“A Table of the Greek and Hebrew Chronologies from Creation to the end of the Jewish War,” 4th Edition. We shall now endeavour to give a short notice of these discoveries. 3. Various cycles which enter into the true system of chronology—Mr. Cuninghame’s discovery of the _trinal fraction_—Its explanation and application by an Algebraic formula—Original form in which it was discovered—Its superiority to the formulæ of the _figurate numbers_—Remarkable instance of its application to Scriptural and other numbers, and to lunar and solar cyclical numbers—Mr. Cuninghame’s definition of the trinal fraction the most correct—The series deduced from its formula possesses curious properties. It has been already shown that according to the will of Him, who (τοὺς ἀιῶνας ἐποίησεν) _constructed the ages_, the septenary cycle, with its multiples and higher powers, and the lunisolar cycles, with their sums, differences and multiples, including the Metonic, the Jubilean, the Prophetic and the Secular, enter into the structure of the true chronology. To these, Mr. Cuninghame adds the Duodenary cycle, and its multiples and higher powers; the Undenary cycle, which is also Lunisolar; the Quinary cycle, which is indicated no less than _four_ times in the formation of man; and the _Trinal fraction_, which alone seems to require explanation. The author was led by circumstances detailed in “The Scientific Chronology” pp. 5–8, to give the name of “Trinal fraction” to the general term of a series of numbers of which each is composed of the root, its square, and unity, that is, in Algebraic language, _n_^2 + _n_ + 1; an expression, in which _n_ may be zero, unity, or any whole number whatever, and giving, by the substitution of 0, 1, 2, 3, &c. as roots, the series itself, namely, 1, 3, 7, 13, 21, 31, 43, 57, 73, &c. To the discovery of this series, as _new in mathematics_, of course, he makes no claim; because, a mere tyro in that science could write out a hundred such in as many minutes; see “Dissertation on the Apocalypse,” fourth edition, pp. 522, 523; but, to the discovery of its application to the cyclical character of the mundane times, he has a decided claim, and we think he has fully substantiated it by a reference to chronological facts.