A scale of shade is used for this system, founded upon the same principles as that already given for the horizontal system. The scale adopted is due originally to Major Lehmann, of the Saxon Infantry; but it has received some modification to adapt it to the requirements of practice. [Fig. 75] shows Lehmann’s scale. It is constructed for every 5°, from a level up to a slope of 45°, which is the steepest slope at which earth will stand. Each division of the scale corresponding to a given slope is subdivided into nine parts, to show the proportions of black to white. For a level, the whole of these spaces are left white; for a slope of 5°, the proportion is one black to eight white; for a slope of 10°, two black to seven white; and so on up to 45°, for which slope we have all black. The longitudinal divisions of the scale below that against the outer edge A B contain the same proportions of black to white, but equally distributed to show the mode of applying it. Thus, in the division o p r s, corresponding to a slope of 5°, the single black space is, in E F G H, divided into two equal parts and distributed; in G H I K, these two parts are again equally divided and distributed; and so on throughout the other longitudinal divisions. If now the scale be cut off along the line L M, the part L M C D will constitute a scale, the graduated edge L M of which will furnish us with a means of marking off the distance between the centres of the shading lines.

Fig. 75.
Lehmann’s Scale of Shade.

[Larger illustration] (48 kB).

To find the proportion of black to white in the foregoing scale for any given slope:—Subtract the given inclination from 45° for a denominator, and put the given inclination for a numerator. In the scale, as drawn in the figure, the variations are by 5°; but it is obvious that a scale may be drawn in the same manner to mark smaller variations, if thought desirable.

In applying this method in the United States’ Coast Survey, it was remarked that “this scale of shade does not represent slopes greater than 45°, thereby limiting the graphic capabilities and effect of the map. It also makes the slopes too dark as they approach the inclination of 45°, and does not well represent slopes of less than 5°, which latter it is often desirable and necessary to express distinctly.” The following modification was therefore made:—

Slope.Proportion of
Black.White.
21⁄2°or23⁄4°110
5°„6°29
10°„11°38
15°„16°47
25°„26°56
35°65
45°74
60°83
75°92

By this scale, the slighter slopes are represented distinctly. For slopes less than 16°, the shades are darker than in Lehmann’s scale; this makes their difference more noticeable. Above 25° the shades are lighter.

A further modification, which for ordinary purposes possesses the advantages of simplicity and facility of application, has been made in England, and very generally adopted. This modification consists in fixing with accuracy only three proportions of black to white for three medium slopes, as follows:—

Slope.Proportion of
Black.White.
Level..all
15°12
221⁄2°11
30°21
45°all..