Articles & Notes
13 September 2024
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Articles & Notes
13 September 2024
Translating scientific Latin texts with artificial intelligence: the works of Euler and contemporaries
The major hindrance in the study of earlier scientific literature is the availability of Latin translations into modern languages. This is particularly true for the works of Euler who authored about 850 manuscripts and wrote a thousand letters and received back almost two thousand more. The translation of many of these manuscripts, books and letters have been published in various sources over the last two centuries, but many more have not yet appeared. Fortunately, nowadays, artificial intelligence (AI) translation can be used to circumvent the challenges of translating such substantial number of texts. To validate this tool, benchmark tests have been performed to compare the performance of two popular AI translating algorithms, namely Google Translate and ChatGPT. Additional tests were accomplished in translating an excerpt of a 1739 letter from Johann Bernoulli to Euler, where he announces that he was sending Euler the first part of his manuscript Hydraulica. Overall, the comparative results show that ChatGPT performed better that Google Translate not only in the benchmark tests but also in the translation of this letter, highlighting the superiority of ChatGPT as a translation tool, catering not only to general Latin practitioners but also proving beneficial for specialized Latin translators.Current Articles
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Translation & Commentary
19 August 2021
Euler's three-body problem
In physics and astronomy, Euler's three-body problem is to solve for the motion of a body that is acted upon by the gravitational field of two other bodies. This problem is named after Leonhard Euler (1707-1783), who discussed it in memoirs published in the 1760s. In these publications, Euler found that the parameter that controls the relative distances among three collinear bodies is given by a quintic equation. Later on, in 1772, Lagrange dealt with the same problem, and demonstrated that for any three masses with circular orbits, there are two special constant-pattern solutions, one where the three bodies remain collinear, and the other where the bodies occupy the vertices of two equilateral triangles. Because of their importance, these five points became known as Lagrange points. The quintic equation found by Euler for the relative distances among the collinear bodies was also found later by Lagrange, and because of that, Euler has also been given credit for the discovery of the three collinear Lagrange points. A practical application of the collinear points for satellite location is also presented.Articles & Notes
19 September 2022
Basel Problem: Historical perspective and further proofs from stochastic processes
In this note, we offer a historical perspective on solutions of the Basel problem. In particular, we have a closer look at some of the less famous results by Euler E41 and provide a review of a selection of the assemblage of earlier proofs. Moreover, we show how to generate further proofs using Karhunen-Lo\`{e}ve expansions of stochastic processes.Articles & Notes
19 August 2021
Euler's Miracle
This article features some genuine Eulerian magic. In 1748, Leonhard Euler considered a modification of the harmonic series in which negative signs were attached to various terms by a rule that was far from self-evident. With his accustomed flair, he determined its sum, and the result was utterly improbable. There are a few occasions in mathematics when the term “breathtaking” is not too strong. This is one of them.Articles & Notes
19 August 2021
Ed Sandifer: An Eulerian Marathoner
Ed Sandifer was the founding secretary of the Euler Society. He published a remarkable quantity of Euler scholarship at the time of Euler’s Tercentenary in 2007.Translation & Commentary
19 August 2021
Euler and the multiplication formula for the Gamma Function
We show that an apparently overlooked result of Leonhard Euler (1707-1783) from [E421] is essentially equivalent to the general multiplication for- mula for the Γ-function that was proven by Carl Friedrich Gauss (1777-1855) in [Ga28].Articles & Notes
22 August 2023