The rational treatment of Geometry has this important disadvantage, that for want of suitable demonstrations it seems impossible to preserve the natural grouping of the facts developed. The study of Rational Geometry, in fact, should always be supplemented by a systematic attempt to array the facts demonstrated according to their subject-matter; for it will hardly be denied that a direct and systematic knowledge of the Properties of Geometrical Figures has an intrinsic value apart from the knowledge of their demonstrations. In pursuing such a retrospective scheme as this in connection with the Second Book of Euclid, I have found that a very comprehensive view of the subject-matter is obtained by adding a Third Mode of Section of a straight line to the two which are already recognised. This third mode of section, for which I have not been able to find a more suitable name than “Circuitous Section,” along with the other two known as Internal and External Section respectively, exhausts the possible modes of section of a line—for three-dimensional space at any rate. From this point of view, the elementary treatment of the subject may be arranged as follows. It will be observed that several important properties of triangles and polygons acquire a new-interpretation as cases of circuitous section.