The strength of analytical methods is generality. It is not quite easy to bring out effectively the essential generality of elementary Analytical Geometry, In that connection the following points appear to be of fundamental importance:

1. (i) If P, Q, R are collinear points, the ratio PR : PQ is positive or negative, according as the directions * PQ, PR are the same or opposite.

(ii) (1) Cartesian coordinates of the representative point P of a given plane are determined by reference to chosen axial lines xOx′, yOy′ in the plane, and the question of sign may be put thus, using (i) : If A, B are the points at unit distance from O on the half-lines Ox, Oy, respectively, and if the parallels through P to the axial lines intersect the x-axis at M and the y-axis at N, the coordinates x, y of P are given by x = OM: OA, y = ON: OB. They are essentially one-valued functions of the position of P.

Euclid does not define the inclination to one another, or the mutual perpendicularity of non-intersecting straight lines. (For brevity I shall omit the adjective straight before lines.)

The following definition is justified by the 10th Proposition of Bk. XI. Def. : The inclination of two non-intersecting lines to one another, or the “ angle between them ” is the angle between two intersecting straight lines which are parallel to them respectively.