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 lines, will fall on one of the lines. Then came Sylvester's four-point problem: to find the probability that four points, taken at random within a given boundary, shall form a re-entrant quadrilateral. Local probability has been studied in England by A. R. Clarke, H. McColl, S. Watson, J. Wolstenholme, but with greatest success by M, W, Crofton of the military school at Woolwich. It was pursued in America by E. B. Seitz; in France by C. Jordan, E. Lemoine, E. Barbier, and others. Through considerations of local probability, Crofton was led to the evaluation of certain definite integrals.

The first full scientific treatment of differential equations was given by Lagrange and Laplace. This remark is especially true of partial differential equations. The latter were investigated in more recent time by Monge, Pfaff, Jacobi, Émile Bour (1831–1866) of Paris, A. Weiler, Clebsch, A. N. Korkine of St. Petersburg, G. Boole, A. Meyer, Cauchy, Serret, Sophus Lie, and others. In 1873 their reseachesresearches [sic], on partial differential equations of the first order, were presented in text-book form by Paul Mansion, of the University of Gand. The keen researches of Johann Friedrich Pfaff (1795–1825) marked a decided advance. He was an intimate friend of young Gauss at Göttingen. Afterwards he was with the astronomer Bode. Later he became professor at Helmstädt, then at Halle. By a peculiar method, Pfaff found the general integration of partial differential equations of the first order for any number of variables. Starting from the theory of ordinary differential equations of the first order in $$\scriptstyle{n}$$ variables, he gives first their general integration, and then considers the integration of the partial differential equations as a particular case of the former, assuming, however, as known, the general integration of differential equations of any order between two variables. His researches led Jacobi to introduce the name "Pfaffian problem." From the connection, observed by Hamilton, between