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 an elaborate structure. There are few great ideas pursued by succeeding analysts which were not suggested by Euler, or of which he did not share the honour of invention. With, perhaps, less exuberance of invention, but with more comprehensive genius and profounder reasoning, Lagrange developed the infinitesimal calculus and placed analytical mechanics into the form in which we now know it. Laplace applied the calculus and mechanics to the elaboration of the theory of universal gravitation, and thus, largely extending and supplementing the labours of Newton, gave a full analytical discussion of the solar system. He also wrote an epoch-marking work on Probability. Among the analytical branches created during this period are the calculus of Variations by Euler and Lagrange, Spherical Harmonics by Laplace and Legendre, and Elliptic Integrals by Legendre.

Comparing the growth of analysis at this time with the growth during the time of Gauss, Cauchy, and recent mathematicians, we observe an important difference. During the former period we witness mainly a development with reference to form. Placing almost implicit confidence in results of calculation, mathematicians did not always pause to discover rigorous proofs, and were thus led to general propositions, some of which have since been found to be true in only special cases. The Combinatorial School in Germany carried this tendency to the greatest extreme; they worshipped formalism and paid no attention to the actual contents of formulæ. But in recent times there has been added to the dexterity in the formal treatment of problems, a much-needed rigour of demonstration. A good example of this increased rigour is seen in the present use of infinite series as compared to that of Euler, and of Lagrange in his earlier works.

The ostracism of geometry, brought about by the master-minds of this period, could not last permanently. Indeed, a