(Godfrey William-de), an eminent mathematician and philosopher, was born at Leipzig in Saxony in 1646. At the age of 15 years, he applied himself to mathematics at Leipzig and Jena; and in 1663, maintained a thesis de Principiis Individuationis. The year following he was admitted master of arts. He read with great attention the Greek philosophers; and endeavoured to reconcile Plato with Aristotle, as he afterwards did Aristotle with Des Cartes. But the study of the law was his principal view; in which faculty he was admitted bachelor in 1665. The year following he would have taken the degree of doctor; but was refused it on pretence that he was too young, though in reality because he had raised himself several enemies by rejecting the principles of Aristotle and the schoolmen. Upon this he went to Altorf, where he maintained a thesis de Causibus Perplexis, with such applause, that he had the degree of doctor conferred on him. He might have settled to great advantage at Paris; but as it would have been necessary to have embraced the Roman Catholic religion, he refused all offers. In 1673, he went to England; where he became acquainted with Mr Oldenburg, secretary of the royal society, and Mr John Collins, fellow of that society. In 1676, he returned to England, and thence went into Holland, in order to proceed to Hanover, where he proposed to settle. Upon his arrival there, he applied himself to enrich the duke's library with the best books of all kinds. The duke dying in 1679, his successor Ernest Augustus, then bishop of Olmütz, showed our author the same favour as his predecessor had done, and ordered him to write the history of the house of Brunswick. He undertook it, and travelled over Germany and Italy in order to collect materials. The elector of Brandenburg, afterwards king of Prussia, founded an academy at Berlin by his advice; and he was appointed perpetual president, though his affairs would not permit him to reside constantly at Berlin. He projected an academy of the same kind at Dresden; and this design would have been executed, if it had not been prevented by the confusions in Poland. He was engaged likewise in a scheme for an universal language. His writings had long before made him famous over all Europe. Beside the office of privy-counsellor of justice, which the elector of Hanover had given him, the emperor appointed him in 1711 aulic counsellor; and the czar made him privy counsellor of justice, with a pension of 1000 ducats. He undertook at the same time the establishment of an academy of science at Vienna; but the plague prevented the execution of it. However, the emperor, as a mark of his favour, settled a pension on him of 2000 florins, and promised him another of 4000 if he would come and reside at Vienna. He would have complied with this offer, but he was prevented by death in 1716. His memory was so strong, that in order to fix any thing in it, he had no more to do but to write it once; and he could even in his old age repeat Virgil exactly. He professed the Lutheran religion, but never went to sermon; and upon his death bed, his coachman, who was his favourite servant, desiring him to send for a minister, he refused, saying, he had no need of one. Mr Locke and Mr Molyneux plainly seem to think that he was not so great a man as he had the reputation of being. Foreigners did for some time ascribe to him the honour of an invention, of which he received the first hints from Sir Isaac Newton's letters, who had discovered the method of fluxions in 1664 and 1665. But it would be tedious to give the reader a detail of the dispute concerning the right to that invention.

LEIBNITZIAN philosophy, or the philosophy of Leibnitz, is a system of philosophy formed and published by its author in the last century, partly in emendation of the Cartesian, and partly in opposition to the Newtonian. The basis of Mr Leibnitz's philosophy was that of Des Cartes; for he retained the Cartesian subtle matter, with the universal plenitude and vortices; and represented the universe as a machine that should proceed for ever by the laws of mechanism, in the most perfect state, by an absolute inviolable necessity, though in some things he differs from Des Cartes. After Sir Isaac Newton's philosophy was published in 1687, he printed an essay on the celestial motions, Act. Erud. 1689, where he admits of the circulation of the ether with Des Cartes, and of gravity with Sir Isaac Newton; though he has not reconciled these principles, nor shown how gravity arose from the impulse. impulse of this ether, nor how to account for the planetary revolutions, and the laws of the planetary motions in their respective orbits. That which he calls the harmonical circulation, is the angular velocity of any one planet, which decreases from the perihelion to the aphelion in the same proportion as its distance from the sun increases; but this law does not apply to the motions of the different planets compared together; because the velocities of the planets, at their mean distances, decrease in the same proportion as the square roots of the numbers expressing those distances. Besides, his system is defective, as it does not reconcile the circulation of the ether with the free motions of the comets in all directions, or with the obliquity of the planes of the planetary orbits; nor resolve other objections to which the hypothesis of the plenum and vortices is liable. Soon after the period just mentioned, the dispute commenced concerning the invention of the method of fluxions, which led Mr Leibnitz to take a very decided part in opposition to the philosophy of Sir Isaac Newton. From the wisdom and goodness of the Deity, and his principle of a sufficient reason, he concluded that the universe was a perfect work, or the best that could possibly have been made; and that other things, which were inconvenient and evil, were permitted as necessary consequences of what was best: the material system, considered as a perfect machine, can never fall into disorder, or require to be set right; and to suppose that God interferes in it, is to lessen the skill of the author, and the perfection of his work. He expressly charges an impious tendency on the philosophy of Sir Isaac Newton, because he asserts, that the fabric of the universe and course of nature could not continue forever in its present state, but would require, in process of time, to be re-established or renewed by the hand of its Former. The perfection of the universe, by reason of which it is capable of continuing forever by mechanical laws in its present state, led Mr Leibnitz to distinguish between the quantity of motion and the force of bodies; and, whilst he owns, in opposition to Descartes, that the former varies, to maintain that the quantity of force is for ever the same in the universe, and to measure the forces of bodies by the squares of their velocities.

This system also requires the utter exclusion of atoms, or of any perfectly hard and inflexible bodies. The advocates of it allege, that according to the law of continuity, as they call a law of nature invented for the sake of the theory, all changes in nature are produced by insensible and infinitely small degrees; so that no body can, in any case, pass from motion to rest, or from rest to motion, without passing through all possible intermediate degrees of motion; whence they conclude, that atoms or perfectly hard bodies are impossible: because if two of them should meet with equal motions, in contrary directions, they would necessarily stop at once, in violation of the law of continuity.

Mr Leibnitz proposes two principles as the foundation of all our knowledge; the first, that it is impossible for a thing to be and not to be at the same time, which, he says, is the foundation of speculative truth: the other is, that nothing is without a sufficient reason why it should be so rather than otherwise; and by this principle, according to him, we make a transition from abstracted truths to natural philosophy. Hence he concludes, that the mind is naturally determined, in its volitions and elections, by the greatest apparent good, and that it is impossible to make a choice between things perfectly like, which he calls indifferibles; from whence he infers, that two things perfectly like could not have been produced even by the Deity: and he rejects a vacuum, partly because the parts of it must be supposed perfectly like to each other. For the same reason he also rejects atoms, and all similar particles of matter, to each of which, though divisible in infinitum, he ascribes a monad (Act. Lipsiae 1698, p. 435.) or active kind of principle, endowed, as he says, with perception and appetite. The essence of substance he places in action or activity, or, as he expresses it, in something that is between acting and the faculty of acting. He affirms absolute rest to be impossible, and holds motion, or a sort of nisus, to be essential to all material substances. Each monad he describes as representative of the whole universe from its point of sight; and after all, in one of his letters he tells us, that matter is not a substance, but a substantium, or phénomène bien fondé. He frequently urges the comparison between the effects of opposite motives on the mind, and of weights placed in the scales of a balance, or of powers acting upon the same body with contrary directions. His learned antagonist Dr Clarke denies that there is a similitude between a balance moved by weights, and a mind acting upon the view of certain motives; because the one is entirely passive, and the other not only is acted upon, but acts also. The mind, he owns, is purely passive in receiving the impression of the motive, which is only a perception, and is not to be confounded with the power of acting after, or in consequence of, that perception. The difference between a man and a machine does not consist only in sensation and intelligence, but in this power of acting also. The balance, for want of this power, cannot move at all when the weights are equal; but a free agent, he says, when there appear two perfectly alike reasonable ways of acting, has still within itself a power of choosing; and it may have strong and very good reasons not to forbear.

The translator of Mosheim's Ecclesiastical History observes, that the progress of Arminianism has declined in Germany and several parts of Switzerland, in consequence of the influence of the Leibnitzian and Wolfian philosophy. Leibnitz and Wolf, by attacking that liberty of indifference, which is supposed to imply the power of acting not only without, but against, motives, struck, he says, at the very foundation of the Arminian system. He adds, that the greatest possible perfection of the universe, considered as the ultimate end of creating goodness, removes from the doctrine of predestination those arbitrary procedures and narrow views with which the Calvinists are supposed to have loaded it, and gives it a new, a more pleasing, and a more philosophical aspect. As the Leibnitzians laid down this great end as the supreme object of God's universal dominion, and the hope to which all his dispensations are directed; so they concluded, that if this end was proposed, it must be accomplished. Hence the doctrine of necessity, to fulfil the purposes of a predestination founded in wisdom and goodness; a necessity, physical and mechanical, in the motions of material and inanimate things, but a necessity moral and spiritual. spiritual in the voluntary determinations of intelligent beings, in consequence of propellent motives, which produce their effects with certainty, though these effects be contingent, and by no means the offspring of an absolute and essentially immutable fatality. These principles, says the same writer, are evidently applicable to the main doctrines of Calvinism; by them predestination is confirmed, though modified with respect to its reasons and its end; by them irresistible grace (irresistible in a moral sense) is maintained upon the hypothesis of propellent motives and a moral necessity; the perseverance of the saints is also explicable upon the same system, by a series of moral causes producing a series of moral effects.