As to the formula itself, its most general form is (_n_^3 + _n_^2 + _n_)/(_n_) as originally discovered by the author; and in this form it is manifestly more simple and general than any of the formulæ of the _figurate numbers_; for, if _n_ be taken equal to zero, in any of the latter, the value of the vanishing fraction is always equal to _zero_; but, in the former, it is equal to _unity_, the first term of the series, and the basis of all numerical calculation. Let us take some other examples of its application: the _sacred_ number 3, is the trinal fraction of unity, and although it includes the higher powers of the root, is only the sum of _three_ units, mysteriously indicating a trinity in unity. The _sacred_ number 7, is the trinal fraction of 2, which is the basis of the binary system of numeration so natural to man. The number 13, is the trinal fraction of 3, and is a lunisolar cycle of years, the hebdomadal measure of the seasons of the year, and the actual number of the tribes of Israel. The number 21 is the trinal fraction of 4, and the product of the sacred numbers 3 and 7, the trinal fractions of 1 and 2. The number 31, is the trinal fraction of 5, the basis of the Quinary scale so incorporated with the human frame, and is the measure of the life of the _first man_. The number 57, is the trinal fraction of the sacred number 7, and three times the Metonic cycle of _nineteen_ years, being an element of the Mundane Times. Lastly, the number 73, is the trinal fraction of 8, a lunisolar cycle of years, and gives, when multiplied by 5, the number of days in the solar cycle. The trinal fraction has been compared also with the formula _n_^2 − _n_ + 1, which is only a particular case of it, namely, where _n_ is negative. It is true, that if in this formula, −1, −2, −3, &c., be taken for values of _n_, it will still give the series of trinal fractions; but it does not therefore follow that the two formulæ are the same; for, if in the latter, _n_ be taken equal to zero, it will give the same result as when _n_ is taken equal to −1! The definition given by Mr. Cuninghame, is therefore the most accurate, simple, and general, and one which can be easily comprehended without any reference to the formulæ of the _Figurate Numbers_. Moreover, the author has shown in the works last referred to, that the series of trinal fractions possess higher properties of science, mathematically, astronomically, and chronologically, than the triangular numbers, from which it is pretended that they have been derived. To some very curious properties and applications of the trinal fractions, the author has added a “Table of the Trinal Fractions from 1 to 85, showing the sums of the Roots and Fractions at each Pentad,” p. 519 of the “Dissertation;” and he has shown how these numbers enter so extensively and so mysteriously into the whole structure of the Mundane Times!