Observatio arithmetica

II. how, conversely, not only is the number a together with the modulus b obtained from a true period, but also the entire series of numbers s, s₁, ... sₙ, and consequently the fraction a a/b is evolved into a continued fraction;

III. Finally, it is evident that, once a and b are known, the remainders r result immediately from the characteristic itself. For if among the terms g₁, g₂, ... gμ, m of them are found to be = d, and the rest = c, then rμ = μa - mb.

9. What we have just set forth is more widely applicable and seems quite worthy of attention. For our series P^x determines the fraction a/b through the mere numbering of the letters c and d, and, if one wishes, provides its evolution into a continued fraction through successive numberings.

This can be stated as follows: by means of the characteristic, the very notion of fractions is reduced to mere and true numbering—which requires items that are numerable but of any kind whatsoever—without any notion of division, which requires items that are divisible, that is, quantities.

Strassburg, 18/1/74.