In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The (...) article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids. (shrink)
The article deals with the Arabic sources of Chr. Clavius in Rome and the six different ways they were used by him in mathematics and astronomy. It inquires especially into his attitude towards al-Farghani, Thabit ibn Qurra, al-Bi[tdotu]ruji, Ibn Rushd, Mu[hdotu]ammad al-Baghdadi, Pseudo-Ibn al-Haytham, Jabir ibn Afla[hdotu], and Pseudo-al-[Tuotu]usi.
Johannes Kepler belonged to a long tradition of inquiring into nature with reference to God. This applies to Ptolemy, N. Copernicus, Chr. Clavius. Kepler's „new kind of poem” is analyzed in five sections which are based on Keplerian key words: Innovation, Hypothesis, Cause, Soul, Picture. Kepler consciously adhered to new questions, new answers, new methods. He relied on a new notion of hypothesis. His celestial dynamics included a celestial psychology whereby he used a visual conception of astronomy.
The object of this article is the study of possibilities and tendencies arising from the use of symbolic language including signs, characters, and symbols in mathematics. Five aspects are discussed: compactness and simultaneity, problem-solving and generalizations, heuristics and progress, mechanisms and calculations, formalism. This is done primarily by looking at three disciplines, which at the same time are of fundamental importance to theoretical physics: classical algebra, calculus, and vector analysis.Mathematical achievements and statements by eminent mathematicians from antiquity to the present (...) show the importance of adequate symbolism for the development of mathematics. The corresponding philosophical discussions are controversial. (shrink)
This article deals with six aspects of analogical thinking in mathematics: 1. Platonism and continuity principle or the “geometric voices of analogy” , 2. analogies and the surpassing of limits, 3. analogies and rule stretching, 4. analogies and concept stretching, 5. language and the art of inventing, 6. translation, or constructions instead of discovery. It takes especially into account the works of Kepler, Wallis, Leibniz, Euler, and Laplace who all underlined the importance of analogy in finding out new mathematical truth. (...) But the meaning of analogy varies with the different authors. Isomorphic structures are interpreted as an outcome of analogical thinking. (shrink)
Ähnlich 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.
In 1661, Kaspar Schott published his comprehensive textbook Cursus mathematicus in Würzburg for the first time, his Encyclopedia of all mathematical sciences. It was so successful that it was published again in 1674 and 1677. In its 28 books, Schott gave an introduction for beginners in 22 mathematical disciplines by means of 533 figures and numerous tables. He wanted to avoid the shortness and the unintelligibility of his predecessors Alsted and Hérigone. He cited or recommended far more than hundred authors, (...) among them Protestants like Michael Stifel and Johannes Kepler, but also Catholics like Nicolaus Copernicus. The paper gives a survey of this work and explains especially interesting aspects: The dedication to the German emperor Leopold I., Athanasius Kircher’s letter of recommendation as well as Schott’s classification of sciences, explanations regarding geometry, astronomy, and algebra. (shrink)
Der Aufsatz gibt einen Überblick über Eulers zwölf bisher unveröffentlichte mathematische Notizbücher, die im Archiv der Akademie der Wissenschaften der UdSSR, Leningrad, aufbewahrt werden. Sie bestehen aus rund 2300 Blatt und behandeln in äußerst unsystematischer Weise alle mathematischen Themen, Naturwissenschaften, und einige andere Fragen. Im Aufsatz werden die erörterten Themen in systematischer Reihenfolge vorgestellt.The article gives an account of Euler's twelve mathematical notebooks, unpublished to date, that are in the archives of the Soviet Academy of Sciences in Leningrad. The folios, (...) some 2300 in number, deal haphazardly with mathematical and scientific topics and other matters. In this paper, the notes are placed in a subject index and discussed. (shrink)
Otto von Guericke's scientific method was based on reason and experimental science. His cosmology was embedded in theology and can be interpreted as a refutation of Descartes's worldview. He used Nicolaus Cusanus's theory of quantities in order to characterize the space. The notion of space has to be distinguished from that of world or heaven. Forces play a crucial role in this respect described by Kircher in his ‘Celestial journey’. Guericke read this work very diligently. In spite of some obvious (...) similarities between Guericke's and Newton's scientific aims and methods there are crucial differences between the scientific convictions and results of these scholars. (shrink)