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. Footnote 35: We have much satisfaction in giving the following abridgement of the remarks of Mr. Birks, on this interesting subject; pp. 371, 372. Thus, he says, “A fresh light is thrown upon the words of the Psalmist, _He appointed the moon for seasons_. A divine ladder of time is set before us, and, as we rise successively from step to step, days are replaced by years, and years by millenia; and these, perhaps, hereafter, in their turn, by some higher unit, from which the soul of man may measure out cycles still more vast, and obtain a wider view of the immeasurable grandeur of eternity. Human science has strained its utmost efforts in calculating the actual motions of the Moon and the Earth; but the determining causes which fixed at first the proportion of their monthly and yearly revolutions have altogether eluded its research. Yet these elements of the natural universe are linked in, by these sacred times and celestial cycles, with the deepest wonders of Providence, and the whole range of Divine prophecy. How glorious, then, must be the inner shrine, lit up with the Shechinah of the Divine Presence, when the approaches themselves reveal such a secret and hidden wisdom!” Footnote 36: See his work “On the Jubilean Chronology of the Seventh Trumpet of the Apocalypse,” pp. 1–3, and 19–26. Footnote 37: