Observatio arithmetica

3. The period P, therefore, contains b letters and among them, as follows from the first notions concerning residues, the letter d in a places.

Given, therefore, the number a and the modulus b, the period P is completely determined, and conversely, given the period P, the number a is completely determined together with the modulus b.

Hence it is clear that the order in which the letters c, d are distributed in the period P is not arbitrary, but is constrained by certain laws, so that an aggregate which results from a true period by permuting the letters c, d in any way cannot be the period of any number a with any modulus b added.

For this reason, we shall call the series (g.) itself the Characteristica of the number a with respect to the modulus b, or, if you please, the characteristic of the fraction $\frac{a}{b}$.

4. Among these very laws concerning the characteristic must be included those which the great Gauss taught regarding quadratic residues (New Proof of an Arithmetical Theorem), from which it can hardly be doubted that the most intimate properties obtaining between numbers a and b lie within the characteristic.

But what we are to expound on this occasion does not look so high, nor does it reach beyond numeration.

However, we have found a theorem that all periods are contained under one single form, and that the most simple, such that one and the same law embraces the distribution of the letters c, d in all characteristics.

5. In this matter, we shall use a notation which indeed does not place before the eyes the symmetry of $Π$ itself just mentioned, but which is derived from the true source of these questions and therefore is to be considered more useful.

If in the characteristic the letter c is to be noted in a continuous series twice or thrice or more often, we write c², c³ instead of cc, ccc, and so on, whence c⁰ will signify that the letter c is to be omitted entirely in such a place.

In the second example, therefore, $Π = c^2dc^2dc^3dc^2dc^2dc^2$ will be the case.

In the same way, we collect the aggregates themselves by writing $(c^n d)^2$ instead of $c^n d c^n d$, and so on generally.

Hence $P = Π dc$ will be the case, if the order of the letters is carefully observed, and the characteristic itself of the number a with respect to the modulus b is this: