3d. By Graphical Scale, that is, a drawn scale. A graphical scale is a line drawn on the map, divided into equal parts, each part being marked not with its actual length, but with the distance which it represents on the ground. Thus:

[Ill.u.s.tration: Fig. 1]

For example, the distance from 0 to 50 represents fifty yards on the ground; the distance from 0 to 100, one hundred yards on the ground, etc.

If the above scale were applied to the road running from A to B in Fig. 2, it would show that the length of the road is 675 yards.

[Ill.u.s.tration: Fig. 2]

=1863. Construction of Scales.= The following are the most usual problems that arise in connection with the construction of scales:

1. Having given the R. F. on a map, to find how many miles on the ground are represented by one inch on the map. Let us suppose that the R. F. is 1/21120.

Solution

Now, as previously explained, 1/21120 simply means that one inch on the map represents 21,120 inches on the ground. There are 63,360 inches in one mile. 21,120 goes into 63,360 three times--that is to say, 21,120 is 1/3 of 63,360, and we, therefore, see from this that one inch on the map represents 1/3 of a mile on the ground, and consequently it would take three inches on the map to represent one whole mile on the ground. So, we have this general rule: To find out how many miles one inch on the map represents on the ground, divide the denominator of the R. F. by 63,360.

2. Being given the R. F. to construct a graphical scale to read yards.

Let us a.s.sume that 1/21120 is the R. F. given--that is to say, one inch on the map represents 21,120 inches on the ground, but, as there are 36 inches in one yard, 21,120 inches = 21,120/36 yds. = 586.66 yds.--that is, one inch on the map represents 586.66 yds. on the ground. Now, suppose about a 6-inch scale is desired. Since one inch on the map = 586.66 yards on the ground, 6 inches (map) = 586.66 6 = 3,519.96 yards (ground). In order to get as nearly a 6-inch scale as possible to represent even hundreds of yards, let us a.s.sume 3,500 yards to be the total number to be represented by the scale. The question then resolves itself into this: How many inches on the map are necessary to represent 3,500 yards on the ground. Since, as we have seen, one inch (map) = 586.66 yards (ground), as many inches are necessary to show 3,500 yards as 586.66 is contained in 3,500; or 3500/586.66 = 5.96 inches.

[Ill.u.s.tration: Fig. 3]

Now lay off with a scale of equal parts the distance A-I (Figure 3) = 5.96 inches (about 5 and 9-1/2 tenths), and divided it into 7 equal parts by the construction shown in figure, as follows: Draw a line A-H, making any convenient angle with A-I, and lay off 7 equal convenient lengths (A-B, B-C, C-D, etc.), so as to bring H about opposite to I. Join H and I and draw the intermediate lines through B, C, etc., parallel to H-I. These lines divide A-I into 7 equal parts, each 500 yards long. The left part, called the Extension, is similarly divided into 5 equal parts, each representing 100 yards.

=3. To construct a scale for a map with no scale.= In this case, measure the distance between any two definite points on the ground represented, by pacing or otherwise, and scale off the corresponding map distance. Then see how the distance thus measured corresponds with the distance on the map between the two points. For example, let us suppose that the distance on the ground between two given points is one mile and that the distance between the corresponding points on the map is 3/4 inch. We would, therefore, see that 3/4 inch on the map = one mile on the ground. Hence 1/4 inch would represent 1/3 of a mile, and 4-4, or one inch, would represent 4 1/3 = 4/3 = 1-1/3 miles.

The R. F. is found as follows:

R. F. 1 inch/(1-1/3 mile) = 1 inch/(63,360 1-1/3 inches) = 1/84480.

From this a scale of yards is constructed as above (2).

4. To construct a graphical scale from a scale expressed in unfamiliar units. There remains one more problem, which occurs when there is a scale on the map in words and figures, but it is expressed in unfamiliar units, such as the meter (= 39.37 inches), strides of a man or horse, rate of travel of column, etc. If a noncommissioned officer should come into possession of such a map, it would be impossible for him to have a correct idea of the distances on the map. If the scale were in inches to miles or yards, he would estimate the distance between any two points on the map to be so many inches and at once know the corresponding distance on the ground in miles or yards. But suppose the scale found on the map to be one inch = 100 strides (ground), then estimates could not be intelligently made by one unfamiliar with the length of the stride used. However, suppose the stride was 60 inches long; we would then have this: Since 1 stride = 60 inches, 100 strides = 6,000 inches. But according to our supposition, 1 inch on the map = 100 strides on the ground; hence 1 inch on the map = 6,000 inches on the ground, and we have as our R.

F., 1 inch (map)/6,000 inches (ground) = 1/6000. A graphical scale can now be constructed as in (2).

Problems in Scales

=1864.= The following problems should be solved to become familiar with the construction of scales:

=Problem No. 1.= The R. F. of a map is 1/1000. Required: 1. The distance in miles shown by one inch on the map; 2. To construct a graphical scale of yards; also one to read miles.

=Problem No. 2.= A map has a graphical scale on which 1.5 inches reads 500 strides. 1. What is the R. F. of the map? 2. How many miles are represented by 1 inch?

=Problem No. 3.= The Leavenworth map in back of this book has a graphical scale and a measured distance of 1.25 inches reads 1,100 yards. Required: 1. The R. F. of the map; 2. Number of miles shown by 1 inch on the map.

=Problem No. 4.= 1. Construct a scale to read yards for a map of R. F.

= 1/21120. 2. How many inches represent 1 mile?

=1865. Scaling distances from a map.= There are four methods of scaling distances from maps:

1. Apply a piece of straight edged paper to the distance between any two points, A and B, for instance, and mark the distance on the paper.

Now, apply the paper to the graphical scale, (Fig. 2, Par. 1862), and read the number of yards on the main scale and add the number indicated on the extension. For example: 600 + 75 = 675 yards.

2. By taking the distance off with a pair of dividers and applying the dividers thus set to the graphical scale, the distance is read.

3. By use of an instrument called a map measurer, Fig. 4, set the hand on the face to read zero, roll the small wheel over the distance; now roll the wheel in an opposite direction along the graphical scale, noting the number of yards pa.s.sed over. Or, having rolled over the distance, note the number of inches on the dial and multiply this by the number of miles or other units per inch. A map measurer is valuable for use in solving map problems in patrolling, advance guard, outpost, etc.

[Ill.u.s.tration: Fig. 4]

4. Apply a scale of inches to the line to be measured, and multiply this distance by the number of miles per inch shown by the map.

=1866. Contours.= In order to show on a map a correct representation of ground, the depressions and elevations,--that is, the undulations,--must be represented. This is usually done by _contours_.

Conversationally speaking, a _contour_ is the outline of a figure or body, or the line or lines representing such an outline.

In connection with maps, the word _contour_ is used in these two senses:

1. It is a projection on a horizontal (level) plane (that is, a map) of the line in which a horizontal plane cuts the surface of the ground. In other words, it is a line on a map which shows the route one might follow on the ground and walk on the absolute level. If, for example, you went half way up the side of a hill and, starting there, walked entirely around the hill, neither going up any higher nor down any lower, and you drew a line of the route you had followed, this line would be a _contour line_ and its projection on a horizontal plane (map) would be a _contour_.

By imagining the surface of the ground being cut by a number of horizontal planes _that are the same distance apart_, and then projecting (shooting) on a horizontal plane (map) the lines so cut, the elevations and depressions on the ground are represented on the map.

It is important to remember that the imaginary horizontal planes cutting the surface of the ground must be the same distance apart. The distance between the planes is called the _contour interval_.

2. The word _contour_ is also used in referring to _contour line_,--that is to say, it is used in referring to the line itself in which a horizontal plane cuts the surface of the ground as well as in referring to the projection of such line on a horizontal plane.

An excellent idea of what is meant by contours and contour-lines can be gotten from Figs. 5 and 6. Let us suppose that formerly the island represented in Figure 5 was entirely under water and that by a sudden disturbance the water of the lake fell until the island stood twenty feet above the water, and that later several other sudden falls of the water, twenty feet each time, occurred, until now the island stands 100 feet out of the lake, and at each of the twenty feet elevations a distinct water line is left. These water lines are perfect contour-lines measured from the surface of the lake as a reference (or datum) plane. Figure 6 shows the contour-lines in Figure 5 projected, or shot down, on a horizontal (level) surface. It will be observed that on the gentle slopes, such as F-H (Fig. 5), the contours (20, 40) are far apart. But on the steep slopes, as R-O, the contours (20, 40, 60, 80, 100) are close together. Hence, it is seen that contours far apart on a map indicate gentle slopes, and contours close together, steep slopes. It is also seen that the shape of the contours gives an accurate idea of the form of the island. The contours in Fig. 6 give an exact representation not only of the general form of the island, the two peaks, O and B, the stream, M-N, the Saddle, M, the water shed from F to H, and steep bluff at K, but they also give the slopes of the ground at all points. From this we see that the slopes are directly proportional to the nearness of the contours--that is, the nearer the contours on a map are to one another, the steeper is the slope, and the farther the contours on a map are from one another, the gentler is the slope. A wide s.p.a.ce between contours, therefore, represents level ground.

[Ill.u.s.tration: Fig. 5]

[Ill.u.s.tration: Fig. 6]

The contours on maps are always numbered, the number of each showing its height above some plane called a datum plane. Thus in Fig. 6 the contours are numbered from 0 to 100 using the surface of the lake as the datum plane.

The numbering shows at once the height of any point on a given contour and in addition shows the contour interval--in this case 20 feet.

Generally only every fifth contour is numbered.

The datum plane generally used in maps is mean sea level, hence the elevations indicated would be the heights above mean sea level.

The contours of a cone (Fig. 7) are circles of different sizes, one within another, and the same distance apart, because the slope of a cone is at all points the same.

[Ill.u.s.tration: Fig. 7]

The contours of a half sphere (Fig. 8), are a series of circles, far apart near the center (top), and near together at the outside (bottom), showing that the slope of a hemisphere varies at all points, being nearly flat on top and increasing in steepness toward the bottom.

[Ill.u.s.tration: Fig. 8]