The analysis of unpublished manuscripts and of the published textbook on mechanics written between about 1730 and 1744 by Euler reveals the invention, application, and establishment of important physical and mathematical principles and procedures. They became central ingredients of an “embryonic” lunar theory that he developed in 1744/1745. The increasing use of equations of motion, although still parametrized by length, became a standard procedure. The principle of the transference of forces was established to set up such equations. Trigonometric series expansions (...) together with the method of undetermined coefficients were introduced to solve these equations approximatively. These insights constitute the milestones achieved in this phase of research, which thus may be characterized as “developing the methods”. The documents reveal the problems Euler was confronted with when setting up the equations of motion. They show why and where he was forced to introduce trigonometric functions and their series expansions into lunar theory. Furthermore, they prove Euler’s early recognition and formulation of the variability of the orbital elements by differential equations, which he previously anticipated with the concept of the osculating ellipse. One may conclude that by 1744 almost all components needed for a technically mature and successful lunar theory were available to Euler. (shrink)

ArgumentJohannes Kepler published hisAstronomia novain 1609, based upon a huge amount of computations. The aim of this paper is to show that Kepler's new astronomy was grounded on methods from numerical analysis. In his research he applied and improved methods that required iterative calculations, and he developed precompiled mathematical tables to solve the problems, including a transcendental equation. Kepler was aware of the shortcomings of his novel methods, and called for a new Apollonius to offer a formal mathematical deduction. He (...) was also in great need of computational power, and his friend and colleague, Wilhelm Schickard, constructed the first prototype of a true mechanical calculator, although it never came into regular use. The article concludes that Kepler's new astronomy was clearly backed up by numerical methods and embedded concepts and challenges of great importance for the future development of numerical analysis. (shrink)

By taking the work and life of the historian of mathematics Heinrich Wieleitner as an example, this study aims to highlight the many interrelations between the historiography of mathematics, mathematics education, and science communication in mathematics.By integrating aspects of the history of media, this case study also explores mathematical public relations work in the 20th century and draws attention to the important persons, institutions and contents. The focus is on the Weimar period, in which the self-understanding of mathematics was challenged (...) in different ways by far-reaching cultural debates. The article demonstrates that as a consequence of a changing media landscape, Weimar culture turned out to be a suitable environment for the successful self-presentation of mathematics. (shrink)

This paper critically assesses the claim by Gavin Menzies that Regiomontanus knew about the Chinese Remainder Theorem (CRT) through the Shù shū Jiǔ zhāng (SSJZ) written in 1247. Menzies uses this among many others arguments for his controversial theory that a large fleet of Chinese vessels visited Italy in the first half of the 15th century. We first refute that Regiomontanus used the method from the SSJZ. CRT problems appear in earlier European arithmetic and can be solved by the method (...) of the Sun Zi, as did Fibonacci. Secondly, we pro-vide evidence that remainder problems were treated within the European abbaco tradition independently of the CRT method. Finally, we discuss the role of recreational mathematics for the oral dissemination of sub-scientific knowledge. (shrink)

Focalisé sur un problème posé par Bernard Frenicle de Bessy vers 1639, sa solution et les réponses de ses correspondants, cet article s'attache à décrire plusieurs registres enchevêtrés de l'expérience du mathématicien: expérimentation sur les nombres empruntée en partie aux sciences de la nature, injonctions d'une pratique collective cimentée par les problèmes et leurs constructions explicites, entraînement personnel de l'attention et du savoir-faire s'articulent ainsi dans les efforts de Frenicle pour contester la suprématie de l'analyse algébrique et dans les modes (...) de conviction qui fondent son travail. Ce récit d'expérience permet en retour d'apprécier les problèmes que posèrent à Frenicle la transmission de ses résultats et, à ses historiens, la restitution de son activité. (shrink)

The discovery of incommensurability by the Pythagoreans is usually ascribed to geometric or arithmetic questions, but already Tannery stressed the hypothesis that it had a music-theoretical origin. In this paper, I try to show that such hypothesis is correct, and, in addition, I try to understand why it was almost completely ignored so far.

In this article, we give a summary of Leonhard Euler’s work on the pentagonal number theorem. First we discuss related work of earlier authors and Euler himself. We then review Euler’s correspondence, papers and notebook entries about the pentagonal number theorem and its applications to divisor sums and integer partitions. In particular, we work out the details of an unpublished proof of the pentagonal number theorem from Euler’s notebooks. As we follow Euler’s discovery and proofs of the pentagonal number theorem, (...) we pay attention to Euler’s ideas about when we can consider a mathematical statement to be true. Finally, we discuss related results in the theory of analytic functions. (shrink)

Gottfried Wilhelm Leibniz is the self-proclaimed inventor of the binary system and is considered as such by most historians of mathematics and/or mathematicians. Really though, we owe the groundwork of today’s computing not to Leibniz but to the Englishman Thomas Harriot and the Spaniard Juan Caramuel de Lobkowitz, whom Leibniz plagiarized. This plagiarism has been identified on the basis of several facts: Caramuel’s work on the binary system is earlier than Leibniz’s, Leibniz was acquainted—both directly and indirectly—with Caramuel’s work and (...) Leibniz had a natural tendency to plagiarize scientific works. (shrink)