Introductory. The theorems of Projective Geometry extended to any number of dimensions can be deduced as necessary consequences of the definitions of a positional manifold. Grassmann's ‘Calculus of Extension’ (to be investigated in Book IV.) forms a powerful instrument for such an investigation; the properties also can to some extent be deduced by the methods of ordinary co-ordinate Geometry. Only such theorems will be now investigated which are either useful subsequently in this treatise or exemplify in their proof the method of the addition of extraordinaries.

Anharmonic Ratio. (1) Any point p on the straight line αα′ can be written in the form ξα + ξ′α′, where the position of p is defined by the ratio ξ/ξ′. If p1 be another point, ξ1α + ξ1′α′, on the same line, then the ratio ξξ1′/ξ′ξ1 is called the anharmonic ratio of the range (αα′, pp1). It is to be carefully noticed that the anharmonic ratio of a range of four collinear elements is here defined apart from the introduction of any idea of distance. It is also independent of the intensities at which α and α′ happen to represent their elements. For it is obviously unaltered if α, α′ are replaced by αα, α′α′, α and α′ being any arbitrary quantities.