equal to its projection rus; that is, any angle formed by two intersecting lines on the surface is truly represented in the stereographic projection. In this projection, therefore, angles are correctly represented and every small triangle is represented by a similar triangle. Projections having this property of similar representation of small parts are called _orthomorphic_, _conform_ or _conformable_. The word orthomorphic, which was introduced by Germain[27] and adopted by Craig,[28] is perhaps the best to use. Since in orthomorphic projections very small figures are correctly represented, it follows that the scale is the same in all directions round a point in its immediate neighbourhood, and orthomorphic projections may be defined as possessing this property. There are many other orthomorphic projections, of which the best known is Mercator's. These are described below. We have seen that the stereographic projection of any circle of the sphere is itself a circle. But in the case in which the circle to be projected passes through V, the projection becomes, for a great circle, a line through the centre of the sphere; otherwise, a line anywhere. It follows that meridians and parallels are represented in a projection on the horizon of any place by two systems of orthogonally cutting circles, one system passing through two fixed points, namely, the poles; and the projected meridians as they pass through the poles show the proper differences of longitude. [Illustration: FIG. 9.] To construct a stereographic projection of the sphere on the horizon of a given place. Draw the circle vlkr (fig. 9) with the diameters kv, lr at right angles; the latter is to represent the central meridian. Take koP equal to the co-latitude of the given place, say u; draw the diameter PoP¹, and vP, vP´ cutting lr in pp´: these are the projections of the poles, through which all the circles representing meridians have to pass. All their centres then will be in a line smn which crosses pp´ at right angles through its middle point m. Now to describe the meridian whose west longitude is [omega], draw pn making the angle opn = 90° - [omega], then n is the centre of the required circle, whose direction as it passes through p will make an angle opg = [omega] with pp´. The lengths of the several lines are op = tan ½u; op´ = cot ½u; om = cot u; mn = cosec u cot [omega]. Again, for the parallels, take Pb = Pc equal to the co-latitude, say c, of the parallel to be projected; join vb, vc cutting lr in e, d. Then ed is the diameter of the circle which is the required projection; its centre is of course the middle point of ed, and the lengths of the lines are od = tan ½(u - c); oe = tan ½(u + c). The line sn itself is the projection of a parallel, namely, that of which the co-latitude c = 180° - u, a parallel which passes through the point of vision. [Illustration: FIG. 10.]