The Foundations of Geometry

Abstract

“OUGHT there not to be a perfect subjective geometry, as well as an applied objective one, the applicability of the former to the latter being a matter to be determined by induction from experiment? It is the object of this book to show that such is the case, to establish the perfect geometry, and to examine the grounds on which we may believe that it applies to the objective space in which we live.” Such, stated in the author's own words, is the object he has proposed to himself, and he appeals with confidence to geometricians to criticize his system. The work is divided into three parts, devoted respectively to (1) a discussion on the logical status of geometry; (2) the development of the author's subjective theory; and (3) an investigation into its application to material space. It is to part (2) that we shall chiefly direct our remarks; for though the author's views, as laid down in part (1) (see pp. 20, 21), on definition as the basis of a deductive system, have an important bearing on his geometrical theory, and seem in opposition to those of at least one school of logicians, they will be best tested by an examination of the definitions of the two concepts “direction” and “sameness of direction” laid down at the beginning of part (2). With much of the author's criticism of Euclid's treatment of the plane we are in agreement, but he seems to lay himself open to the chief objections urged against his predecessors. With regard to part (2), granting the assumptions tacitly made at the outset, we are willing to admit the formal accuracy of most of the proofs of the propositions in the “subjective theory,” and the possibility of supplying it, without any serious derangement of the system, to those which seem to want it. But we do take serious exception to the way in which the foundations of the new edifice are laid. The system is based partly on three axioms, explicitly stated, as to (1) the possible transference of geometrical figures, (2) the possible extension of a straight line, (3) the three-way extension of space; and partly on what are styled, not inaptly, the “implicit” definitions of position and direction. With that of “position” (used in the sense of “fixed point”) we are not much concerned, merely remarking that in it the word position is used to explain position. This defect, under which Bardolph's more famous definition of “accommodated” also labours, could probably be easily rectified. But so much depends on that of “direction” and “sameness of direction,” that we give it in full:—

The Foundations of Geometry.

By E. T. Dixon. (Cambridge: Deighton, Bell, and Co., 1891.)