An introduction to astronomy: designed as a text-book for the use of students in college. By Denison Olmsted ...

PARALLAX. 29
earth. The doctrine of parallax teaches how to reduce observations made at any place on the surface of the earth, to such
as they would be if made from the center.
82. The angle AEC is called the horizontal parallax, which
may be thus defined. Horizontal parallax is the change of
position which a celestial body, appearing in the horizon as
seen from the surface of the earth, would assume if viewed
from the earth's center. It is the angle subtended by the
earth's radius, when viewed perpendicularly from the body.
If we consider any one of the triangles represented in the
figure, ACG for example,
Sin AGC: Sin GAZ (= Sin GAC):: AC: CG;
Sin GAZ x AC Sin GAZ..Sin Parallax= O G OG
CG CG
Hence the sine of the angle of parallax, or (since the angle
of parallax is always very small)* the parallax itself varies as
the sine of the zenith distance of the body directly, and the distance of the body from the center of the earth inversely. Parallax, therefore, increases as a body approaches the horizon
(but increasing with the sines, it increases much slower than
in the simple ratio of the distance from the zenith), and diminishes, as the distance from the spectator increases. Again,
since the parallax AGC is as the sine of the zenith distance,
let P represent the horizontal parallax, and P' the parallax
at any altitude; then,
P,
P': P:: sin zenith dist.: sin 900.~. P= -
sin zen. dist.
Hence, the horizontal parallax of a body equals its parallax
at any altitude, divided by the sine of its distance from the
zenith.
83. From observations, therefore, on the parallax of a body
at any elevation, we are enabled to find the angle subtended
* The moon, on account of its nearness to the earth, has the greatest horizontal parallax of any of the heavenly bodies; yet this is less than 1~ (being 57'),
while the greatest parallax of any of the planets does not exceed 30". The difference in an arc of 10, between the length of the arc and the sine, is only 0".18.