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A modern look at a neglected summation formula by Euler

In Euler derives the approximate summation formula This primitive result does not appear to be well known to modern readers, and we will try to show by examples that it deserves more attention. After translating this paper into English and reflecting on its contents, we thought it desirable to present Euler’s work in a form convenient for today’s readers. Because of inadequate notations for functions, sums, and definite integrals, the modern reader is likely to find Euler’s writing difficult, even in translation. Also, in Euler’s time, mathematicians were familiar with properties of the conic sections that have largely been ignored today.