that within the limits of fact which must serve as a basis for this inference Euclidean and non-Euclidean straight lines are not to be distinguished from each other. Hence with K finite but very large as compared with terrestrial measurements and with different laws of optics the same facts might be accounted for in a manner quite as simple as they are at present. On the other hand, the discovery of a parallax in the case of the most distant star or of a negative parallax for any star cannot be regarded as disproving Euclid. Such anomalies, if not too numerous, could be accounted for in a manner which is much less expensive by making suitable changes in the received laws of optics; or even by assuming a slight strain in the ether itself in those particular regions of space. While therefore an actual discovery of any such facts would certainly prove interesting and suggestive this alone could not be accepted as a conclusive proof or disproof of Euclid's validity; for the very conditions which would render this discovery possible must be found to involve certain assumptions which might very well be withdrawn. The facts, therefore, which count most either for or against Euclid must be found if possible within the realm of direct observation where such assumptions as to what takes place beyond this realm do not need to be made. The problem then is at bottom a psychological one which must be decided for the most part by experiment. What are the sensory contributions which should be taken into account in the formation of the abstract conception of space? If we confine ourselves to the contributions of vision alone our geometry is not Euclidean; it is projective, and so far as its space conception is concerned is not to be distinguished from the double or single elliptic systems of metrical geometry. But if we consider the larger realm of spatial experiences in which sensations of motion, direction and touch are also involved, all the known facts inevitably suggest the parallel postulate. Whether they shall continue to do so as tested by the most refined experimental analysis of man's ability to discriminate under the most favorable conditions slight variations in the size of angles, in the length of lines, and in the latter's departure from ideal straightness remains to be determined. As already stated, the author has undertaken an experimental investigation of this sort, at the suggestion of Professor Ladd. Unfortunately the data thus far obtained are not sufficient to justify any positive statement as to what the outcome will probably be. Obviously a very great number and variety of experiments need to be performed by a large number of individuals to obtain results that can be regarded as significant. CHAPTER VI. RESULTING IMPLICATIONS AS TO THE NATURE OF SPACE. In the present chapter it shall be our purpose to point out certain implications as to the nature of space which seem to result from denying the necessary validity of the parallel postulate and from the consequent possibility of non-Euclidean geometries. As a preparatory step in this direction it is necessary to get a clear understanding of precisely what is meant by non-Euclidean spaces. The question therefore recurs, are these so-called 'spaces after all anything more than ingenious logical constructs which one cannot even think of except as finite forms in Euclidean spaces? In framing a reply to this question we must guard against certain errors which almost inevitably result from the introduction of Euclidean analogies of corresponding non-Euclidean conceptions. Spherical and pseudo-spherical surfaces upon which non-Euclidean straight lines are represented as curves in ordinary Euclidean space of three dimensions are, in reality, very different from |