Elements of Algebra

CHAP. II. ADDITIONS. 495 the wheel, and, besides, as the periods of the planets are incommensurable, or, at least, cannot be represented, with any exactness, but by very large numbers, we must content ourselves with an approximation; and the difficulty is reduced to finding ratios expressed in smaller numbers, which approach the truth as nearly as possible, and nearer than any other ratios can, that are not expressed in greater numbers. Huygens resolves this question by means of continued fractions as we have done; and explains the manner of forming those fractions by continual divisions, and then demonstrates the principal properties of the converging fractions, which result from them, without forgetting even the intermediate fractions. See, in his *Opera Posthuma*, the Treatise entitled *Descriptio Automati Planetarii*. Other celebrated mathematicians have since considered continued fractions in a more general manner. We find particularly in the Commentaries of Petersburgh (Vol. IX. and XI. of the old, and Vol. IX. and XI. of the new), Memoirs by M. Euler, full of the most profound and ingenious researches on this subject; but the theory of these fractions, considered in an arithmetical view, which is the most curious, has not yet, I think, been cultivated so much as it deserves; which was my inducement for composing this small Treatise, in order to render it more familiar to mathematicians. See, also, the Memoirs of Berlin for the years 1767, and 1768. I have only to observe farther, that this theory has a most extensive application through the whole of arithmetic; and there are few problems in that science, at least among those for which the common rules are insufficient, which do not, directly or indirectly, depend on it. John Bernoulli has made a happy and useful application of it in a new species of calculation, which he devised for facilitating the construction of Tables of proportional parts. See Vol. I. of his *Recueil pour les Astronomes*. *** CHAP. II. Solution of some curious and new Arithmetical Problems. Although the problems, which we are now to consider, are immediately connected with the preceding, and depend on