m

Ն

k

length of the base; having done which, place yourself at b, and measure the angle, a b g; protract this angle. Next measure from b towards k, noting the distance of d (prolongation of hl); then fix k, and measure on to g, marking the point, s; observe the angle, g kf, which lay down. Now return to b, and measure to e (prolongation of fl); observe the angle, leb, and fix the point, l, by measurement from e; join dl, of which 7 h is a continuation; then measure on to f, and join ƒk. Next measure from e to p, along the base line, and observe the angle, ip e, and measure from p to i and h; join ei, and continue the line towards m. Measure from p to a, observe the angle rap, and find the points rand m. Thus far no notice has been taken of the flêches; but, while at g, the angle, k go, should have been observed, and the prolongation of i h, intersecting go, fixes the point o. The directions of the faces of this flêche are determined by v and s. For the flêche, n, its salient is determined by an angle taken at ƒ, intersecting the prolongation of rm, while the right face falls on lf at w. We have, lastly, only to ascertain the lengths of the faces of these two advanced works, and the task is complete.

Either the sextant or compass may be used for taking the angles, but if the ground is tolerably level, the former instrument will give them most correctly. The linear measurement should be performed with a chain and measuring tape, or, if these are not at hand, a rope may be used. Pacing is not suited to such an operation, and can only be admissible when other means cannot be employed.

A plan of any military work requires to be accompanied by profiles, or cross vertical sections, showing the height and thickness of its parapets and ramparts, width, and depth of the ditches, steepness of the slopes of the work, &c.; but this part of the business belongs to levelling, and will be found in its proper place. [See Levelling.]

TAKEN

WITH A SEXTANT, ΤΟ THE CORRESPONDING HORIZONTAL ANGLES-PORTABLE TRIGONOMETRY.

THE instrument proper for measuring horizontal and vertical angles, in common trigonometrical operations, is a theodolite; but, after all the care that may have been bestowed in correcting the line of collimation, telescope level, &c., it seldom happens that the elevations or depressions shown by the instrument are correct. It is, therefore, always advisable to determine the error, or how much the elevations or depressions are too great or too little. This may be done in the following manner :—

P

Let C be the centre of the earth, S R an arc on its surface, A the place of the telescope when the theodolite stands in the vertical line C A, B the place of the telescope when it stands in the vertical line C B, A G (perpendicular to A C) the horizontal line at A drawn to meet C G, and B O (at right angles to B C) the horizontal line at B.

S

B

R

Then, if the telescope at B be directed to a mark or object at A, the elevation of that object above the horizontal line, B O, is the angle, O B A; and when the telescope is at A, and directed to an object at B, its depression below the horizon, A G, will be the angle, G A B.*

* These problems from Dalby's Mathematics.

Let S D=R B, and R P=S A. Then because the triangles, A P C, D B C, are isosceles, and the angles, CAG, C BO, right ones, the angle C A P+angle P A G =a right angle; but the angle C A P+ half the angle A C P, also make a right angle; therefore, the angle, PA G, or its equal, D B O, is equal to half the angle, C.

Now, the depression or angle, G A B=G A P+P A B (or A B D); or GAB=PAG+DBO+OBA; but PA G +D BO=angle C. Therefore the depression, G A B= angle C+elev. O B A; or depr. G A B+ elev. O B A= angle C+twice the elev. O B A. Therefore the elevation and depression together, lessened by the angle, C, is equal to twice the elevation; consequently, half the difference between the sum of the elevation and depression, and the angle, C, is the elevation.

Now, whatever be the error in elevation or depression, their sum will be constant; for one is always diminished by the same quantity that the other is augmented: hence the preceding rule gives the true elevation, except the angle, C, be greater than the elevation or depression together, in which case the said half difference is the true depression of the highest of the two points or objects, A B.

And when the observations are both elevations, or both depressions, their difference is constant, and half the difference between the angle, C, and that constant difference, will be the true elevation of the highest of the two points, A, B, if the angle, C, be the less, but equal to the true depression of that highest point or object when it is the greater.

Should both the reciprocal observations be depressions (or both elevations), and equal to each other, the vertical heights, S A and R B, are equal; and the true depressions will be half the angle, C.

EXAMPLE.

The following observations were made with a theodolite, for determining the error in the vertical angles taken with that instrument.

Two marks, A and B, were set up exactly at the same height above the ground as the height of the telescope; and at A the depression of B, or the angle, GA B, was 24'; and at B the elevation of A, or the angle, O B A,= 12'. The distance of the stations or arc, S R, was 2600 yards, which, allowing 69 miles to a degree, gives 1'28 of a degree nearly, the angle, C.

Then,

24' + 12' 1'.28

2

= 17'36, or about 171', the

true elevation or angle, O B A; consequently, 17-12 5 is the error, or what the altitudes shown by the instrument were too little, or the depressions too great.

A distance of 600 or 700 yards, however, is sufficient for trying a common theodolite; in which case the angle, C, may be neglected, and the verticals, S A and R B, considered as parallels: the expressions then become more simple. Thus, if one observation be an elevation=17', and the other a depression=13', then half their sum=15′ is the true elevation or depression; and 17'-15'-2' is what the instrument gives elevations too great. If both are elevations or both depressions, half the difference is the true elevation of one station and the true depression of the other.

A base for trigonometrical operations is sometimes measured on sloping ground; it must then be reduced to the corresponding horizontal line, if horizontal angles at its extremities are taken with a theodolite.

Suppose A B is a base of 300 yards, O B a theodolite, and let the height of the staff, A R, be equal to O B, the