[Illustration: FIG. 17.] The simple conical projection is therefore arrived at in this way: imagine a cone to touch the sphere along any selected parallel, the radius of this parallel on paper (Pp, fig. 17) will be r cot [phi], where r is the radius of the sphere and [phi] is the latitude; or if the spheroidal shape is taken into account, the radius of the parallel on paper will be [nu] cot [phi] where [nu] is the normal terminated by the minor axis (the value [nu] can be found from ordinary geodetic tables). The meridians are generators of the cone and every parallel such as HH´ is a circle, concentric with the selected parallel Pp and distant from it the true rectified length of the meridian arc between them. This projection has no merits as compared with the group just described. The errors of scale along the parallels increase rapidly as the selected parallel is departed from, the parallels on paper being always too large. As an example we may take the case of a map of South Africa of the same range as that of the example given in (a) above, viz. from 15° S. to 35° S. Let the selected parallel be 25° S.; the radius of this parallel on paper (taking the radius of the sphere as unity) is cot 25°; the radius of parallel 35° S. = radius of 25° - meridian distance between 25° and 35° = cot 25° - 10[pi]/180 = 1·970. Also h = sin of selected latitude = sin 25°, and length on paper along parallel 35° of [omega]° = [omega]h × 1.970 = [omega] × 1.970 × sin 25°, but length on sphere of [omega] = [omega] cos 35°, 1.970 sin 25° hence scale error = ------------- - 1 = 1.6%, cos 35° an error which is more than twice as great as that obtained by method (a). _Bonne's Projection._--This projection, which is also called the "modified conical projection," is derived from the simple conical, just described, in the following way: a central meridian is chosen and drawn as a straight line; degrees of latitude spaced at the true rectified distances are marked along this line; the parallels are concentric circular arcs drawn through the proper points on the central meridian, the centre of the arcs being fixed by describing one chosen parallel with a radius of [nu] cot [phi] as before; the meridians on each side of the central meridian are drawn as follows: along _each_ parallel distances are marked equal to the true lengths along the parallels on sphere or spheroid, and the curve through corresponding points so fixed are the meridians (fig. 18). [Illustration: FIG. 18.]