where the numerators represent the numbers of years, and the denominators the numbers of lunations, necessary to bring the sun and moon again into the same relative position, very nearly at the same point of time in the tropical year. Of these ratios, some have been long known; the _fourth_ is the Greek cycle called _Octaëteris_, discovered B.C. 600, and is a very rude approximation: the _sixth_ is the famous cycle re-discovered by _Meton_ B.C. 432, but probably known to the Hebrews from the earliest ages, as the lives of Seth, Methuselah and Noah are exact multiples of this cycle, as well as the _Antediluvian age_ itself, and is a remarkably near approximation to the truth; _four times_ this ratio gives the period of _Calippus_, which was rectified by the omission of _one_ day in _seventy-six_ years. These approximations, however, are much inferior in accuracy to the higher terms of the series; from which, in fact, any number of approximate ratios may be deduced by the following principle:—If a series of fractions be all equal to each other, the sum or difference of the numerators and denominators of any pair will constitute a new fraction equal to each; and the same is true of fractions whose numerators and denominators are equi-multiples of those of any of the given or derived fractions. Hence, from the terms of the preceding series, we derive the following additional ratios, whose degree of approximation, of course, depends on that of the fractions of which they are composed. From the _eighth_ and _ninth_ ratios, by addition, we obtain the ratio ¹⁰⁴⁰⁄₁₂₈₆₃, which is so remarkably correct, that the approximation is within about _three-quarters of an hour_ of the truth. From the _sixth_ and _seventh_ ratios, by subtraction, we obtain, the ratio ³¹⁵⁄₃₈₉₆ = ¹²⁶⁰⁄₁₅₅₈₄, which shows that the prophetic period of 1260 years is a scientific lunisolar cycle. From the two new ratios thus obtained, by addition, we have a third new ratio ²³⁰⁰⁄₂₈₄₄₇, which proves that the prophetic period of 2300 years is another scientific cycle, in which the approximation is nearly _twelve hours_.