LIOUVILLE, JOSEPH. - THE EXISTANCE OF TRANSCENDENTAL NUMBERS PROVED.

Remarques... à des classes très-étendues de quantités dont la valeur n'est ni rationelle ni méme réductible à des irratonnelles algébriques... à un passage du livre des principes ou Newton... (Séance du Lundi 13 mai 1844). (+) Nouvelle démonstration d'un théorème sur les irrationnelles algébriques, inséré dans le Compte rendu de la derniere séance. (Séance du Lundi 20 mai 1844). 2 papers.

(Paris, Bachelier), 1844 4to. No wrappers. In: "Comptes Rendus Hebdomadaires des Séances de L'Academie des Sciences", Tome 18, No 20 a. 21. Pp. (883-) 898 a. pp. (899-) 960. (Entire issues offered). Liouville's papers: pp. 883-885 a. 910-911.

First printing of the papers in which Liouville proved the existence of transcendental numbers.

"J. Liouville..... invented a method (1844) for constructing any one of an extensive class of transcendental numbers. His numbers were the first to be proved transcendental; Hermite's proof of the transcendence of e ... followed in 1873; F. Lindemnn's for pi in 1882."(Bell "Development of Mathematics", p. 275).

Liouville showed that not only are these numbers irrational, they are always transcendental. Thus he was the first to prove the existence of transcendental numbers and also that there were an infinite number of them. Liouville suggested that e was transcendental, and provided the first example of a provably transcendental number, now known as Liouville’s constant.

Parkinson "Breakthroughs", 1844 M.

Order-nr.: 49077