being over 440 contour; the third length 410 is over 432; the fourth length 436 is over 404; the fifth length 432 is over 408 west of Willoughby Run; the sixth length 428 is over 422; the seventh length 424 is under 432, hence it must have pierced the ground somewhere; 1⁄2 its length, being 426, reaches midway between the contours 424 and 428, hence here is where it pierces.

So the point where any line pierces the ground, starting from a given point with a given slope, can be determined.

To find the visible horizon from a point is to determine the line separating all the visible from the invisible portions of the ground with regard to that point. To do this a number of radiating lines are drawn from the point of observation and along these the portions of ground which are visible are determined by the methods explained above. The points determined as separating the visible and invisible are then joined, forming the horizon, which if the map were large enough would be a closed line perhaps with loops.

In a limited portion of a country it is evident that there may be several visible horizons.

Problem 8: What was the visible horizon of Lane's 7th N. C. at 4 P. M., when in depression on south edge of map 22 inches east of Willoughby Run, height 400? By drawing radiating lines and determining what portions of them are over seen and what over unseen ground, the horizon will be found to run irregularly along the ridge through the word Hagerstown almost up to the hill 480 east of J. Forney's, then back again along Seminary Ridge.

Problem 9: If the horizon is to be determined from a high hill, as Benner's Hill, height 452, the problem becomes more complicated. A point of Seminary Ridge south of Q. McMillan's is just visible, McMillan's being hidden by the ridge west of East Cemetery Hill, the south part of which is hidden by ridge of East Cemetery Hill; the line down the latter extends to contour 428, when it divides: one part going west to the creek, crosses, then tends southwest up the slope to ridge

at about 448, down ridge, across col, over hill in town, down ridge towards Stevens' Run; another part winds off from 428 around to the southeast, off the map, on again, down the ridge to 408, across Rock Creek, etc. There are other portions.

To calculate the height of a point just visible.

Problem 10: An object at i (Fig. 193) is just visible from k, over the point j; what is the height of the object? The distance of j from k is 928 feet and its fall 1 contour (4 feet). The distance of i from k is 1,533 feet; hence by similar triangles its fall is 928:1533::4:x=66 feet; but i is 5 contours or 20 feet below k, while the top of the object is only 6.6 feet below k; hence the object is 20-6.6-13 4 feet high. Or, since the slope of the line from k toj is 4 on 928, or 1 on 232, then the fall from j to the object on this line will be as many feet as 232 is contained times in 605-2.6 feet. Now i being 4 contours or 16 feet below j, and the top of the object 2.6 feet below j, the object must be 16-2.6-13.4 feet high.

Problem 11: The ridge of a house in Gettysburg on the 3d contour below the number 412 is just visible from contour 480 of East Cemetery Ridge; how high is the house? The intervening hill is distant 2,300 feet from the observer and 36 feet lower; the house is 2,860 feet distant; hence by similar triangles 2300:2860::36:x=44.8 feet lower than the observer. But the contour on which the house stands (420) is 60 feet lower; therefore the house is 60-44.8-15.2 feet high.

Problem 12: Where and what is the steepest gradient on the Hagerstown road? Applying the scale of horizontal equivalents it is found on the east slope of Seminary Ridge and just west of Willoughby Run, about 7.5 degrees er 1 on about 7.6.

CHAPTER XXI.

COPYING MAPS.

COPYING SAME SIZE.-Maps may be copied the same

size-

1st. By fastening a piece of tracing-paper or tracing-linen on the map and tracing over the lines with a pencil, or with pen and ink. If tracing-linen is used, the lines are drawn on the glazed side. and if to be tinted, the colors are applied on the back.

2d. By fastening the drawing-paper on the map and holding both against a window, or a pane of glass in a frame, so situated as to receive a strong light on the back then trace the lines.

3d. By transfer paper as previously described.

4th. By dividing the map to be copied into a certain number of equal squares or rectangles, with sides from 1⁄2 inch to 2 inches, depending upon the amount of detail to be copied and the accuracy required. The paper on which the copy is to be made is then divided into squares or rectangles of the same size, which are numbered the same on both.

The points where the different prominent lines on the map, as roads, rivers, etc., intersects the sides of the different squares are marked on the corresponding squares on the paper to contain the copy and then properly joined. These being drawn in, the other details are "sketched in" in their proper positions. For greater accuracy within the squares, points may be located by perpendiculars from the sides, or by the intersection of arcs from two corners of the square.

In copying contours, draw on the map lines of greatest slope on water-sheds and in water-courses; draw these on the

copy, marking on them where the contours cut, and then join them with the proper curves.

Instead of defacing the map by drawing the squares on it, a pane of glass with the proper size squares ruled on it, or a frame with a fine silk thread stretched from side to side forming the proper size squares, can be laid on the map.

5th. By photography in the regular way, or by making the map translucent with oil or otherwise, then placing the printed side next the sensitive side of a plate and exposing to light, etc., thus obtaining a negative from which any number of copies may be made in a variety of ways.

6th. By using the pantograph. This is an instrument

consisting of four pieces of wood about -inch thick, 2-inch wide, and from 18 to 36 inches long, fastened together with movable joints so as to always form a parallelogram. To use it, a tracing-point is made to travel over the outlines of the map; a pencil is so connected with the tracing-point that it is always in a straight line with the tracing-point and with a fixed center, and always at a distance from that center bearing a given constant ratio to the distance of the tracing-point from the center; the pencil draws the outlines of a copy of the map in the given ratio.

REDUCING AND ENLARGING MAPS.-Maps may be enlarged or reduced by photography, or with the pantograph, but the only method usually available for officers is by means of squares or rectangles. The original will be divided as explained, and the paper to contain the copy will be prepared with the same number of squares or rectangles having sides

bearing the required ratio to those of the original. The details are then copied to the enlarged or reduced scale.

The distinction must be thoroughly understood between enlarging to two, three, or any other times the size (area), and to two, three, or any other times the scale. Likewise for reductions. Thus a map six inches square containing 36 square inches enlarged to two times the size or area will contain 72 square inches and will be 8.485 inches square; while the same map enlarged to two times the scale will contain 144 square inches and be 12 inches square.*

* In enlargements in terms of size or area, the relation of the sides is found as follows: n inches on the map will be represented on the copy double the size by n√, and on the copy three times the size by ny, etc. In reductions, n inches on the map will be represented on the copy one-half the size by and on the copy one-third the size

by, etc.