Eberhard Knobloch

De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis

Prop. 1. est lemma, cujus ope triangula ex puncto fixo A incipientia transmutantur in rectangula MNF rectae AMN per punctum fixum transeunti normaliter applicata. fig. 1. 2.

Leonhard Eulers Mathematische Notizbücher

Zusammenfassung Der Aufsatz gibt einen Uberblick uber Eulers zwolf bisher unveroffentlichte mathematische Notizbucher, die im Archiv der Akademie der Wissenschaften der UdSSR, Leningrad, aufbewahrt werden. Sie bestehen aus rund 2300 Blatt und behandeln in auserst unsystematischer Weise alle mathematischen Themen, Naturwissenschaften, und einige andere Fragen. Im Aufsatz werden die erorterten Themen in systematischer Reihenfolge vorgestellt.

Analogy and the Growth of Mathematical Knowledge

High esteem for the mathematical discoveries and rigorous argumentation of the ancient mathematicians like Euclid or Archimedes has always been accompanied by astonished speculation about how after all they had found their results, which were subsequently demonstrated in such an exemplary way that they became paradigms of rigorous argumentation. Thus we find that Kepler as well as Leibniz appealed to Archimedes in the context of justification. What is more, Leibniz justified his differential calculus (LMG 5, 350) by saying that the difference from the style of Archimedes consists only in the expressions (expression), which in his method are more direct and more appropriate to the art of inventing (art d’inventer). His differential calculus is only a new kind of notation, novum notationis genus (Leibniz 1714, 404).

THE INFINITE IN LEIBNIZ'S MATHEMATICS­ THE HISTORIOGRAPHICAL METHOD OF COMPREHENSION IN CONTEXT

G. W. Leibniz published only a very small part of his papers and manuscripts. Up until now we know only 15% of his mathematical writings; their publication will probably fill 35 volumes of the Academy edition of his works and letters. The first volume, edited by my collaborator W. S. Contro and myself, appeared three years ago (Leibniz 1990), after many years of rational printing planning.

das Wesen der reinen Mathematik verherrlichen

Zusammenfassung. C.G.J. Jacobi gehört zu den prägenden Gestaltern der Mathematik in der ersten Hälfte des 19. Jahrhunderts. Dies gilt für seine Forschungs- und Lehrtätigkeit, aber auch für sein Mathematikverständnis allgemein. Mit seiner Konzeption der Mathematik als einer autonomen, reinen, d.h. erfahrungs- und anwendungsunabhängigen Mathematik grenzt er sich insbesondere explizit gegen die zeitgenössische französische Tradition ab. Im Kontext dieser Wissenschaftsauffassung versucht er, Antworten auf die Fragen nach dem Grund des Fortschritts der Mathematik und ihrer Anwendbarkeit zur Beschreibung der Realität zu formulieren. Im vorliegenden Beitrag werden Jacobis diesbezügliche Anschauungen und ihre Veränderungen dargestellt und vor dem Hintergrund der zeitgenössischen Mathematik und Philosophie analysiert. Im Mittelpunkt steht dabei das ausführlichste von Jacobi erhaltene Dokument zum Themenkomplex: eine lateinische Rede, die er zum Eintritt in die Königsberger philosophische Fakultät im Jahre 1832 hielt. Diese Rede wird hier erstmals in deutscher Übersetzung wiedergegeben und ausführlich kommentiert.

Von riemann zu lebesgue—Zur entwicklung der integrationstheorie

Abstract This paper analyzes certain phases of the development of modern integration theory by emphasizing the motivation of mathematicians for their work and their own estimations of their proceedings. The following seven points are discussed: 1. Introductory sketch of the history to be covered; 2. Cauchy and Dirichlet paving the way for Riemann; 3. Riemann; 4. Darboux and Hankel continuing the work of Riemann; 5. Evaluation of Hankel and Riemann; 6. Borels measure theory; 7. Lebesgues idea of the integral.

Symbolik und Formalismus im mathematischen Denken des 19. und beginnenden 20. Jahrhunderts

In his famous Formulario mathematico the Italian mathematician Guiseppe Peano (1858–1932) wrote “Every advance in mathematics corresponds to the introduction of ideographic signs or symbols.” Alfred North Whitehead and Bertrand Russell, using the symbolism invented by Peano in their own Principia Mathematica, extended the applicability of symbolic logic and argued that deductive thought could be extended to areas beyond mathematics. Similarly, David Hilbert and Wihelm Ackermann, in the introduction to their Foundations of Theoretical Logic, appealed to Leibniz to indicate what could be achieved in mathematics through a formalized logic. Clearly, symbolism leads to the advance of techniques and methods of mathematical proofs. Hilberts proof theory actually can be traced back to Leibniz. This paper analyzes the roots of Logicism and Formalism, especially in the abstract algebra and vector analysis of the 19th and early 20th centuries. The paper is divided into six parts: (1) heuristics and principles of order, (2) symbolic algebra and formal mathematics, (3) Hamilton and the theory of quaternions, (4) operational calculus, (5) algorithmization, and (6) universalization.

The mathematical studies of G.W. Leibniz on combinatorics

Abstract Leibniz considered the “ars combinatoria” as a science of fundamental significance, much more extensive than the combinatorics of today. His only publications in the field were his youthful Dissertatio de Arte Combinatoria of 1666 and a short article on probability, but he left an extensive (hitherto unpublished and unstudied) Nachlass dealing with five related topics: the basic operations of combinatorics, symmetric functions in connection with theory of equations, partitions (additive theory of numbers), determinants, and theory of probability and related fields. This paper concentrates on the first and third topics as they appear in published sources and the Nachlass. It shows that Leibniz was in possession of many results not published by other mathematicians until many decades later. These include a recursion formula for partitions of n into k parts (first published by Euler in 1751), the Stirling numbers of the second kind (first published in 1730), and several special cases of the general formula for partitions that was published only in 1840 by Stern.

Die Berliner Gewerbeakademie und ihre Mathematiker

This article is a study of the development of the Berliner Gewerbeakademie, of the mathematical curricula at this institution, and in particular, of E. B. Christoffel’s appointment and his occupations at the Gewerbeakademie. Chr. P. W. Beuth, the reknown reformer and organizer of Prussian polytechnical education, founded the “Technisches Institut” in 1821. This institution changed names twice, and in 1879 (then called Gewerbeakademie) it merged with the Bauakademie to constitute the “Technische Hochschule Charlottenburg”.

Erkundung und Erforschung: Alexander von Humboldts Amerikareise

ZusammenfassungÄhnlich wie Adalbert Stifters Erzähler im Roman „Nachsommer” verband A. v. Humboldt auf seiner Amerikareise Erkundung und Erforschung, Reiselust und Erkenntnisstreben. Humboldt hat sein doppeltes Ziel klar benannt: Bekanntmachung der besuchten Länder, Sammeln von Tatsachen zur Erweiterung der physikalischen Geographie. Der Aufsatz ist in fünf Abschnitte gegliedert: Anliegen, Route, Methoden, Ergebnisse, Auswertung.AbstractIn a similar way as Adalbert Stifter’s narrator in the novel “Late summer” A. v. Humboldt combined exploration with research, fondness for travelling with striving for findings during his travel through South America. Humboldt clearly indicated his double aim: to report on the visited countries, to collect facts in order to improve physical geography. The treatise consists of five sections: object, route, methods, results, evaluation.ResuméHumboldt de même que le narrateur dans le roman “Fin de l’été” d’Albert Stifter combinait l’exploration avec la recherche, le goût des voyages avec les efforts pour des découvertes. Humboldt indiqua clairement sa double fin: faire connaître les pays visités, accumuler des faits afin de développer la géographie physique. Le traité consiste en cinq sections: l’objet, la route, les méthodes, les résultats, l’évaluation.