Erdmann's conclusion, in the second chapter, is that Congruence is probable, but cannot be verified in the infinitesimal; that its truth involves the actual existence of rigid bodies (though, by the way, we know these to be, strictly speaking, non-existent), that rigid bodies are freely moveable, and do not alter their size in rotation (Helmholtz's Monodromy); that the axiom of three dimensions is certain, since small errors are impossible; and that the remaining axioms of Euclid--those of the straight line and of parallels--are approximately, if not accurately, true of our actual space (pp. 78, 83). He does not discuss how Congruence, on the above view, is compatible with the atomic theory, or even with the observed deformations of approximately rigid bodies; nor how, if space, as he assumes, is homogeneous, rigid bodies can fail to be freely moveable through space. The axioms are all lumped together as empirical, and it appears, in the following chapters, that Erdmann regards their empirical nature as sufficiently proved by their applicability to empirical material (cf. pp. 159, 165)--a strange criterion, which would prove the same conclusion, with equal facility, of Arithmetic and of the laws of thought. =83.= The third chapter, on the philosophical consequences of Metageometry, need not be discussed at length, since it deals rather with space than with Geometry. At the same time, it will be worth while to treat briefly of Erdmann's criterion of apriority. On this subject it is very difficult to discover his meaning, since it seems to vary with the topic he is discussing. Thus at one time (p. 147) he rejects most emphatically the Kantian connection of the _à priori_ and the subjective[97], and yet at another time (p. 96) he regards every presentation of external things as partly _à priori_, partly empirical, merely because such a presentation is due to an interaction between ourselves and things, and is therefore partly due to subjective activity, partly due to outside objects. Hence, he says, the distinction is not between different presentations, but between different aspects of one and the same presentation. This seems to return wholly to the Kantian psychological criterion of subjectivity, with the added disadvantage that it makes the distinction, like that of analytic and synthetic, epistemologically worthless. And yet he never hesitates to pronounce every piece of knowledge in turn empirical. The fact seems to be, that where he wants a more logical criterion, he adopts a modification of Helmholtz's criterion for sensations. If space be an _à priori_ form, he says, no experience could possibly change it (p. 108); but this Metageometry has proved not to be the case, since we can intuit the perceptions which non-Euclidean space would give us (p. 115). I have criticised this argument in discussing Helmholtz; at present we are concerned with Erdmann's criterion of apriority. The