must take place through inassignable passages and never by a leap. This also is the reason why nature suffers no hard non-elastic bodies. In order to show this, let us pretend that a hard non-elastic globe proceeds to strike against a similar globe at rest: after the impact it is necessary that the two globes rest, in which case the law of the conservation of force is violated, or that there be some motion and that the globe which was at rest receive it, being unable to be taken as immovable, although even if it should feign to be such, the striking body (in order to preserve the force) would nec essarily be reflected suddenly backward. This is a forbidden (defendu) change, since it would be by a leap, a body which proceeds from a certain side being obliged to abate its motion, even to rest, before beginning to proceed gradually further and further backward. But the globe which is struck being obliged to receive motion, there will also be a change by a leap, the struck globe which was at rest being obliged to receive a certain degree of velocity suddenly, not being pliable so as to receive it gradually and by degrees. It being also manifest, as is necessary, either that the globe striking passes suddenly to rest, which would be already a change by a leap, or that if this striking globe retains a certain velocity, the struck globe which was at rest receives suddenly an amount which is not less than that of the striking globe, since the globe struck must either stop the striking globe, or go before it. Thus the striking globe passes suddenly from velocity to rest, or at least the struck globe passes suddenly from rest to a certain degree of velocity, without passing through the intermediate degrees, which is contrary to £he law of continuity, which admits no change by a leap in nature. I have also many other reasons all of which concur in banishing hard non-elastic bodies, but this is not the place to enlarge upon them.
But it is necessary to admit, that although bodies must be thus naturally elastic in the sense which I have just explained, nevertheless the elasticity often appears insufficient in the masses or bodies which we employ, even if these masses should be composed of elastic parts and should resemble a sack full of hard balls which would yield to a moderate impact, without leaving the sack, as we see in the case of soft bodies or those which yield without recovering themselves sufficiently. The reason is that the parts are not sufficiently united therein to trans fer their change to the whole. Whence it conies that in the impact of such bodies a part of the force is absorbed by the small parts which compose the mass, without this force being given to the whole; and this must always happen when the pressed mass does not recover per fectly; although it also happens that a mass shows itself more or less elastic according to the different manner of the impact, witness the water itself which yields to a moderate impression, and makes a can non-ball rebound.
670 LEIBNITZ'S CRITIQUE OF LOCKE Now when the parts of the bodies absorb the force of the impact, as a whole, as when two pieces of rich earth or of clay come into collision, or in part, as when two wooden balls meet, which are much less elastic than two globes of jasper or tempered steel; when, I say, some force is absorbed by the parts, it is as good as lost for the absolute force, and for the respective velocity, that is to say, for the third and the first equation, which do not succeed, since that which remains after the impact has become less than what it was before the impact, by reason of a part of the force being turned elsewhere. But the quantity of prog ress, or rather the second equation, is not concerned therein. And even the motion of this total progress remains alone, when the two bodies proceed together after the impact with the velocity of their com mon centre, as do two balls of rich earth or clay. But in the semi- elastics, as two wooden balls, it happens still further that the bodies mutually depart after the impact, although with a weakening of the first equation, following this force of the impact which has not been absorbed. And in consequence of certain experiments touching the degree of the elasticity of this wood, we might predict what should happen to the balls which should be made of it in every kind of col lision or impact. But this loss of the total force, or this failure of the third equation, does not detract from the inviolable truth of the law of the conservation of the same force in the world. For that which is absorbed by the minute parts is not absolutely lost for the universe, although it is lost for the total force of the concurrent bodies.
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 [Front, the Latin] PART I From the time we made mention of the founding of a New Science of Dynamics, many distinguished men in various places have asked for a fuller explication of this doctrine. Since, therefore, we have not i Gerhardt, Leibniz, math. Schrift., II., 2 [Vol. 6], pp. 234-246. Dutens, Leibnit. op. om., 3, 315-324. First published in the " Acta Eruditor. Lips.," ill APPENDIX 671 yet leisure to compose the book,1 we will give in this place those things which may kindle some light, which perhaps will return to us with interest, if, indeed, we shall elicit the opinions of those who unite energy of thought with elegance of speech, whose judgments also we openly confess will be acceptable to us, and we hope useful in the set ting forward of the work. We have elsewhere suggested that there is in corporeal things something besides extension, nay, prior to ex tension, namely, the force itself of nature everywhere implanted by its Author, which consists, not in the simple faculty with which the schools seem to have been content, but, besides, is provided with a tendency (conatu) or effort (m'su) which will have its full effect unless impeded by a contrary tendency (conatu). This effort often appears to the senses, and in my judgment is known everywhere in matter by the reason, even when it does not appear to the sense. But if now this force must not be assigned to God through a miracle, it is certainly necessary that this force in bodies themselves be produced from the body itself, nay, that it constitute the inmost nature of bodies, since to act is a mark of substances, and extension means nothing else than the continuation or diffusion of the already presupposed struggling and withstanding, that is, resisting substance, so far is it from being itself able to produce substance. Nor is it necessary, because every corporeal action arises (est} from motion, and motion itself does not exist unless from motion, either in the body already before exist ing or impressed from something external to it (aliunde). For motion (just as time) never exists, if you reduce the thing to d/cpi- fltiav, because a whole never exists, when it has not coexisting parts. And nothing is so real in itself, as that momentary increment (momen- taneum) which must be constituted in a force striving for change. To this, therefore, returns whatever there is in corporeal nature besides the object of geometry or extension. And by this method, in fact, regard is had at the same time for both the truth and the doctrine of the ancients. And as our age has freed from contempt the atoms of Pemocritus, the ideas of Plato, and the tranquillity of the Stoics in the best nexus of things, so now the traditions of the Peripatetics concern ing forms or entelechies (which deservedly seemed enigmatical and scarcely rightly perceived by the authors themselves) will be referred to intelligible notions, so that we think it necessary rather to explain the philosophy thus received by so many ages, so that it may be con sistent (where this is permitted) and to illustrate and then increase it with new truths, than to destroy it.
And this kind of studies seems to me especially suited both to the intelligence (prudenticc^) of the teacher and to the profit of the learners, 1 For the work referred to, cf. Gerhardt, Leibniz, math. Schrift., II., 2 672 LEIBNITZ'S CRITIQUE OF LOCKE so that we may not seem more desirous of destroying than of building, nor be tossed between perpetual changes of doctrine, daily uncertain because of the pride of audacious geniuses, but at length the human race, the lust of sects being curbed (which the inane glory of change [novandi~\ stimulates), the certain dogmas established, without stum bling, not less in philosophy than in mathematics, will make further advance, since in the writings of distinguished men, ancient and mod ern (if you take away entirely those things in which they speak too severely against others), there is wont to be very much that is true and good, which deserves to be rescued and to be distributed into the pub lic treasury. And would that men preferred to do this rather than spend their time in censures by which they only appease their own vanity. But somehow very many even hostile views do not displease us certainly, whom fortune has so favored in certain new views of ours, that friends often bade us think of these only, and each view is considered according to its own value, although diverse; the reason of which perhaps is that in discussing many things we have learned to despise nothing. But now let us return to our subject.
A ctive force (which with some you call not ill power — virtus) is two fold, namely primitive, which exists in every corporeal substance per se (since I think a wholly quiescent body abhorrent to the nature of things), or derivative, which by a limitation as it were of the primi tive, resulting through the conflicts of bodies with each other, is vari ously exercised. And the primitive force indeed (which is nothing- else than €j/reA.e'x£ia 17 irpuTrj) corresponds to Ike substantial soul or form, but indeed for this reason pertains only to general causes, which cannot suffice for the explanation of phenomena. And so we agree with those who deny that forms must be employed in handing down the particular and special causes of sensible things: to point out which is worth while, lest, while we lead them as it were back again to the open fountains of things, at the same time we seem to desire to return to the 'vain repetitions' (battolocjias) of the vulgar school. Meanwhile a knowledge of them is necessary to correct philosophiz ing, nor may any one think he is master of the nature of body, unless he has turned his mind to such things and understood that that crass notion of corporeal substance is imperfect, not to say false, and depends upon the imagination alone and was introduced inconsider ately some years since by an abuse of the corpuscular philosophy (in itself excellent and most true), as indeed is evident by this argument which does not entirely exclude cessation and rest from matter, and cannot bring forward reasons for the laws controlling the derivative force of nature. In like manner passive force also is twofold, either primitive or derivative. And indeed the primitive force of enduring or resisting constitutes that very thing which is called primary matter, if you rightly interpret it, in the schools, by which it happens that body APPENDIX 673 is not penetrated by body, but forms an obstacle to it, and it is en dowed at the same time with a certain laziness, so to speak, that is, repugnance to motion, and does not indeed suffer itself to be set in motion unless by the somewhat broken force of the active body. Whence afterwards the derivative force of enduring variously exhibits itself in secondary matter. But it is our part now to proceed farther, having removed those general and primitive forces and substituted those by which we are taught that because of form every body always acts and because of matter every body always endures and resists, and in this doctrine of derivative forces and resistances to investigate how far bodies prevail by various efforts or again variously resist; for the laws of actions, which are known not only by reason, but are confirmed also by sense itself through phenomena, are adapted to these.
Derivative force therefore, by which bodies in action act mutu ally on each other or mutually suffer from each other, we under stand in this place as 110 other than that which is connected with motion (i.e. local), and in turn tends to produce further local motion. For we admit that through local motion other material phenomena can be explained. Motion is a continual change of place, and thus requires time. Yet the movable element (mobile) existing in motion, as it has motion in time, so in any moment whatever has velocity, which is so much the greater as more space is run over and less time is expended. Velocity taken in connection with direction is called conatus; but impetus is the product of the mass of the body into the velocity, and its quantity is so much that the Cartesians are wont to call it the quantity of motion, namely, the mo mentary increment (momentaneam'), although, speaking more accurately, the quantity of motion itself, existing forsooth in time, arises from the aggregate of the impetuses (equal or unequal) existing in the mov able element in the given time multiplied in order into the time. We, nevertheless, in discussing with these have followed their fashion of speaking. Nay even as (not inconveniently for the doctrinal use of speaking) we can distinguish the accession which now is made from the accession already made or to be made, as an increment of accession or element; or as we may distinguish the present descent from the descent already made, which it increases; so we can discern and call Motion the momentary or instantaneous element of motion diffused by the motion itself through a period of time; and so that which is com monly ascribed to motion is called the quantity of motion. And although in the use of terms we are compliant (faciles) in accord with an ac cepted interpretation, nevertheless it especially behooves us to be care ful in their use lest we be caught by their ambiguity.
Moreover, as the estimate of motion through a period of time is made from infinite impulses, so in turn the impulse itself (although 674 LEIBNITZ'S CRITIQUE OF LOCKE a momentary thing) is made from infinite degrees successively im pressed upon that same movable body (mobile), and has a certain ele ment from which, unfolded, nothing but infinity can arise.
Conceive a tube AC (Fig. 4) to revolve with a certain fixed uniform velocity in the horizontal plane of this page about an immovable centre C, and a ball B, existing in the cavity of the tube, to be freed from its chain or impediment, and to begin to be moved by the centrifugal force; it is manifest that the attempt in the beginning to depart from the centre by which, namely, the ball in the tube tends towards its extremity A, is infinitely small in respect to the impulse which it already has from the rotation, or by which with the tube itself, the ball B tends from the place D towards (D), its distance from the centre being retained. But with the continuance for some time FIG. 4. of the centrifugal impression proceeding from the rotation, from its own progress there must arise in the- ball a certain complete centrifugal impulse (Z))(Z>), comparable with the impulse of rotation D(D). Hence it is evident that the effort is twofold, elementary to be sure, or infinitely small, which I call, also, solicitation, and formed by the continuation or repetition of elementary efforts; that is, the impulse (impetum) itself, although I do not, for that reason, mean that these mathematical entities are really so found in nature, but only that they are useful in making accurate estimates by mental abstraction.
Hence force, also is twofold: the one elementary, which T call also dead, because motion (motus) does not yet exist in it, but only a solicitation to motion (solicitatio ad motum), such as that of the ball in the tube, or of the stone in the sling, even while it is held still by the chain; the other, however, is ordinary force, united with actual motion, which I call living. And an example of dead force indeed is the centrifugal force itself, and likewise the force of gravity or centripetal force, the force also by which the tense elastic body (elastrum) begins to restore itself. But in percussion, which arises from a heavy body falling already for some time, or from a bow restoring itself for some time, or from a similar cause, the force is living force, which has arisen from an infinite number of con tinued impressions of dead force. And this is what Galileo meant, when in his enigmatical manner of speaking he spoke of the infinite force of percussion, namely if compared with the simple effort of gravity. But although the impulse (impetus) is always united with living force, yet we shall show below that these two are different.
Living force in any aggregate of bodies again can be known as APPENDIX 675 twofold, namely total, or partial; and partial again is either re spective or directive, that is, either proper or common to the parts. Respective (respective?) or proper (propria) force is that by which the bodies comprised in the aggregate can act among themselves mutually; directive or common force is that by which, besides, this aggregate can act outside itself. But I call it direct, because the total force of direction is preserved intact in this partial force. But this alone would remain, if suddenly the aggregate were imagined to congeal by the intercepted motion of its parts among themselves. Whence from respective and directive taken together total absolute force is composed. But these things will be better understood from the rules to be propounded below.
The ancients, as far as known, had a science of dead force alone, and this it is which is commonly called Mechanics, treating of the lever, block, inclined plane (where belongs the wedge and the spiral), the equilibrium of liquids, and similar things, in which they treat in turn only of the first tendency (conatu) of the bodies among themselves, before they received an impulse by acting. And although the laws of dead force can in some fashion be transferred to living force, yet there is need of great caution, as even they may have been deceived, for this reason, who confounded force in general with the quantity produced by the multiplication of the mass into the velocity, because they understood that force is dead in the regular theory of these. For this thing happens there for a special reason, as we already long ago suggested, since (for example), in dif ferent descending weights, in the very beginning of motion at least the descents themselves or the quantities of the spaces gone through in the descent, certainly infinitely small or elementary hitherto, are proportional to the velocities or to the efforts to descend. But the progress being made and living force having arisen, the acquired velocities are no longer proportional to the spaces already run over in the descent, by which, nevertheless, we have shown formerly and shall further show-, that the force must be estimated, but only to the elements of these velocities. Galileo began to discuss concerning living force (although under another name, nay, I should rather say, concept) and was the first to explain how by the acceleration of descending \veights motion arises. Descartes rightly distinguished velocity from direction, and saw even in the conflict of bodies that follow by which the former conditions are least changed. But he did not rightly estimate the least change, while he changes the direction alone or the velocity alone, since the change moderated by mixing would be obtained from both: but how this must come about escaped him, because to him, intent at that time upon modal manifestations rather than upon realities, phenomena so heterogeneous did not seem capable of being compared 676 LEIBNITZ'S CRITIQUE OF LOCKE and modified by union, so that we shall say no more of his other errors in this doctrine.
Honoratus Fabri,1 Marcus Marci,2 Joh. Alph. Borellus,3 Ignatius Baptista Pardies,4 and Claudius de Chales,5 and other men very acute in learning have published things not to be despised on motion, but nevertheless have not shunned these capital errors. Huygens,6 the first that I know who has adorned our age with splendid discoveries, seems to me in this argument also to have reached the pure and un adulterated truth and to have freed this doctrine from paralogisms, 2 Johannes Marcus Marci von Kronland, 1595-1(567, a German physician, mathematician, and physicist, published his De proportions mot us, sea rer/ula sphygmica ad celeritatem et tarditatem pulsuum, ex illius motu pondcribus geometricis librato, absque errore metiendam, Pragse, 1039, a remarkable work on the theory of impact, preceding by thirty years the researches of Wallis, Wren, and Huygens. This is probably the work to which Leibnitz here refers. — Tit.
3 Giovanni Alfonso Borelli, 1(508-1(579, a distinguished Italian physician and mathematician, the founder of the iatromathematical theory of medicine, was Professor of Mathematics at Pisa and Naples. He was the author of several medical and mathematical works, among which were Tlieorlca mecHceorum planetarium, ex causis physicis deducta, Florent., 1(566; L>e motu animalium, Rome, 1080-1(581; De vi percussionis, Lugd. Bat., 1(586; and De Motionibas naturalibuA, a gravitate pendentibus, Lugd. Bat., 1(>86. The two latter works are probably the ones referred to by Leibnitz. For an account of his views, cf. Lasswitz, Gesch. d. Atoinistik, Hamburg u. Leipzig, 1890, Vol. 2, pp. 300-;!28.
4 Ignace Gaston Pardies, 1030-1673, a French Jesuit and geometer, was Professor of Philosophy, and afterwards of Mathematics, at Paris. In his correspondence with Newton, he sought to explain the dispersion of light as a diffraction by aid of the assumption that the transmission of light depends upon a wave-movement. He intended to write a great work on Mechanics, of Avhich only Pts. 1 and 2, Discours du mouvement local, Paris, 1670, and La statique ou la science des forces mouvantes, Paris, 1673, appeared. His CEiirres de mathe'matiques, etc., 4th ed., appeared a La Haye, 1710. Cf. Lasswitz, Gesch. d. Atoniistik, 2, 340. Leibniti refers to Pardies in a communication to H. Fabri, cf. Gerhardt, Leibniz, math. Schrift., II., 2 [Vol. 6], 81, 84; Gerhardt, Leibniz, philos. Schrift., 4, 244, 247. —TR.
5 Claude Francois Milliet Deschales, 1621-1678, a French Jesuit and dis tinguished mathematician and physicist, published his Cursus sen mundus mathematicus, 3 tomi, 1st ed., Lugduni, 1074, 2d ed., Ki90. Leibnitz refers to him briefly in a communication to H. Fabri, Gerhardt, Leibniz. philos. Schrift., 4, 245; Leibniz, math. Schrift., II., 2 [Vol. <>], 81. Brief account of his views is given in Lasswitz, Gesch. d. Atomistik, 2,487-190. — TR.
6 Cf. ante, p. 150, note 3. The doctrine of Huygens, here referred to, is found in his De motu corporum ex percussione (1(>(>9), which was first published in 1703; Opera reliqua, Ainstel., 1728, Vol. 2; CEiivres completes, La Haye, 1888 sq. His correspondence with Leibnitz is found in Gerhardt, Leibniz, math. Schrift., I., 2 [Vol. 2], 1-208. References to the doctrine of motion occur on pp. 140, 184. An account of Huygens' physical and mechanical views is given in Lass witz, Gesch. d. Atomistik, 2, 341-397. — Tit.
APPENDIX 677 APPENDIX 677 by certain rules formerly published. Wren x also, Wallis 2 and Mariotte,8 men excellent in these studies though diverse in method, demonstrated nearly these same rules. But concerning the causes, nevertheless, opinion is not the same; whence men distinguished in these studies do not always admit the same conclusions. And indeed the true sources of this science have not yet, as is evident, been dis closed. Nor indeed is what seems certain to me admitted by all: that rebounding or reflection springs only from elastic force, that is, from internal resistance to motion. Xor has any one before us ex plained the notion itself of forces, which thing hitherto has disturbed the Cartesians arid others, who, even for this reason, could not com prehend that the sum of the motion or impulse (which they regard as the quantity of the forces) can appear different after the encounter from before, because for this very reason they believed the quantity of the forces to be changed.
From me, still a youth, and at that time constituting the nature of body, with Democritus and his adherents in this matter, Gassendi and Descartes, in inert mass alone, there escaped a little book " Hypothesis Physica " 4 by title, in which I set forth a theory of motion, at the same time abstract (abstractam) from the system and concrete (concretam) for the system,5 which beyond the merit of its 1 Sir Christopher Wren, 1G31-1723, best known as the architect of St. Paul's, London, was distinguished at Oxford for his knowledge of geometry and applied m;ithematics. In IGliO he was elected Savilian Professor of Astronomy at Oxford. Newton, in his Principia, ed. 1713, p. 19, speaks highly of his work as a geometrician. Leibnitz refers to him in the Hypoth. phys., cf. Gerhardt, Leibniz, math. Schrift., II., 2 [Vol. G], 26, 29, 30, 75; Phllos. Schrift., 2 John Wallis, 1616-1703, an eminent English mathematician, appointed Savilian Professor of Geometry, at Oxford, Ki49, published his Mechanica, sive de Motu Tractatus Geometric.ux, 3 parts, 1669-1671. His complete works were published, Oxford, 1695-1699, 3 vols., fol. The correspondence between Leibnitz and Wallis is found in Vol. 3 of this ed., and also in Gerhardt, Leibniz, math. Schrift., I., 4 [Vol. 4], 1-82. — TR.
3 Cf. ante, p. 121, note 4. An elaborate treatise on the percussion of bodies, De la percussion ou choc des corps, probably the one to which Leibnitz here refers, is found in first volume of his QSuvres, Leyden, 1717. — TR.
4 Cf. Gerhardt, Leibniz, math. Schrift., II., 2 [Vol. 6], 17-80; Gerhardt, Leib niz, jihilos. Schrift., 4, 177-240; Dutens, Leibnit. op. om., 2, Pt. II., 1-48. — TR.
5 The compressed and somewhat obscure text is explained by the titles of two essays forming the Hypothesis Physica nova. The title of the first essay is: Theoria mot us concreti sen Hypothesis de rationibus phsenomenornni nostri Orbis; that of the second is: Theoria motus abstracti sen Rationes motuum universales a sensu et phsenomenis independentes. Cf. Gerhardt's Einleitung, Leibniz, ph'dos. Schrift., 4, 9-12, and especially the portion, there quoted, of a letter to Foucher, found also in op. cit., 1, 415; Erdmann, Leibnit. op.philos., 117; F. de Careil, Lettres et Opuscules inedits de Leibniz, Paris, 1854, 119-120; Ttntens, Leibnit. op. om.,2, Pt. I.,242; transl. Duncan, Philos. Wks. of Leibnitz, 678 LEIBNITZ'S CRITIQUE OF LOCKE mediocrity I see has pleased many distinguished men. I there estab lished, having assumed such a conception of body, the fact that every striking body gives its impulse (conatwn) to the receiving or directly opposing body as such. For since, in the moment of attack, the receiving body undertakes to go forward and thus run away with itself, and that impulse (on account of the indifference then believed by me of the body to motion or rest) must have its effect wholly in the receiving body, unless hindered by a contrary impulse, nay, even if hindered by it, since it is so necessary that these different impulses should be adjusted among themselves; it was manifest that no cause could be given why the striking body should not attain the effect towards which it tends, or why the receiving body should not receive the entire impulse of the striking body, and so the motion of the receiving body was composed of its own pristine impulse, and of the newly received or foreign impulse. From which then 1 was showing that if mathematical notions alone, — magnitude, figure, place, — and the change of these, or in the moment itself of impact (concursus) the impulses to change were perceived in the body, no theory of meta