=35.= But what are projective coordinates, and how are they introduced? This question was not touched upon in Cayley's Memoir, and it seemed, therefore, as if a logical error were involved in using coordinates to define distance. For coordinates, in all previous systems, had been deduced from distance; to use any existing coordinate system in defining distance was, accordingly, to incur a vicious circle. Cayley mentions this difficulty in a note, where he only remarks, however, that he had regarded his coordinates as numbers arbitrarily assigned, on some system not further investigated, to different points. The difficulty has been treated at length by Sir R. Ball (Theory of the Content, Trans. R. I. A. 1889), who urges that if the values of our coordinates already involve the usual measure of distance, then to give a new definition, while retaining the usual coordinates, is to incur a contradiction. He says (op. cit. p. 1): "In the study of non-Euclidean Geometry I have often felt a difficulty which has, I know, been shared by others. In that theory it seems as if we try to replace our ordinary notion of distance between two points by the logarithm of a certain anharmonic ratio[46]. But this ratio itself involves the notion of distance measured in the ordinary way. How, then, can we supersede our old notion of distance by the non-Euclidean notion, inasmuch as the very definition of the latter involves the former?" =36.= This objection is valid, we must admit, so long as anharmonic ratio is defined in the ordinary metrical manner. It would be valid, for example, against any attempt to found a new definition of distance on Cremona's account of anharmonic ratio[47], in which it appears as a metrical property unaltered by projective transformation. If a logical error is to be avoided, in fact, all reference to spatial magnitude of any kind must be avoided; for all spatial magnitude, as will be shown hereafter[48], is logically dependent on the fundamental magnitude of distance. Anharmonic ratio and coordinates must alike be defined by purely descriptive properties, if the use afterwards made of them is to be free from metrical presuppositions, and therefore from the objections of Sir R. Ball.