211. To determine the point of intersection of two visual lines.

This is most readily done by three persons, two of whom take their stations in the lines, at some distance from the point of intersection, and, looking along their lines respectively, signal the third until he ranges in both lines. A stake may then be driven at the point of intersection.

This operation may readily be performed by two persons. First, let them run out one of the lines, and stretch a cord or the chain across the course of the other. One of them then taking his station in the second line can signal the other to his proper position.

212. To run a line towards an invisible intersection.

one of the modes to be pointed out. (See Arts. 227–229.) Divide BD in F, so that BF: FD :: AP: PC; that is,

Problem 1.—To draw a perpendicular to a given line from a given point in it.

213. (a.) When the Point is accessible. This may be done on the ground by the methods described in Arts. 88, 89, and 90, using the chain for a pair of compasses to sweep the circles, or by the following methods:

and, with 80 links for a radius, sweep an arc cutting the former in E. CE will be perpendicular to AB.

=

Any other distances, in the same ratio as the above, will answer. Thus, DC might be 30, CE 40, and DE 50. With these numbers no circles need be struck. Lay off DC 30 links; fix the end of the chain at D, and the end of the ninetieth link at C: then, taking the end of the fiftieth link, stretch both parts of the chain equally tight, and set a stake at the point of intersection.

These numbers are very convenient when short perpendiculars are required; but when the line is run to some distance the greater lengths are preferable.

therefore DCE 30°, and ACE = 90°.

=

216. (6.) When the Point is inaccessible.

FE need not be taken equal to DF. If unequal, GH will be determined by the proportion EF: FD:: FG : GH.

(c.) If the line is inaccessible, trigonometrical methods must be employed.

Problem 2. To let fall a perpendicular to a line from a point without it.

(a.) When the point and line are both accessible.

will be the foot of the perpendicular.

Describe the semicircle ECA. Then, if CF is perpendicular to AB, EC is a mean proportional between AE

(b.) If the point is remote or inaccessible.

218. First Method.-In AB (Fig. 86) take any convenient points A and D; erect the perpendicular FDE, making FD DE; range out AE,

=

and EC cutting AB in H, and FH intersecting AE in G: then GBC will be perpendicular to AB.

For, by construction, the triangles ADE and ADF, as also FDH and EDH, are equal in all respects. Hence, AFG and AEC, having two angles and the included side of one equal to two angles and the included side of the other, are equal in all respects; therefore AG = AC. Finally, ABC and ABG have two sides and their included angles respectively equal, whence B is a right angle.

For the perpendiculars to the three sides of a triangle from the opposite angles intersect in the same point.

221. The preceding methods will apply in all the cases

enumerated. They are, however, only to be considered as substitutes for the neater and more accurate methods by the use of the theodolite or transit. Measurements such as those directed above, when they are intended to determine the direction of an important line, require to be made with scrupulous accuracy; for every deviation will be magnified as we proceed. An error of two or three inches, which would be a matter of but little importance in a line of a chain long, would cause a deviation of from twelve to twenty feet if the line were prolonged to a mile.

In the absence of a transit or theodolite, the following simple instruments, either of which can be constructed by any one having a moderate degree of facility in the use of tools, will enable the surveyor to lay out perpendiculars with readiness and considerable accuracy.

222. The Surveyor's Cross. This consists of a block of wood four or five inches in diameter, with two saw-cuts across its centre precisely at right angles. An auger hole should be made at the bottom of each saw-cut, to afford a larger field of view. The block is fastened to the top of a staff about eight or ten inches long. It should turn freely but firmly on the head of the staff.

Instead of saw-cuts, four wires may be set upright at the extremities of perpendicular diameters; but, as these are likely to be deranged, the other form is better.

223. To erect a perpendicular with the cross, set it up at the point at which the perpendicular is to be drawn, and turn it round till one of the cuts ranges with the given line; then, looking through the other cut, the surveyor can direct his assistant to set a stake in the required perpendicular.

If the point is out of the line, take a station as near as the eye can judge to the position of the foot of the perpendicular, and, having set the cross so that one cut may range with the given line, look through the other, and see how far the line of sight misses the given point. Move the cross that distance and test it again. A few trials will determine the proper position.