The Garden of Archimedes
 A Museum for Mathematics

The foundations of calculus

In 1734, the Irish bishop and philosopher George Berkley published a pamphlet called The Analyst , written in the form of "a discourse addressed to an infidel mathematician" (probably Halley the astronomer) in which he strongly and in a timely fashion attacks the foundations of calculus. Berkley hits upon, with great accuracy, the main weakness of Newton's construction, as well as that of the "foreign mathematicians", and of Leibniz' assistants. The matter involved both the definitions of fluxions and infinitesimal increments.
To calculate the fluxion of a function $f(x)$ one in fact needs to calculate the incremental relation $\frac{f(x+e\dot{x})-f(x)}{e}$ and then let $e=0$. But dividing by $e$ the tacit assumption that is made is that $e$ is not nought, and thus once the division has been carried out it is not, therefore, correct, as Berkley observes, to let $e=0$.The lack of an accurate theory of limits makes it difficult to go beyond the paradox.
The other definition Newton gave, based on the prime and ultimate ratios, considers the relation not when it equals zero, or when it is different from zero, but in the exact moment in which the characteristic triangle disappears and collapses into one point. Berkley easily makes a joke out of this even more obscure formulation:
This is utterly unconceivable, he says. And yet there are some people who are capable, while expressing disappointment in front of the proposition of any mystery, do not make any problems, as far as they are concerned, in sipping a gnat or swallowing a camel
And further on:
And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?
Berkley's criticism is not limited to the pure and simple attack to the foundations of calculus; it discusses the reasons of the success of the new analysis in facing and solving problems with an amplitude previously unimaginable. He therefore tries to explain how such surprising results can be the outcome of such precarious principles. For Berkley, these successes are due to a mutual "cancelling out" of the errors. Let us imagine, for instance, that we want to find the tangent to a curve $y(x)$, that is its subtangent, the segment AP. Since the triangles ABP and BRT are proportional, then $AP=BP \cdot BR/TR=ydx/TR$. Now, says Berkley, the first mistake that we make is in writing $SR=dy=y(x+dx)-y(x)$ instead of TR, a mistake that cannot be ignored if $dx$ is different from zero. On the other hand, after having made the necessary simplifications, in the quotient $ dx/dy$ let $dx=0$: a second mistake which compensates the first and brings us to the correct result $AP=y/y'$.
The reciprocal cancellation of both errors therefore brings us to the right results and one achieves "if not science, at least truth"; a thesis that will again be taken up by Lazare Carnot in the Refléxions sur la métaphysique du calcul infinitesimal written in 1797, where he substitutes the accidental elision, with a theory of the need to compensate errors to give the calculus a solid basis.
Berkley's thesis conditioned the course of English analysis in a non indifferent manner; if mathematicians who were not so powerful defended the theory with superficial and repetitive considerations, others engaged themselves in a deep effort of failed attempts to eliminate the controversial fluxions from analysis. Among these was Colin MacLaurin, who published a Treatise of fluxions in 1742 in defense of Newton's method. Here he attempts a systematic exposition in rigorously geometrical terms of the theory of fluxions avoiding infinites and infinitesimals, prime and ultimate ratios, based on instantaneous velocity.
The situation on the European continent, where premature discussions on the principles had only marginal influence, whilst all fields of the sciences and particularly Physics are invaded by the new analysis, was very different. This leads to a simultaneous and unprecedented flourishing of calculus methods and techniques, which for a certain period involved the field of mathematics as a whole.
Gradually, though, in the bulk of results achieved over the course of about half a century, the weaknesses of the theory begin to appear, in particular, as far as the relations between series of functions and continuities are concerned. These became unacceptable after the treaty of Fourier on the propagation of heat, to the point that the Academy of Berlin felt the need to offer a prize for the best work to be carried out on the foundations of calculus.
D'Alembert expresses his position in various articles of the Encyclopédie. At the entry "limit" he explains that "the theory of limits is the basis of the true metaphysics of differential calculus".
Lagrange sustained a different position in his Note sur la métaphysique du calcul infinitésimal, which he later developed in his Théorie des fonctions analytiques (1797) which is considered the most complete attempt of rigorous arrangement of the analysis before the final intervention of Cauchy. Here he starts with a premise that gives a detailed criticism of the founding principles of calculus based on infinitesimals and tries to avoid the obstacles by using the development in series. The "derivative" (the term which appears here for the first time) is introduced developing the function $f$ in the point $x_o$, $f(x)=a_o +a_1 (x- x_o)+a_2 (x-x_o)^2+...$, the coefficients $a_o$, $a_1$, $a_2$, ... will depend on $x_o$; among these $a_o$ is the value of the function in the point $x_o$, while, by definition, the coefficient $a_1$ is said to be the derivative of $f$ in $x_o$ and is indicated with the symbol $f'(x_o)$.
The formulation of Lagrange was criticised and overturned by the revision of the issue made independently by Augustin Louis Cauchy and by Bernard Bolzano.
In 1817, Bolzano published a pamphlet Rein analytischer Beweis des Lehrsatzes, in which, in a rigorous manner, he introduced certain concepts such as the continuity of functions, the convergence of sequences and of series, the higher extreme, with the intention of offering a demonstration of the theory of zeroes. Nevertheless, the contributions of Bolzano remained almost unknown and were only re-discovered later on.
Quite different, instead, is the influence of the works of Cauchy, which determine a bifurcation point in the history of infinitesimal calculus and determine in great part of the course of the theory from then on. In 1821, Cauchy published the first of three treatises for the benefit of his course students, the Cours d'analyse de l'École Polytechnique. According to the vision that d'Alembert had proposed, the concept of limit is put at the basis of all the constructions of analysis. By means of this concept Cauchy settles the controversial notion of infinitesimal and that of infinite, he defines the continuity of a function and he studies the convergence of series and sequences. In the following Résumé des leçons sur le calcul infinitesimal (1823), the theory of limits is applied to infinitesimal calculus. Even though the terminology of Lagrange is kept, the "derivative" is now rigorously defined as the limit of the incremental ratio, thus proving the various theorems of calculus.

|    an extract from Lagrange    |    and one from Bolzano    |    the limit in Cauchy    |   

An extract from the Theorie des fonctions analytiques of Lagrange.

The starting point is the theoretical construction of Lagrange and the expansion of any function into a series,$f(x)$, in which by the function of one or more quantities he intends "every expression of calculus in which these quantities appear in any fashion, with or without other quantities that are considered as having given and constant values, whilst the quantities of the function can assume any possible value". The series obtained by substituting $x+i$instead of $x$, will be a series of powers in which "no fraction power of $i$can be found". Having secured the general form of expansion of every function, Lagrange studies the meaning of every term, and from this expansion obtaining the "derivative" functions, in particular.

* Exhibit VI. 1

Joseph Louis Lagrange
Théorie des fonctions analytiques
After these general considerations on the expansion of functions, let us consider in particular the formula

\begin{displaymath}f(x+i)=f(x)+pi+qi^2+ri^3+\mbox{ecc.}\end{displaymath}

and let us try to find out how the derivative functions $p$, $q$, $r$, etc. depend on the primitive function $f(x)$.
For this let us suppose that the indeterminate $x$ becomes $x+o$, $o$ being any indeterminate quantity and independent from $i$; one can see that $f(x+i)$ becomes $f(x+i+o)$ and at the same time that one can obtain the same result by simply putting $i+o$ instead of $i$ in $f(x+i)$. Therefore the result should be the same whether $i+o$ is put in the series $f(x+i)=f(x)+pi+qi^2+ri^3+\mbox{ecc.}$ instead of $i$, or $x+o$ is inserted instead of $x$. [...]
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An extract from Rein analitischer Beweis des Lehrsatzes by Bolzano.

* Exhibited page VI.2

Bernard Bolzano
Rein analytischer Beweis des Lehrsatzes
If in a succession of quantities $F_1(x)$, $F_2(x)$, $F_3(x)$, ..., $F_n(x)$, ..., $F_{n+r}(x)$, the difference between the nth term $F_n(x)$ and every following term $F_{n+r}(x)$, at a certain distance from the nth , maintains itself smaller than every given quantity, by taking a $n$ sufficiently big, there will always be a certain constant quantity and only one towards which the terms of this sequence tend towards more and more, and to which they can approach as much as desired if the series is lengthened sufficiently far.
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The limit in Cauchy.

In the introduction to Cours d'analyse Cauchy criticises the use of "reasonings taken from the generality of algebra", in a veiled dispute with Lagrange. He in fact states:
Though reasonings of this kind are generally accepted, especially in the transition from the converging series to the diverging series and from real quantities to imaginary expressions, they cannot be considered, in my opinion, more then inductive reasonings capable of anticipating truth, but finding little agreement with the accuracy of approach boasted by the mathematical sciences. One must, moreover, observe that they tend to attribute an indefinite extension to algebraic formulas, whilst most of these formulas actually exist under certain specific conditions and when the quantities they contain have certain values [...] Therefore before carrying out the sum of any series, I was compelled to examine the conditions under which they converged. Thus I have established general rules that I feel deserve some attention.
As already stated, the fundamental point of the construction of Cauchy becomes the definition of the limit.

* Exhibit VI.3

Augustin Luis Cauchy
Cours d'analyse
When the values of a given variable take up different values successively and approach a fixed value indefinitely, such that it ends up being very slightly distinct from it, this latter will be called the limit of all the others. Thus, for example, an irrational number is the limit of the different fractions that give it a more and more approximate value. In geometry, the surface of a circle is the limit towards which the polygons inscribed in it tend, whilst the number of edges they have grows more and more, etc.
When the successive numerical values of the same variable decrease indefinitely, such as to become less than a given number, this number becomes what is called an infinitesimal or an infinitesimal quantity. The limit of this type of variable is zero.
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