The end of the seventeenth and the beginning of the eighteenth centuries were rendered illustrious, as we have already seen, by the mathematical discoveries of two of the greatest men who have ever enlightened the world. A slight sketch of the improvements which the theory of mechanics owes to Newton has been just given; those which it owes to Leibnitz, though not equally important nor equally nume rous, are far too conspicuous to be passed over in silence. So far as concerns ge neral principles they are reduced to three,—the argument of the sufficient reason,—the law of continuity,—and the measurement of the force of moving bodies by the square of their velocities; which last, being a proposition that is true or false according to the light in which it is viewed, I have supposed it placed in that which is most favourable.
With regard to the first of these,—the principle of the sufficient reason,—according to which, nothing exists in any state without a reason determining it to be in that state rather than in any other,—though it be true that this proposition was first distinct ly and generally announced by the philosopher just named, yet is it certain that, long before his time, it had been employed by others in laying the foundations of science. Archimedes and Galileo had both made use of it, and perhaps there never was any attempt to place the elementary truths of science on a solid foundation in which this principle had not been employed. We have an example of its application in the proof usually given, that a body in motion cannot change the direction of its motion, abstraction being made from all other bodies, and from all external action ; for it is evident, that no reason exists to determine the change of motion to be in one direction more than another, and we therefore conclude that no such change can possibly take place. Many other .instances might be produced where the same principle appears as an axiom of the clearest and most undeniable evidence. Wherever, indeed, we can pronounce with certainty that the conditions which determine two different things, whether magnitudes or events, are in two cases precisely the same, it cannot be doubt ed that these events or magnitudes are in all respects identical.
However sound this principle may be in itself, the use which Leibnitz sometimes made of it has tended to bring it into discredit. He argued, for example, that of the particles of matter no two can possess exactly the same properties, or can perfectly resemble one another, otherwise the Supreme Being could have no reason for employ ing one of them in a particular position more than another, so that both must ne cessarily be rejected. To argue thus, is to suppose that we completely understand the
manner in which motives act on the mind of the a postulate that seems but ill suited to the limited sphere of the human understanding. But, if Leibnitz has mis applied his own principle and extended its authority too far, this affords no ground for rejecting it when we are studying the ordinary course of nature, and arguing about the subjects of experiment and observation. In fact, therefore, the sciences which as pire to place their foundation on the solid basis of necessary truth, are much indebted to Leibnitz for the introduction of this principle into philosophy.
Another principle of great use in investigating the laws of motion, and of change in general, was brought into view by the same author,—the law of Continuity,—accord ing to which, nothing passes from one state to another without passing through all the intermediate states. Leibnitz considers himself as the first who made known this law ; but it is fair to remark, that, in as much as motion is concerned, it was dis tinctly laid down by Galileo,' and ascribed by him to Plato. But, though Leibnitz was not the first to discover the law of continuity, he was the first who regarded it as a principle in philosophy, and used it for trying the consistency of theories, or of supposed laws of nature, and the agreement of their parts with one another. It was in this way that he detected the error of Descartes's conclusions concerning the collision of bodies, showing, that though one case of collision must necessarily graduate into another, the conclusions of that philosopher did by no means pass from one to another by such gradual transition. Indeed, for the purpose of such detections, the knowledge of this law is extremely useful; and I believe few have been much occupied in the investigations either of the pure or mixed mathematics, who have not often been glad to try their own conclusions by the test which it fur nishes.
Leibnitz considered this principle as known a priori, because if any saltus were to take place, that is, if any change were to happen without the intervention of time, the thing changed must be in two different conditions at the same individual instant, which is obviously impossible. Whether this reasoning be quite satisfactory or not, the con formity of the law to the facts generally observed, cannot but entitle it to great autho rity in judging of the explanations and theories of natural phenomena.