Elements of Algebra

494 ADDITIONS. CHAP. I. creasing, and the other decreasing, towards the true ratio in question; in this manner we shall have Fractions less than the ratio of the circumference to the diameter. $\frac{3}{1}$, $\frac{25}{8}$, $\frac{47}{15}$, $\frac{69}{22}$, $\frac{91}{29}$, $\frac{113}{36}$, $\frac{135}{43}$, $\frac{157}{50}$, $\frac{179}{57}$ $\frac{201}{64}$, $\frac{223}{71}$, $\frac{245}{78}$, $\frac{267}{85}$, $\frac{289}{92}$, $\frac{311}{99}$, $\frac{333}{106}$, $\frac{689}{219}$ $\frac{1043}{332}$, $\frac{1398}{445}$, $\frac{1753}{558}$, $\frac{2108}{671}$, $\frac{2463}{784}$, &c. Fractions greater than the ratio of the circumference to the diameter. $\frac{4}{1}$, $\frac{7}{2}$, $\frac{10}{3}$, $\frac{13}{4}$, $\frac{16}{5}$, $\frac{19}{6}$, $\frac{22}{7}$, $\frac{355}{113}$, $\frac{104345}{33215}$ $\frac{312689}{99532}$, $\frac{1146408}{364913}$, $\frac{5419351}{1725033}$, $\frac{85563208}{27235615}$, $\frac{165707065}{52746197}$ $\frac{411557987}{131002976}$, $\frac{1480524883}{471265707}$, &c. Each fraction of the first series approaches nearer the truth than any other fraction whatever, expressed in simpler terms, and the error of which consists in being too small; and each fraction of the second series likewise approaches nearer the truth than any other fraction, which is expressed in simpler terms, and the error of which consists in its being too large. These series would become very long, if we were to continue them as far as we have done that of the principal fractions before given. The limits of this work do not permit us to insert them at full length; but they may be found, if wanted, in Chap. XI. of Wallis's Algebra. (Oper. Mathemat.) SCHOLIUM. 22. The first solution of this problem was given by Wallis in a small treatise, which he added to the posthumous works of Horrox, and it is to be found in his Algebra as quoted above; but the method of this author is indirect, and very laborious. That which we have given belongs to Huygens, and is to be considered as one of the principal discoveries of that great mathematician. The construction of his planetary automaton appears to have led him to it: for, it is evident, that, in order to represent the motions and periods of the planets exactly, we should employ wheels, in which the teeth are precisely in the same ratios, with respect to number, as the periods in question; but as teeth cannot be multiplied beyond a certain limit, depending on the size of