Plane trigonometry, by S.L. Loney.

HEIGHTS AND DISTANCES.

41

This table being at 0 and horizontal and the telescope being initially
pointing in the direction OM, the latter can be made to rotate in a
vertical plane until it points accurately towards P. A graduated scale
shews the angle through which it has been turned from the horizontal,
i.e. gives us the angle of elevation MOP.
Similarly, if the instrument were at P, the angle NPO through which
the telescope would have to be turned, downward from the horizontal,
would give us the angle NPO.
The instrument can also be used to measure angles in a horizontal
plane.
44. The Sextant is used to find the angle subtended by any two
points D and E at a third point F. It is an instrument much used on
board ships.
Its construction and application are too complicated to be here
considered.
45. We shall now solve a few simple examples i:
heights and distances.
Ex. 1. A vertical flagstaff stands on a horizontal plane; from a point
distant 150 jeet from its foot the angle of elevation of its top is found to be
30~; find the height of theflagstaff.
Let MP (Fig. Art. 42) represent the flagstaff and O the point from
which the angle of elevation is taken.
Then OM= 150 feet, and / MOP= 30~.
Since PMO is a right angle, we have
MP 1
= tan MOP= tal 30~= (Art. 33)., b y e n of te s e r, 503e
Now, by extraction of the square root, we have
^.3 = 1-73205....
Hence MP = 50 x 1-73205... feet = 86-6025... feet.
Ex. 2. A man wishes to find the height of a church spire which stands
on a horizontal plane; at a point on this plane he finds the angle of
elevation of the top of the spire to be 45~; on walking 100 feet toward the
tower he finds the corresponding angle of elevation to be 60~; deduce the
height of the tower and also his original distance from the foot of the
spire.