Augustin Cauchy, Many mathematicians of the day believe that your largest accomplishments were in the field of complex functions. You are known for your book that you mentioned in your second letter called Cours d'Analyse, where you stress the importance of "rigor" in analysis. It is in this book where you defined Cauchy's Criterion for Convergence. Since your book is written in French, I will recite a translated version. It states, "[it] is that a necessary and sufficient condition for convergence of a sequence {x n } is that it be a Cauchy sequence, where a Cauchy sequence is defined to be for each e > 0, there is a positive integer N such that | x n - x m | < e for all m > N and all n > N." (Cauchy). It is extremely difficult to find what value N the series converges to, and you realized that it is not necessary to find the best or most exact value for N. You need only find any value that works. You were not the first mathematician...
Dear Augustin Cauchy, Your mathematical accomplishments have had an enormous impact in the fields of math and science. More concepts and theorems have been named after you than any other mathematician alone, and to this day, you have written the second largest number of papers of any mathematician, only second to Leonard Euler. Some areas you have had influence in are complex functions, optics, elasticity, group theory, mathematical physics and astronomy, hydrodynamics, and differential equations. Even though you are not most well-known for your work in algebra, I first heard of you in an Abstract Algebra course. In the field of algebra, your early work is the basis of group theory, and you have contributed to finding the inverse of a matrix, creating theorems on determinants formed by subdeterminants, and you proved a generalization of one of Ruffini's theorems. Even though it took some time after you for abstract group theory to form and for it to move alway from the term subs...