balls, and more and more approach each other in turn by the con tinually increased pressure; that, moreover, by this thing the motion itself is weakened, the force itself of the effort being carried over into the elasticities (elastra) of the bodies, until at length they are reduced to rest; but then at length the elasticity of the bodies restoring them, they themselves rebound from each other in turn with a retrograde motion begun again from rest and continually increasing; at length with the same velocity with which they ap proached each other, regained but turned in the opposite direction, they recede in turn from each other and return into the positions KArB which coincide with the positions }A^B, if the bodies are supposed equal and of equal velocity. Thence it is already evident how no change takes place by a leap, but the progress being gradually dimin ished and at length reduced to rest, then at length a regress arises. So that as from one figure another is not made (as from a circle an oval) unless through innumerable intermediate figures, nor is there any passing over from a place to a place or from a time to a time unless through all intermediate places and times, so not from motion is rest produced, and much less an opposite motion, unless through all the intermediate degrees of motions. And since this principle APPENDIX 687 is of so great consequence in nature, I wonder it is so little thought of. From these considerations follows what Descartes had attacked in his letters, and now also certain great men are unwilling to admit, that all reflection arises from elasticity, and a reason is given of many remarkable experiments which indicate that a body bends before it is propelled, as Mariotte has very beautifully illustrated. Finally, that especially wonderful principle follows from these considerations, that no body is so poor but that it has elasticity, and so is pervaded by a fluid still more subtile; and then that there are no elements of bodies, and that neither the most fluid matter nor certain solid globules of the second element, exact and durable, are to be granted, but that analysis proceeds to infinity.
It is consistent with this law of continuity, excluding leap from change, that a case of rest can be cons^dered_ JS-JL special, case of motion, namely, as an evanescent or very small motion, and a case i>!' equality can be considered as a case of evanescent inequality. Whence the consequence is, that such laws of motions must be assigned that there be no need for peculiar rules for aad at r£sj, but these rules spring from the rules of bodies unequal and moving of themselves, or, if we wish to enounce peculiar rules for rest and equality, we must be careful lest we assign such as do not agree with the hypothesis which considers rest as the last motion or equality as the last inequality, otherwise \ve shall violate the harmony of things and our rules will not agree among themselves. This new system of testing our own or others' rules, I published first in the " Nouvelles de la Republique des Lettres," July, 1687t article S,1 and called it a general principle of order (principium ordinis gen- erale), springing from the notion of the infinite artfHffie ^ cofttintnim, acTcfing to it the axiom, that from orderly data the results also are orderly (datis ordinatis etiam qucesita sunt ordinata). I expressed the principle universally thus: If a case approaches a case continually in the data and at length disappears in itself, the results of the cases must also approach each other continually in the things sought for, and at length cease in turn in themselves. (Si casus ad casum continue accedat in datis tandemque in ipsum evanescat, necesse est ut etiam eventus casuum sibi con tinue accedant in quwsitis tandemque in se invicem desinant.) Precisely as in geometry the case of the ellipse approaches continually the case of the parabola, in proportion as one focus remaining another more and more remote is regarded as assumed, until in the case of another focus infinitely removed the ellipse passes into the parabola. Whence all the rules of the ellipse must be verified in the parabola (taken as an ellipse whose other focus is infinitely distant). Whence the radii falling parallel into a parabola can be conceived as coming from 1 Cf. Gerhardt, Leibniz, philos. Schrift., 3, 51-55; Erdmann, 104-106. — TK.
LEIBNITZ'S CRITIQUE OF LOCKE another focus or tending towards it. Since, therefore, in the same way a case in which the body A runs against B in motion can be varied continually, so that the motion of A itself remaining, the motion of B itself is regarded as less and less, until at length it is regarded as disappearing in rest and thence again grows in the con trary direction; I say the result of the attack, but rebounding into what whether in A itself or in B itself, continually approaches by both motions the result of the attack which exists in the case of B at rest, and in it finally ceases; and thus the case of rest both in the data (in datis) and in the result or that which is sought (qucesitis) is the limit of the cases of motion in a straight line, or the common limit of direct and continuous motion, and thus as it were a special example of either. With regard to this touchstone (lydium lajridem), brought over by me from geometry to physics, when I examined the Cartesian rules of motions, it happened, wonderful to say, that a cer tain hiatus or leap showed itself utterly abhorrent to the nature of things, for in expressing quantities by lines, and taking for abscissas (pro ahscissis) the motions of B itself before the encounter, the data, and for ordinates (pro ordinatim applicatis) the motion of the same after the encounter, the results sought for, anTl by drawing the line through the extremities of the ordinates (ordinatarum), according to the precept of the rules of Descartes, this line was not a continuum, but something wonderfully gaping and leaping in a certain absurd and unthinkable manner. And when on that occasion I observed also that the rules of Rev. Father Malebranche did not bear this examina tion in all things, the distinguished man having considered the matter again according to his candor, declared publicly that from this an occasion had arisen for him to change his rules, for which reason, also, he published a brief pamphlet. Although it must be confessed, that because he had not yet directed his attention sufficiently to the use of this new system, he has left something now also not yet suffi ciently in all things complete (quadrant).
From what has been said, this wonderful principle also follows, that the passion of every body is spontaneous, or arises Jrom intemal.^fQrce, alllttmyli u/ioii I'.i-lci-nal occasion. I understand here, however, passion proper, which arises from percussion, or \vliicli remains the same, whatever hypothesis at length is assigned, or to whatever at length we ascribe absolute rest or motion. For since the percussion is the same, to whatever at length true motion corresponds, it follows that the result of the percussion is distributed equally between both, and thus both act equally in the encounter, and thus half the result arises from the action of the one, the other half from the action of the other; and since half, also, of the result or passion is in one, half in the other,4 it is sufficient that we derive the passion which is in one from the action also which is in itself, and we need no influence of the one upon APPENDIX 689 the other, although by the action of one an occasion is furnished the other for producing a change in itself. Certainly, while A and 7> meet, the resistance of the bodies, united with their elasticity, causes them to be compressed because of the percussion, and the compression is equal in each and according to whatever hypothesis, as the experi ments show also, if any one conceives two inflated balls to meet, whether both are in motion, or each is at rest, even if the one at rest be sus pended from some thread, in order that it may most easily recede; for, always provided the velocity of approach or the respective velocity be the same, the compression, or the intensity of the elasticity, will be the same and equal in both. Then the balls A and B, restoring them selves by the force of their own violent, namely, compressed and con fined, elasticity, mutually repel each other by turns, and spread out, as it were, in an arc, and, with a force equal on both sides, each is driven back by the other, and so, not by the force of the other, but by its own force, it recedes from that one. But what is to be understood in the case of inflated balls is to be understood in the case of every body so far as it is passive in percussion, namely, that the rebounding and leaping apart arises from the elasticity in itself, that is, from the motion of the permeating etherial fluid matter, and thus from force internal, or proceeding from within. I understand, however, as I have said, that the proper motion of bodies is separated from the common, which can be ascribed to the centre of gravity; whence their proper motion is so to be conceived (to be conceived, I say, by the way of hypothesis) as if they were produced on board a ship, which would have the motion of their common centre of gravity, but they them selves on board ship were so moved that from the composite common motion of the ship or centre, and their own proper motion, the phenomena are preserved. From what has been said, also, it is under stood that the action of bodies is never ivithout reaction, and both are equal to each other, and directly contrary.
Since, then, only force, and thence nascent effort exists in any mo ment (for motion never truly exists, as we have explained above), and every effort tends in a straight line, it follows that all motion is rectilinear, or composed of rectilinears. Hence, already it not only fol lows that those bodies which move in a curved, line, try always to proceed in the straight line tangent to it, but also, what any one least expects, hence arises the true notion of stability (Jirmitatis). For, if we suppose that some one of those bodies which we call stable (although in truth nothing is absolutely stable or fluid, but everything has a certain degree of stability or fluidity, by us, however, it is named from a predominant regard for our senses) circulates abniifr \\g\ nw^ "an+™ the parts will attempt to fly away by the tangent, nay, they will really begin to fly away; but since this separation of themselves from each other in turn disturbs the motion of the encircling body, hence they are repelled, or 690 LEIBNITZ'S CRITIQUE OF LOCKE again crowded together towards themselves in turn, as if there were in the centre, a magnetic force of attraction, or as if there were in the parts themselves a centripetal force, and therefore the revolution will arise from the composition of the effort (nisu) to recede from the rectilinear by the tangent and the centripetal effort (conatu). And thus it remains that all curvilinear motion arises from the composition among themselves of rectilinear efforts, and at the same time it is understood that this crowding together by the encircling body is the cause of all stability. Otherwise it could not be that all curvilinear motion was composed of nothing but rectilinear motions. Whence, also, again, we have a new and not less than before unexpected argu ment against atoms. Moreover, nothing could be devised more incon sistent with things than that stability be sought from rest, iortrue rest never exists in bodies, nor from rest can anything arise but resf; bu(;)Illioii'_;-h.! ami I! are mutually at rest, with themselves, if not in fact, at least relatively (although this never occurs exactly, for no body preserves exactly the same distance from another, however small the time), and although whatever is once at rest will always be at rest unless a new cause is added, nevertheless it does not follow, for this reason, that because B resists the impelling body, it resists, also, the one separating it from another, so that certainly, as the resistance of B itself is overcome, or B itself driven forward, at the same time A follows. But were the attraction, which is not given in nature, but from the primitive stability (firmitate), explained through rest or something similar, it would assuredly follow. And so stability, also, should not be explained unless by the crowding together produced by the encircling body. For pressure alone does not sufficiently explain the matter, as if the separation of B itself from A itself only is im peded, but it is to be understood that in fact they separate from each other in turn, that moreover one is again impelled to the other by the encircling body, and thus, from the composition of the two motions, this conservation of the conjunction is produced. And so those who conceive in bodies certain tablets or insensible layers (for example, of two polished marbles, which are exactly applied to each other), whose separation is made difficult on account of the resistance of the ambient body, and hence explain the stability of two sensible bodies, although very often they speak truly, yet when they suppose some stability in the layers again, they do not give the last reason for stability. From these considerations, also, it can be understood why I cannot in this thing continue (stare) in certain philosophic opinions of certain great mathematicians, who, besides the fact that they admit vacant space, and seem not to shrink back from attraction, consider motion, also, as an absolute thing, and hasten to prove it from the revolution and the centrifugal force which has thence arisen. But since the revolu tion also arises only from the composition of rectilinear motions, it APPENDIX 691 follows, if the equivalent of the hypotheses is sound in rectilinear motions, however assumed, that it will be sound in the curvilinears.
From what has been said it can also be understood that the common motion in many bodies does not change their actions among themselves, since the velocity with which they approach each other in turn, and thus the force of the encounter by which they act on each other in turn, is not changed. Whence the remarkable experiments follow which Gassendi mentions in his letters on motion impressed by a trans ferred motor, that he might satisfy those who seemed to themselves to be able to infer the rest of the earth's sphere from the motion of projectiles. Nevertheless, it is certain that, if any [persons] are borne in a large ship (closed, if agreeable, or certainly so constituted that the external phenomena cannot be observed by the travellers), and if the ship is moved, although with great velocity, yet quietly or uniformly, they themselves will have no principle by which to dis tinguish (from those things, namely, which take place on shipboard) whether the ship is at rest or moves, even if by chance they play ball on the ship, or practice other movements. And this fact must be noted in favor of those whose belief accords with the not rightly understood notion of the Copernicans, that according to these, things projected from the earth into the air are carried off (abripi) by the air with the gyrating earth, and thus the motion of the bottom fol lows, and fall back upon the earth just as if this were at rest; a view which is properly judged insufficient, since the very learned men who make use of the Copernican hypothesis conceive, rather, that some thing on the surface of the earth is moved with the earth, and, just as if discharged from a bow or hurling machine (tormento), carries with itself the impetus made by the gyration of the earth, together with the impetus made by the projection. Thence, when their double motion is the one common with the earth, the other peculiar to the projection, it is no wonder that the common motion changes nothing. Meanwhile, it is not to be disguised that, if the projectiles could be driven so far, or if the ship were conceived so large, and borne with such velocity, that before the descent the heavy earth or ship described an arc perceptibly different from a straight line; a distinction would be discovered, because then, indeed, the motion of the earth or ship (because circular) does not remain common to the motion which was impressed upon the missile by the gyration of the ship or earth (because rectilinear). And in the effort of heavy bodies towards the centre, external action is added, which can no less produce a diversity of phenomena, than if the compass were kept closed on the ship, which would certainly indicate a variation of the ship. As often, however, as the question concerns the equivalence of hypotheses, all things must be united which concur in the phenomena. From these considera tions, also, it is understood that any composition of motions or reso- 092 LEIBNITZ'S CRITIQUE OF LOCKE lution, whatever, of one motion into two or more can safely be employed, concerning which, nevertheless, a certain very clever man in the works of Wallis had hesitated, not without reason. For the matter certainly deserves proof, and cannot (as is done by many) be assumed as in itself known.
VI ON THE RADICAL ORIGIN OF THINGS1 November 23, 1697 [From the Latin] Besides the world or the aggregate of finite things, there is a Unique Being who rules, not only as the mind in me, or rather as I in my body rule myself, but also in a much higher manner. For this unique sovereign of the universe not only rules the world, but also frames or fashions it, and is superior to the world and, so to speak, outside the world (extramundanuni), and thus is the ultimate reason of things. For the sufficient reason of existence can be found neither in any single thing, nor in the entire aggregate and series of things. Let us suppose that there was an eternal book of the Elements of Geometry, one copy always made from another, it is evident that, although we can account for the present book by the past, whence it has been copied, nevertheless we never, by assuming in the past as many books as we please, come to the complete reason, for we may always wonder why, from all time, such books have existed; that is to say, why these books were written, and why so written. What is true of books is likewise true of the different states of the world, for the following state has in a measure been copied from the preceding (although according to certain laws of change), and so to whatever extent you go back into anterior states, you will never find in these states the full reason; that is to say, why any world exists rather than none, and why such an one.
Therefore, although you imagine the world to be eternal, since, nevertheless, you assume only a succession of states, and do not find in any one of these whatever the sufficient reason, nay more, since by assuming any number you please you do not advance even the least towards accounting for them, it is evident that the reason must be sought elsewhere. For in eternal things we must understand that, 1 Gerhardt, Leibniz, philos. Schrift., 7, 302-308; Erdmann, 147-150; Janet, APPENDIX 693 even if there were no cause, yet there is a reason, which in persisting things is the necessity itself or the essence; but in a series of chang ing things, if we suppose that this series proceeds from a prior series eternally, the reason would be the prevalence of inclinations, as we shall soon see, in which, that is to say, the reasons do not necessitate (with an absolute or metaphysical necessity so as to imply the con trary), but incline. From these considerations it is evident that we cannot escape the ultimate extramundaue reason of things or God by assuming the eternity of the world.
The reasons therefore of the world lie concealed in something outside the world, different from the chain of circumstances or series of things, whose aggregate constitutes the world. "We must then come from physical or hypothetical necessity, which determines the posterior states of the world from the prior states, to something which is of absolute or metaphysical necessity, the reason for which cannot be given. For the present world is physically or hypothetically but not absolutely or metaphysically necessary. In fact, having assumed that it is what it is, it follows that henceforth things must be what they are. Since, therefore, the ultimate root must be in something which is metaphysically necessary, and since there is no reason of the existing unless from the existing, it is therefore necessary that a unique being exist of metaphysical necessity, or whose essence is existence, and that thus something exists different from the plurality of beings or the world, which we have admitted and shown not to be of metaphysical necessity.
But that we may explain a little more distinctly how temporal, contingent or physical truths originate in eternal, or essential or metaphysical truths, we must first know, that, by the very fact itself that something rather than nothing exists, there is some demand for existence in possible things or in possibility itself or essence, or (so to speak) a stretching forth to existence, and, to sum it up in a word, that essence per se tends to existence. Whence it hereafter follows, that all possible things, or those expressing essence or pos sible reality, with equal right tend to essence (essentiam l) in pro portion to the quantity of essence or reality, or in proportion to the degree of perfection which they involve; for perfection is nothing- else than quantity of essence.
But from this we see most clearly that from the infinite combina tions of possible things and possible series there stands forth one through which the greatest quantity of essence or possibility is brought through to existence. There is always, in fact, in things a 1 The reading according to both Gerhardt and Erdmann. Janet's French version reads: " 1'existence." The argument would seem to require the read ing " existeutiam." — TR.
694 LEIBNITZ'S CRITIQUE OF LOCKE principle of determination which is to be sought from the maximum or minimum, so that beyond question the greatest effect is manifested with, so to speak, the least expense. And here time, place, or, in a word, the receptivity or capacity of the world can be considered as the expense or ground upon which, as conveniently as possible, it must be built, while the varieties of the forms correspond to the proportion of the building and the number and elegance of the rooms. And it is as in certain games when all the places at a table must be filled according to certain laws, where, unless you employ a certain skill, hindered finally by the unfavorable places, you will be compelled to leave vacant more places than you were able or desired to do. There is, however, a certain method by which the greatest possible space is most easily filled. If therefore, for instance, we assume that it has been decreed that there be a triangle, though with no other accidentally determining condition, the result is an equilateral triangle; and if it is assumed that we must proceed from point to point, though nothing determines the road beyond, the easiest and the shortest way will be chosen; thus having once assumed that being prevails over non-being, or that there is a reason why something rather than nothing existed, or that from possibility a transition must be made to act, it follows hence, in the absence of any further determination, that the quantity of existence is as great as possible in proportion to the capacity of the time and place (or the order of possible existence), just as tiles are so laid that in the proposed area as many as possible may be contained.
From these considerations we understand already in a wonderful way how, in the very original formation of things, a certain Divine Mathematics or Metaphysical Mechanics is employed, and the deter mination of the greatest amount of existence has place. Thus the right angle is the determinate of all the angles in Geometry, and liquids placed in different media arrange themselves in the most capacious form, namely, the spherical; but especially in general Mechanics itself, when many heavy bodies struggle with each other, such a motion at length arises, through which the greatest descent on the whole is accomplished. For if all possibilities with equal right tend to existence according to the measure of reality, so all weights by equal right tend to descend in proportion to the measure of gravity, and as here the motion appears, in which is contained the greatest possible descent of the heavy bodies, so there the world appears, through which is realized the greatest possible production of possibilities.
And so also we already have physical necessity from metaphysical: for although the world is not metaphysically necessary, so that the contrary implies contradiction or logical absurdity, it is nevertheless physically necessary or determined so that the contrary implies APPENDIX 695 APPENDIX 695 imperfection or moral absurdity. And as possibility is the source of essence, so perfection or the degree of essence (through which the greatest number of things are compossible) is the source of exist ence. Whence it is at the same time evident how freedom exists in the Author of the world, although he does all things deter minately because he acts from the principle of wisdom or perfection. Indif ference certainly arises from ignorance, and the greater one's wisdom the more he is determined to the most perfect.
But (you will say) this comparison of a certain determining meta physical mechanism with the physical one of heavy bodies, although it seems elegant, nevertheless is wanting in this because the struggling heavy bodies truly exist, but the possibilities or essences before or besides existence are imaginary or fictitious, therefore no reason of existence can be sought in them. I reply that neither these essences nor the eternal truths which they call from them are fictitkms, but exist in a certain so to speak region of ideas, namely in God himself, the source of every essence and of the existence of the rest. That we do not seem to have spoken gratuitously, the existence itself of an actual series of things indicates. For since reason is not found in this series, as we showed above, but must be sought in meta physical necessities or eternal truths; moreover, since existences cannot exist unless from existences, as already we maintained above, eternal truths must have existence in a certain absolute or meta physically necessary subject, that is in God, through whom these things, which otherwise would be imaginary, are (to speak barbar ously but significantly) realized.