Page:Philosophical Review Volume 4.djvu/465

449 But the problem still remains in the occasional opposition between Altruism and Egoism, and the practical solution can only be found in the concrete case. In general, 'self-assertion' and 'self-denial' will be found to be synonymous.

In the present article Professor Delbœuf, after some interesting biographical remarks in which he explains that his views, though enunciated thirty-five years ago, are only now attracting attention, passes from criticism to a systematic statement of his own remodelling of geometry. Its chief principles, as already indicated in the earlier articles, are as follows, (1) Geometry is the science of the determinations of extension or figures; extension is the indefinite sum of the places that bodies can occupy; space is extension regarded as not determined or limited; and place is what remains of a body when abstraction is made of its matter. (2) Geometrical space must consequently be distinguished from the various physical spaces, and is an ideal constituted by abstraction. (3) The characteristic assumption which distinguishes geometrical from all other real or ideal spaces is its homogeneity, i.e., the fact that in all its parts, whatever their magnitude, it is capable of receiving the same determinations. From this follow the ideal universality of geometrical propositions and the general validity of geometrical demonstrations. The infinite extension and divisibility of geometrical space are equally consequences of its homogeneity. In itself, however, this infinite capacity of extension and contraction is self-contradictory, for it implies that there are as many parts in the less as in the greater, in the part as in the whole. Hence it is evident that geometrical space is imaginary, ideal, abstract, and simplified so as to render possible the study of real figures which could not otherwise be treated. (4) In geometrical space the form is independent of the size of a figure, and hence similar figures are possible in it ad infinitum. Chapter III contains a list of the axioms employed by geometry, which are fully stated and classified as logical, arithmetical, and algebraical.

F. C. S. S.