Thus, to conclude: Geometry requires, if it is to be _practically_ possible, some body or bodies which are either rigid (in the dynamical sense), or known to undergo some definite changes of shape according to some definite law. (These changes, we may suppose, are known by the laws of Physics, which have been experimentally established, and which throughout assume the truth of Geometry.) One or more such bodies are necessary to applied Geometry--but only in the sense in which rulers and compasses are necessary. They are necessary as, in making the Ordnance Survey, an elaborate apparatus was necessary for measuring the base line on Salisbury Plain. But for the _theory_ of Geometry, geometrical rigidity suffices, and geometrical rigidity means only that a shape, which is possible in one part of space, is possible in any other. The empirical element in practice, arising from the purely empirical nature of physical rigidity, is comparable to the empirical inaccuracies arising from the failure to find straight lines or circles in the world--which no one but Mill has regarded as rendering Geometry itself empirical or inaccurate. But to make Geometry await the perfection of Physics, is to make Physics, which depends throughout on Geometry, forever impossible. As well might we leave the formation of numbers until we had counted the houses in Piccadilly. Erdmann. =74.= In connection with Riemann and Helmholtz, it is natural to consider Erdmann's philosophical work on their theories[92]. This is certainly the most important book on the subject which has appeared from the philosophical side, and in spite of the fact that, like the whole theory of Riemann and Helmholtz, it is inapplicable to projective Geometry, it still deserves a very full discussion. Erdmann agrees throughout with the conclusions of Riemann and Helmholtz, except on a few points of minor importance; and his views, as this agreement would lead one to expect, are ultra-empirical. Indeed his logic seems--though I say this with hesitation--to be incompatible with any system but that of Mill: there is apparently no distinction, to him, between the general and the universal, and consequently no concept not embodied in a series of instances. Such a theory of logic, to my mind, vitiates most of his work, as it vitiated Riemann's philosophy[93]. This general criticism will find abundant illustration in the course of our account of Erdmann's views.