These, as we shall see hereafter, are retained, in actual mathematical work, by all metrical Metageometers, even when they believe, like Riemann and Helmholtz, that no axioms are philosophically indispensable. =24.= _Helmholtz_, the historically nearest follower of Riemann, was guided by a similar empirical philosophy, and arrived independently at a very similar method of formulating the axioms. Although Helmholtz published nothing on the subject until after Riemann's death, he had then only just seen Riemann's dissertation (which was published posthumously), and had worked out his results, so far as they were then completed, in entire independence both of Riemann and of Lobatchewsky. Helmholtz is by far the most widely read of all writers on Metageometry, and his writings, almost alone, represent to philosophers the modern mathematical standpoint on this subject. But his importance is much greater, in this domain, as a philosopher than as a mathematician; almost his only original mathematical result, as regards Geometry, is his proof of Riemann's formula for the infinitesimal arc, and even this proof was far from rigid, until Lie reformed it by his method of continuous groups. In this chapter, therefore, only two of his writings need occupy us, namely the two articles in the _Wissenschaftliche Abhandlungen_, Vol. II., entitled respectively "Ueber die thatsächlichen Grundlagen der Geometrie," 1866 (p. 610 ff.), and "Ueber die Thatsachen, die der Geometrie zum Grunde liegen," 1868 (p. 618 ff.). =25.= 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: (1) _As regards continuity and dimensions._ In a space of _n_ dimensions, a point is uniquely determined by the measurement of _n_ continuous variables (coordinates). (2) _As regards the existence of moveable rigid bodies._ Between the 2_n_ coordinates of any point-pair of a rigid body, there exists an equation which is the same for all congruent point-pairs. By considering a sufficient number of point-pairs, we get more equations than unknown quantities: this gives us a method of determining the form of these equations, so as to make it possible for them all to be satisfied.