course on the theory may be asserted to be the first to place calculus on a firm and rigorous foundation. Operational methods Antoine Arbogast (1800) was the first to separate the symbol of operation from that of quantity in a differential equation. Francois-Joseph Servois (1814) seems to have been the first to give correct rules on the subject. Charles James Hargreave (1848) applied these methods in his memoir on differential equations, and George Boole freely employed them. Hermann Grassmann and Hermann Hankel made great use of the theory, the former in studying equations, the latter in his theory of complex numbers. Integrals Niels Henrik Abel seems to have been the first to consider in a general way the question as to what differential equations can be integrated in a finite form by the aid of ordinary functions, an investigation extended by Liouville. Cauchy early undertook the general theory of determining definite integrals, and the subject has been prominent during the 19th century. Frullani integrals, David Bierens de Haan's work on the theory and his elaborate tables, Lejeune Dirichlet's lectures embodied in Meyer's treatise, and numerous memoirs of Legendre, Poisson, Plana, Raabe, Sohncke, Schlömilch, Elliott, Leudesdorf and Kronecker are among the noteworthy contributions. Eulerian integrals were first studied by Euler and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows: although these were not the exact forms of Euler's study. If n is a positive integer: but the integral converges for all positive real and defines an analytic continuation of the factorial function to all of the complex plane except for poles at zero and the negative integers. To it Legendre assigned the symbol , and it is now called the gamma function. Besides being analytic over positive reals ℝ+, also enjoys the uniquely defining property that is convex, which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject Lejeune Dirichlet has contributed an important theorem (Liouville, 1839), which has been elaborated by Liouville, Catalan, Leslie Ellis, and others. Raabe (1843–44), Bauer (1859), and Gudermann (1845) have written about the evaluation of and . Legendre's great table appeared in 1816. Applications The application of the infinitesimal calculus to problems in physics and astronomy was contemporary with the origin of the science. All through the 18th century these applications were multiplied, until at