The methods of the Bernoullis and of Taylor, were held, at the time of their invention, to be most complete and exact. Several imperfections, however, belong to them. They do not apply to problems involving three or more properties; nor do they extend to cases involving differentials of a higher order than the first: for instance, they will not solve the problem, in which a curve is required, that with its radius of curvature and evolute shall contain the least area. Secondly, they do not extend to cases, in which the analytical expression contains, besides x, y, and their differentials, integral expressions; for instance, they will not solve the second case proposed in James Bernoulli's Programma if the Isoperimetrical condition be excluded; for then the arc s, an integral, since it =<math>\int \!dx \sqrt(1+\frac{dy^2}{dx^2})</math>, is not given. Thirdly, they do not extend to cases, in which the differential function, expressing the maximum should depend on a quantity, not given except under the form of a differential equation, and that not integrable; for instance, they will not solve the case of the curve of the quickest descent, in a resisting medium, the descending body being solicited by any forces whatever.

The method applied by Newton to the grounding of the Infinitesimal Calculus, and which since the beginning of this century has been recognised by the best mathematicians as the only one that furnishes sure results, is the method of limits. The method consists in this, viz., instead of considering a continuous transition from one value of a quantity to another, from one position to another, or, speaking generally, from one determination of a concept to another, one considers in the first place a transition through a finite number of intervals and then allows the number of these intervals to increase so that the distances of two successive points of division all decrease infinitely.

There is another point... and that is the method of demonstration by geometrical figures. In the first solution of Isoperimetrical problems, the Bernoullis use diagrams and their properties. Euler, in his early essays, does the same; then, as he improves the calculus he gets rid of constructions. In his Treatise [footnote: Methodus inveniendi, &c.], he introduces geometrical figures, but almost entirely, for the purpose of illustration: and finally, in the tenth volume of the Novi Comm. Petrop. as Lagrange had done in the Miscellanea Taurinensea, he expounds the calculus, in its most refined state, entirely without the aid of diagrams and their properties. A similar history will belong to every other method of calculation, that has been advanced to any degree of perfection.

This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles. ...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals.

In the method of indivisibles, lines were conceived to be made up of points, surfaces of lines,and solids of surfaces; and such suppositions have been employed by several ingenious men for proving the old theorems, and discovering new ones, in a brief and easy manner. But as this doctrine was inconsistent with the strict principles of geometry, so it soon appeared that there was some danger of its leading them into false conclusions: therefore others, in the place of indivisible, substituted infinitely small divisible elements, of which they supposed all magnitudes to be formed; and thus endeavoured to retain, and improve, the advantages that were derived from the former method for the advancement of geometry.

On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements. ...there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect...

I came across the mathematical writings of Torricelli... which... I read in... 1651... where... he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met; for what holds for most... concerning the circle... usually had by polygons with an infinite number of sides, and... the circumference by... an infinite number of infinitely short lines... could.., it seemed to me, with... changes, be... adjusted to other problems; and... by that means examine... Euclid, Appolonius and especially... Archimedes. ...I began to think ...whether this might bring ...light to the quadrature of the circle.

The second point to be considered is the method adopted by Spinoza for setting forth his philosophy; it is the demonstrative method of geometry as employed by Euclid, in which we find definitions, explanations, axioms, and theorems. Even Descartes made it his starting-point that philosophic propositions must be mathematically handled and proved, that they must have the very same evidence as mathematics. The mathematical method is considered superior to all others, on account of the nature of its evidence; and it is natural that independent knowledge in its re-awakening lighted first upon this form, of which it saw so brilliant an example. The mathematical method is, however, ill-adapted for speculative content, and finds its proper place only in the finite sciences of the understanding. In modern times Jacobi has asserted (Werke, Vol. IV. Section I. pp. 217-223) that all demonstration, all scientific knowledge leads back to Spinozism, which alone is a logical method of thought; and because it must lead thither, it is really of no service whatever, but immediate knowledge is what we must depend on. It may be conceded to Jacobi that the method of demonstration leads to Spinozism, if we understand thereby merely the method of knowledge belonging to the understanding. But the fact is that Spinoza is made a testing-point in modern philosophy, so that it may really be said: You are either a Spinozist or not a philosopher at all. This being so, the mathematical and demonstrative method of Spinoza would seem to be only a defect in the external form; but it is the fundamental defect of the whole position. In this method the nature of philosophic knowledge and the object thereof, are entirely misconceived, for mathematical knowledge and method are merely formal in character and consequently altogether unsuited for philosophy. Mathematical knowledge exhibits its proof on the existent object as such, not on the object as conceived; the Notion is lacking throughout; the content of Philosophy, however, is simply the Notion and that which is comprehended by the Notion. Therefore this Notion as the knowledge of the essence is simply one assumed, which falls within the philosophic subject; and this is what represents itself to be the method peculiar to Spinoza's philosophy.

The second point to be considered is the method adopted by Spinoza for setting forth his philosophy; it is the demonstrative method of geometry as employed by Euclid, in which we find definitions, explanations, axioms, and theorems. Even Descartes made it his starting-point that philosophic propositions must be mathematically handled and proved, that they must have the very same evidence as mathematics. The mathematical method is considered superior to all others, on account of the nature of its evidence; and it is natural that independent knowledge in its re-awakening lighted first upon this form, of which it saw so brilliant an example. The mathematical method is, however, ill-adapted for speculative content, and finds its proper place only in the finite sciences of the understanding. In modern times Jacobi has asserted (Werke, Vol. IV. Section I. pp. 217-223) that all demonstration, all scientific knowledge leads back to Spinozism, which alone is a logical method of thought; and because it must lead thither, it is really of no service whatever, but immediate knowledge is what we must depend on. It may be conceded to Jacobi that the method of demonstration leads to Spinozism, if we understand thereby merely the method of knowledge belonging to the understanding. But the fact is that Spinoza is made a testing-point in modern philosophy, so that it may really be said: You are either a Spinozist or not a philosopher at all. This being so, the mathematical and demonstrative method of Spinoza would seem to be only a defect in the external form; but it is the fundamental defect of the whole position. In this method the nature of philosophic knowledge and the object thereof, are entirely misconceived, for mathematical knowledge and method are merely formal in character and consequently altogether unsuited for philosophy. Mathematical knowledge exhibits its proof on the existent object as such, not on the object as conceived; the Notion is lacking throughout; the content of Philosophy, however, is simply the Notion and that which is comprehended by the Notion. Therefore this Notion as the knowledge of the essence is simply one assumed, which falls within the philosophic subject; and this is what represents itself to be the method peculiar to Spinoza's philosophy.

At the Stourbridge Fair in 1663, at age twenty, he purchased a book on astrology, “out of a curiosity to see what there was in it.” He read it until he came to an illustration which he could not understand, because he was ignorant of trigonometry. So he purchased a book on trigonometry but soon found himself unable to follow the geometrical arguments. So he found a copy of Euclid’s Elements of Geometry, and began to read. Two years later he invented the differential calculus.

The method of demonstration, which was invented by the author of fluxions, is accurate and elegant; but we propose to begin with one that is somewhat different; which, being less removed from that of the antients, may make the transition to his method more easy to beginners (for whom chiefly this treatise is intended), and may obviate some objections that have been made to it.

At the time the book of Marquis de l'Hôpital had appeared, and almost all mathematicians began to turn to the new geometry of the infinite [that is, the new infinitesimal calculus], until then little known. The surprising universality of the methods, the elegant brevity of the proofs, the neatness and speed of the most difficult solutions, a singular and unexpected novelty, all attracted the mind and there was in the mathematical world a well marked revolution [une révolution bien marquée.

Sur un nouveau genre de calcul, 1826.

Awaiting me upon my return to Strasbourg were Henri Cartan and the course on "differential and integral calculus," which was our joint responsibility. ... One point that concerned him was the degree to which we should generalize Stokes' formula in our teaching. ... In his book on invariant integrals, Elie Cartan, following Poincare in emphasizing the importance of this formula, proposed to extend its domain of validity. Mathematically speaking, the question was of a depth that far exceeded what we were in a position to suspect. ... One winter day toward the end of 1934,1 thought of a brilliant way of putting an end to my friend's persistent questioning. We had several friends who were responsible for teaching the same topics in various universities. "Why don't we get together and settle such matters once and for all, and you won't plague me with your questions any more?" Little did I know that at that moment Bourbaki was born.