examples, which will chiefly consist of those applicable to the usual cases that occur in practical astronomy and navigation.

Before proceeding to the application of these theorems, it will be necessary to show the method of correcting astronomical observations obtained by the usual instruments.

Of the Method of correcting Astronomical Observations for the Effects of Refraction, Parallax, Semidiameter, and Dip of the Horizon.

The method of computing the refraction is shown in the explanation of Table XVII. It is always to be added to the apparent zenith distance, or subtracted from the altitude. The parallax of the sun answering to the time of the year, and Z. D. or altitude may be taken from Table XVI. and is always to be subtracted from the Z. D. or added to the altitude. If the horizontal parallax of the moon or a planet be known, the parallax in altitude may be found by adding to the log. secant of the altitude the proportional logarithm of the horizontal parallax, the sum will be the prop. log. of the parallax in altitude. The semidiameter of the sun may be taken from Table XV., that of the moon from the Nautical Almanac, and they must be applied by addition or subtraction according to the limb observed, in order to reduce an observation to the centre.

PROBLEM I.

Given the latitude of the place, the sun's altitude and declination, to find the time and the azimuth.

At the Observatory of Edinburgh, on the Calton-hill, in latitude 55° 57′ 21′′ N., on the third of June, 1826, the following observations of the sun's lower limb were taken in the morning; required the time and azimuth, the barometer being at 29.56 in., and the thermometer at 64° F.?

1. Now in the figure, (page 87), there are given OP the latitude, and consequently ZP the colatitude, PK the polar distance, and ZK the zenith distance, the place of the sun being K near the prime vertical, as being most advantageous to determine the time with accuracy, or the three sides of the triangle KPL; to find the angle ZPK the time, and the angle PZK the azimuth from the southern meridian PEP'. This, therefore, is solved by means of theorem IX. The polar distance in most problems in astronomy and navigation is reckoned from the elevated pole of the same name with the latitude.

Now the latitude being 55° 57′ 21′′, the colatitude is 34° 2′ 39′′

2. To find the azimuth or the angle KZP, the point K being that in which the circles n m and ZIN cut each other, there are given the three sides of the triangle KPZ.

South in north latitude, or from the North in south latitude. This problem is very useful in navigation for the purpose of finding the variation of the compass, which is the difference between the true and observed amplitude or azimuth.

To determine this, let the observer be supposed to look directly from the centre of the card towards the point representing the true azimuth; then if the observed azimuth is to the left of the true azimuth, the variation is easterly, but if to the right it is westerly to the amount of the difference between them.

Thus let the true azimuth be S.
Observed

Variation

Or about 24 points westerly.

89° 47' E. 62 25

27 22 West.

Ex. 3. Required the time of rising and setting of the sun on the top of the Calton-hill at Edinburgh, in latitude 55° 57′ 20′′ N. and longitude 3° 10′ W. on the 1st of June, 1828, the height of the eye above the sea being 350 feet?

To latitude 56° Ñ. and declination 22° N. nearly, the approximate time of setting by the table is about 8h 28m, and rising 32h 3m.

Approximate time of rising 3h 32m

Setting

Long. in time, West

+ 13

8h 28m + 13

: 8 41

Setting 22° 8' 34" N.

N

The height of the eye being greater than any in the table (XI.) the dip must be computed by the rule, page 40.

Constant logarithm

3.53441

Height of the eye, 350 feet log.

2.54407

3 26 45 of upper limb.

Rising In the same manner, the time of setting of the sun's upper will be found to be 8h 33m 40s.

limb

If the semidiameter had been omitted, the computed time would have been that of the centre. The exact time of the rising and setting of the centre may be found by taking the mean times of the rising and setting of the upper and lower limbs, which, with a sea horizon, may serve to find the error of a clock when a better cannot be obtained.

These results for time and variation have been deduced strictly from the solution of the spherical triangle formed by the data, but they may be found more readily by rules derived from it, as may be seen in various books on navigation and nautical astronomy.

When tables which have proportional parts annexed to them are used, the following method may be advantageously employed for determining the time.

Rule. When the latitude of the place and the declination are of the same name, let their difference, but, if of contrary names, let their sum be taken. Under this difference or sum place the zenith distance, and let the half sum and half difference of these be taken; then add together the secant of the latitude, the secant of the declination, the sine of the half sum, and the sine of the half difference; half the sum of these four logarithms will be the sine of half the hour angle or time from noon, from which the apparent and mean time may be obtained as formerly.

June 2d, 19 5 18.50 P. M., the same as before.

In the above computation the several proportional parts are set down and summed all together, which renders the operation somewhat more easy when our tables are employed.

For the practice of these problems, the following rules derived from the foregoing principles are subjoined.

1. To find the Time.-Set down the true altitude, polar distance, and latitude. Find half the sum of these three, and the difference between the altitude and half sum. Then to the log. cosecant of the polar distance add the log. secant of the latitude, the cosine of the half sum, and the sine of the difference or remainder, half the sum of these four logarithms will be the log. sine of half the hour-angle from the meridian.

2. To find the Azimuth.-Set down the polar distance, the true altitude and latitude, and find half their sum, and the difference between this half sum and the polar distance. Now to the log. secant of the altitude add the secant of the latitude, the cosine of the half sum, and the cosine of the difference, half the sum of these four logarithms will be the log. sine of half the azimuth from the meridian to be reckoned from the south in north latitude, and from the north in south latitude. When a table of reduced versines is given, the sum of these four logarithms will be the log. versine of the hourangle or azimuth respectively.

In the case of determining the time, it would be convenient to estimate it according to the astronomical method of reckoning, namely, from noon to noon, through the whole 24 hours. In using the table of reduced versines, therefore, the time must be taken from the top of the page if the observation be made in the afternoon, but from the bottom if in the forenoon.

3. To find the Time by an Altitude of a Star.-To the time of observation add the longitude in time if west, but subtract it if east, and the result will be the estimated Greenwich time. To this time find the sun's right ascension, and the star's right ascension and de