Euler's series for sine and cosine: an interpretation in nonstandard analysis (Q6639165)

This article examines how the sine and cosine series in Chapter VIII of Euler's Introductio in analysin infinitorum can be understood through nonstandard analysis, a modern area of mathematics. It explains how Euler's ideas, including infinitesimals and infinite quantities, can be clearly interpreted using hyperreal numbers.\N\NThe paper also places Euler's work in a rich historical context, linking it to Ptolemy's chord theory, which treated sine as a geometric property, and Newton's pioneering work on infinite series. This approach highlights the innovative nature of Euler's contributions, especially his derivation of series for \(\sin v\), \( \cos v\) and \(e^{iv}= \cos v +i \sin v\). These results bridged classical trigonometry and emerging analytical methods. By situating Euler's work within this historical framework, the authors provide a deeper understanding of the evolution of trigonometric and analytical ideas.\N\NBeyond its historical focus, the article reinterprets Euler's work as a key connection between classical and modern mathematics. Using tools from nonstandard analysis, the authors uncover implicit lemmas and ordered field principles that support Euler's arguments. They also revisit Euler's assumptions, such as as \(\sin z \approx z\) and \(\cos z \approx 1\) for infinitesimal \(z\), validating them within the hyperreal number framework. These insights reveal the consistency and lasting importance of Euler's methods in contemporary mathematics.\N\NFor the entire collection see [Zbl 1516.01006].