physical notions being held, namely, of power active (actricis) in form and of inactivity (ignavlie), of of resistance to motion in matter, and so if it were necessary for the concourse of events to be determined by the geometrical composition of impulses alone, as we have ex plained: then it must follow that the impulse of the striking body, even the least, is impressed upon the entire receiving body, although the greatest, and so the greatest body at rest is dragged away by the striking body, however small, without any retardation of this body, since indeed no repugnance, but rather indifference of it to motion, is contained in such a notion of matter. Whence it would not be anv more difficult to impel a large quiescent body than a small one, and thus there would be action without reaction, and there could be no estimate of power, since anything whatever could be proved by any thing whatever. And since these things, and many others of the same kind, are opposed to the order of things and conflict with the principles of a true metaphysic, I thought at that time therefore (and indeed truly) that the most wise Author of things in the construction of the system shunned those things which would follow of themselves from the mere laws of motion derived from pure geometry.
But afterwards, having examined things more deeply, I saw in what the systematic explanation of things consisted, and regarded that former hypothesis of the notion of the body as incomplete, and both by other arguments and also by this itself it was proved that in body there must be placed something besides magnitude and impene trability, whence the consideration of forces arises, by adding the met aphysical laws of which to the laws of extension those very rules of motion are produced which I called systematic; namely, that every APPENDIX 679 change takes place gradually, and every action is accompanied with reaction, and new force is not produced without loss of the former, and so the one dragging always is retarded by the one dragged, and neither more nor less power is contained in the effect than in the cause. And since this law is not derived from the notion of mass, it must necessarily follow from another thing, which is in bodies, namely, from the force itself, which certainly always preserves its own quan tity the same, although it is employed by different bodies. Hence therefore, besides considerations purely mathematical, and subject to the imagination, I have concluded that certain considerations meta physical, and perceptible by the mind alone, must be admitted, and a certain principle superior to the material mass, and, so to speak, a formal addition, since indeed all the truths of corporeal things can not be concluded from logical and geometrical axioms alone, namely, from great and small, whole and part, figure and position, but some others must be added from cause and effect, and action and passion, by which the reasons of the order of things are saved. Whether we call that principle form, or evTeA.e'x«a, or force, does not matter, pro vided we remember that it is intelligibly exhibited through the notion of forces alone.
But although to-day certain distinguished men, seeing this very thing, that the common notion indeed of matter is not sufficient, fetch in God OTTO yi«7Xav»7s> and take away all force of acting from things, a kind of Mosaic philosophy as it were (as Fludd once called it), I cannot assent. For although I admit that it has been very clearly perceived by them that there is no proper influx of one cre ated substance into another, if the thing is driven to metaphysical strictness, and I confess even freely that all things always proceed from a continuous creation by God; yet I think there is no natural truth in things, the reason of which is to be sought immediately in the divine action or will, but that always in the things themselves something has been placed by God, whence all their predicates are explained. It is certainly evident that God has created not only bodies, but also souls, to which correspond the primitive entelechies. But these things will be demonstrated elsewhere by their own proper reasons more profoundly drawn out.
Meanwhile, although I admit an active principle superior to mate rial notions and, so to speak, vital everywhere in bodies, yet I do not therefore here agree with Henry More and other men, distinguished for piety and genius, who so make use of a certain Archaeus 1 or hylar- chic principle even for the management of phenomena (ad phenomena procuranda), as if forsooth all things cannot be explained mechani cally in nature, and as if those who undertake this seem to make way 1 Cf. Neio Essays, ante, p. 67, note 3. — TR.
680 LEIBNITZ'S CRITIQUE OF LOCKE with incorporeal things, not without suspicion of impiety; or as if with Aristotle it is necessary to imagine intelligences in the revolving orbs, or the elements must be said to be driven up or down by their own form, by a short (compendiosa) but useless method of teaching: with these, 1 say, I do not agree, nor has that philosophy pleased me any more than that theology of certain ones, who so believed that Jupiter thundered or snowed that they even branded the searchers after proper causes with the crime of atheism. In my opinion the temperament is the best which satisfies both piety and science, so that we admit that all corporeal phenomena indeed can be sought from efficient mechanical causes, but we know that the mechanical laws in the universe are derived from higher reasons, and so \ve make use of a higher efficient cause only in establishing general and remote things. But these once established, as often as afterwards we treat of the near and special efficient causes of natural things, we give no place to souls or entelechies, no more than to the idle faculties or inexplicable sympathies, since the primal and most universal efficient cause itself, must not intervene unless as far as the ends are regarded, which the divine wisdom had in so ordering things, that we neglect no occasion of singing his praise and the most beau tiful hymns.
And in truth final causes (as I have shown by a wholly remarkable example of an optical principle, with the strong approval of the very celebrated Molyneux in his " Dioptrics ") are repeatedly employed with large result even in special physics, not only that we may admire the more the most beautiful works of the supreme Author, but also that we may sometimes in this way divine what through the way of efficient causes not equally or only hypothetically are manifest. Thus far perhaps philosophers have not yet sufficiently observed this use. And it must be maintained in general, that everything in things can be explained in two ways: through the kingdom of power or efficient causes, and through the kingdom of wisdom or through final causes; God regulating bodies as machines after the manner of an architect according to the laws of magnitude or mathematical laws, and indeed for the use of souls; but souls, capable of wisdom, as his own fellow- citizens and sharers of a certain society with himself, after the manner of a leader, nay of a father rather according to the laws of goodness or moral laws for his own glory, both kingdoms everywhere inter penetrating, yet un confused and undisturbed the laws of each, so that at the same time both in the kingdom of power the great est and in the kingdom of wisdom the best is obtained. But we propose in this place to establish the general rules of productive forces, which we can then use in the explanation of special efficient causes.
Next I came to the true, and indeed precisely the same, estimate APPENDIX C81 of forces, by the most different ways: one indeed o priori, from the simplest consideration of space, time, and action (which I elsewhere will explain), the other a posteriori, namely, by estimating the force by the effect which it produces in consuming itself. For I under stand here not any effect, but that for which force must be expended or in which it must be consumed, which you can call, for that reason, violent, such as that effect is not, which a heavy body employs in run ning through a perfectly horizontal plane, because in such an effect however produced it always retains the same force, although also in this very effect rightly treated, so to speak, as harmless, we have followed this our method of estimating, but now it is laid aside by us. Further I chose that effect of the violent effects which is especially capable of homogeneity or division into similar and equal parts, such as exists in the ascent of a body possessed of weight: for the elevation of a heavy body two or three feet is precisely double or triple the eleva tion of the same heavy body one foot; and the elevation of a doubly heavy body one foot is precisely double the elevation of a single heavy body to the height of one foot; whence the elevation of a doubly heavy body three feet is precisely six times the elevation of a simple heavy body one foot, supposing namely (at least for the sake of teaching, although perhaps in truth the matter is otherwise constituted, but the error here nevertheless is imperceptible), that the heavy bodies gravitate equally in the greater or less distance from the horizon. For in an elastic body homogeneity has not with equal ease a place. When therefore I wished to compare bodies different or endowed with different velocities, I easily saw that, if the body A is single and the body B is double, but the velocity of each equal, the force of that one is simple, of this double, since, in short, whatever is placed in that once is placed in this twice. For in B there is a body twice the equal and equivalent of A itself, and nothing besides. But if the bodies A and C are equal, but the velocity in A is simple and in C double, I saw that, not in short what is in A, is doubled in C, since the velocity indeed is doubled, yet not also the body. And T saw that here an error has been made by those who believed that the force itself is doubled by that reduplication of modality (modalitatis) alone; as already I once observed and sug gested, and that the true and not hitherto (although after so many Elements of universal Mathematics have been written) handed-down art of estimating consists in this, that finally it attains to something homogeneous; that is, a reduplication accurate and of all kinds, not only of modes, but also of things. Of which method no other better or more remarkable specimen could be given than that which is exhibited in this argument itself.
In order, therefore, to obtain these results, I considered whether these very two bodies A and C, equal in magnitude but different in 682 LEIBNITZ'S CRITIQUE OF LOCKE velocity, could produce some effects equal to their causes and homo geneous among themselves. For thus those things which by them selves could not easily be compared, by their effects at least might be compared accurately. I assumed, moreover, that the effect must be equal to its cause if it is produced by the expenditure or consumption - C of the entire force: in which case ,V it matters not in how much time it is produced. Let us suppose, there- H C R to be heavy, and to convert their pIQ 5 force into an ascent, which will hap pen, if at the moment in which they have their said velocities, A simple. B double, are known to exist at the extremities of the vertical pendulums PA, EC. But it is evident from the demonstrations of Galileo and others that, the body A with a velocity as 1 at the highest ascending above the hori/.on HR to the height of one foot 2A H, the body C with a velocity as 2 can surely ascend (at the highest) to the height 2CR of four feet. Whence it already follows that a heavy body having a velocity as 2 is in power four times as much as the one having a degree of velocity as 1, since by the expenditure of all its force it can accomplish, in short, four times as much. For raising a pound (that is, itself — id est se ipsum) four feet, in short raises four times one pound one foot. And in the same manner it is inferred generally that the forces of equal bodies are as the squares of the velocities, and thence the forces of bodies in general are in reason composed of the simple of the bodies and the doubles of the velocities.
I have confirmed the same things to absurdity (namely, to per petual motion) by bringing back the contrary opinion, generally received, especially among the Cartesians, according to which forces are believed to be in reason composed of bodies and velocities: which method, indeed, I used repeatedly to define a posteriori two states unequal in force, and to distinguish the greater at the same time from the less by a certain mark. And since in substituting the one for the other, perpetual mechanical motion or an effect more powerful than the cause does not arise, those states are not in the least equivalent to themselves, but that which was substituted for the other was more powerful because it has caused something greater to be performed. Hut I assume as certain that nature never substitutes things unequal to the forces themselves, but the complete effect is always equal to the full cause; and, in turn, those things which are equal to the forces, with safe reckoning can be substituted by us for them with the freest supposition, as if we made that substitution in act, and thus APPENDIX 683 with no fear 1 of perpetual mechanical motion. But if, therefore, it were true, as men generally persuade themselves, that a heavy body A as 2 (for so now we assume it) endowed with a velocity as 1, and a heavy body C as 1 endowed with a velocity as 2, are equivalent to each other, we ought to be able to substitute with impunity the one for the other. But this is not true. For let us assume that A as 2 has acquired a velocity as 1 in the descent 2AVA from the height 2ATI less than a foot; and now in ^A itself or in the existing horizon, let us substitute instead of it the equivalent (as they wish) weight (pondus) C as 1 with a velocity as 2, which ascends as far as to C, or to the height of four feet. And so by the descent alone of the weight A of two pounds from the height of one foot 2AH, and having sub stituted its equivalent, we have accomplished the ascent of one pound four feet, which is double the former. Therefore we have gained just as much force, or we have produced perpetual mechanical mo tion, which is certainly absurd. And it does not matter whether by the laws of motions we can actually accomplish this substitution; for between equivalents indeed substitution can safely be made.
Although, indeed, we have thought out various plans by which it will be accomplished actually so nearly as we wish, that the entire force of the body A will be transferred to the body C, before at rest, but which now (A itself being brought to rest) is alone put in motion. Whence it will happen, that, instead of a weight of two pounds of a velocity as 1, would succeed one pound of a velocity as 2, if these were equivalent; whence we have shown that an absurdity arises. For these things are not indeed worthless, nor do they consist in logomachies, but are of the greatest use in comparing machines and motions. For if any one has force from water or animals or from other cause, by which a heavy body of a hundred pounds is kept in constant motion, so that within a fourth part of a minute of time it can complete a horizontal circle of a diameter of thirty feet; but another maintains that a double weight in its place, in the same time, uniformly accomplishes only half the circle with less expendi ture, and reckons that to you as if it were a gain; be it known that you are deceived and caught by half of the forces. But now having put to flight errors, let us set forth a little more distinctly in the second part of this hastily thrown-off production (Schediasmatis) the true and truly to be admired laws of nature.2 1 Gerhardt reads, " motu," evidently a typographical error. Dutens, Leibnit. op. om., 3, 324, reads, " metu," which the translation follows. — TR.
2 Dutens (Leibnit. op. om., 3, 3'24) adds: " proponemus, mense Maio exhi- benda," i.e. to be presented in the month of May. The article, however, never appeared in print, but remained in Ms. among Leibnitz's papers, and was first printed by Gerhardt. in his edition of Leibnitz's mathematical writings. A translation of the article is herewith given. — TK.
684 LEIBNITZ'S CRITIQUE OF LOCKE ESSAY ON DYNAMICS IN DEFENCE OF THE WONDER FUL LAWS OF NATURE IN RESPECT TO THE FORCES OF BODIES, DISCLOSING THEIR MUTUAL ACTIONS AND REFERRING THEM TO THEIR CAUSES1 [From the Latin] PART II The nature of body, nay, of substance in general, not being suffi ciently known, had brought it about (a fact we have already touched upon) that certain distinguished philosophers of our time, since they placed the notion of body in extension alone, were compelled to have recourse to God in order to explain the union between the soul and the body, nay, also, the communication of bodies with each other. For it must be confessed that it is impossible for a mere extension, involving only geometrical notions, to be capable of action and passion: and so this alone seemed to be left to them, that, when man thinks and undertakes to move his arm, God, as it were, by a primeval compact, moves his arm instead of he himself; and on the other hand, when the motion exists in the blood and spirits,2 God excites percep tion in the mind. But these very things, since they are foreign to a correct method of philosophizing, ought to admonish the authors that they are resting upon a false principle, and that they have not assigned rightly the notion of body, from which such things followed. We have shown, therefore, that there is in every substance a force of action and, if it is created, of passion also, that the notion of extension is in itself not complete, but that a relation to something which is extended, whose diffusion or continued replication it makes known, and so the substance of a body is presupposed, which involves a power of acting and resisting, and everywhere exists as a corporeal mass, and the diffusion of this is involved in extension. Whence one day we shall kindle a new light, also, for the explanation of the union of the soul and the body. But now we must show how from thence ^follow wonderful and extremely useful practical theorems, pertaining to Dynamics, that is, the science which teaches the rules especially of corporeal forces.
It must be known before all things that force, indeed, is some thing truly real, even in created substances; but space, time, and motion have something of a rational entity, and are true and real, iGerhardt, Leibniz, math. Schrift., II., 2 [Vol. C], pp. 246-254; aus d. Manuscript, der Konigl. Biblioth. zu Hannover.
2 Cf. Bacon's theory of the Spiritus, a brief account of which is given by Lasswitz, Gesch. d. Atomistik, 1, 431-432. — TR.
APPENDIX 685 not of themselves, but since they involve divine attributes, — im mensity, eternity, operation, or the force of created substances. Hence it already follows that there is no vacuum in space or time; that motion, moreover, separated from force, or where in it only the geo metrical notions, — magnitude, figure, — and the variation of these are considered, is in truth nothing else than a change of position, and so motion, as far as (quoad) phenomena are concerned, consists in a mere relation, which also Descartes acknowledged, when he defined transla- tibn~Trom the neighborhood of one body into the neighborhood of another. But in deducing his consequences he forgot his definition, and determined the rules of motions as if motion was something real and absolute. So, therefore, we must consider, if an indefinite num ber of bodies are in motion, that from the phenomena it cannot be deduced in which of them absolute determinate motion orrest exists, but to any one you please taken from these can be attributed rest,""] provided that the same phenomena come forth. Hence it follows (a result which Descartes did not notice) that the uniformity of the hypotheses is changed neither by the encounters (concurs;**) of bodies with each other; and besides, that such rules of motions must be assigned that the respective nature of motion may remain intact, nor from the event after the encounter can it be divined through the phenomena where before the encounter there had been rp^f, or determinate abso lute motion. Whence the rule of Descartes does not at all accord with the facts, by which he asserts that a body at rest can in no way be driven from its place by another smaller body, and other things of this sort. Than which nothing is more remote from the truth. It follows, also, from the relative nature of motion, that the action of bodies against each other by turns or percussion is the same, provided they approach each, other with the same velocity; that is, the same appearance remaining in the given phenomena, whatever at length be the true hypothesis, or to whichever at length we rightly ascribe motion or rest, the same event appears in the phenomena sought or resulting, even in respect to the action of bodies among themselves. And this also is what we find by experience (experimur), that we shall feel the same pain, whether our hand runs against a stone at rest, suspended, if you please, from a thread, or the stone with the same velocity runs against our hand at rest. Meanwhile, we speak thus, according as the thing demands, for a more suitable and simpler explanation of the phenomena, precisely as in spherics we employ the motion of the"! primum mobile, and in the theory of the planets we must use the I Copernican hypothesis, so that already these disputes, urged on with ' so much erfort(iif w"Kich even theologians were implicated), straight way disappear. For although force is something real and absolute, nevertheless motion pertains to the class of relative phenomena, and ~~ truth is looked for not so much in phenomena as in causes.
686 LEIBNITZ'S CRITIQUE OF LOCKE From our notions also of body and forces this principle arises, that what happens in substance can be known to happen spontaneously and in an orderly manner. With this is connected the principle that no change takes place by a leap. This posited, it follows also that atoms cannot be allowed. That we may seize the force of this conclusion, let us assume that the bodies A and B (Fig. 6) meet together and that ^A comes to 2A, and again VB to 2B, and thus meeting in 2A2B are reflected from 2A to SA and from 2/j to 3B. But supposing that atoms exist, that is, bodies extremely hard and thus inflexible, it is evi dent that the change takes place by a leap or momentary incre ment (rnomentaneam), for direct motion in the moment (momenta) itself of the encounter becomes retrograde unless we can assume that immediately after the encounter the bodies rest, that is, lose their force, which thing, besides the fact that it would otherwise be absurd, would continue again the change by a leap, a momentary increment (rnomentaneam), namely, from motion to rest, and not however a transition by intermediate steps. And so we must know, if the bodies A and B (Fig. 7) meet and come from -^A^B into the place YIG 7. of concourse 2A2B, that there they are gradually compressed like two inflated