PROPOSITIONS OF INCIDENCE

Propositions of incidence in three dimensions. The entities with which we deal in the first instance we call by the names, point, line, and plane. These are any objects which are subject to the following laws of combination, which we call the Propositions of Incidence, together with another law, explained below (Sect. II). It is provisionally assumed that these laws are self-consistent and, when properly explained, are sufficient to enable the reader to form a clear impression whether any statement made in regard to these entities is a consequence, or not, of the fundamental laws. It is also very frequently assumed provisionally, when, in the course of a geometrical construction, two points are obtained by certain rules and it is desired to continue the construction with the help of the line joining these points, that these points do not coincide. In many cases it may be easy to shew that the coincidence of the points would involve an undesired limitation in the given points of the figure. But there may be other cases in which it would be consistent with the assumed fundamental propositions to assume either that the two points always coincide, or that they do not always coincide. In taking the latter alternative we should then be neglecting possibilities which, even if special, may quite well be worth examination.