with the above data, the time-piece may be corrected as follows:

In the triangle ZP S, we have S the position of the sun at the time of observation,

ZP the complement of the latitude,

Z S, the zenith distance, or the complement of the observed altitude;

And PS, the polar distance which is the complement of the declination at the time of the observed altitude.

The three sides of the triangle being thus given, the angle Z P S, usually called the "hour angle," that is, the difference between the times of the sun's being at S and at S' on the meridian, is obtained.

The angles of a spherical triangle, in which the sides are given, may be obtained by the following formula (THOMPSON'S Trigonometry, page 33),

Sin. A = sin. (
√

or as it may be expressed,

s − b) sin. (s—c) r2

sin. b sin. c

Sin. A=

r

sin. (s—b) sin. (s-c). see Emata

sin. b

sin.c

By taking the logarithms of both members of the equation, we find

log. sin. A = ///

10-log. sin. b+10 −log. sin. c +

log. sin. (sb) + log. sin. (s−c).

In a similar manner we find

log. cos. A = {10- log. sin. b + 10-log. sin. c +

and log. tan. A =

log. sin. s + log. sin. (s — a),

10 - log. sin. s +10 - log. sin. (s - a)

+ log. sin. (s—b) + log. sin. (s−c) This last formula is perhaps the most convenient for practice.

Example:-Given the apparent altitude of the sun's lower limb in lat. 54° 36', April 4, 1823, equal to 29° 24', at ten minutes past nine in the morning, by a clock, to find the error of the clock.

(See fig. p. 284) Co-altitude SZ = 60

21 35

s-SZ= 29

47

2010-log. sin. s

0.0000016

47 45 +10-log. sin. (s—a)0·3037217

s-ZP 54 45 20 s-SP= 5 36 14

"

+ log. sin. (s—b) 9.9120612 +log. sin. (s-c) 8.9897025

2)19-2054870

ZPS=21 49 57 log. tan. ZPS = 9-6027435

To reduce this to time, we have (one hour being

54: 12-54 39, to be subtracted

288

SEMIDIAMETERS OF THE SUN FOR THE DIFFERENT MONTHS THROUGHOUT THE YEAR.

PARALLAX OF THE SUN ON THE FIRST DAY OF EACH MONTH, THE MEAN HORIZONTAL PARALLAX BEING 8.60".

AUGMENTATION OF MOON'S SEMIDIAMETER ACCORDING TO HER INCREASE IN ALTITUDE.

The Moon's horizontal semidiameter is found in page 3 of each month in the Nautical Almanac, for every day at mean noon and midnight at Greenwich; and the Sun's in page 2 for every mean noon.

Moon's app. Altitude.

Horizontal Semidiameter.

14' 30" 15' 0" 15' 30" 16' 0" 16 30"

17' 0"