__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.

*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).

- Imaginary expressions

- Calculus of finite differences

- Differential and integral calculus

- The application of integral calculus to geometry

*Cours d'analyse*, the first part of which (titled

*Analyse Alg*

*éb*

*rique*) 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:

*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)*,

*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)

*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 Alg*

*éb*

*rique*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).

## Works Cited

__Augustin-Louis Cauchy A Biography__. Translated by Frank Ragland. Ann Arbor, MI: Springer-Verlag New York Inc, 1991.

__God Created the Integers: The Mathematical Breakthroughs that Changed History__. Philadelphia: Running Press Book Publishers, 2005.

__The origins of Cauchy’s rigorous calculus__. Cambridge: The Massachusetts Institute of Technology, 1981.