Teaching the Fundamental Theorem of Calculus: A Historical Reflection - Integration from Cavalieri to Darboux

As remarked by Bressoud [5], the statement of the FTC has two equivalent formulations:

Theorem 1 (FTC, Derivative version).

If \(f:[a,b]\rightarrow\mathbb{R}\) is a continuous function (\({-\infty\,{\rm{<}}\,a\,{\rm{<}}\,b\,{\rm{<}}\,\infty}\)), then \[\frac{d}{dx}\int_{a}^{x}f(u)\,du=f(x)\quad\quad\quad\quad(1)\] for each \(x\in [a,b]\) (if \(x=a\) or \(x=b,\) derivatives are taken to be unilateral).

Theorem 2 (FTC, Antiderivative version).

If \(f:[a,b]\rightarrow\mathbb{R}\) is a continuous function (\({-\infty\,{\rm{<}}\,a\,{\rm{<}}\,b\,{\rm{<}}\,\infty}\)), and \(F:[a,b]\rightarrow\mathbb{R}\) is an antiderivative of \(f\) on \([a,b]\) (that is to say, \(F\) is continuous on \([a,b]\) and \(F^{\,\prime}(x)=f(x)\) for each \(x\in (a,b)\)), then \[\int_{a}^{b}f(u)\,du=F(b)-F(a).\quad\quad\quad\quad(2)\]

The definition of the integral as a limit of Riemann sums is due to Augustin-Louis Cauchy (1789-1857). According to R. Laubenbacher and D. Pengelley [29, p. 139], his superbly elegant definition appeared in 1823, two years after the publication of his Cours d'Analyse. The definition appears in Cauchy's Resumé des Leçons sur le calcul infinitesimal in his Oevres Complètes, published in 1899 [8, Vingt et unième leçon, Intégrales Définies, p. 122]. Cauchy started with a function \(y=f(x)\), continuous “with respect to the variable \(x\) between two finite limits” (his language), and then proved (see note 2.1) that the corresponding Riemann sums converge to a value which he took as the definition of the integral of \(y=f(x)\) on the given interval (see note 2.2).

The first published statement of the FTC was authored by Isaac Barrow [9, Lecture X, Section 11, p. 117] and published around 1674. Isaac Barrow, Isaac Newton, Gottfried Leibniz, and James Gregory are all credited with having proved the FTC independently of each other (see note 2.3). It goes without saying that, in contrast to Cauchy's times, during the second half of the 17th century (the time period in which calculus was invented) there was, at best, an incipient and rudimentary notion of function, with no way to express the continuity of “curves” (roughly corresponding to the graphs of the functions of today) and no standard mathematical notation either for the integral or for the derivative. According to D. T. Whiteside [44, p. 196] the 17th century in mathematics was a period of “rapid advance using valid but tenuously defined concepts as a basis for a rich and varied technical achievement.” One of the greatest of such achievements was that of inventing notations to adequately represent the mental images of the objects of mathematical discourse and their relations [44]. This, in turn, was concomitant with the conceptual mathematical advance that culminated, eventually, in Cauchy's 19th century definition of the integral.

It is worth noticing that during the latter part of the 17th century, motion and geometric transformations were taken to be central ideas in the generation of curves, planar areas, volumes and general surfaces (see, for example, Barrow [9, Lectures I-III, pp. 35-52]). For instance, Bonaventura Cavalieri (1598-1647), a disciple of Galileo Galilei, believed that a surface consisted of an indefinite number of equidistant parallel straight lines, and a solid, in turn, of a set of equidistant parallel planes. These constituted for Cavalieri the linear and planar “indivisibles” [2, p. 124]. In the case of plane figures or solids, a regula was a line or a plane that moved parallel to itself from an initial point until it coincided with another such line or plane. This was the central idea of Cavalieri's theory of indivisibles. Thus, in the case of areas bounded by curves, these were often conceived of as being generated by the corresponding regula viewed as ordinates of variable length and infinitesimal width that moved from an initial position to a final one. This image of a moving ordinate generating an area corresponds to what we call today an “area function.”

For a non-negative continuous function defined on a closed bounded interval, the corresponding area function assigns the value of the area “swept out” by a moving ordinate as its abscissa moves from an initial to a final point in the interval. If \(X_{u}^{v}\) represents the area generated from an initial position of the ordinate with abscissa \(u\) to a final one with abscissa \(v\), then it is clear that \(X_{u}^{v}\) has the following additive property: \[X_{u}^{w}=X_{u}^{v}+X_{v}^{w}\quad{\rm{whenever}}\quad u\,{\rm{<}}\,v\,{\rm{<}}\,w.\] This discussion applies also to other quantities that can be described by one parameter that can take the form of an abscissa, an ordinate, an area, or even the abscissa or the ordinate of a changing point on a curve (e.g. the generating curves of Barrow [9, pp. 35-46, Lectures I and II]).

The property of additivity is closely related to Leibniz' introduction of the \(\int\,\) notation. According to F. Cajori [7, p. 207], in a manuscript written in 1673, Leibniz used Cavalieri's notation to establish the relation: \[ \overline{{\rm{omn.}}\overline{{\rm{omn.}} l}\,\,\frac{l}{a}}=\frac{\,\,\,\,\overline{{\rm{omn.}} l}^{\,\,\,2}}{2a}\, ,\] where \(a\) and \(l\) were, respectively, “infinitesimal elements” of the abscissa and the ordinate associated with a subnormal [7, p. 207], and \({\rm{omn.}}\) (which stood for “omnia” or “all”) was the symbol used by Cavalieri to sum infinitesimal elements such as the regula. Also, the line over \({\rm{omn.}}\) was used in place of our modern parenthesis (see note 2.4). In this manuscript Leibniz remarked: “It will be useful to write \(\int\) for \({\rm{omn.}}\), as \(\int l\) for \({\rm{omn.}} l\), that is, the sum of the \(l\)'s” (see [7, p. 207]). According to F. Cajori [7], since the symbol of summation \(\int\,\) raises dimension, Leibniz concluded that the difference symbol \(d\) would lower it. It is transparent that the operator \({\rm{omn.}}\) (and thus, the operator \(\int\,\)) has the property of additivity as defined above. According to F. Cajori [6, p. 39], the limits of integration were added to the integral sign much later, by Joseph Fourier in 1832. Before then, limits were indicated in words or in symbols within parenthesis or brackets next to the integral sign.

It is noteworthy that all four of the proofs of the FTC mentioned above use, in an essential way, the additive property of area functions. The proofs also use, in various guises, the fact that an ordinate of a curve that moves infinitesimally from a given one, can be taken to be the same as the original ordinate. This statement appears explicitly in Newton's 1669 proof of the FTC (see [23, p. 185]) and also in Section I of L'Hospital's Analyse des infiniment petits pour l'intelligence des lignes curves [34, p. 3, I, Demande ou Supposition], where it was stated as a postulate (see note 2.5), and we shall refer to it as L'Hospital's Postulate:

Grant that two quantities whose difference is an infinitely small quantity may be taken indifferently for each other: or (which is the same thing) that a quantity which is increased or decreased only by an infinitely small quantity may be considered as remaining the same.

If we make seventeenth century curves correspond to today's graphs of functions, Newton's and L'Hospital's statements say, in fact, that all curves are continuous (see page 3 for Newton's proof of the FTC). Not surprisingly, these two properties are all that is needed in order to have the theory of the integral including its usual applications, without ever mentioning a Riemann sum. The theory of the elementary integral as developed by S. Saks [37], H. Hahn and A. Rosenthal [24, 38], and H. Levi [33], and expounded by S. Lang [28, p. 213] and L. Gillman [20] is, in our view, a direct descendant of the ideas and intuitions that nurtured the mathematics of the seventeenth century. Also, it is our contention that the historical development of the integral can be used to make an argument for a teaching advantage in presenting the integral as an “area function” based on modern versions of the additive and continuity properties mentioned above in connection to Newton's proof of the FTC and L'Hospital's Postulate about infinitely close ordinates. We discuss this point in our Concluding Remarks on page 8.

Notes for page 2:

2.1. Whether this is a valid proof or not has been the subject of many discussions. Cauchy's definition of continuity [4, Section 2.2, p. 26] would seem to correspond to our definition of uniform continuity, especially if we take at face value his statement in terms of infinitesimals. In order to show that the corresponding Riemann sums converged to a real number, he used his definition of continuity, which was ready made for the proof. He did not consider (as we do today) continuity at individual points, so that our notion of uniform continuity on intervals was the only notion of continuity that he used.

It is also worth noticing that Cauchy took subdivisions of the given interval, not necessarily uniform; multiplied the lengths of the resulting subintervals by the value of the function at the left endpoint; and added all such products to get his Riemann sums.

2.2. For a commentary on Cauchy's motivation for his definition of the integral using a special case of a Riemann sum, we refer the reader to V. Katz [26, p. 177]. Katz also remarks that Joseph-Louis Lagrange used area functions in proving the FTC in the setting of functions having power series expansions [26, p. 635].

2.3. This independence is questioned in Child [9, p. 201]; see Guicciardini [23, p. 170] for different view.

2.4. With \(a=1\), the modern version of this formula, for a function \(y\) of \(x\), would be: \[\int_{a}^{b}y\bigg(\int_{a}^{x}y\,du\bigg)dx=\frac{1}{2}\bigg[\int_{a}^{b}y\,dx\bigg]^{2}.\quad\quad\quad\quad(3)\] The formula is obtained by a simple application of the method of integration by parts.

2.5. Our wording is taken from Edmund Stone's translation of Analyse des infiniment petits [35, p. 3, Postulate I].