Augustin-Louis Cauchy

Augustin-Louis Cauchy was born to a motivated middle-class family. Born on August 21, 1789 in Paris, he was baptized in the parish church of Saint-Roch. His name comes from his father Louis-François and the month he was born in (Belhoste 1). Louis-François himself being a learned man (Belhoste 2), and spent quite a bit of time and attention on Augustin’s studies due to the loss of his position and protector due to the Great Revolution which had caused them to flee from Paris to Louis-François’ country house in Arcueil . The Reign of Terror brought hard times for the Cauchy family. Life in Arcueil was an exile (Hawking 636) with food being scarce and Augustin contracting small pox (Belhoste 3). The stress and torment of his parents must have affected the young Cauchy at least at a subconscious level. Indeed, the fact that his birth almost coincided with the Cauchy family’s fleeing from Paris did not go unnoticed in his family (Belhoste 1). Belhoste admires Cauchy’s parents calling them “a very strict and pious mother and a very open and hard-working father” (8). Cauchy remained obedient and had great respect for his parents, especially his father. He yearned to be close to his family, and where possible, took steps to ensure he stayed in Paris where he would have weekly family luncheons with his parents and siblings. After the luncheon, Cauchy and his father often studied Hebrew together (Belhoste 8). Belhoste called his family “very pious” (Belhoste 14). Indeed, it was his father’s letter which encouraged Augustin to prove one of Fermat’s Theorems that started his work which would take the next several years (Belhoste 29).

Perhaps because of his upbringing, Augustin, perhaps unwittingly, invited antipathy from quite a people of secular inclination at Cherbourg (where he had his first position as a full engineer). Augustin was a very motivated and hardworking person, who would not complain about the awful living conditions and work load of Cherbourg where he would have to get up at four in the morning every single day (Belhoste 20) and “supervise teams of men, make decisions, endure hardships of the construction site, and spend long hours outside in all kinds of weather” while learning his way about the aristocracy and social relationships (Belhoste 21).

Grabiner calls Cauchy’s achievements “outstanding” (9). He cautions that Bernhard Bolzano discovered the same ideas at around the same time. At a surface level, it seems to be a story similar to the beginning of modern calculus – when Newton and Leibniz independently invented modern calculus. Cauchy carried the same authority for being at École Polytechnique in Paris that Newton carried with the Royal Society. However, unlike Leibniz-Newton story which tore apart mathematicians across the English Channel for decades, Bolzano was virtually unheard of until after death (9). Thus, Cauchy had a very important role in transmitting the basis of analysis. Even Neils Abel, who went so far to call Cauchy a “bigoted Catholic” (Wachsmuth) had an “almost religious conversion” and changed his perspective of math on reading Cauchy’s Cours d’analyse (Grabiner 13). Thus, it may be that despite being very prolific, Cauchy’s lasting legacy lies in his inspiring a whole generation (and beyond) of mathematicians and engineers to think in a way very different from the eighteenth century.

Grabiner speculates that Cauchy's important discovery of reducing calculus to an algebra of inequalities might have “developed gradually” (77). Cauchy pioneered in understanding and applying the concept of limit to calculus – first by treating limit, continuity and convergence in Cours d'analyse (published in 1821) and then giving his theory of the derivative of derivative and integral in Calcul infinitésimal two years later (77). The concept of limit is also the basis of the definitions of continuity and convergence in Cours d'analyse. In this book, in addition to defining the sum of a convergent series, Cauchy went on to state the Cauchy criterion and prove it as a necessary condition for convergence. Although he did not prove that it was also a sufficient condition, he did go on to state it with examples. (97- 98).

According to Belhoste, Cauchy was zealous and rather innovative (64). He devised a four section plan in analysis for École Polytechnique consisting of:

1. Imaginary expressions
   
2. Calculus of finite differences
   
3. Differential and integral calculus
   
4. The application of integral calculus to geometry
   

The first three sections were meant for the first year (also known as second division) while the last section was meant for the second year (also known as the first division). Not much is known about the details of his lesson plans. We might find it interesting that section two consisted of “calculus of finite differences, integrals of first and higher orders, analogy between powers and differences, simplest notation about integration of some finite difference equations, and interpolation formula” while the third section was modeled on the second section. It broke traditional rules by drawing parallels between two calculi rather than trying to stress on the differences as it was customary then (62). His plans were struck down and there were no significant changes to the analysis course as the institute reopened on January 17, 1817 as the commission viewed analysis as “only a tool—albeit an indispensable one” and contended that École ws founded to train engineers, not mathematicians(63). Cauchy tried to incorporate his plans to his studies as much as he could, although the administration was not fond of what it called “an uncalled-for extravagance” in teaching “pure mathematics” and asked Cauchy to adhere to the official syllabus (65). His brightest students loved his teaching and one of them later wrote:

We all found that this professor was extremely energetic, good mannered, and tireless. I often heard him repeat and review, for several hours on end, whole lessons that we had not understood clearly; we would then become impressed by the elegant clarity of his analysis, an analysis dry and tedious. Indeed, M. cauchy had the genius of Euler, Lagrange, Laplace, Gauss, and Jacobi, and his love for teaching, which bordered on pure zeal, brought with it a kindness, a simplicity, and warmth of heart that he retained until the end of his life (9). (64)

The obstacles Cauchy faced in trying to teach analysis is significant in the sense that it was probably one of the most significant factors in his writing Cours d'analyse, the first part of which (titled Analyse Algébrique) was published in June 1821 which Belhoste calls “a landmark in the history of analysis”. A quick look at what the book has to offer is due here. According to Belhoste, Cauchy defines the limit and the infinitesimal (building on top of the limit concept) in the opening accounts, without wasting any time (66). Cauchy starts off with real functions in chapter one, defining a function “as a variable that can be expressed by means of one or several other variables, which he called 'independent variables'. If the function is multivalued, Cauchy denoted it f((x)).” Cauchy goes back to look at infinitesimal in chapter two and defines continuity of a rel function of one variable based on the concept of infinitesimal (68). It is worthwhile to repeat the definition here:

Let f(x) be a function of the variable x and suppose that for each value of x between two given bounds, this function constantly takes one finite value. If, from a value of x between these bounds, one attributes to the variable x an infinitely small increment α, the function itself will recieve as an increment the difference

f(x+α) — f(x),

which will depend on the same time on the new variabl and on the value of x. This being granted, the function f(x) will be a continuous function of the variable x between the two assigned bounds if, for each value of x between the bounds, the numerical value of the difference f(x+α) — f(x) decreases indefinitely with α. In other words, the function f(x) remains continuous with respect to x between the given bounds, if, between these bounds, an infinitely small increment in the variable always produces an infinitely small increment in the function itself (19). (68)

Cauchy then goes on to state the intermediate-value theorem. Belhoste notes that Cauchy 's use of the same infinitesimal, α, and the confusion between uniform continuity and simple continuity as “one of the most serious flaws in Cauchy's development of the course.” He also remarks that the ideas in Analyse Algébrique were not new. Cauchy himself learned calculus from the method of limits (68). However, the limit theory was seen as to be lacking in truthfulness, vigor, applicability by the Counseil d'Instruction which renounced it in 1811 to reinstate the former method of infinitesimals (69).

Cauchy took the theory of series in chapter six.

Works Cited

Belhoste, Bruno. Augustin-Louis Cauchy A Biography. Translated by Frank Ragland. Ann Arbor, MI: Springer-Verlag New York Inc, 1991.

Hawking, Stephen. God Created the Integers: The Mathematical Breakthroughs that Changed History. Philadelphia: Running Press Book Publishers, 2005.

Grabiner, Judith V. The origins of Cauchy’s rigorous calculus. Cambridge: The Massachusetts Institute of Technology, 1981.

Wachsmuth, Bert G et. al. "10.6. Cauchy, Augustin (1789-1857)" Interactive Real Analysis, ver. 1.9.5. Mar 26, 2009. Bert G. Wachsmuth. Mar 28, 2007