THE ancient Greeks determined various areas and volumes by a method known as that of exhaustion; but they had no integral calculus properly so called, any more, than (pace Prof. Burnet) they had a differential calculus, although they were familiar enough with the idea of a locus described by the motion (or flow) of a point. Even Fermat missed the analytical method devised by Barrow, Newton, and Leibniz. This was so rapidly developed as to assume a form which (except in notation) remained practically unaltered for a century and a half. The reason of this quiescence- a sort of dormant vitality was the neglect of function-theory, or, rather, its non-existence. The appearance of Fourier's work on the theory of heat compelled mathematicians to study the properties of trigonometrical series, and the conditions under which they could be used for the representation of so-called arbitrary functions. Dirichlet and Riemann shed a flood of light upon the matter; and Riemann gave a definition of a definite integral which could be applied to functions more general than those that could be integrated according to the older (say Newtonian) definition. In particular, the function to be integrated might have a finite number of isolated discontinuities in the range of integration; isolated, that is, in the sense of being separated by finite intervals. Thus a new type of integrals, the Riemann integrals, had come under observation.