SECTION XX.

APPLICATION OF TRIGONOMETRY TO MEASURING HEIGHTS AND DISTANCES - REDUCTION OF OBLIQUE ANGLES, TAKEN WITH A SEXTANT, TO 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 :

Let C be the centre of the earth, SR an arc on its surface, A the place of the telescope when the theodolite stands in the ver

tical line CA, B the place of the telescope when it stands in the vertical line CB, AG (perpendicular to AC) the horizontal line at A drawn to meet CG, and BO (at right angles to B C) the horizontal line at B.

A

D

S

B
R

C

P

G

Then, if the telescope at B be directed to a mark or object at A, the elevation of that object above the horizontal line, BO, is the angle, OBA; 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.*

Let SD=RB, and RP=SA. Then because the triangles, APC, DBC, are isosceles, and the angles, CAG, CBO, right ones, the angle CAP+ angle PAG=a right angle; but the angle CAP + half the angle ACP, also make a right angle; therefore the angle, PAG, or its equal, DBO, is equal to half the angle, C.

Now, the depression or angle, GAB = GAP + PAB (or ABD); or GAB=PAG+DBO+OBA; but PAG + DBO = angle C. Therefore the depression, GAB = angle C + elev. OBA; or depr. GAB + elev. OBA = angle C + twice the elev. OBA. 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, AB. 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, SA and RB, are equal; and the true depression 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, GAB, was 24'; and at B the elevation of A, or the angle, OBA, = 12'. The distance of the stations or arc, SR, was 2600 yards, which, allowing 694 miles to a degree, gives 1.28 of a degree nearly, the angle, C.

24′ + 12′- 1'-28 Then, = 17′36, or about 174, the 2 true elevation or angle, OBA; consequently, 174'-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, SA and RB, 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 AB is a base of 300 yards, OB a theodolite, and let the height of the staff, AR, be equal to O B, the height of the instrument; also suppose HOR, the angle

parallel to HO, will be the horizontal base, corresponding

to the measured base, AB.

Now, the angles, HOR, BAC, being equal, we have,

As radius

Το Α Β 300

log. 10.000000

log. 2.477121

So is cosine of 5° (angle, BAC) log. 9-998344

Το ΑC, 298.9

log. 2.475465

The difference of AB and AC is only 1.1 yards. Therefore a reduction of this kind seems unnecessary when the measured base is inclined to the horizon in a small angle, except the operation is intended to produce a very accurate result.

To find the distances, AO, BO, from the stations, A and B, to the inaccessible object, O.

A base, A B, was measured of 730 feet, the ground being nearly level; and having set up marks at A and B, the horizontal angles at those stations, taken with the theodolite,

A=57° 12′

A

0

B

were

B=24 45, whence the distances, AO, BO,

are

required.

The angle at O, or supplement of the angles, A and B,

is 98° 3'. And the calculation will be,

So is sine of angle A, 57° 12′ log. 9-924572

So is sine of angle B, 24° 45′ log. 9.621861

Το ΑΟ, 308.6 feet

log. 2.489485

By construction. - Take A B = 730 from any convenient scale of equal parts; and, by means of a protractor, make the angle at A=57° 12', and the angle at B=24° 45': then the distances, A O and BO, may be measured by the same scale from which AB was taken.

Wanting to know the breadth (DO) of a river.

A base, AB, of 400 yards was measured along the bank, and at the extremities, A and B, angles were taken to an object, O, on the opposite side.

B

Angle OBA=37° 40′ Namely, Angle OAB=59 15.

0

D

A

Hence the breadth, OD,

is required.