1. During the present year there has appeared at intervals, in the Comptes Rendus of the French Academy, quite a series of communications by M. E. de Jonquieres, on the subject of those periodic continued fractions which are the equivalents of the square roots of integers. These communications have attracted attention, both on account of the number of results given in them, and because, as a writer in the Bulletin des Sciences Mathématiques says, of their interesting and profound character. To any one really intimate with the bibliography of the subject, this cannot but be a little surprising. It is true that the number of so-called theorems is great; but the very special character of a number of them, the fact that they are just such theorems as may be obtained by experiment and induction, and the want of demonstrations of them as evidence that the author was in possession of a mathematical theory of the subject, are points that have been too much overlooked. Further, and what is more important, many of the theorems are not new, and there is a sense in which the epithet “new” cannot fairly be applied to any of the earlier ones, because of the existence of a widely general theorem in which they are directly included, or from which they may with readiness be deduced.
A characterization of rational numbers by p-adic Ruban continued fractions
