Is Continuity of Space Necessaey to Euclid's Geometry?

DEDEKIND has said that continuity of space is not essential to the constructions of Euclid and has defined a discontinuous space in which, " so far as he sees," the constructions can be made.* To prove this it is only necessary to show that the constructions will never ]ead out of the given discontinuous space if we start in it—in other words, if we start with a figure of the discontinuous space, the construction will lead always to other figures of that space. If the constructions can be made in a given space the theorems will be true in it. I t is not safe to assume in the demonstration that discontinuity may not render invalid principles which are postulated for continuous space ; the question of the truth of the postulates in a given discontinuous space can be decided only by a knowledge of the properties of the space. For instance, in a space consisting of the points whose rectangular coordinates with a given system of axes are rational in a given unit of distance, not all figures can be moved about ; parts of a figure may disappear entirely as the result of an allowable change in the position of other parts ; thus a square of unit side may be constructed in this space, but if its diagonal is brought into coincidence with one of the axes, one