On Calculus (-1799)

"[…] it is not necessary that these hypotheses should be true, or even probably; but it is enough if they provide a calculus which fits the observations […]" (Andrew Osiander, "On the Revolutions of the Heavenly Spheres", 1543)

"Even though these are called imaginary, they continue to be useful and even necessary in expressing real magnitudes analytically. For example, it is impossible to express the analytic value of a straight line necessary to trisect a given angle without the aid of imaginaries. Just so it is impossible to establish our calculus of transcendent curves without using differences which are on the point of vanishing, and at last taking the incomparably small in place of the quantity to which we can assign smaller values to infinity." (Gottfried W Leibniz, [letter to Varignon], 1702)

"The calculus is to mathematics no more than what experiment is to physics, and all the truths produced solely by the calculus can be treated as truths of experiment. The sciences must proceed to first causes, above all mathematics where one cannot assume, as in physics, principles that are unknown to us. For there is in mathematics, so to speak, only what we have placed there […] If, however, mathematics always has some essential obscurity that one cannot dissipate, it will lie, uniquely, I think, in the direction of the infinite; it is in that direction that mathematics touches on physics, on the innermost nature of bodies about which we know little." (Bernard Le Bovier de Fontenelle, "Elements de la géométrie de l'infini", 1727)

"This method of subjecting the infinite to algebraic manipulations is called differential and integral calculus. It is the art of numbering and measuring with precision things the existence of which we cannot even conceive. Indeed, would you not think that you are being laughed at, when told that there are lines infinitely great which form infinitely small angles? Or that a line which is straight so long as it is finite would, by changing its direction infinitely little, become an infinite curve? Or that there are infinite squares, infinite cubes, and infinities of infinities, one greater than another, and that, as compared with the ultimate infinitude, those which precede it are as nought. All these things at first appear as excess of frenzy; yet, they bespeak the great scope and subtlety of the human spirit, for they have led to the discovery of truths hitherto undreamt of." (Voltaire, "Letters on the English", 1733)

"And yet in the calculus differentialis, which method serves to all the same intents and ends with that of fluxions, our modern analysts are not content to consider only the differences of finite quantities: they also consider the differences of those differences, and the differences of the differences of the first differences. And so on ad infinitum. That is, they consider quantities infinitely less than the least discernible quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding infinitesimals, and so on without end or limit. Insomuch that we are to admit an infinite succession of infinitesimals, each infinitely less than the foregoing, and infinitely greater than the following. As there are first, second, third, fourth, fifth, &c. fluxions, so there are differences, first, second, third, fourth, &c., in an infinite  progression towards nothing, which you still approach and never arrive at. And (which is most strange) although you should take a million of millions of these infinitesimals, each whereof is supposed infinitely greater than some other real magnitude, and add them to the least given quantity, it shall never be the bigger. For this is one of the modest postulata of our modern mathematicians, and is a cornerstone or ground-work of their speculations."  (George Berkeley, "The Analyst: A Discourse Addressed to an Infidel Mathematician...", 1734)

"What concerns us most here is the metaphysics of the differential calculus. This metaphysics, of which so much has been written, is even more important and perhaps more difficult to explain than the rules of this calculus themselves: various mathematicians, among them Rolle, who were unable to accept the assumption concerning infinitely small quantities, have rejected it entirely, and have held that the principle was false and capable of leading to error. Yet in view of the fact that all results obtained by means of ordinary Geometry can be established similarly and much more easily by means of the differential calculus, one cannot help concluding that, since this calculus yields reliable, simple, and exact methods, the principles on which it depends must also be simple and certain." (Jean LeRond D'Alembert, "Differentiel" ["Differentials", 1754)

"Those quantities that depend on others in this way, namely, those that undergo a change when others change, are called functions of these quantities. This definition applies rather widely and includes all ways in which one quantity could be determined by another." (Leonhard Euler, "Foundations of differential calculus, with applications to finite analysis and series", 1755)

"I see with much pleasure that you are working on a large work on the integral Calculus [...] The reconciliation of the methods which you are planning to make, serves to clarify them mutually, and what they have in common contains very often their true metaphysics; this is why that metaphysics is almost the last thing that one discovers. The spirit arrives at the results as if by instinct; it is only on reflecting upon the route that it and others have followed that it succeeds in generalising the methods and in discovering its metaphysics." (Pierre-Simon Laplace [letter to Sylvestre F Lacroix] 1792)

"It appears that Fermat, the true inventor of the differential calculus, considered that calculus as derived from the calculus of finite differences by neglecting infinitesimals of higher orders as compared with those of a lower order [...] Newton, through his method of fluxions, has since rendered the calculus more analytical, he also simplified and generalized the method by the invention of his binomial theorem. Leibnitz has enriched the differential calculus by a very happy notation." (Pierre-Simon Laplace, "Exposition du système du monde" ["Exposition of the System of the World"], 1796)