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Leonhard Euler was the leading eighteenth-century critic of Isaac Newton's projectile theory of light. Euler's main criticisms of Newton's views are surveyed, and also his alternative account according to which light is a wave motion propagated through the aether. Important changes are identified as having occurred between 1744 and 1746 in Euler's thinking about the way in which a wave such as he supposed light to be is propagated through a medium. Paradoxically, in view of (...) class='Hi'>Euler's overtly anti-Newtonian stand, these amount to Euler abandoning his early, Malebranchian notions about the physical basis of wave propagation, in favour of the ideas set out by Newton in Book II of his Principia. (shrink)

Leonhard Euler (1707–1783) published two lunar theories in 1753 and 1772. He also published lunar tables in 1745, 1746, and—anonymously—in 1750. There are notebook records, unpublished manuscripts, and manuscript fragments by Euler reflecting the development of his lunar theories between about 1725 until about 1752. These documents might be used to reconstruct Euler’s theory on which he based his calculations of those lunar tables and to analyze the development of his lunar theories within this time span. (...) The results of this analysis will be published here in three parts representing three stages of Euler’s research on this topic: First approaches (about 1725–1730), developing the methods (about 1730–1744), and the breakthrough (about 1744–1752). In this part, I analyze Euler’s manuscripts and, predominantly, Euler’s records of his first two notebooks written between 1725 and 1730. I found that his early theoretical approach is coined by his development of analytical (rational) mechanics of punctiform bodies moved by central forces. He tried to describe the Moon’s motion in terms of two simultaneously acting centripetal forces, Huygens’ centrifugal theorem, and associated osculating radii. (shrink)

Euler ’s wave theory of light developed from a mere description of this notion based on an analogy between sound and light to a more and more mathematical elaboration on that notion. He was very successful in predicting the shape of achromatic lenses based on a new dispersion law that we now know is wrong. Most of his mathematical arguments were, however, guesswork without any solid physical reasoning. Guesswork is not always a bad thing in physics if it leads (...) to new experiments or makes the theory coherent with other theories. And Euler tried to find such experiments. He saw the construction of achromatic lenses, the explanation of colors of thin plates and of the opaque bodies as proof of his theory. When it came to the fundamental issues, the correctness of his dispersion law and the prediction of frequencies of light he was not at all successful. His wave theory degenerated, and it was not until Augustin Fresnel introduced transverse waves and an elaborate notion of interference that the wave theory again progressed. (shrink)

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)

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)

This essay explores ethical foundations for meeting the digital challenge via a case study of the work, life, and virtues of the greatest mathematician and natural scientist of the eighteenth century, Leonhard Euler. By biography and history one can learn of the gifts of human strength, practices, good will, dependence on others, and friendships which made possible Euler’s own astonishing corpus of work and that of many other scientists, mathematicians, engineers, and technologists. Digital technology results from a (...) combination of science (demonstrable knowledge and method), technology, engineering, and art (forms and artifacts of making and expressing), and mathematics (abstract numerical, algebraic, geometrical, formal, and digital concepts, rules, representations, and logics). Joint reflection on the biographical, historical, and natural sources of mathematics and the digital is essential for any humanization or ascesis in response to the perils and promises of digital technology for human thriving. As ethics enable and embody an ethos, so technologies are means and manifestations of a telos. Thus, thought and action for thriving through the digital needs to contemplate and conciliate the ends of humans and of the digital. (shrink)

Die Forschung zu und über Leonhard Euler verweist oft darauf, dass das Entstehen und Etablieren neuer Zeichenpraktiken zum dominierenden Legitimationsmuster der übersichtlichen Mathematik in der Zeit der Aufklärung wird. Bei genauerem Hinsehen fällt jedoch auf, dass die konkreten Formen der Erscheinung und Durchsetzung dieser Praktiken mit ihren symbolischen Dimensionen über Jahrzehnte kaum ins Zentrum der Aufmerksamkeit gerückt wurden. Gerade diesen Formen der Wissensentfaltung und -vermittlung versuchen die hier versammelten Aufsätze eine zentrale Position zukommen zu lassen. Denn im Falle (...) Eulers beschränkt sich die Problematik solcher Erscheinungen nicht auf das mathematische Abstrahieren der zu lösenden Aufgaben, sind diese doch mit den graphischen Abstrahierungen vernetzt, deren Erscheinungskriterien untersucht werden. (shrink)

Leonhard Euler (1707–1783) treated in his study „De motu vibratorio tympanorum [6], published in 1766, the theory of the vibrating rectangular and circular membrane mathematically in such a comprehensive way that little more had to be added, considering today's standards. However, he omitted to interprete his results physically. Therefore his uncomparable work found little recognition, especially in the field of musical accoustics.

Investigamos a lógica de Leonhard Euler com ênfase no papel dos “diagramas de Euler”. Concluímos que os referidos diagramas constituem um instrumento intuitivo, embora não sistemático, para determinar validade na silogística tradicional, isto é, a silogística munida de implicações conversacionais. Nisto, contrastam-se com os diagramas de Venn que constituem um instrumento sistemático, porém menos intuitivo, para determinar validade numa silogística mais voltada para os fundamentos da matemática moderna.: We investigate the logic of Leonhard Euler, giving (...) emphasis to the role of “Euler diagrams”. We conclude that these diagrams are an intuitive, non-systematic instrument for determining validity in the traditional syllogistic, that is, the syllogistic furnished with conversational implications. In this regard, they contrast with Venn diagrams which are a systematic, less intuitive instrument for determining validity in a syllogistic more appropriate for the foundation of modern mathematics. Keywords: History of Logic. Leonhard Euler. Euler Diagrams. Venn Diagrams. (shrink)

ABSTRACT We are interested in the case of Friedrich Christoph Mayer, who in the 1720s, while at the Imperial Academy of Sciences in St. Petersburg, developed a system of the aurora borealis, as well as a mathematical method for calculating the height of the aurora from the geometrical characteristics of the auroral arc. Mayer, encountering a major contradiction in his system which placed the aurora at the height of the clouds, whereas his mathematical method led to an altitude a hundred (...) times higher, never applied his method to concrete cases to deduce the height of the aurora, and quickly lost interest in their detailed description, a task that was nevertheless assigned to him at the St. Petersburg Observatory. Jean-Jacques Dortous de Mairan suggests that Mayer’s abandonment was due to his lack of confidence in observations. We set Mayer’s case against that of Leonhard Euler who, working with Mayer and being aware of the great height of the aurora, later developed a system of the aurora borealis that was compatible with the observational fact. We put forward possible hypotheses to explain Mayer’s disinterest in observing the aurora and in the mathematical method he himself had developed. (shrink)