if z1 is the co-latitude of the centre of the map, z the co-latitude of any other point, [alpha] the difference of longitude of the two points, A the azimuth of the line joining them, and c the spherical length of the line joining them, then the position of the intersection of any meridian with any parallel is given (on the spherical assumption) by the solution of a simple spherical triangle. Thus-- let tan [theta] = tan z cos [alpha], then cos c = cos z sec [theta] cos (z - [theta]), and sin A = sin z sin [alpha] cosec c. The most useful case is that in which the central point is the pole; the meridians are straight lines inclined to each other at the true angular differences of longitude, and the parallels are equidistant circles with the pole as centre. This is the best projection to use for maps exhibiting the progress of polar discovery, and is called the _polar equidistant projection_. The errors are smaller than might be supposed. There are no scale errors along the meridians, and along the parallels the scale error is (z / sin x) - 1, where z is the co-latitude of the parallel. On a parallel 10° distant from the pole the error of scale is only 0.5%. _General Theory of Zenithal Projections._--For the sake of simplicity it will be at first assumed that the pole is the centre of the map, and that the earth is a sphere. According to what has been said above, the meridians are now straight lines diverging from the pole, dividing the 360° into equal angles; and the parallels are represented by circles having the pole as centre, the radius of the parallel whose co-latitude is z being [rho], a certain function of z. The particular function selected determines the nature of the projection. [Illustration: FIG. 23.] Let Ppq, Prs (fig. 23) be two contiguous meridians crossed by parallels rp, sq, and Op´q´, Or´s´ the straight lines representing these meridians. If the angle at P is d[mu], this also is the value of the angle at O. Let the co-latitude Pp = z, Pq = z + dz; Op´ = [rho], Oq´ = [rho] + d[rho], the circular arcs p´r´, q´s´ representing the parallels pr, qs. If the radius of the sphere be unity, p´q´ = d[rho]; p´r´ = [rho] d[mu], pq = dz; pr = sin z d[mu]. Put [sigma] = d[rho]/dz; [sigma]´ = [rho]/sin z, then p´q´ = [sigma]pq and p´r´ = [sigma]´pr. That is to say, [sigma], [sigma]´ may be regarded as the relative scales, at co-latitude z, of the representation, [sigma] applying to meridional measurements, [sigma]´ to measurements perpendicular to the meridian. A small square situated in co-latitude z, having one side in the direction of the meridian--the length of its side being i--is represented by a rectangle whose sides are i[sigma] and i[sigma]´; its area consequently is i²[sigma][sigma]´.