Euler and combinatorial calculus

Abstract

Leonhard Euler is one of the creators of methods of calculus. Together with rigorous mathematical (deductive) proofs, Euler extensively used empirical methods (induction) and “likely” arguments. Euler developed methods of calculus largely with the purpose of solving particular mathematical and applied problems. Euler made a large contribution to various domains of mathematics, including combinatorics. The combinatorial results of Leonhard Euler, which are somewhat aside of the main scope of his work, have made it possible to better evaluate both the versatility of his talent and his inclination to use analytical methods. The term “combinatorial mathematics” is understood differently by different authors. Since the ancient Greeks, the principal mathematical abstraction has been infinity (which appears in the infinite series of positive integers, the unbounded divisibility of a line interval, the unboundedness of space, etc.). Somewhat exaggerating, one might say that, unlike almost all other domains of mathematics, which are related, in some way, to studying properties of infinite objects (unbounded or consisting of infinitely many elements), combinatorial calculus studies problems related to properties of objects finite in all senses. In what follows, we give a brief review of Euler’s studies in the directions which are now called graph theory, the theory of Latin squares, and enumerative combinatorics. We discuss methods used by Euler and some more recent results in these directions related to Euler’s work.