48° 12′, and C A B = 89° 18′; at B the angle A B C was 46° 14', and A B D 87° 4′; it is required from these data to compute the distance between C and D ?

C

D

From the angle C A B take the angle D A B, the remainder, 41o 6′, is the angle CAD. To the angle D BA add the angle D A B, and 44° 44', the supplement of the sum, is the angle A D B. In the same way the angle A C B, which is the supplement of the sum of C A B and C B A, is found to be 44o 28'.

Hence in the triangles A B C and A B D we have,

Then, in the triangle C A D, we have given the sides CA and A D, and the included angle C A D, to find C D; to compute which we proceed thus:

The supplement of CAD is the sum of the angles ACD and ADC; ACD+AD C

hence

2

= 69° 27', and, by proportion, we have,

ACD+ADC
2

ACDAD C

69° 27'

12.599294

: tan

2

EXAMPLE III.

To determine the altitude of a lighthouse, I observed the elevation of its top above the level sand on the seashore to be 15° 32′ 18′′, and measuring directly from it along the sand 638 yards, I then found its elevation to be 9° 56′ 26′′; required the height of the lighthouse?

C

Let C D represent the height of the lighthouse above the level of the sand, and let B be the first station, and A the second; then the angle C B D is 15° 32′ 18′′, and the angle CAB is 9° 56′ 26′′; therefore the angle A C B, which is the difference of the angles CBD and CA B, is 5° 35′ 52′′.

Wanting to know the height of a steeple, I measured 210 yards from the bottom of it, and then found the elevation of its top above the level of my instrument to be 33° 28′ 40′′; required its height, the instrument standing five feet above the ground?

Let CE represent the steeple, D E the ground, and AD the instrument. Draw A B parallel to D E, then A B and DE will be equal, and BE will be equal to A D, the height of the instrument.

Now in the right angled triangle ABC, there are given A B, 210 yards, and the angle B A C 33° 28′ 40′′. Hence

A

B

D

E

EXAMPLE V.

Coming from sea, at the point D, I observed two headlands, A and B, and inland at C a steeple, which appeared between the headlands ; I found from a map that the headlands were 5.35 from each other, that the distance from A to the steeple was 2.8 miles, and from B to the steeple 3.47 miles; and I found with a sextant that the angle ADC was 12° 15′, and the angle B D C 15o 30'; required my distance from each of the headlands, and from the steeple.

BY CONSTRUCtion.

On A C describe the segment of a circle to contain the angle ADC (Prob. 12. Geo.;) and on B C describe the segment of a circle to contain the angle BD C, and these circles will intersect in D, the place of the ship.

BY CALCULATION.

Let ADB be the segment of a circle described on A B to contain the sum of the two given angles. Join DC and let it meet the circumference of the circle in E, and

join AE, BE. Then the angles E A B, E DB, being in the same segment, are equal to each other; and so, for the same reason, are the angles A B E and A D E. Hence in the triangle A B E, all the angles and the side A B are given to find the side A E; and as all the sides of the triangle ABC are given, the angle B A C may be computed; and the difference of the angles BAC and BAE is the angle C A E, which, therefore, becomes known.

Now, in the triangle C A E, the two sides A C and A E, and the included angle CA E, being known, the angle C may be determined; and hence, as the angle A D C is given, the angle C A D, and consequently B A D, the difference of C A D and C A B. Hence all the angles of the triangle A D C and the side AC are given, whence A D and C D become known; and A B being known, as well as the angles B A D, BD, A of the triangle A B D, the side B D is also determined.

The computation at length is as follows:

The elevation of a spire at one station was 23°50′17′′, and the horizontal angle at this station between the spire and another station was 93° 4′ 20′′; the horizontal angle at the latter station between the spire and the first station was 54° 28.36", and the distance between the two stations 416 feet, required the height of the spire ?

Let CD be the spire, A the first station, and B the second; then the vertical angle CA D is 23° 50′ 17′′; and as the horizontal angles C A B and C B A are 93° 4′ 20′′, and 54° 28′ 36′′ respectively; the angle A C B, the supplement of their sum, is 32° 27' 4".

To find A C.

B