for, the breadth of hachure is twice that of the interval; and between these limits and that corresponding to hachures 2 mm. in length, the breadths are graded by eye,-the space remaining constant.

A working-scale in accordance with these rules and conventions may be constructed as follows:

Assuming the scale of the map as T, and the equidistance 40 feet, first ascertain the limiting slope of the 2 mm. hachures.

The equidistance reduced to the scale of the map is

0.05 in.

The tangent of the required slope is therefore = 0.625, and the slope is 32°. From 32° to 45° the space is mm. = 0.02 in.; and the space for slopes less than 32° is found cot. of the slope X equidistance from the formula

These spaces are then laid off in groups, a convenient number in each group, as shown in the rectangles of Fig. 58; e.g., for 25°, lay off ab = 10 X 0.022 in. = 0.22 in., and subdivide it into 10 equal parts. The hachures are then drawn in the spaces according to the modifications of the rules above given the black extending from the vertical lines of subdivision in the same direction throughout the scale. Thus, since the ratio of black to white for the slope of (26° 30', very nearly) is 1, the breadth of hachure for 25° is a trifle less, and for 30° a trifle greater, than the space; for 45°, the breadth is the space, and the breadths diminish regularly from 45° to 30°, and from 25° to 1° where the hachures are fine lines as already described.

A scale of inclination (par. 80) may then be attached, as shown in Fig. 53, or constructed separately; and with this and the scale of shade, the hachures for the different slopes are at once determined.

115. The Operation of Shading in the Vertical System with Vertical Illumination is illustrated in the right half of Plate XI.

The preliminary work (par. 101) having been performed, the hachuring is begun at the summit of the principal elevation, and the work is extended from left to right, as indicated at a, Fig. 62, so that the hachure last drawn shall always be at the left of the pen-point and the interval readily observed.

Having completed the upper row or band of hachures about the summit, the shading is extended downwards by a second row, and so on to the base of the slope. The other elevations are similarly shaded, and finally the slopes of the general surface.

In order to produce good work, certain rules must be observed in drawing hachures.

1. The hachures must follow lines of greatest descent.

2. They are always drawn towards the draughtsman, and the paper should be so disposed as to facilitate this operation.

3. Shades of different intensities are blended into each other, and the limiting hachures of a slope are blended into the level surface. Thus, as in Fig. 62, the hachures for the slopes at b and c having been determined by the scale of shade, they are gradually increased in breadth in working between these points. Similarly, in working down the slopes, the rows corresponding to different degrees of declivity are blended by tapering the extremities of the hachures as along dd or ee. The upper and lower rows are also blended with the level surfaces, as at ƒ and g. The shading is sometimes blended at the bases of slopes by shortening alternate hachures, as shown at g'; but an irregular outline produces the best effect.

4. The hachures of any row should begin at an imaginary line joining the lower extremities of the preceding row, as at h--neither below it, as at i, nor above it, as at k; and as a general rule they should break intervals, although no special effort should be made to this effect.

5. To avoid too much splaying of hachures, a free use of guiding-lines (par. 100) is necessary in hachuring the parts of slopes defined by sharply curved contours. As shown in Fig. 63, the hachures in the vicinity of mm' and lo', are made quite short as compared with the breadth of zone, and the numerous guiding-lines assure their direction.

At p and q is shown the manner of rounding sharp curves by occasionally shortening the hachures; and at r, an application of the foregoing rules to the hachuring of a col, -see also d, par. 100.

In rounding sharp curves, hachures for steep slopes are also tapered, as at s.

In the French system, in order to obtain the proper shade when the hachures diverge considerably, the spaces are measured on an auxiliary contour midway between the principal contours, as at t.

For regular spacing of hachures, a roulette or small toothed wheel, the intervals between the teeth corresponding to the number of hachures per inch, is very convenient for marking off the spaces.

The requisites for good work are a steady hand, good instruments and material, facility in drawing guiding-lines, in blending the different grades of shade along as well as across the zones and in preserving uniformity of shade for like slopes throughout the map; all of which necessitates a careful observance of the rules and considerable practice.

116. A rigid adherence to the scales of shade in hachuring would evidently result in representing slopes strictly in accordance with their steepness; and with the working-scale attached to the map, it would be simply necessary to refer to it, to ascertain the exact inclination at any point; but, aside from the great labor and skill required for such precise work, the reference itself requires a practised eye; and for practical purposes and general use, maps so constructed, with hachures only to represent the slopes, may be said to be inappropriate.

If, however, the contours are in the finished map, and hachuring is employed simply as an accessory to give relief to the forms of ground, exactness of representation is afforded by the contours, much latitude is permitted in hachuring, whereby the labor and skill required are very much lessened, and a map is produced which is not only correct in the delineation of forms, but also attractive in its nicety of detail.

117. Plate IX. affords an example of Hill-shading by the Vertical System, with Oblique Illumination. The operation of shading is similar to that described in par. 115, with a further gradation of the hachures in conformance with the general rule given in par. 98.

118. The relative merits of the horizontal and vertical systems are difficult to determine.

It may be said that the former is harder to learn, for fine work it is slower of execution, and it has the disadvantage that communications, which usually follow the general level, are liable to be confounded with the hachures. In the latter, the hachures, as a rule, break intervals throughout the map; therefore no continuous lines exist to cause confusion of detail, besides, the vertical system appears best suited to engraving, since the finest maps of recent date are engraved in accordance with it.

119. Representation of Rock-surfaces.-The outlines of rock-surfaces and rugged localities are indicated in pencil during the plotting of the contours; and in order that the particular formations may be approximately represented, sketches or verbal descriptions of them should be found in the field-notes. Characteristic lines, extending in the general directions of the main combs and crevices, give an idea of the formation and furnish a basis or frame-work for the shading. Vertical or oblique illumination is used with either system of hill-shading. The latter is simpler, more effective, and when well executed is much more pleasing to the eye, the former requiring so much black on the steep surfaces as to rob other details of their values. The regular hachuring ends at the pencilled outlines above described, abruptly, if the hachured slope is steep, or with fine hachures touching the outlines here and there if this slope is gentle. The rock-shading is effected by groups of hachures, adjacent groups having different directions; and while these are being applied, a variety of harmonious forms, as well as the proper intensity of shade to bring them out, will suggest themselves.

Fig. 39, Plate VIII., contains outline rock-formations. Plate VI., examples of rockshading with vertical illumination; and Plate XII., examples with both vertical and oblique illumination.

It mars a map very much to employ criss-cross lines, or regular blocks of lines oblique to each other, to represent this feature. The study and copying of good examples, or a little practice in sketching rock-forms in outline, and in constructing a rough plan of such outlines, would soon give facility in drawing pleasing signs for them. Variety of shape and a careful consideration of the direction of the light are essentials to good work.

120. Contours Combined with Brush Hill-shading (Plate XIII.).—This system, as employed at the French School of Application, is as follows:

The same principle applies as in the horizontal and vertical systems,-viz., that the intensity of shade is proportioned to the degree of declivity; it is therefore necessary to establish a corresponding law of gradation.

Vertical Illumination is first considered.

32

16 8 4 2

1.

a. Six elementary shades are used, having the intensities or values represented by the fractions 3,,,,and, corresponding to the slopes 1, 1, 1, 1, and, which they respectively represent. These values taken in the above order are also designated by the numbers 6, 5, 4, 3, 2, and I. No. 6 is termed the normal tone or shade, and its value corresponds to that of a series of parallel lines and spaces of equal breadth, as shown in Fig. 64. This figure also shows the order of arrangement of the different shades. The normal shade is obtained by trial with India ink, or other shading material used, and brush,-the correct shade being produced, when, at a short distance from the eye, it appears of the same intensity as the above series of parallel lines. No. 5 is obtained by adding to 6 a quantity of water equal to that it already contains, 4 is obtained from 5 in a similar manner, and so on to number 1; in each case doubling the quantity of water used in the next preceding shade.

b. A working-scale could be formed by filling these rectangles with the elementary shades prepared as already described; but to conform to the method in practice, it is constructed by superposed shades as indicated in "Superposition of Shades,"--see figure. The rectangles from 1 to 6 inclusive are shaded with No. 1; those from 2 to 6 inclusive, again with No. 1; from 3 to 6, with No. 2: from 4 to 6, with No. 3; and so on, sensibly doubling the intensity for each increase in the degree of declivity.

Although the shades for but few slopes are given in the scale, it is readily seen that by the addition of more or less water to the elementary shades, any intermediate shade required can be produced.

c. The application of the working-scale is thus illustrated: Let a plane be moved tangent to the surface of the ground while preserving a constant inclination to the horizon, of say. The line of contact, termed a curve of equal shade, thus formed, would be represented by shade No. 3, corresponding to the inclination of; while the inclination on one side of this curve would be greater and on the other less than. The curve of equal shade, being similarly determined, and represented by No. 2, a zone is defined bounded by these two curves; and it is apparent that in shading this zone, the shade should be graded from one curve to the other.

Supposing that all the curves of equal shade,,,..... 1 are traced upon the map; then, conforming to the working-scale, all slopes steeper than would be covered with shade No. 1, graded to zero at the edges of level ground; shade No. I would then again be applied to slopes steeper than, and graded to zero at curve; No. 3, to slopes steeper than 1, and graded to, and so on for the rest of the slopes.

The intensities of shade so produced conform to the scale, and a true representation of the configuration can thus be obtained.

If irregular slopes intervene between these curves of equal shade, the shades corresponding to them are capable of exact determination as follows:

64

Let be such a slope; the shade value required is++, and shades 3, 2 and I are correspondingly superposed. As a ready means of obtaining these different values the following table is prepared :

It is observed that a slope of 45° is represented by the superposition of all the shades. The result, although darker than No. 6, is by no means black; in fact, all the shades produced form simply a parallel series having certain relative values, as in many other forms of scales

of shade.

d. The next step is to trace the curves of equal shade upon the contoured map, which is readily effected from the consideration that the slope along a normal between adjacent contours is uniform, and that normals of equal length between adjacent contours indicate equal degrees of declivity. Thus, in Fig. 65 the normals n, n,..... being of equal length mark slopes of equal declivity about the watercourse ab, and the dotted line drawn through their middle points is a curve of equal shade. The points n' and n" are located approximately as follows: n' is evidently between c and d; and since the inclination is less rapid from the middle point of cd towards c than toward d, n' will be nearer the lower contour; and, similarly, n" is nearer e than f. The points n, n, .. can be located by measurement upon the map; e.g., to find the curve of equal shade-this ratio representing practically the tangent of 70,-take in the dividers the horizontal equivalent of 7° from the scale of inclination (par. 80), and mark by dots, as in the figure, the different points where adjacent contours are separated by this distance. No sensible error arises from considering the normals as straight where they may be slightly curved.

e. In par. d above, an undulating surface is assumed; but where the forms of ground are more pronounced, they are considered geometrically.

Thus the spur (Fig. 66) may be regarded as a conoidal surface. Draw the parallels gh and ik tangent respectively to the 40 and 20 contours; the line Im joining the points of tangency is sensibly the line of contact of the plane of these tangents, with the surface, and the line op drawn through the middle point of lm, and perpendicular to the tangents, will have the same inclination as the plane.

In practice an opening, qporst, is made in paper or cardboard, the parallel edges qp and or being separated by the perpendicular edge op horizontal equivalent, according to the scale of the map, of the slope for which the curve of shade is desired; and the middle point u of op is marked.

This instrument is so applied to adjacent contours that qp and or are tangent to them, and ol appears equal to pm; a point is marked immediately below u for one point of the required curve; and so on for the other points.

Curves are similarly determined for ravines or re-entrants, the instrument in any case being so placed that the convexity of the contours within the opening shall be towards the edges qp and or.

ƒ. The hill-features requiring no shade are a summit, the base of a cup-shaped depression or hollow, and a col. The white spaces marking them are slightly exaggerated to make them more conspicuous; and the shades beginning here are gradually increased to their proper values at the nearest curves.

g. In practice the curves of equal shade, which are represented by light pencil-dots, are not traced with mathematical accuracy; and after considerable experience in reading con