In the making of the map thus far, one and only one of the three factors mentioned above has been taken into the reckoning: the factor of direction, namely; and the resulting map is drawn to an unknown scale. It is drawn to some scale, of course; there is some ratio between its distances and the distances at which the objects stand apart, but the ratio is unknown. It may be determined: the distance from B to C may be measured, and the distance b-c on the map may be measured, and the ratio of the two distances ascertained. That ratio is the scale to which the map is drawn. Thus the second factor, that of distance, enters in. It may be reckoned with from the beginning.
Suppose the two points B and C, above mentioned, to be signal towers on a straight stretch of railway, and the point A to be the chimney of a house standing by the side of a wagon road which crosses the railroad at C. The map-maker, having at B set down the data described above, in proceeding to C, paces the distance from B to C, and finds it to be, e.g., 3,500 feet. He has previously determined what the scale of his map is to be: say, 1 inch to 1000 feet. He then carefully lays off on ray b-C 3½ inches from the point b, and thus he fixes point c. He then sets up his drawing-board at C; but, instead of shifting the ruler freely upon the paper, he sights from point c to distant object A and brings the edge of the ruler into coincidence with the line of sight. He draws along the edge of the ruler the ray c-A, which, intersecting the previously drawn ray b-A, gives him the point a.
The railroad from b to c may be indicated thus,
and the highroad from c to a represented by two closely spaced parallel lines. (The conventional signs for various features of topography may be found on the back of a U. S. Geological Survey quadrangle.) On the way from B to C there may be a bridge, crossing a stream. The map-maker, pacing the distance, will, without stopping or interrupting the swing of his stride, note the number of paces from B to the bridge, as well as from B to C. He will then have the figures, and can accurately place the bridge upon his map.
He now has a map of a length of railroad and of a length of intersecting highway, drawn to the known scale of 1″ = 1000′.
And, be it noted, this has been accomplished without visiting the point A at all.
Suppose now there be a haystack D, and a tree on a hilltop E, situated with respect to the points already considered thus:
They may be mapped in like manner. The map-maker goes successively to any two of the three points A, B, and C from which the object to be plotted (D or E) is visible; he sets his board at each place, levels it, and turns it until the ray on the map from the point where he stands to another point lies directly in the line of sight to that other point in the landscape. Having so oriented his board, he draws at his successive stations rays in the direction of the object to be mapped (D or E.) The point d or e where those rays intersect will be the mapped location of the object.