Note 169 — Introductory material

To begin with, I must point out that my axiom is not quite equivalent to Euclid's. Euclid's axiom states that two straight lines cannot enclose a space, i.e., cannot have more than one common point. Now if every two points, without exception, determine a unique straight line, it follows, of course, that two different straight lines can have only one point in common—so far, the two axioms are equivalent. But it may happen, as in spherical space, that two points in general determine a unique straight line, but fail to do so when they have to each other the special relation of being antipodes. In such a system every pair of straight lines in the same plane meet in two points, which are each other's antipodes; but two points, in general, still determine a unique straight line. We are still able, therefore, to obtain distances from unique straight lines, except in limiting cases; and in such cases, we can take any point intermediate between the two antipodes, join it by the same straight line to both antipodes, and measure its distance from those antipodes in the usual way. The sum of these distances then gives a unique value for the distance between the antipodes.