Translated and annotated by
Ian Bruce


This was the last book published by Euler in his lifetime in 1783, and consists of a number of papers presented to the St. Petersburg Academy of Sciences. A general summary of this work can be found in Ronald S. Calinger's Leonhard Euler, pp. 529-30. It should be observed that such works were not written to educate aspiring mathematicians, as were his calculus and analytical books, but rather took the form of what we would now call research papers.

            Click here for the 1st  Chapter : On series in which the products from two contiguous terms constitute a given progression.

This is rather a long paper, in which two methods are introduced for finding the general term of a special kind of series of the form a, ab, bc, cd, de, etc. , where the series or rather sequence has the terms designated by A, B, C, D, etc., where the first term a may not be known initially. This gives rise to solutions in terms of infinite products that may be solved either by integrals [E223] , or by continued fractions , originally investigated in E122. Most of the work consists of setting up products of quadratic equations of various kinds which can be recast as continued fractions, using the comparison of coefficients to show how this is done in these cases.

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