Ridge and Valley Lines.-Ridge lines are lines which part water falling upon them, and from which it flows on opposite sides. A valley line is the opposite of a ridge line, and is indicated by the watercourse which runs in it.

Sketching and Inking in Contours.-A good guide is to observe that the contour lines are perpendicular to the ridge or valley lines. On a long slope or hill sketch first the top and bottom contours, and the middle one; then interpolate the others. Always remember that two contours can never run into one another except on a vertical face, and that if a contour runs entirely round a hill or hollow, it will come back to its commencing point.

In a contoured map, if the levels of the contours are not marked against them, there may be some doubt as to which are the highest and lowest and which are the ridges and which the valleys.

Watercourses indicate the slopes. Hatchings on the under sides of the contour lines as if water were flowing off will also remove the ambiguity. A good way to ink in contours is to use burnt sienna, making every fifth contour line thicker and darker than the others. The contours are, however, often inked in with Indian ink the same as the rest of the map.

Calculation of Contents from Contour Lines.-The cubic contents of a hill to be excavated may be conveniently calculated from contours, the contents of each part between successive contours being approximately equal to the average of the areas of the upper and lower surfaces multiplied by the vertical distance apart of the contours. This method is also applicable to finding the contents of a hollow to be filled in, a reservoir, &c.

Delineation of Ground by Hatchings.-This system is quick and effective but not accurate. The hatchings may be guided by contour lines lightly drawn in. The hatchings should be drawn exactly perpendicular to the contour lines. When the contours are very far from each other, draw in intermediate contours. Hatchings in adjacent lines should "break joint," so

as to show the position of the contour lines, which are lightly drawn in to guide the hatchings and then erased.

Delineation of Ground by Shades from Light Falling Vertically.-Consider the two slopes ac and ad, Fig. 129, then we have

ad COS a1

ac COS 2

Now as the longer the slope is the less lighted it will be, since from Fig. 129 it is seen that the same amount of vertical light falls on each slope, and therefore the amount of light is inversely proportional to the length of the slope, or which is the same thing, directly proportional to the cosine of the angle of the slope. Tables may therefore be prepared giving the relative amounts of light for different slopes. In practice the difference of shade is usually exaggerated. The different shades may be put on with Indian ink or sepia, the tints being made light for flat slopes and heavy for steep slopes, a slope of 60° being quite dark, and one of 30 a shade about half-way between that and white, and so on.

b

Fig. 129. Shading Slopes.

The shading may also be done by contour lines--that is, by interpolating in contour lines, and making them more numerous on steep slopes and less numerous on flat slopes. This, however, is apt to confuse the map.

The most usual method of shading is to make the thickness and distance between the hatchings proportional to the angle of the slope. The required degree of shade may be made either by varying the thickness of the hatching lines or their distance apart, or both.

French Method. In this method the degree of slope is shown by varying the spaces between the centres of the hatchings. The rule is "The distance between the centres of the hatchings equals in. plus of the denominator of the fraction denoting the slope (ie., the tangent of the angle of the

1

slope) in in." The lines are made fine for flat slopes and thicker for steeper slopes. Only slopes from to inclusive are shown by this method.

In this

German Method or Lehmann's Method. method there are nine shades for slopes from o° to 45°, the first is white and the last black. For intermediate slopes the ratio of white to black is as follows:

:

black angle of slope

For example, for a slope of 30°—

white 45° 30°

15

I

30 2

Slopes steeper than 45° are shown by short thick lines parallel to the contour lines. The angle of slope may always be found from a contoured map, since the tangent of its angle is equal to the vertical height between contours divided by the horizontal distance between them.

Delineation of Ground by Shades from Light Falling Obliquely.-The light is supposed to come from the upper left hand corner of the map, and those slopes facing the light have a light shade, and those on the opposite side a dark shade. This method is effective but not accurate.

By drawing a map with contour lines, however, and shading it from oblique light, it may be made both effective and accurate.

Correction of Levels for Curvature.-The line given by the level is tangent to the surface of the earth, as AC, Fig. 130, where AE represents the earth's surface and o its centre.

Thus we have

AC2 = CE (CE + 2EO).

As EC is very small compared with the earth's diameter 2EO, we may put AC2 = EC × 2EO, and we get

AE being the true level line, EC is the correction for curvature, which is therefore equal to the square of the distance divided by the earth's diameter. Taking the earth's diameter

as 7,916 miles, the correction for 1 mile would

be

A

The correction for 10 chains is in. The correction is proportional to the square of the distance and the effect of curvature is to make distant points appear lower than they really are. The effect of curvature and also of refraction is eliminated by placing the instrument midway between the two points the difference of whose levels is required.

Fig. 130. Correction for Curvature.

The effect is to make objects

The error due to refraction that due to curvature, and its

Refraction. -Rays of light coming through the atmosphere are refracted or curved downwards. appear higher than they really are. is thus in a contrary direction to amount is on an average about of the error due to curvature. It, however, varies with the state of the atmosphere, and is somewhat uncertain. See also pages 185, 370.

Other Instruments: Plumb Line Levels.-The A level is shown in Fig. 131. It is so made that when the plumb line

A

Fig. 131.-The A Level.

Fig. 132.-Plumb Line Level.

is adjusted to the mark on the cross-piece, the feet of the level are at the same height, and a line joining them is horizontal.

Another form of the plumb-line level is shown in Fig. 132. In this the cross-piece is at right angles to the plumb line, and is therefore horizontal when the plumb line coincides with its mark. These forms are not convenient for producing a line. For this purpose the last form is inverted. By sighting along the crosspiece we get a level line when the vertical piece is plumbed with the plumb line.

Reflecting Levels. These instruments are made on the principle that a ray of light which strikes a reflecting plane at right angles is reflected back in the same direction. When the eye is reflected in a plain mirror the line joining the eye and its

A

D

B

Fig. 133.
Reflecting Level.

Fig. 134.
Reflecting Level.

image is perpendicular to the mirror, and if the mirror is vertical this line is horizontal, and may therefore be used for finding points at the same level as the eye. The first form (Fig. 133) is a rhomb of lead about 2 in. in the side and 1 in. thick. On one side, the shaded part of Fig. 133, is a mirror. The right hand part of the rhomb is cut off as shown in the figure, and a wire AB is stretched across it. To use the instrument, hold it up by the string D, with the mirror opposite the eye, so that the eye is seen bisected in the mirror by the wire AB. Then look through the opening at в and any point in line with the eye and wire will be on the same level with them. The instrument is made to hang vertical by means of the weight instrument as made by Stanley.

shown. Fig. 134 shows this The second form is a hollow