are reciprocal to their masses or weights, or when they have the same quantity of motion.
But we must know that this equality of force in this case arises from another principle, for generally absolute force must be estimated by the violent effect which it can produce. I call the effect violent which consumes the force of the agent, as, for example, to give such a velocity to a given body, to raise a given body to such a height, etc. And we can conveniently estimate the force of a heavy body by the product of the mass or of the weight multiplied by the height to which the body might rise by virtue of its motion. Xow two bodies being in equilibrium, their heights to which they might rise or from which they might descend are reciprocal to their weights, or rather the products of the heights by the weights are equal. And it happens only in the case of equilibrium or of dead force, that the heights are as the velocities, and that thus the products of the weights by the velocities are as the products of the weights by the heights.2 This, I say, happens only in the case of dead force, or of the infinitely small motion which I am accustomed to call solicitation, which takes place 1 The French is: " Mais justement de cette chose qu'il fallait." — TR.
2 On the margin of the manuscript Leibnitz has remarked: "Thus it is astonishing that Descartes has avoided so well the rock of velocity taken for force, in his little treatise on Statics or dead force, where there was some danger, having reduced all to weights and heights, when it was indifferent, and that he has abandoned the heights for the velocities in the case where he should have done wholly the contrary; that is to say, when he discusses percussions or living forces which must be measured by weights and heights." — Gcrhardt's Note. — TR.
660 LEIBNITZ'S CRITIQUE OF LOCKE when a heavy body tries to commence movement, and has not yet conceived any impetuosity; and this happens precisely when bodies are in equilibrium, and, trying to descend, are mutually hindered. But when a heavy body has made some progress in descending freely, and has conceived some impetuosity or living force, then the heights to which this body might attain are not proportional to the veloci ties, but to the squares of the velocities. And it is for this reason that in case of living force the forces are not as the quantities of motion or as the products of the masses by the velocities.
Nevertheless it is noticeable, and has contributed to the error,1 that two bodies unequal in absolute living force, — for it is of this I speak, — but whose quantity of motion is equal, can stop each other, which fact has made men believe them absolutely equal in force; as, for example two bodies A of mass 3 velocity 2, and B of mass 2 velocity 3. For although A is absolutely weaker than B, A being able to raise a pound only 12 feet, if B can raise a pound 18 feet; nevertheless in the concourse they can stop each other, the reason of which is that bodies are hindered only according to the laws of dead or static force. For being elastic as we suppose, they act between themselves only by dead forces or according to the equilibrium in the concourse, that is to say, by inassignable changes, because in pressing, resisting, and continually weakening each other more and more until they come to rest, they destroy one another at each moment only by the infinitely small motion, or dead force, equal on both sides; now the quantity of dead force is estimated according to the laws of equilibrium by the quantity of motion, infinitely small in truth, but whose continual repetition exhausts at last the whole quantity of motion of the two bodies, which being supposed equal in both bodies, each quantity of motion is exhausted in the same time, and consequently the two bodies are reduced to rest in the same time by the pressures of their elasticities, which, restoring themselves afterwards, reproduce the motion. It is (in) this continual diminution of the quantity of motion according to the equilibrium in the concourse of the two elasticities that the cause of this paradox consists, that two absolute unequal forces, but which have the quantities of motion equal, must stop each other because this happens in a relative action where the contest takes place only according to the quantities of motion infinitely small continually repeated.
Now it is found by reason and by experiment, that it is living abso lute force, or that which is estimated by the violent effect it can pro duce, which is preserved, and nowise the quantity of motion. For if this living force could ever be augmented, the effect would be more 1 The French text reads: " Cependant il est remarquable et a contribuer a 1'erreur," etc. The reading should be: " et a contribue," etc. — TR.
APPENDIX 661 powerful than the cause, or rather the perpetual mechanical motion, that is to say, which could reproduce its cause and something more, which is absurd. But if the force could be diminished, it would perish at last entirely; for never being able to increase, and being able never theless to diminish, it would always go more and more into decay, which is without doubt contrary to the order of things. Experiment con firms it also, and we shall find always that if bodies should convert their horizontal into ascending motions, they could always raise on the whole the same weight to the same height before or after the impact, supposing that no force has been absorbed in the impact by the parts of the bodies, when these bodies are not perfectly elastic, without speaking of that which the medium, the base, and other cir cumstances absorb. But as this is a thing which I have sufficiently explained before, I will not repeat it here.
Now I am very happy to give still another turn to the matter and to show further the conservation of something approaching more the quantity of motion, namely, the conservation of moving action (I'action motrice). Here then is the general rule that I establish. Whatever changes may take place between concurrent bodies, of whatever num ber, there must ahvai/s be in the concurring bodies between themselves alone the same quantity of moving action in one and the same interval of time. For example, there must be during this hour as much moving action in the universe or in the given bodies, acting between themselves alone, as there will be during any other hour whatever.
To understand this rule, it is necessary to explain the estimate of moving action {I'action motrice), wholly different from the quantity of motion, in the manner that the quantity of motion has been wont to be understood as has been explained above. Now in order that the moving action may be estimated, we must first esti mate the formal effect of motion. This formal or essential effect of motion consists in that which is changed by the motion, namely, in the quantity of the mass which is transferred, and in the space or in the length through which this mass is transferred. There is the essential effect of motion, or that which finds itself changed: for this body was there, now it is here: the body is so much and the distance is so much. I conceive in order to greater facility that the body is moved so that each point describes a straight line equal and parallel to that of every other point of the same body. I mean also a motion uniform and continuous. This assumed, the formal effect of motion is the product of the mass which is transferred multiplied by the length of the removal, or rather the formal effects are in reason com posed of the masses and the lengths of the removal, so that a body, as 2, being transported the length of 3 feet, and another body, as 3, being transported the length of 2 feet, the formal effects are equal. It is necessary carefully to distinguish what I here call \h& formal effect, or LEIBNITZ'S CRITIQUE OF LOCKE that essential to motion, from that which I called above the violent effect. For the violent effect consumes the force and is exercised upon something without; but the formal effect consists in the body in motion, taken in itself, and does not consume the force, and even conserves it rather, since the same translation of the same mass must always be continued, if nothing from without prevents; it is for this reason that the absolute forces are as the violent effects which con sume them, but nowise as the formal effects.
Now it will be easier to understand what moving action (Faction motrice) is: it must then be estimated not only by the formal effect which it produces, but also by the vigor or velocity with which it pro duces it. We wish to transport 100 pounds to a distant place; that is the formal effect which is demanded. One desires to do it in one hour, another in two hours; I say that the action of the first is double that of the second, being doubly quick with reference to an equal effect. I suppose always continual and uniform motion. We may say also that a body, as 3, being transported the length of 5 feet, in 15 minutes, is the same action as if a body, as 1, were transported the length of one foot in one minute.
This definition of moving action (V action motrice) is justified suffi ciently a priori because it is manifest that in a purely formal action taken by itself, as is here that of a moving body considered by itself, there are two points to examine, — the formal effect or that which is changed, and the promptness of the change; for it is very manifest that that which produces the same formal effect in less time is the more active. But if any one is obstinately bent upon disputing with me this definition of moving action, it would suffice me to say that I am free to call moving action what I just explained, provided that nature justifies afterwards the reality of this nominal definition, which will be when I shall show that it is precisely this whose quantity nature conserves.
Now since moving action is that which comes by multiplying the formal effect by the velocity, I wish to give more distinctly the esti mate of velocity. We know that when two movable bodies run over uniformly the same space in unequal times, the velocity of that one which runs over it in less time will be the greater, in proportion as the time is shorter. Thus the spaces gone over being equal, the velocities are reciprocally proportional to the times. But if the times were equal, the velocities would be as the spaces gone over. For one body in motion having gone over a foot in one minute, and the other two feet, it is manifest that the velocity of the second is double. Thus the velocities are in reason composed of the direct of the spaces gone over and of the reciprocal of the times employed. Or what is the same thing, to estimate the velocity, we must take the space and divide it by the time. For example, A accomplishes 4 feet in 3 APPENDIX seconds and B 2 feet in 1 second; the velocity of A will be as 4 divided by 3, namely as f, and the velocity of B will be as 2 divided by 1, namely as 2, so that the velocity of A will be to that of B as |- to 2, that is to say, as 2 to 3.
Now the question is to verify the conservation of the moving action (I' action motrice). I can give its general demonstration in a few words, because 1 have already proved elsewhere that the same force is conserved, and because at bottom the exercise of force or the force taken at the time is action, the abstract nature of force consisting only in that. Thus since the same force is conserved, and since action is the product of the force by the time, the same action will be con served in equal times. But I wish to verify it by the detail of the laws of motion established by experiment and commonly received. I shall content myself with one example; but we shall find it the same in every other example we might choose. And indeed we could see at once the general reason of it, by making the calculation in abstracto, or in general and by letters, without employing any particular numbers, liut to suit the intelligence of everybody I prefer to give an example in numbers.
Let there be a right angle LMN (Fig. 3), whose sides ZM, I/TV may E <E be prolonged at discretion. Let a straight line AM be taken, so that prolonged beyond the point M it would cut the angle LMN into two equal parts. We might consider ^AM as the hypotenuse of a square whose side may be called 1. This being so, I suppose that the body, A,1 1 We take no account here of the thickness of the bodies, which we suppose inconsiderable. — Leibnitz's Note.
664 LEIBNITZ'S CRITIQUE OF LOCKE being in the place t/l at the moment 1, A goes from the point v-l to the point M, during the time 1, 2, and there meets at the moment 2 the two bodies B and C, which had been in repose during the 1, 2, which is known in the figure in that their place is designated by jfi and by 2B, as also by jC1 and by 2C. Now the body A meeting the two bodies in M at the moment 2, being in Af or 2A, will drive them forward and come to rest in M, a point which will also be 3A and 4A, because A will remain there during the times 2, 3 and 3, 4, as I suppose the two mutually equal, and to the times 1, 2. But B will go towards L from the moment 2 during the time 2, 3 with a velocity as 1, and will meet at the moment 3 the body D, which had before gone in front of it during the times 1, 2, from the place ^D to the place 2D, and during the times 2, 3, from the place 2D to the place 3Z>, with a velocity as J. Now B, meeting D at the moment 3, will give it the velocity 3-D4/); that is to say, in the times 3, 4, tZ) will reach 4D, and during that time, B will go from 3B to 4B with the velocity sBtB. It will be the same on the other side, where C, pushed by A in the moment 2, will go towards N with the velocity 1, and will meet, at the moment 3, the body E, which goes against it, having gone before, during the times 1, 2, from the place ^E to the place 2E, and during the times 2, 3 from the place 2E to the place,E, with a velocity as f. Now C, meeting E at the moment 3, will give it the velocity SE±E; that is to say, that in the times 3, 4, it comes from SE to 4E. And during this time, C will go from 3C to 4C with the velocity 3C4C.
The register of the masses and velocities follows.
During the times 1, 2 the velocities of the bodies A, B, C, D, E are During the times 2, 3 the velocities of the bodies A, B, C, D, E During the times 3, 4 the velocities of the bodies A, B, C, D, E are 0, |, J-, f, ^, where it is to be remarked that the body C, instead of advancing, reflects backward with the velocity ^.
The justification of these numbers will be found in the rules or equations which we shall assign farther on.
Let us now make the calculation of the moving actions (actions matrices) during the times equal between them — 1, 2; 2, 3; 3, 4. During the times 1, 2.
A is in mass 1, the length of the transfer -^A2A is v/2. Then multiplying one by the other, the formal effect is ^/2. The velocity comes from dividing the length ^/2 by the time 1, which makes V-- And multiplying the effect by the velocity, the moving action is 2.
B and C are at rest during this time in ^B, 2B, or jC, 2C, conse quently their moving action is 0.
APPENDIX 665 D is in mass 2, the length of the transfer £, the formal effect 2 by £ or 1. The length £ being divided by the time 1, the velocity J arises, and the effect multiplied by the velocity is 1 by J, or J, which is the action of D.
E is in mass J, the length of the transfer f, consequently the effect §. Now the length f divided by 1 gives the velocity f, which, multiplied by the effect, furnishes f the action of E.
A is at rest, and its action is 0.
B is in mass 1, the length of the transfer 1 (namely, 2B3B), the formal effect 1; the length 1 divided by the time 1 gives the velocity 1, which, being multiplied by the effect 1, 1 arises, which is the action of B.
C; the calculation is the same in regard to C and there arises the same action 1.
D has the same action as in the preceding time; namely, £.
E likewise has the same action as in the preceding time; namely, f.
And the sum of all the moving actions of the bodies A, B, C, D, E, during the times 2, 3, is 0 + 1 + 1 + i + | = ff> as before. Finally, during the time 3, 4.
A is at rest, and its action is 0.
B is in mass 1, the length of the transfer, namely, SB4B, is ^, conse quently the effect is |. The same length, -*, divided by the time 1, gives } for the velocity, which multiplied by the effect, | arises, the action of B.
C is in mass 1, the length of the transfer sCtC is ^, consequently the formal effect is ^. For it matters not here when we seek absolute things, whether C advances by 3C4C, or reflects backward, as it does in fact. The same length, i, divided by the time 1 gives the velocity^, which, multiplied by the effect, there arises ^ as the action of C.
D is in mass 2, the length of the transfer aD±D is f, consequently the effect is f. The same length divided by the time 1 is f, or the velocity, which multiplied by the effect, there arises ff, which is the action of D.
E is in mass £, the length of the transfer is -1/-, the effect £. The same length divided by the time 1 is -x^-, that is to say, the velocity, which, multiplied by the effect, produces f f- for the action of E.
And the sum of all the moving actions of the bodies A, B, C, D, E, during the time 3, 4, is as in each one of the preceding times.
666 LEIBNITZ'S CRITIQUE OF LOCKE I have followed in this calculation the general method, for as the moving actions are not only equal in equal times, but proportional to the times in unequal times, I have divided the space by the time, in order to have the velocity; but when the time is always the same, as here, a-nd thus we can take it as unity, the division by the time changes nothing, and consequently for the velocity we can take the number of the length of the transfer, the velocities being as the spaces: whence it is manifest that the effect being the product of the mass and the space, and the velocity being as the space, the action is as the product of the mass by the square of the space of the transfer (we mean a horizontal transfer in falling bodies), or as the product of the mass by the square of the velocity. Now, I shall prove, further on, in the 3d equation, that the sum of these products of the masses by the squares of the velocities is conserved in the concourse of the bodies. Con sequently, it is proved that the moving action is conserved, without speaking of other proofs by which I have shown elsewhere that the forces are conserved, and that the forces are as the products of the masses by the squares of the velocities, while the actions are as the products of the forces by the times, so that if we did not know elsewhere this estimate and conservation of force, we might learn it here, in finding by the calculation in detail, or even in general, by the 3d equation, further on, that the moving action is conserved; now it is clear that the moving actions are in reason composed of the forces and the times, and the times being the same, the moving actions are as the powers or forces.
But shall we be astonished whence comes this success, which will never fail, however intricate may be the example which we may choose? It may be proved a priori, independently of the rules of motion received; and this is what I have shown many times in dif ferent ways. But here I shall show that it is proved by these very rules of percussion which experience has justified, and whose rationale we may give by the method of a boat, as Huygens has done, and in many other ways, although we are always obliged to assume something non-mathematical, which has its source higher. But I shall reduce the whole to three equations very simple and beautiful, and which contain all which concerns the central concourse of two bodies in one and the same straight line.
Conspiring velocities of the body a before the impact v after x.
b y I call these conspiring velocities, because I suppose they all tend from the side whence proceeds the centre of gravity common to the two bodies. But if perchance any velocity proceeds really in the contrary direction, then the letter which expresses the conspiring velocity sig nifies a negative quantity. But we shall always take the body a as a APPENDIX 067 APPENDIX 067 body whose velocity is really conspiring, or proceeds from the side of the centre of gravity before the impact, and also in such a manner that the body a follows and does not precede the common centre of gravity. Thus the signs do not vary in v, but they may vary in y, z, x. Here, now, are our three equations: — I. Lineal equation, which expresses the conservation of the cause of the impact, or of the relative velocity and v — y signifies the relative velocity between the bodies before the impact with which they approach, and z — x signifies the relative velocity with which they depart after the impact. And this relative velocity is always the same in quantity before or after the impact, supposing that the bodies are very elastic, which this equation states. It is necessary only to remark that while the signs vary in the explica tion of the detail, this general rule will embrace all the particular cases. This also occurs in the following equation: — II. Plane equation, which expresses the conservation of the common or total progress of the two bodies I call progress here the quantity of motion which proceeds from the side of the centre of gravity, so that if the body b, for example, should proceed in the contrary direction before the impact, and thus its con spiring velocity y be negative or be expressed by — (?/), understand ing by (y) mass (molem), or that which is positive in y, then the progress of a will be av, the progress of b will be —b(y). And the total progress will be av — b(y), which is the difference of the quanti ties of motion of the two bodies. If the bodies a and b proceed from one and the same side before and after the impact, these letters, v, y, x, z, signify only conspiring velocities real or affirmative, and conse quently in this case it appears by this equation that the same quantity of motion will be conserved after and before the impact. But if the bodies a and b should proceed in a contrary direction before the impact and in the same direction after the impact, the difference of the quantity of motion before the impact would be equal to the sum of the quantity of motion after the impact. And there will be other similar variations according to the variation of the signs of the letters III. Solid equation, which expresses the conservation of the total absolute force or of the moving action aw + byy = axx + bzz.
This equation has this excellence, that all the variations of the signs which can arise only from the diverse direction of the velocities y, x, z, y cease, by the fact that all the letters which express these veloci- LEIBNITZ'S CRITIQUE OF LOCKE ties mount here to the square. Now — y and + y have the same square + yy, so that all these different directions of y produce noth ing more. And it is also for that reason that this equation gives something absolute, independent of the relative velocities, or of the progressions from a certain side. The question here concerns only the estimating of masses and velocities, without troubling ourselves from what side these velocities arise. And this it is which satisfies at the same time the rigor of the mathematicians and the wish of the philoso phers, — the experiments and reasons drawn from different principles. Although I put together these three equations for the sake of beauty and harmony, nevertheless two of them might suffice for our needs. For, taking any two of these equations, we can infer from them the remaining one. Thus, the first and the second give the third in the following manner. By the first, we shall have v + x= y + z; by the second, we shall have a, v — x = b, z — y; and, multiplying one equation by the other, according to the corresponding sides, we shall have a, v — x, v + x = b, z — y, z + y, which makes aw — axx — bzz — byy, or the third equation. In the same way, the first and the third give the second; for a, vv — xx — b, zz — yy, which is the third, divided by the first v + x = z + y, side by side, we shall have a, vv — xx,:, v + x = b, zz — yy,:, z + y, which makes a, v — x — b, z — y, that is, the second equation. Finally, the second and the third equation give the first. For the third a, vv — xx — b, zz — yy divided by the second, namely, which makes v + x = z + y, according to the first equation.
I would add only one remark, which is that many distinguish between hard and soft bodies, and the hard themselves as elastic or not, and build thereupon different rules. But we may take bodies naturally as hard-elastic, without however denying that the elas ticity must always come from a fluid more subtile and penetrating, whose motion is disturbed by the tension or by the change of the elasticity. And as this fluid must be composed itself in its turn of little solid bodies, elastic between themselves, we see well that this replication of solids and of fluids continues to infinity. Now this elasticity of bodies is necessary to nature, in order to obtain the execution of the grand and beautiful laws which its infinitely wise author has proposed, among which not the least are these two laws of nature which I first made known, the first of which is the law of the conservation of absolute force or of moving action in the uni verse, with some other absolutely new conservations which depend upon it and which T will explain some day, and the second is the law of continuity, in virtue of which, among other effects, every change APPENDIX 609