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* LEIBNITZ. 104 LEICESTER. are objects of ratioiuil or incessary knowledge. The validity of rational knowledge is guaranteed by the principle of contradiction; that of empirical knowledge by the principle of suffi- cient reason, which Leibnitz was the first to introduce into a system of philosophy. In other wordsj necessary truths are analytical, contingent truths synthetical. The latter must have authentication from without: an adequate reasdu nuist be given for their validity. The fcjrmer are authenticated by the fact that it is impossible to think their opposites. The changes that take place at the same time in various monads liave no influence on each other. There is no interaction. But there is a prcestablished harmony of such sort that presentations in one lUdiKid correspond to those in another. The rela- tion b<'tween any two monads is likened to that between two clocks keeping perfect time. They do not inlluence each other's movements, but they kee]i together. This correspondence is due to "the fact that God. the monad of monads, created all other monads in such a way that in their subsequent course of development their changes should harmonize. These monads are immortal. In choosing to create this world of monads, God selected the best of all po.ssible worlds, (iod's wisdom gave Him an infinite range of choice ; His goodness determined the selection ]le nuxle. This is Leibnitz's peculiar optimism, which does not assert that everything is perfectly pood, but that the world as a whole is the best of all possible worlds. In this way Leibnitz sought to justify God in creating a woi-ld with evil in it. This is Leibnitz's theodicy. His mathematical work is worth special treat- ment. He began his work on the calculus (q.v. ) alxmt the time of liis settling in Hanover in 107G, and two years later he had developed it into a fairly complete discipline. H was not. however, until six years after this that he published (1G84) anything upon the subject. Two years earlier (1G8'2) he and Mencke founded the Acfa EriulitorKm. and it was in this celebrated jour- nal that most of his mathematical memoirs ap- peared (l(i82-!)21. The first one on the new cal- culus was his Xnra Mrtliodun pro Minimis et Maximis (lfi84). Newton (q.v.) had known and used the principles of the tluxional calculus as early as lOOo, and had made them public, al- though not in print, in 16G9. Leibnitz had access to certain letters of Newton'.s in 1676. He also had the opportunity of knowing of the theory when he was in London in 1673, and with the mathematical acquaintances he met there it might be expected that the new theory woidd be discussed. There is, however, no exact evidence that he knew anything of Newton's discovery at the time he began his own work. It s!u)iild, how- ever, be slated that the genns of the theory of the calculus are to be found in the works of Format, Wallis, and Cavalieri, all of which were well known at that time in the mathematical world. The essential dilTerences in the two systems lie in the notation and the method of attack. New- ton used .r where Leibnitz used dx. and based his treatment on the notion of velocity of material substances where tlie latter proceeded from the concept of the infinitesimal. As mathematics developed, the dilTerenlial notation proved greatly superior to the f!ixi(innl. and in the first quarter of the nineteenth centiirv it was adopted in Eng- land, as it had been adopted a hundred years earlier on the Continent. With this change of notation the so-called tluxional calculus dis- appeared and the ditlerential calculus took its place. The further mathematical work of Leibnitz was not of a high order. His contributiims to analytic geometry were noteworthy only for lay- ing the foundation (1602) for the theory of envelopes, and for introducing the terms 'co- ordinates' and "axes of coordinates.' He con- tributed a little to the theory of mechanics, but his work was often inaccurate, and he made no great discoveries. In addition to Leibnitz's works already re- ferred to, special mention should be made of Hyslcme nouvcau de la nature (1605) ; Principcs de la nature et de hi yracc (1719); Nuuveaiix essiiis nur I'eiitendement humuin (no,*)); and A Collection of Learned Papers Which Passed Between the Late Mr. Leibnitz and Dr. Clarke in the Years 1115 and 1716 (London, 1717). His Latin and French philosoi)liical works have been many times collected, edited, and published. The publishing of a complete edition of all Leib- nitz's works was undertaken by Pertz. This edition, as it now stands, contains 4 vols, of histoi-y (Hanover, 1843-47); 7 vols, of nuithe- matics. edited by Gerhardt (Berlin and Halle, 1840-03) ; but of the philosophical portion only one volume appeared. In the Jonrnal of Speculative Philosophy are to be found transla- tions of the Monadolof/ie, and many of the lesser writings; and some of the important philosophi- cal works have been translated liy G. jl. Duncan (New Haven, 1890) ; the ]ouveinia- exsais by A. G. Langley (London, 1894) ; The Monndolofi)/ and' Other Philosophical Writint/s, Eng. by R. Latta (Oxford, 1898). The literature on Leibnitz is immense. The following works deserve special mention: Dill- mann, Eine ncue Darstellnnfj der Lcihnizschen Monadenlehre (Leipzig, 1.S91); Feiierbach, /><(»■- slellunij, Entwickeluny und Kritik der Lcihniz- schen Philosophic (Ansbach, 1837); Nourrisson, /,« philosophic de Leibniz (Paris, 1860); R. Zimniermann, Leibnizs Monudoloyic (Vieniui, 1S47); Jlerz, Leibnitz (London, 1884); Har- nack. Leibnizs Bedeutuny in der (Irsctiichtr der MathciHUlik (Stade. 1887). See, also, lir.lVding, History of Modern Philosophy (London, 1000 j, and Dewey, Leibnitz's New Essays Concerning the Human Understandiny (Chicago, 1888). Many biographies of Leibnitz have been written. Among these may be named Guhrauer, Gottfried Wilheim Freihcrr von Leibniz ( Breslau, 1842- 40; Eng. trans. Boston, 1S4.5) ; Plleiderer, 0. V. Leibniz als Patriot, StniilsnKiiin und Bildirnfistriiger (Leipzig, 1870). LEICESTER, les'ter. The capital of Leices- tershire, F.ngland, on the Soar, 100 miles north- northwest of London (Jlap: England, E 4). It is a well-l>uilt town, with spacious and regular streets, interesting municipal Iniildings. educa- tional and benevolent institutions, and numerous churches, one of which. Saint Nicholas, is partly built of bricks from the old Roman wall. It is a progressive municipality, and owns or maintains an excellent water-supply, gas and electric-light plants, markets, abattoirs, baths, batliing-places, libraries, technical schools, an art gallery, arti- snns' dwellings, garden allotments, four parka, eight recreation-grounds, and two public gj'mna- siums, three sewage farms, an isolation hospital.