the infinitely small quantities he used. Taylor's theorem He was the first to prove Taylor's theorem rigorously, establishing his well-known form of the remainder. He wrote a textbook (see the illustration) for his students at the École Polytechnique in which he developed the basic theorems of mathematical analysis as rigorously as possible. In this book he gave the necessary and sufficient condition for the existence of a limit in the form that is still taught. Also Cauchy's well-known test for absolute convergence stems from this book: Cauchy condensation test. In 1829 he defined for the first time a complex function of a complex variable in another textbook. In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods; thus one of his theorems was exposed to a "counter-example" by Abel, later fixed by the introduction of the notion of uniform continuity. Argument principle, stability In a paper published in 1855, two years before Cauchy's death, he discussed some theorems, one of which is similar to the "Principle of the argument" in many modern textbooks on complex analysis. In modern control theory textbooks, the Cauchy argument principle is quite frequently used to derive the Nyquist stability criterion, which can be used to predict the stability of negative feedback amplifier and negative feedback control systems. Thus Cauchy's work has a strong impact on both pure mathematics and practical engineering. Published works Cauchy was very productive, in number of papers second only to Leonhard Euler. It took almost a century to collect all his writings into 27 large volumes: (Paris : Gauthier-Villars et fils, 1882–1974) His greatest contributions to mathematical science are enveloped in the rigorous methods which he introduced; these are mainly embodied in his three great treatises: Le Calcul infinitésimal (1823) Leçons sur les applications de calcul infinitésimal; La géométrie (1826–1828) His other works include: Exercices d'analyse et de physique mathematique (Volume 1) Exercices d'analyse et de physique mathematique (Volume 2) Exercices d'analyse et de physique mathematique (Volume 3) Exercices d'analyse et de physique mathematique (Volume 4) (Paris: Bachelier, 1840–1847) Analyse algèbrique (Imprimerie Royale, 1821) Nouveaux exercices de mathématiques (Paris : Gauthier-Villars, 1895) Courses of mechanics (for the École Polytechnique) Higher algebra (for the ) Mathematical physics (for the Collège de France). Mémoire sur l'emploi des equations symboliques dans le calcul infinitésimal et dans le calcul aux différences finis CR Ac ad. Sci. Paris, t. XVII, 449–458 (1843)