THE INTEGRATION OF RATIONAL FUNCTIONS

(LEIBNITZ, G.W.) G.G.L. [= GOTTFREDUS GUGLIELMUS LEIBNITZ el. LEIBNIZ].

Specimen Novum Analyseos pro Scientia Infiniti, circa Summas & Quadraturas. [In: Acta eruditorum anno MDCCII publicata].

Lipsiae [Leipzig], Gross, Frisch and Groschuf, 1702. 4to. Entire volume present. Bound in a nice contemporary full vellum with handwritten year and an old library-label to spine. A bit of soiling. Internally evenly brownspotted throughout, occasionally with heavier brownspotting (not the Leibnitz-article). Discreet stamp ti title-page and to first leaf of text. Pp. 210-219 + one folded plate ("Regulae universales pro fractionibus numeratoris indeterminati..."). [Entire volume: (2), 566 pp. + 9 folded engraved plates.

First edition of Leibnitz' important work on the integration of rational fractions, the "New specimen of the Analysis for the Science of the infinite about Sums and Quadratures", in which Leibnitz investigates the factorization of real polynomials and thus comes close to the "fundamental theorem of algebra" (the theorem that states that every non-constant single-variable polynomial with complex coefficients has at least one complex root, and that field of complex numbers is algebraically closed), which was not attempted proven until 1746 (by d'Alembert, whose proof was incomplete), and which was not actually proven before the 19th century (by Argand, and more fully by Gauss).

The present work constitutes the first printing of Leibniz' new examples of calculating with infinite numbers, a most fundamental work for the further development within algebraic equations and integral formulae.

Apart from Leibnitz' "Scientia infiniti", the present volume also includes an essay by Halley on Hooke's barometer (pp. 180-83, one engr. Plate depicting a barometer).

Order-nr.: 41371