Fig. 3
Now lay off with a scale of equal parts the distance A-I ([Figure 3]) = 5.96 inches (about 5 and 91/2 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 = 11/3 miles.
The R. F. is found as follows:
R. F. 1 inch/(11/3 mile) = 1 inch/(63,360 × 11/3 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: