Gottfried Wilhelm Leibniz invented the calculus independently of Newton and in a better notation, designed a calculating machine that multiplied, worked out binary arithmetic, made contributions to geology, jurisprudence, linguistics, physics and library science, and spent forty years as a court servant to the House of Hanover writing a genealogical history he never finished. He published one book of philosophy in his lifetime. The system has had to be reconstructed from essays, letters and manuscripts, of which the edition is still incomplete three centuries later.
Two principles organise everything. The principle of contradiction governs necessary truths. The principle of sufficient reason — that nothing is so without a reason why it is so and not otherwise — governs contingent ones, and does most of the work. From it he argues that God, having conceived infinitely many possible worlds, actualised the one with the greatest variety of phenomena produced by the simplest laws. This is the best of all possible worlds, not because it contains no evil, but because any world without those evils would be worse overall. Voltaire's Candide made the doctrine a joke, which it is not: Leibniz's Theodicy is a serious attempt to reconcile evil with omnipotence by distinguishing what God wills from what he permits as a condition of a greater good.
The metaphysics is stranger. Reality consists of monads — simple, indivisible, immaterial substances, each a perceiving point of view on the whole universe. They have no windows: nothing enters or leaves, and no monad acts on another. What looks like causation is the unfolding of each monad's own nature in a sequence God synchronised at creation, the pre-established harmony, which he offered as a better solution than Malebranche's constant divine intervention. Bodies and space are well-founded appearances arising from relations among monads rather than substances in their own right, and his relational account of space, argued against Newton's absolute space in the correspondence with Clarke, was vindicated two centuries later.
The New Essays, a chapter-by-chapter reply to Locke's Essay, defends innate ideas as dispositions and coins the distinction that nothing is in the intellect that was not first in the senses — except the intellect itself. Withheld when Locke died, it shaped German philosophy through Wolff and reached Kant.