The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and **integral calculus** of an analogous method depending wholly on the serial development of algebraical functions.

The general method of constructing formulae of this kind involves the use of the **integral calculus** and of the calculus of finite differences.

If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an algebraical function of the abscissa, the application of the **integral calculus** gives the area of the figure.

The establishment of these formulae involves the use of the **integral calculus**.

It will thus be seen that Legendre's works have placed him in the very foremost rank in the widely distinct subjects of elliptic functions, theory of numbers, attractions, and geodesy, and have given him a conspicuous position in connexion with the **integral calculus** and other branches of mathematics.