Note 34 — Chapter IV · CYCLICAL CHARACTER OF THE MUNDANE TIMES.
Reading edition
The rules for the determination of the approximating ratios of any two numbers by the method of Continued Fractions, are very clearly explained, on Algebraical principles in Hind’s Algebra, 5th edition, pp. 273–292; and, on Arithmetical principles in Thomson’s Arithmetic, 16th edition, pp. 241–246. We subjoin the operation, as much abridged as possible, by which the preceding series of fractions was obtained. Reducing the above lengths of the year and the month into seconds, we have the numbers 31556929·7 and 2551442·87; hence,
2 255144287 3155692970 12
187923052 3061731444
2 67221235 93961526 1
53480582 67221235
1 13740653 26740291 1
12999638 13740653
1 741015 12999638 17
402383 12597255
338632 402383 1
&c.
Quotients, 12, 2, 1, 2, 1, 1, 17, 1, 1, &c.
Ratios, ¹⁄₁₂, ²⁄₂₅, ³⁄₃₇, ⁸⁄₉₉, ¹¹⁄₁₃₆, ¹⁹⁄₂₃₅, ³³⁴⁄₄₁₃₁, ³⁵³⁄₄₃₆₆, ⁶⁸⁷⁄₈₄₉₇, &c.
Unmathematical readers will find a much more simple, but of course, a more lengthened explanation, of these ratios in Mr. Cuninghame’s “Synopsis of Chronology,” pp. 6–10, and 54–60.