If the ground is perfectly level, and you have a looking-glass, lay it down and level it by setting on it a basin full of water; retire till you see the top of the tree reflected in it, then if your distance from the mirror equals the height of your eye, the distance from the mirror to the tree will be equal to its height. In perfectly calm weather the basin of water will do without the mirror, or a shallow pool or river will give an approximation; but, as the ground is always depressed where water settles, there will be some uncertainty about the height of the eye, which will more or less vitiate the observation, and this will also be the case if thirsty animals rush in to disturb it, as in our sketch. Or if the sun or moon is shining, set up a stick, and watch till its shadow is equal to its height, or note when your own shadow equals your height, and the height of the tree and the length of its shadow will also be equal. But, as it may not be always convenient to wait for this moment, the height of the tree may be found by proportion. If the stick is 5ft. and its shadow 7ft., then if the shadow of the tree be 70ft., its height will be 50ft.; or if in looking at its reflection in the mirror, the height of your eye be 5ft., and the distance 8ft., then if the distance from the mirror to the tree be 80ft., its height is 50ft. In either of the first two methods the same rule must be observed; the paper may be folded to a greater angle if you cannot get far enough from the tree, or a smaller one if you must go farther, and the same with the elevation of the rifle. In these cases, carefully measure the base and perpendicular of your smaller angle, and say, “as the base of the small angle is to its perpendicular, so is the distance from the tree to its height.”
Thus, as in Fig. 4 on next page, if the distance between the two rests is 2ft. and the elevation of the rifle 1ft., the distance from the tree must be equal to double its height.
All these observations will apply to any object of which the highest point is perpendicular to the accessible base, such as a precipice, the wall of a fort, or the gable end of a house, but not to the peak of a mountain, two or three miles beyond its base, nor to the pitched roof of a house seen sideways, nor to the spire of a church, or flagstaff on the central tower of a castle, unless the doors of these buildings be opened so that you can continue to measure your base to a point exactly beneath that which you have taken the angular height. Still, if the base be not accessible, it is not impossible to measure the height, for the distance of the object may be taken by any of the plans for ascertaining the breadth of a river, or any of the above methods may be performed twice over, as in Fig. 5;
first, at any convenient distance, b, and secondly, at a measured distance, c, nearer to or farther from the object; and the easiest way of obtaining the result is to lay down on paper the obtained angles, d, e, f, and g, h, i, in due proportion to the measured distance, b, c, between them; then from them to protract the angle, d, g, a, and continuing the base line, find on it the point j, from which a perpendicular would meet the top of the object, a. The distance, b, c, being known, that of the base, b, j, and the height of the tree, j, a, will be best found by measurement of equal parts, but bear in mind that the result can only be an approximation to truth, for every additional operation involves an increase of possible error.