Note 25 — Introductory material
Reading edition
In the first of these, which is chiefly philosophical, Helmholtz gives hints of his then uncompleted mathematical work, but in the main contents himself with a statement of results. He announces that he will prove Riemann's quadratic formula for the infinitesimal arc; but for this purpose, he says, we have to start with Congruence, since without it spatial measurement is impossible. Nevertheless, he maintains that Congruence is proved by experience. How we could, without the help of measurement, discover lapses from Congruence, is a point which he leaves undiscussed. He then enunciates the four axioms which he considers essential to Geometry, as follows: