Elements of Algebra

492 ADDITIONS. CHAP. I. the two series $\frac{4}{1}$, $\frac{169}{41}$, and $\frac{33}{8}$, $\frac{140}{34}$; for it is evident, that it could fall only between those last fractions, between which, on account of the number 3, which is above the fraction $\frac{140}{41}$, there may be inserted two intermediate fractions, which will be $\frac{202}{49}$, and $\frac{371}{90}$; whence it appears, that it would have been more exact, if in the Gregorian reformation they had only intercalated ninety days in the space of three hundred and seventy-one years. If we reduce the fraction $\frac{400}{97}$, so as to have for its numerator the number 86400, it will become $\frac{86400}{20929}$, which estimates the tropical year at $365^\text{d}. 5^\text{h}. 49'. 12''$. In this case, the Gregorian intercalation would be quite exact; but as observations make the year to be shorter by more than $20'$, it is evident that, at the end of a certain period of time, we must introduce a new intercalation. If we keep to the determination of M. de la Caille, as the denominator 97 of the fraction $\frac{400}{97}$ lies between the denominators of the fifth and sixth principal fractions already found, it follows, from what we have demonstrated (Art. 14), that the fraction $\frac{161}{39}$ will be nearer the truth than the fraction $\frac{400}{97}$; but as astronomers are still divided with regard to the real length of the year, we shall refrain from giving a decisive opinion on this subject; our only object in the above detail is to facilitate the means of understanding continued fractions and their application: with this view, we shall also add the following example. 21. *Example 2.* We have already given, in Art. 8, the continued fraction, which expresses the ratio of the circumference of the circle to the diameter, as it results from the fraction of Ludolph; so that we have only to calculate, according to the manner taught in the preceding example, the series of fractions, converging towards that ratio, which will be 3, 7, 15, 1, 292, 1, 1, $\frac{3}{1}$, $\frac{22}{7}$, $\frac{333}{106}$, $\frac{355}{113}$, $\frac{103993}{33102}$, $\frac{104348}{33215}$, $\frac{208341}{66317}$, 1, 2, 1, 3, 1, $\frac{312689}{99532}$, $\frac{833719}{265381}$, $\frac{1146408}{364913}$, $\frac{4272943}{1360120}$, $\frac{5419351}{1725033}$, 14, 2, 1, 1, $\frac{80143857}{25510582}$, $\frac{165707065}{52746197}$, $\frac{245850922}{78256779}$, $\frac{411557987}{131002976}$, 2, 2, 2, $\frac{1068966896}{340262731}$, $\frac{2549491779}{811528438}$, $\frac{6167950454}{1963319607}$, 2, 1, 84, $\frac{14885392687}{4738167652}$, $\frac{21053343141}{6701487159}$, $\frac{1781366216538}{567663097468}$,