Twierdzenie Bézouta o przecięciu krzywych algebraicznych w pracach Eulera

Abstract

In the paper an early history of the Bézout theorem on algebraic curves and effective methods in elimination theory is presented. The hypothesis, stated in 1665 by Newton, on the ”intersection number” of algebraic curves is given. Effective methods on eliminations of one variable in the system of algebraic variables come from Euler’s papers: Demonstration sur le nombre des points, ou deux lignes des ordres quelconques peuvent se couper (Euler, 1750), Nouvelle methode d’eliminer les quantites inconnues des equations (Euler, 1766) and the chapter De intersectiones curvarum from monography Introductio in analysin infinitorum (Euler, 1748). Finally, Bézout’s result from the paper Reserchers sur le degré des équations résultantes... (Bezout, 1765) is given.