N this tract only the outlines of the subject are dealt with. Accordingly I have endeavoured to avoid reasoning dependent upon the mere wording and on the exact forms of the axioms (which can be indefinitely varied), and have concentrated attention upon certain questions which demand consideration however the axioms are phrased. Every group of the axioms is designed to secure the deduction of a certain group of properties. For the most part I have stated without proof the leading immediate consequences of the various groups. Also I have ignored most of the independence theorems, as being dependent upon mei'e questions of phrasing, and have only investigated those which appear to me to embody the essence of the subject; though, as far as I know, no formal line can be drawn between these two classes of theorems.