Considering equations of infinitely small partial differences as equations of finite partial differences in which nothing is neglected, we are able to throw light upon the obscure points of their calculus, which have been the subject of great discussions among geometricians. It is thus that I have demonstrated the possibility of introducing discontinued functions in their integrals, provided that the discontinuity takes place only for the differentials of the order of these equations or of a superior order. The transcendent results of calculus are, like all the abstractions of the understanding, general signs whose true meaning may be ascertained only by repassing by metaphysical analysis to the elementary ideas which have led to them; this often presents great difficulties, for the human mind tries still less to transport itself into the future than to retire within itself. The comparison of infinitely small differences with finite differences is able similarly to shed great light upon the metaphysics of infinitesimal calculus. It is easily proven that the finite _n_th difference of a function in which the increase of the variable is _E_ being divided by the _n_th power of _E_, the quotient reduced in series by ratio to the powers of the increase _E_ is formed by a first term independent of _E._ In the measure that _E_ diminishes, the series approaches more and more this first term from which it can differ only by quantities less than any assignable magnitude. This term is then the limit of the series and expresses in differential calculus the infinitely small _n_th difference of the function divided by the _n_th power of the infinitely small increase. Considering from this point of view the infinitely small differences, we see that the various operations of differential calculus amount to comparing separately in the development of identical expressions the finite terms or those independent of the increments of the variables which are regarded as infinitely small; this is rigorously exact, these increments being indeterminant. Thus differential calculus has all the exactitude of other algebraic operations.