The Early History of Calculus Problems


Anyone teaching Calculus has presumably studied Calculus at some point in the past and is probably aware that homework problems assigned today are identical to to the ones we faced as students. How far back can this go? Certainly not beyond 1696, when Guillaume François Antoine, Marquis de L'Hospital (often written L'Hôpital) published Analyse des infiniment petits, pour l'intelligence des lignes courbes, a work known as the very first Calculus textbook; to put that date in perspective: Newton's Principia appeared in 1687. In fact some of today's problems are close, if not identical, to those appearing in this 17th-century text (the Infiniment petits did not have a list of exercises at the end of each section, but L'Hospital illustrates his presentation with many worked-out examples). This column presents five specimens from Section III: "The use of the calculus of differences to find largest and least values."
   L'Hospital's text only covers differential calculus (and, as its title suggests, concentrates on applications to analytic geometry). There is no companion volume on integral calculus; here is the Marquis' explanation: "M. Leibnitz having written me that he was working on the subject in a Treatise which he entitles De Scientiâ infiniti, I did not want to risk depriving the Public of such a beautiful Work which must include all the most intriguing facts about the inverse tangent Method, about rectifications of curves, about quadrature of the spaces they enclose, about those of the surfaces of the bodies they describe, about the dimensions of those bodies, about the location of centers of gravity, &c. I am only making this matter public because he asked me to do so in his Letters ... ." Unfortunately De Scientiâ infiniti, if written, was never published.
   Notation: Much of L'Hospital's notation, derived from Leibnitz, has remained standard, although following Descartes he writes $xx$ instead of $x^2$. The concepts of function and derivative were still in the future. He analyzes a curve from its equation by calculating the relation between the differences $dx$ and $dy$, and locates maxima and minima by looking for points where $dy=0$ (it is understood that $dx\neq 0$).
   Historical note: the text of the Infiniment petits I worked with (a 1768 edition available online, with plates here), once belonged to John Quincy Adams, and bears copious annotations in his hand. It is now in the Boston Public Library.

Example I

Supppose that $x^3+y^3=axy$ (AP $= x$, PM $= y$, AB $= a$) gives the nature of the curve MDM. [Find the point E where the height is largest].


This image and those below appear courtesy of the Boston Public Library.

Example VI

Among all the cones that can be inscribed in a sphere, determine which one has the largest lateral area.


Example VII

It is asked, among all the parallelepipeds equal [in volume] to a given cube $a^3$, and that have for one of their edges a given line $b$, which one has the smallest surface.

Example XI

A traveler leaving location C to go to location F must cross two regions separated by the straight line AEB. We suppose that in the region on the side of C, he covers distance $a$ in time $c$, and that on the other, on the side of F, distance $b$ in the same time $c$: we ask through which point E on the line AEB he should pass, so as to take the least possible time to get from C to F.


Example XII

Let there be a pulley F which hangs freely at the end of a cord CF attached at C, with a plumb bob D suspended by the cord DFB which passes over the pulley F and which is attached at B, in such a way that the points C, B are situated on the same horizontal line. One supposes that the pulley and the cords have no weight; and one asks at what place the plumb bob D, or the pulley F, will rest.


[My translations. -TP]