ings cut out of them, and black horse-hairs stretched from Fig. 108. top to bottom of the openings. A small telescope must then be used for ranging these hairs in line. In a hasty survey, straight twigs, with their tops split to receive a paper folded as in the figure, may be used.

A

Fig. 109.
B

H

D F

E

C

(171) By Perpendiculars. The straight line, AB in the figure, is supposed to be stopped by a tree, a house, or other obstacle, and it is desired to prolong the line beyond this obstacle. From any two points, as A and B, of the line, set off (by some of the methods which have been given) equal perpendiculars, AC and BD, long enough to pass the obstacle. Prolong this line beyond the obstacle, and from any two points in it, as E and F, measure the perpendiculars EH and FG, equal to the first two, but in a contrary direction. Then will G and H be two points in the line AB prolonged, which can be continued by the method of the last article. The points A and B should be taken as far apart as possible, as should also the points E and F. Three or more perpendiculars, on each side of the obstacle, may be set off, in order to increase the accuracy of the operation. The same thing may also be done on the other side of the line, as another confirmation, or test, of the accuracy of the prolonged line.

link, and draw the chain tight, pulling equally on each part, and put a pin at the point thus found, C, in the figure. An equilateral triangle will thus be formed, each side being 33 links. Prolong the line AC, past the obstacle, to some point, as D. Make another

equilateral triangle, DEF, as before, and thus fix the point F. Prolong DF, to a length equal to that of AD, and thus fix a point G. At G form a third equilateral triangle GHK, and thus fix a point K. Then will KG give the direction of AB prolonged.

(173) By symmetrical triangles. Let AB be the line to be prolonged. Take any conve

Fig. 111.

which will be a

The symmetri-
Several other

the obstacle. Find, by ranging, the intersection, at E, of DB and AC. From C, measure, on CA', the distance CE' CE. Then range out DC and B'E' to their intersection in P, required point in the direction of AB prolonged. cal points are marked by corresponding letters. points should be obtained in the same manner. In this, as in all similar operations, very acute intersections should be avoided as far as possible.

(174) By transversals. Let AB be the given line. Take any two points C and D, such that the line CD will pass the obstacle. Take another point, E, in the intersection of CA and DB. Measure AE, AC, CD, BD and BE. Then the distance from D to P, a point in the required prolongation, will be CDX BDX AE DP

BEX AC-BD × AE

Other points in the prolongation may be obtained in the same manner, by merely moving the single point C, in the

E

Fig. 112.

P

D

line of EA; in which case the new distances CA and CD will alone require to be measured.

CDX BD

If AE be made equal to AC, then is DP =

BE-BD

If BE be made equal to BD, then is DP =

CDX AE
AC-AE

The minus sign in the denominators must be understood as only meaning that the difference of the two terms is to be taken, without regard to which is the greater.

where in the line CF; a sixth stake, H, at the intersection of CB and DG prolonged; and a seventh, K, at the intersection of CA and EG prolonged. Finally, range out the lines DE and KH, and their intersection at P, will be in the line AB prolonged.

Measure from E to A, and Range out the lines FC and DE

DE produced will clear the obstacle. onward, an equal distance, to F. to their intersection in G. Range out FD and CE to intersect in H. Measure GH. Its middle point, P, is the required point in the line of AB prolonged. The unavoidable acute intersections in this construction are objectionable.

B. TO INTERPOLATE POINTS IN A LINE. (177) The most distant given point of the line must be made as conspicuous as possible, by any efficient means, such as placing there a staff, bearing a flag; red and white, if seen against woods, or other dark back-ground; and red and green, if seen against the sky.

A convenient and portable signal is shown in the figure.

The figure represents a disc of tin, about six inches in diameter, painted white and hinged in the middle, to make it more portable. It is kept open by the bar, B, being turned into the catch, C. A screw, S, holds the disc in a slit in the top of the pole.

Another contrivance is a strip of tin, which has its ends bent horizontally in contrary directions. As the wind will take strongest hold of the side which is concave towards it, the bent strip will continually revolve, and thus be very conspicuous. Its upper half should be painted red and its lower half white.

A bright tin cone set on the staff, can be seen at a great distance when the sun is shining.

178) Ranging to a point, thus made conspicuous, is very simple when the ground is level. The surveyor places his eye at the nearest end of the line, or stands a little behind a rod placed on it, and by signs moves an assistant, holding a rod at some point as nearly in the desired line as he can guess, to the right or left, till his rod appears to cover the distant point.

(179) Across a Valley. When a valley, or low spot, inter

Fig. 116.

B

Ꮓ

venes between the two ends of the line, A and Z in the figure, a rod held in the low place, as at B, would seldom be high enough to be seen, from A, to cover the distant rod at Z. In such a case, the surveyor at A should hold up a plumb-line over the point, at arm's length, and place his eye so that the plumb-line covers the rod at Z. He should then direct the rod held at B to be moved till it too is covered by the plumb-line. The point B is then said to be "in line" between A and Z. In geometrical language, B has now been placed in the vertical plane determined by the vertical plumb-line and the point Z. Any number of intermediate points can thus be "interpolated," or placed in line between A and Z.

(180) Over a Hill. When a hill rises between two points and prevents one being seen from the other, as in the figure, (the upper

Fig. 117.

of which shows the hill in "Elevation," and the lower part in "Plan"), two observers, B and C, each holding a rod, may place themselves on the ridge, in the line between the two points, as nearly as they can guess, and so that each can at once see the other and the point beyond him. B looks to Z, and by signals puts C