Leibniz makes the charge, which he constantly renewed, that the laws of nature of the Cartesians and Newton's law of gravitation were really only formulations of perpetual miracles. To make his case he had to define miracle. Because the notion of miracle involves, at least for Leibniz, the notion of law as that to which a miraculous event is an exception, and because accordingly the criteria for miracles become inversely the criteria for laws, the entire polemic throws valuable light on Leibniz's conception of law.

We may, to begin with, observe that there are three types of law for Leibniz. First, there is the law of the whole universe, sometimes also referred to as the concept of the universe. Second, there are certain architectonic principles like the law of continuity and the law of determination by maxima and minima. These laws or principles govern not only the first kind of law, the law of the universe, but also the third kind of law, the laws of nature; for example, the principle of the conservation of force in mechanics, or the principle of the most determined path in optics.

The law which governs the whole universe is the same in kind as the concept of the individual, as for example that of Julius Caesar or Alexander the Great. The concept of the individual is a law analogous to the law of a mathematical series, differing from the latter, however, in that it is a temporal series. Given the law and the starting point of the series it should be possible to deduce all the successive predicates of the subject or, if you like, deduce all the successive events in its history. Leibniz calls this kind of law a "law of order" (G IV, 518; L 493). He says, "When we say that each monad, soul, mind, has received a particular law, we must add that it is only a variation of the general law which rules the universe; and that it is just as a city appears differently according to the different points of view from which we look at it" (G IV, 553-4). The law of the individual is then a perspectival variant of the law of the universe, a law from which every event in the universe can be deduced. The law of the universe, like the laws of its individual members, is a law of order. It will appear moreover that it is logically impossible for the universe not to be subject to order. Section VI of the Discourse on Metaphysics is headed, "God does nothing out of order, and it is impossible even to feign events which are not regular." Leibniz explains: "Let us suppose, for example, that someone makes a number of marks on paper quite at random, as do those who practise the ridiculous art of geomancy. I say that it is possible to find a geometrical line, the notion of which is constant and uniform according to a certain rule, such that this line passes through all these points, and in the same order as the hand had marked them. And if someone drew in one stroke a line which was now straight, now circular, now of another nature, it is possible to find a notion or rule, or equation common to all the points of this line, in virtue of which these changes must occur. And there is no face, for example, the outline of which does not form part of a geometrical line and cannot be traced in one stroke by a certain movement according to rule. But when a rule is very complex what conforms to it passes for irregular." Leibniz provides another example in the Theodicy of the inescapable nature of order. "One may propose a succession or series of numbers perfectly irregular to all appearance, when the numbers increase and diminish variably without the emergence of any order; and yet he who knows the key to the formula, and who understands the origin and the structure of this succession of numbers, will be able to give a rule which, being properly understood, will show that the series is perfectly regular, and that it even has excellent properties" (par. 242). Up to this point we have the architectonic principle of continuity at work. Now the architectonic principle of determination by maxima and minima comes into play. Leibniz continues in the Discourse on Metaphysics, "Thus one can say that in whatever way God has created the world, it would always have been regular and in a certain general order. But God has chosen the one that is most perfect, that is to say the one that is at the same time the simplest in hypotheses and the richest in phenomena, as a geometrical line might be, of which the construction was easy and the properties and effects very admirable and of great extent." Or, as he puts it else-where "that the maximum effect should be achieved by the minimum outlay." (G VII, 303, L 487).

One may note in passing an anomalous difference in the status of the two principles. That the universe should be without discontinuities is not a matter of divine choice. Discontinuities are simply impossible. That the law of the universe should produce the most by the least is a matter of divine choice and is contingent. Elsewhere Leibniz treats the law of continuity as contingent also. Another thing we may take note of is that when Leibniz speaks of the law of the universe, the universe in question is the total aggregation of monads, spirits and minds. Let us call it the metaphysical universe in order to distinguish it from the universe of natural phenomena. The latter are governed by the laws of motion.

The next section of the Discourse on Metaphysics, number VII, is headed, "That miracles are in conformity with the general order, although they are counter to subordinate maxims…." He explains: "Now since nothing can be done which is not in order, one can say that miracles are as much in order as natural operations, so called [i.e. natural] because they are in conformity with subordinate maxims…. As regards general or particular wills … one can say that God does everything according to his most general will, which is in conformity with the most perfect order which he has chosen; but one can also say that he has particular wills which are exceptions to the said subordinate maxims, for the most general of the laws of God which rules the whole sequence of the universe has no exceptions." Section XVII is headed "Example of a subordinate maxim or law of nature. In which it is shown that God always conserves regularly the same force but not the same quantity of motion against the Cartesians and several others." An exception to that law of nature, the conservation of force, would be a miracle. But that same miraculous event would not be an exception to the law of the universe, for exceptions to it, as we have seen, are impossible. From the law of the universe, and for that matter from the law of any individual in it, it would be possible to deduce not only those events in its history which conform to the laws of nature like the rules of collision, but also exceptions to them, that is to say, those miracles which are necessary for this to be the best of all possible worlds, or as he puts it in the Theodicy, (par. 248) "God ought not to make choice of another universe, since he has chosen the best and has only made use of miracles necessary thereto."

Leibniz has two related definitions of a miracle and both are continually used in his polemics. First, a miracle is that which exceeds the power of creatures, or that which cannot be explained by the nature of creatures. "Power" (or "force") and "nature" are equivalent expressions. "The nature inherent, in created things is nothing but the force to act and be acted on" (De ipsa Natura, par. 9). Second, a miracle is that which is not conceivable by, or explicable to, the created mind. Although miracles are included in the law of the universe, as well as in the law of each individual in it, and are perfectly intelligible to God who knows his reasons for them, finite minds are incapable of knowing the law of the universe, just as they are incapable of having the complete concept of any individual, and therefore they are incapable of understanding a miracle.

With his two definitions of, or criteria for, a miracle, Leibniz rejects absolutely any mere constant conjunction or uniformity conception of the laws of nature, such as he found in the Cartesians and in Bayle defending them, and also in Clarke's defence of Newton. The issue arises first in connection with the Cartesians' occasionalist account of the relation of mind and body, but it is then almost immediately extended to the occasionalist account of all secondary causes. Bayle in his article on Rorarius remarked, "Further-more it seems to me that this able man [Leibniz] dislikes the Cartesian system because of a false assumption, for one cannot say that the system of occasional causes makes the action of God intervene by a miracle (deus ex machina)) in the reciprocal dependence of body and soul. For since God's intervention follows only general laws, he does not therein act in an extraordinary way." In replying to this Leibniz after discussing the mind and body connection goes on to say, "But let us see whether the system of occasional causes does not in fact imply a perpetual miracle. Here it is said that it does not, because God would act only through general laws according to this system. I agree, but in my opinion that does not suffice to remove the miracles. Even if God should do this continuously, they would not cease being miracles, if we take the term not in the popular sense of a rare or wonderful thing, but in the philosophical sense of that which exceeds the powers of created beings. It is not enough to say that God has made a general law, for besides the decree there is also necessary a natural means of carrying it out, that is all that happens must be explained through the nature which God gives to things. The laws of nature are not so arbitrary and indifferent as many people imagine. For example, if God were to decree that all bodies should have a tendency to move in circles and that the radii of the circles should be proportional to the magnitude of the bodies, one would either have to say that there is a method of carrying this out by means of simpler laws, or one would surely have to admit that God must carry it out miraculously, or at least through angels charged expressly with this responsibility…. It would be the same if someone said that God had given natural and primitive gravities to bodies by which each tends to the centre of its globe without being pushed by another body, for in my opinion such a system would need a perpetual miracle, or at least the help of angels." (G IV, 520-1; L 494-5). To Arnauld Leibniz writes; "Strictly speaking God performs a miracle whenever he does something that exceeds the forces which he has given to creatures and maintains in them. For instance, if God were to cause a body which had been set in a circular movement, by means of a sling, to continue to move freely in a circle when it had been released from the sling, without being impelled or checked by anything at all, that would be a miracle, for according to the laws of nature it should continue along in a straight line in a tangent; and if God were to decree that that should always occur, he would be performing natural miracles, since this movement is not susceptible of a similar explanation. Likewise one must say that if the continuation of the movement exceeds the force of the bodies, it must be said according to the accepted concept, that the continuation of the movement is a true miracle, whereas I believe that bodily substance has the force to continue its changes according to the laws that God has placed in its nature and maintains there." (G II, 93) Or again, in commenting on the occasionalist Lami's objection to his system: "If God wills that a body tend of itself in a straight line, that will be a law of nature, but if he wills that of itself it goes in a circular or elliptical line that will be a continual miracle." (Rob. 373).

The second criterion of the miraculous is that it is inconceivable to created minds. Thus "A miracle is a divine action which transcends human knowledge; or more strictly which transcends the knowledge of creatures" (C 508). By conceivable or intelligible Leibniz means quite simply that and that only which is susceptible of mechanical explanation. To Lady Masham he writes, "It is well to consider that the ways of God are of two kinds, the ones natural, the others extraordinary or miraculous. Those which are natural are always such as a created mind would be able to conceive … but the miraculous ways lie beyond any created mind. Thus the operation of the magnet is natural, being entirely mechanical or explicable, although we are still perhaps not in the position of explaining it perfectly in detail, for want of information; but if anyone maintains that the magnet does not operate mechanically and that it does it all by pure attraction from a distance, without any intermediary, and without visible or invisible instruments, that would be something inexplicable to any created mind, however penetrating and informed it should be; and in a word it would be something miraculous," (G III, 353) or again in the New Essays he says "Matter cannot naturally attract … nor of itself proceed in a curved line, because this cannot be explained mechanically, for that which is natural must be capable of being distinctly conceived" (Pref.) Not even God could get us to understand a non-mechanical explanation, as Leibniz points our in a letter to Hartsoeker. "Thus the ancients and moderns, who avow that weight is an occult quality are right if they mean by that that there is a certain mechanism unknown to them, by which bodies are pushed towards the centre of the earth. But if it is their opinion that the thing is done without any mechanism by a simple primitive quality or by a law of God which produces this effect without using any intelligible means, it is an irrational occult quality which is so occult that it is impossible that it can ever become clear, even if an angel, to say nothing of God himself, wanted to explain it" (G III, 519).

With Newton and his spokesman, Clarke, Leibniz came up against a rival conception of the natural and the miraculous, not the less forcefully stated because of the insulting tone in which Leibniz in the opening letter of the exchange with the Englishmen makes the charge that Newton's God has created a world so imperfect that he must resort to miracles to keep the whole machine going, while Leibniz's God with the law of the conservation of force has no need of any such extraordinary interventions. "And I hold," says Leibniz, "that when God works miracles, he does not do so in order to supply the wants of nature, but of grace." (Leibniz, I, 4). From this point on miracles are a major topic throughout the correspondence generating, perhaps, the most heated mutual scorn in the entire exchange.

Clarke's conception of a miracle comes out first in his second letter in which he maintains that the distinction between the natural and the supernatural has no significance in relation to God, but only to human ways of conceiving things. For God nothing is more miraculous than anything else. But for us the natural is the usual or frequent or regular, the supernatural the unusual. Says Clarke, "The raising of a human body out of the dust of the earth, we call a miracle; the generation of a human body in the ordinary way we call natural; for no other reason but because [of] the power of God…. The sudden stopping of the sun (or earth,) we call a miracle; the continual motion of the sun (or earth,) we call natural; for the very same reason only, of the one's being usual and the other unusual. Did a man usually arise out of the grave, as corn grows out of seed sown, we should certainly call that also natural; and did the sun (or earth,) constantly stand still, we should then think that to be natural, and its motion at any time would be miraculous" (Clarke, V, 107-109). It is evident that for Clarke it is not the kind of causes involved which determines what is natural or supernatural. "The means by which two bodies attract one another, may be invisible and intangible, and of a different nature from mechanism; and yet, acting regularly and constantly, may well be called natural." (Clarke, IV, 45). To which Leibniz replies: "He might as well have added inexplicable, unintelligible, precarious, groundless and unexampled" (Leibniz, V, 120). Growing tired of being told repeatedly that the natural is that which can be explained by the natures of creatures, Clarke finally protests that "The terms, nature, and powers of nature, and course of nature and the like are empty words; and signify merely that a thing usually or frequently comes about." (Clarke, V, 107-109). As for Leibniz's calling attraction a miracle or an occult quality Clarke finds this most unreasonable "after it has so often been declared,"—and here Clarke quotes several of Newton's disclaimers, which he assumes that Leibniz will have read—"that by that term we do not mean to express the cause of bodies tending towards each other, but barely the effect, or phenomenon itself, and the laws or proportions of that tendency discovered by experience; whatever be or be not the cause of it." (Clarke, V, 110-116). If Leibniz had lived long enough to read these remarks the dispute should have moved away from miracles to the question of the relation of laws and causes, but again there would have been no meeting of minds.

I want now to consider Leibniz's conception of miracles and laws in relation to his celebrated doctrine of possible worlds of which this world is one. To Arnauld (G II, 40) he says, "If this world were only possible, the individual concept of a body in this world, containing certain movements as possibilities, would also contain our laws of motion (which are free decrees of God) but also as mere possibilities. For as there exists an infinite number of possible worlds, there exists also an infinite number of laws, some peculiar to one world, some to another, and each possible individual of any one world contains in the concept of him the laws of his world."…

Leibniz has effectively excluded as laws of any possible world such uniformities as the tendency of all bodies of themselves to move in circles, or to tend to the centre of their globes without being pushed there by other bodies, to cite examples with which he responds to Bayle. These would be miracles in all possible worlds according to Leibniz because (a) they are not consequences of the natures which God gives things and (b) they are not conceivable or explicable or intelligible, and these are the criteria of the natural as opposed to the supernatural. Are we then left with any other possible laws of nature for other worlds than those operating in this world? To begin with, can God give bodies natures other than that vis viva which he conserves in this world? It would appear not. Leibniz rejects as occult and unintelligible any other candidates for the nature of a body. In any case if we are talking as Leibniz does to Arnauld about the bodies in different possible worlds, these will all, if they are to be bodies, by definition share the same nature. Given, then, that all possible bodies have this nature, and that only mechanical explanations are intelligible, can there be laws of motion which are possible alternatives for other worlds to those operative in this world? What Leibniz calls "the most universal and inviolable" (G III, 45) law of nature, i.e. the conservation of force, he also regards as "the foundation of the laws of motion," (De ipsa Natura par. 4) and indeed claims to "reduce all mechanics" to it (G II, 62). What we are asking, then, is whether there is such a foundational law, other than the conservation of force, peculiar to each of an infinitude of possible worlds and to which all the laws of motion peculiar to each of those worlds can be reduced? The answer would seem clearly to be no, for Leibniz says "I call extraordinary every operation of God demanding something other than the conservation of the nature of things." (A 185). Since the nature of bodies is the same in all possible worlds, i.e. force, an exception to the law of the conservation of force would be a miracle in all possible worlds. It would appear then that Leibniz's conception of the miraculous commits him, contrary, of course, to his deepest intentions, to holding that the laws of motion in this world are the same for all possible worlds, the laws, that is to say, of Leibnizian mechanics. Not only would Euclid's Elements be a textbook in all possible worlds, but so also would the elements of Leibnizian mechanics.

As an addendum I should like briefly to consider the relation to possible worlds of another set of laws, those of optics, without reference, however, to the miraculous, but to Leibniz's use of the two architectonic principles, that of order or continuity and that of determination by maxima and minima. In the Tentamen Anagogicum he combines them to produce the principle of the most determined or unique path for light rays in order to find the laws of reflection or refraction. The unique path is that which has no twin or other path symmetrical with it. All other paths have twins. Implicit in Leibniz's use of this principle is another which he enuciates in his correspondence with Clarke, namely "When two things which cannot both be together, are equally good; and neither by themselves, nor by their combination with other things, has the one any advantage over the other; God will produce neither of them." (Leibniz, IV, 19). God is helpless in choosing between twins; hence if he is to choose at all he must choose the most determined or unique path. The concept of unique determination is purely spatial. If, then, there can be different laws of optics for different worlds, it can only be a consequence of the possibility of different kinds of space for these worlds. Leibniz denies this possibility. "Why," asks Bayle, "has matter precisely three dimensions? Why should not two have sufficed for it? Why has it not four?" To which Leibniz replies, "the ternary number is not determined by reason of the best but by geometrical necessity because geometers have been able to prove that only three straight lines perpendicular to one another can intersect at one and the same point." (Theodicy, par. 355). It looks then, as if the laws of optics will join those of motion as applying in all worlds, with, however, this possible qualification. Leibniz indicates to Arnauld that there are bodies and motion in all worlds. Does he believe that there is light in all worlds? By the following speculative steps we must, I think, come to the conclusion that he does. Leibniz says, "For by the individual concept of Adam I mean, to be sure, a perfect representation of a particular Adam who has particular individual conditions and who is thereby distinguished from an infinite number of other possible persons who are very similar but yet different from him "as every ellipse is different from the circle, however much it approximates to it" (G II, 20). In other words the existing Adam is the last term upon which an infinite series of possible Adams converges in the same way as an infinite series of ellipses converges on the circle. If the law of the individual is only a perspectival variant of the law of the universe, there must be a similar series of possible worlds converging on this world. If that is the case for the metaphysical worlds of individuals or monads, it is, perhaps, not too much to suppose that it is the case also for its phenomenal counterpart, the physical world, and consequently there should be ever increasing degrees of illumination in the series of possible worlds converging on this world. If so, Snell's law of refraction must apply in all possible worlds.