great importance in Weierstrass' theory of functions. It became necessary to prove that a trigonometric series representing a continuous function converges uniformly. This was done by Heinrich Eduard Heine (1821–1881), of Halle. Later
researches on Fourier's series were made by G. Cantor and
Du Bois-Reymond.
As compared with the vast development of other mathematical
branches, the theory of probability has made very
insignificant progress since the time of Laplace. Improvements
and simplifications in the mode of exposition have been made by A. De Morgan, G. Boole, A. Meyer (edited by E. Czuber),
J. Bertrand. Cournot's and Westergaard's treatment
of insurance and the theory of life-tables are classical. Applications
of the calculus to statistics have been made by L. A. J.
Quetelet (1796–1874), director of the observatory at Brussels;
by Lexis; Harald Westergaard, of Copenhagen; and Düsing.
Worthy of note is the rejection of inverse probability by the
best authorities of our time. This branch of probability had been worked out by Thomas Bayes (died 1761) and by Laplace
(Bk. II., Ch. VI. of his Théorie Analytique). By it some logicians have explained induction. For example, if a man,
who has never heard of the tides, were to go to the shore of the Atlantic Ocean and witness on successive days the rise of the sea, then, says Quetelet, he would be entitled to conclude that there was a probability equal to that the sea would rise next day. Putting , it is seen that this view rests upon the unwarrantable assumption that the probability of a totally unknown event is , or that of all theories proposed for investigation one-half are true. W. S. Jevons in his Principles of
Science founds induction upon the theory of inverse probability, and F. Y. Edgeworth also accepts it in his Mathematical
Psychics.
The only noteworthy recent addition to probability is the subject of "local probability," developed by several English