Abstract
Suppose that, the function \( y=f(x), \) being continuous with respect to the variable x between two finite limits \( x=x_0, \) \(x=X, \) we denote by \( x_1, x_2, \dots , x_{n-1} \) new values of x interposed between these limits, which always go on increasing or decreasing from the first limit up to the second.
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- 1.
Within this lecture, Cauchy will define his definite integral in a manner very similar to the more modern definition of Georg Friedrich Bernhard Riemann (1826–1866) later in the 19th century. Riemann will build upon Cauchy’s early work on the definite integral and will remove the restric tion Cauchy imposes here for f(x) to be continuous.
- 2.
Notice Cauchy specifically does not state f(x) is the derivative of some other function. He makes a point of distancing himself from differentiation in this entire lecture to show the integral is independent of the derivative.
- 3.
The 1899 edition has f(x).
- 4.
Today’s partition, i.e. \( \{ x_0, x_1, x_2, \dots , x_{n-1}, X \}.\)
- 5.
Recall Cauchy’s definition of an average from his Cours d’analyse Note II, “We call an average among several given quantities a new quantity contained between the smallest and the largest of those that we consider.” Cauchy’s definition allows for an infinite number of averages for the same set of values.
- 6.
Cauchy duplicates this theorem with a complete proof at the end of his Cours d’analyse in Note II as Theorem XII. Theorem XII is one of his interesting average theorems and has been included in its entirety in this book as part of Appendix B. He is using the Corollary III result of this theorem with \(\alpha _i = \Delta x_i\), \(a_i=f(x_i),\) and \(b_i=1\) here.
- 7.
To justify this result, Cauchy needs to use the Intermediate Value Theorem. It is for this reason he requires the continuity of f(x).
- 8.
Cauchy takes care in noting that the partition is becoming finer and finer in the sense that every subinterval becomes successively smaller, not just that more subintervals are being created.
- 9.
It is here where Cauchy makes what some regard as a slight mistake, or at least is not clear in his description. He claims if the elements \( x_1-x_0, x_2-x_1, \dots , X-x_{n-1} \) are all sufficiently small, then each of the \(\varepsilon _j\)’s are very close to zero. In the limit of increasing n, this argument technically requires f(x) to be uniformly continuous, and not just continuous (in the modern sense), as there is no guarantee all of the \(\varepsilon _j\)’s can be made small for a given \(n. \ \) Cauchy’s own view of continuity from Lecture Two seems to fall somewhere between the two modern versions of simple and uniform continuity, and so, this aspect of his argument is not entirely clear. Some argue his claims are fine, and indeed are consistent with his earlier definitions; others argue his claims are not justified. The debate goes on.
- 10.
It is unlikely that at the time Cauchy wrote this lecture he had worked out all the details of this analysis, especially given his possible misconceptions surrounding the concepts of uniform continuity and completeness, but one can tell he had the right idea in mind about the proof that would eventually develop over time. We know today the type of sequence he is constructing here is called a Cauchy sequence (informally, we can say sequences which have the property in which its terms become arbitrarily close together if one looks out far enough in the sequence are Cauchy sequences) and that this type of sequence always converges to some value, which is precisely what Cauchy has claimed here.
- 11.
Cauchy makes it very clear he is not defining the definite integral to be a sum. Instead, he is defining the definite integral to be the limit of a sequence of sums.
- 12.
Most of the advancements in calculus throughout the 18th century by pioneers such as Jakob Bernoulli (1654–1705), Johann Bernoulli (Jakob’s younger brother and lifelong rival), the prolific Leonhard Euler, and many, many others, were mainly limited to differential calculus due to the mistaken belief integration was merely the inverse operation of differentiation, and so, did not warrant any special consideration. By the year 1800, integration had certainly settled into a secondary position of relevance behind differentiation. However, in the early 1800s, Joseph Fourier (1768–1830) changed the integral landscape forever. His representation of an arbitrary function required the calculation of coefficients defined as definite integrals,
$$\begin{aligned} a_n=\frac{1}{\pi }\int _{-\pi }^{\pi }{f(x)\cos {(n x)} dx} \ \ \ \ \ \ \ \ \text {and} \ \ \ \ \ \ \ \ b_n=\frac{1}{\pi }\int _{-\pi }^{\pi }{f(x)\sin {(n x)} dx}, \end{aligned}$$to construct his now famous Fourier series,
$$\begin{aligned} f(x)=\frac{a_0}{2}+\sum ^{\infty }_{n=1}{\big (a_n\cos {(n x)}+b_n\sin {(n x)} \big )}. \end{aligned}$$An integral theory independent of differential calculus was clearly needed. Cauchy is one of the first mathematicians to take on this challenge and seriously consider the question of the existence of the definite integral. Here, he has demonstrated the definite integral does exist without ever invoking the concept of the derivative. Cauchy has shown the definite integral is important on its own, and his proof in this lecture is one of the highlights of his 1823 text.
- 13.
Leibniz, in the late 1600s, had defined the integral as the sum of infinitely many infinitesimal summands (the sum itself). This is why he chose the symbol of a large S to denote his sum. His large S gradually evolved into the \(\int \) symbol for integration we use today.
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Cates, D.M. (2019). DEFINITE INTEGRALS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_21
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DOI: https://doi.org/10.1007/978-3-030-11036-9_21
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