ANTICIPATING CANTOR'S "DIAGONAL ARGUMENT".

BOIS-REYMOND, PAUL DAVID GUSTAV DU.

Über asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichunge.

Leipzig, B. G. Teubner, 1875. 8vo. Bound in recent full black cloth with gilt lettering to spine. In "Mathematische Annalen", Volume 8., 1875. Entire volume offered. Library label pasted on to pasted down front free end-paper. Small library stamp to lower part of title title page and verso of title page. Title page missing a small piece of paper to the right margin, not affecting text. Very fine and clean. Pp. 363-414. [Entire volume: IV, 576 pp.].

First printing of Paul du Bois-Reymond's important paper in which he anticipate Cantor's famous "diagonal argument". Although Cantor proved that the real numbers are uncountable one year earlier he did not find the much clearer diagonal argument until some years later.

"In 1875, he described dense sets under the title 'pantachisch,' from the Greek for 'everywhere.' He later claimed, against Georg Cantor, priority in their discovery. In his textbook, Hobson awarded the laurels to du BoisReymond. Although Cantor presented his first diagonal proof to the public in On an elementary question of set theory [Cantor 1891], Paul du Bois-Reymond had been there well before him, having published a plainly diagonal argument in an 1875 article on approximation by in?nitesimals. [P. du Bois-Reymond 1875] Arguably his greatest invention was the Infinitärcalcül or infinitary calculus, an original, nonCantorean account of infinite and infinitesimal sizes as first-class
entities that represent not the extents of collections but the rates of growth of real-valued functions." (McCarty, David Hilbert and Paul du Bois-Reymond: Limits and Ideals).

Order-nr.: 47136