How did Augustin-Louis Cauchy make calculus more rigorous?
This suggests that the same constant (Ï) appears in the formula for the circumference, 2Ïr, and in the formula for the area, Ïr2. As the number of pieces increases (from left to right), the ârectangleâ converges on a Ïr by r rectangle with area Ïr2âthe same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
Augustin-Louis Cauchy revolutionized calculus by replacing intuitive or dynamic conceptsâwhich often relied on vague notions of motionâwith a rigorous foundation based on the concepts of continuity and limit.12
During the 1820s, Cauchy rejected the algebraic approach of Joseph-Louis Lagrange, which incorrectly assumed that every function could be expressed as a power series. Instead, Cauchy defined a function as continuous at a point if small changes in input result in arbitrarily small changes in output. Using this definition, he established a precise framework for differentiation and integration:1
- Differentiation: He defined the derivative as the limit of the quotient as tends toward , without referencing motion.1
- Integration: He defined the integral as the limit of the sum of rectangular areas approximating the region under a graph, independent of differential calculus.1
By proving the reciprocal relationship between these two processes, Cauchy provided the first rigorous foundation for calculus, forming the core of modern mathematical analysis.12