In an essay in the Berliner Monatsschrift (May 1796, pp. 395–96), among other examples of the fanaticism that may be induced by attempts to philosophize about mathematical objects, I also attributed to the Pythagorean number-mystic the question: “Why is it that the ratio of the three sides of a right-angled triangle can only be that of the numbers 3, 4, and 5?” I had thus taken this proposition to be true; but Professor Reimarus refutes it, and shows (Berliner Monatsschrift, August, no. 6) that many numbers, other than those mentioned, can stand in the ratio in question.

So nothing seems clearer than that we find ourselves embroiled in a truly mathematical dispute (of a kind that is, in general, almost unheard of). But this quarrel amounts only to a misunderstanding. Each party takes the expression in a different sense; so soon as a mutual understanding is reached, the dispute vanishes, and both sides are correct. Now proposition and counter-proposition are related as follows:

R. says (or at least thinks his proposition thus): “In the infinite multitude of all possible numbers (considered at large) there exist, in regard to the sides of the right-angled triangle, more ratios than that of the numbers 3, 4, and 5.”