The objections which have been raised against the infinitesimal method are based on the false supposition that the errors due to neglecting infinitely small quantities during the actual calculation will continue to exist in the result of the calculation.--CARNOT, L. _Réflections sur la Métaphysique du Calcul Infinitésimal (Paris, 1813), p. 215._ =1908.= A limiting ratio is neither more nor less difficult to define than an infinitely small quantity.--CARNOT, L. _Réflections sur la Métaphysique du Calcul Infinitésimal (Paris, 1813), p. 210._ =1909.= A limit is a peculiar and fundamental conception, the use of which in proving the propositions of Higher Geometry cannot be superseded by any combination of other hypotheses and definitions. The axiom just noted that what is true up to the limit is true at the limit, is involved in the very conception of a limit: and this principle, with its consequences, leads to all the results which form the subject of the higher mathematics, whether proved by the consideration of evanescent triangles, by the processes of the Differential Calculus, or in any other way. --WHEWELL, W. _The Philosophy of the Inductive Sciences, Part 1, bk. 2, chap. 12, sect. 1, (London, 1858)._ =1910.= The differential calculus has all the exactitude of other algebraic operations.--LAPLACE. _Théorie Analytique des Probabilités, Introduction; Oeuvres, t. 7 (Paris, 1886), p. 37._ =1911.= The method of fluxions is probably one of the greatest, most subtle, and sublime discoveries of any age: it opens a new world to our view, and extends our knowledge, as it were, to infinity; carrying us beyond the bounds that seemed to have been prescribed to the human mind, at least infinitely beyond those to which the ancient geometry was confined.--HUTTON, CHARLES. _A Philosophical and Mathematical Dictionary (London, 1815), Vol. 1, p. 525._ =1912.= The states and conditions of matter, as they occur in nature, are in a state of perpetual flux, and these qualities may be effectively studied by the Newtonian method (Methodus fluxionem) whenever they can be referred to number or subjected to measurement (real or imaginary). By the aid of Newton’s calculus the mode of action of natural changes from moment to moment can be portrayed as faithfully as these words represent the thoughts at present in my mind. From this, the law which controls the whole process can be determined with unmistakable certainty by pure calculation.--MELLOR, J. W.