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.

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.

But, because the Method of Infinitesimals is much in use, and is valued for its conciseness, I thought it was requisite to account explicitly for the truth, and perfect accuracy of the conclusions that are derived from it; the rather, that it does not seem to be a very proper reason that is assigned by Authors, when they determine what is called the Difference (but more accurately the ) of a Quantity, and tell us, That they reject certain Parts of the Element, because they become infinitely less than the other parts; not only because a proof of this nature may leave some doubt as to the accuracy of the conclusion, but because it may be demonstrated that those parts ought to be neglected by them at any rate, or that it would be an error to retain them.

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.

As professor in the Polytechnic School [autumn of 1858] in Zurich I found myself for the first time obliged to lecture upon the elements of the differential calculus and felt, more keenly than ever before, the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must certainly approach a limiting value, I had recourse to geometric evidences. Even now such resort to geometric intuition in a first presentation of the differential calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny. For myself this feeling of dissatisfaction was so overpowering that I made the fixed resolve to keep meditating on the question till I should find a purely arithmetic and perfectly rigorous foundation for the principles of infinitesimal analysis.

SALV. I will now say something which may perhaps astonish you; it refers to the possibility of dividing a line into its infinitely small elements by following the same order which one employs in dividing the same line into forty, sixty, or a hundred parts, that is, by dividing it into two, four, etc. He who thinks that, by following this method, he can reach an infinite number of points is greatly mistaken; for if this process were followed to (37) eternity there would still remain finite parts which were undivided.

Indeed by such a method one is very far from reaching the goal of indivisibility; on the contrary he recedes from it and while he thinks that, by continuing this division and by multiplying the multitude of parts, he will approach infinity, he is, in my opinion, getting farther and farther away from it. My reason is this. In the preceding discussion we concluded that, in an infinite number, it is necessary that the squares and cubes should be as numerous as the totality of the natural numbers [tutti i numeri], because both of these are as numerous as their roots which constitute the totality of the natural numbers. Next we saw that the larger the numbers taken the more sparsely distributed were the squares, and still more sparsely the cubes; therefore it is clear that the larger the numbers to which we pass the farther we recede from the infinite number; hence it follows [83] that, since this process carries us farther and farther from the end sought, if on turning back we shall find that any number can be said to be infinite, it must be unity. Here indeed are satisfied all those conditions which are requisite for an infinite number; I mean that unity contains in itself as many squares as there are cubes and natural numbers [tutti i numeri].

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.

As a public teacher of mere striplings, I am often amazed by the facility and absence of resistance with which the principles of the infinitesimal calculus are accepted and assimilated by the present race of learners. When I was young, a boy of sixteen or seventeen who knew his infinitesimal calculus would have been almost pointed at in the streets as a prodigy, like Dante, as a man who had seen hell. Now-a-days, our Woolwich cadets at the same age, talk with glee of tangents and asymptotes and points of contrary flexure and discuss questions of double maxima and minima, or ballistic pendulums, or motion in a resisting medium, under the familiar and ignoble name of sums.

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.

Descend from the infinite to the infinitesimal. Long before . . . observation had begun to penetrate the veil under which Nature has hidden her mysteries, the restless mind sought some principle of power strong enough and of sufficient variety to collect and bind together all parts of a world. This seemed to be found, where one might least expect it, in abstract numbers. Everywhere the exactest numerical proportion was seen to constitute the spiritual element of the highest beauty.

Brook Taylor... in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. ...Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity.

As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious.

You know my method. It is founded upon the observation of trifles.