Plane trigonometry with practical applications, by Leonard E. Dickson.

32

TRIGONOMETRY

[Art. 22

the refracted ray BC makes with this perpendicular QBP is the
angle of refraction. Experiments show that the quotient
sin i
- index of refraction
sin r
has the same value, approximately 4/3, whatever be the angle of
incidence i. Hence, if i = 30~,
3 3
sin r = sin 30~ =- r = 22~.
4 8'
The index of refraction from air to crown glass is about 3/2.
To find the index of refraction from water to air, let the bottom of
the tank (Fig. 21) containing the water be horizontal and transparent. For the same ray of light passing through the bottom
into the air, let i' be the angle of incidence and r' the angle of
refraction. Then i' and r are equal, being alternate angles, while,
as shown by experiments, r' = i, so that the ray CD after passing
through the water, with its upper and lower surfaces parallel, is
exactly parallel to the ray AB before entering the water. Thus
the index of refraction from water to air is
sin i' sin r 3
sin r' sin i 4
and is the reciprocal of the index of refraction from air to water.
EXAMPLE. A pebble lies at the bottom of a pool of water 4 feet deep.
How far below the surface will the pebble appear to be to a man above if
his line of vision makes an angle of 5~ with the perpendicular to the water?
Solution. The ray of light CBA (Fig. 21) proceeds from the pebble
C to the eye at A, the angle of incidence being r and the angle of refraction
i = 5~. Thus
sin r 3
sin 5 = sin 5~ = 0.0872, sin r = 0.0654, r = 3~45'.
In triangle QBC, adj. side BQ is 4 ft., or 40 tenths of a foot. Table VI
shows that when adj. side is 40 and the angle is 3~ or 4~, the opp. side is
2.09 or 2.80 respectively, and by interpolation is 2.62 for 3~45'. Thus
QC = BH = 2.62. Since L GBH = 900-i = 85~, the opp. side HG is 30
(tenths) by Table VI. Hence the answer is 3 ft.