density, and temperature of the air, and is hence very variable. In England the amount is nearly zero for observations in the zenith, and about 57·82 seconds for altitudes of 45°, and about 33 minutes for the horizon. It is generally obtained from tables, and modified by observation; hence the need of recording the meteorological condition at times of astronomical observation.

The correction for Parallax is the reduction of vertical angles observed on the earth's surface to a position at the centre of the earth, a mean terrestrial position convenient for reference. This correction, if necessary, is always to be added to the observed altitude. In observations of the stars it is inappreciable, but in those of the sun and planets the small angles by which the parallax is measured may be used without the intervention of their sines, though the correct formula for the parallax (p) at any altitude (A), is

sin psin H x cos A,

where H is the horizontal parallax, which is given in the Nautical Almanac for the sun for every seven or ten days, and for the planets for every day.

In observations of the moon, which is so much nearer the earth, the increase and diminution of the earth's radius at various latitudes has its effect on the parallax of observations taken at such latitudes. The equatorial horizontal parallax (P) for the moon is hence given in the Nautical Almanac for every Greenwich mean noon and midnight; while for any latitude (7) the true horizontal parallax must be computed by the formula

H= P(1_sin2/)

where c is the denominator of the fraction representing the compression of the earth at that latitude. Tables

exist which effect this computation; there is, however, the further necessity of modifying these results to suit the altitude, in accordance with the former formula.

The correction for Semi-diameter. - Altitudes being generally observed to the upper or lower edge or limb of the larger heavenly bodies, the angular value of the semi-diameter must be added or subtracted to reduce the observation to the centre.

The semi-diameters are all given in the Nautical Almanac for every day; but in the special case of the moon the semi-diameter (s) given is that when she is in the horizon; this has to be further corrected for the altitude of observation, as she draws nearer the observer as she mounts in the sky, by adding to it the following amount =s2(0000019 x sin A') where A' is the altitude already corrected for everything up to and inclusive of parallax.

These corrections must be applied to all similar observations of altitude whose values are utilised in calculation. Apparent altitudes differ from true altitudes in being uncorrected for refraction and parallax. In the foregoing method of determining latitude, the altitude is dependent on the accuracy of a single observation at an exact time; it is better therefore to have two instruments and two observers acting simultaneously as checks on each other.

When, however, all the work is done by one observer, who does not wish his results to depend on a single meridional altitude, he may take several altitudes at a few minutes before and a few minutes after the meridional passage, and then deduce a mean altitude for a moment differing by a small interval of time from the mean time

A small set of tables for correcting altitudes is given in the author's 'Pocket Logarithms' and reproduced in Section IV.

of apparent noon.

A small extra-meridional correction

has then to be applied, which is based on the approximate latitude (L) and the declination (D), and is equal to cos L. cos D versin h

sin (L+D) sin I"

where h is the hour-angle for the mean interval. In the event of the celestial object being the sun, moon, or any planet, the allowance for change of declination in the interval must also be made.

A second method of determining the latitude consists in observing the greatest and least altitudes of a circumpolar star; the mean of these is the altitude of the pole which is the latitude of the place. If, however, the circumpolar star is one whose declination is given in the Nautical Almanac, and whose polar distance can thus be obtained, a single altitude at its inferior passage may be sufficient, in which case the latitude is equal to the sum of the corrected altitude and the polar distance.

A third method of determining the latitude, applicable to northern latitudes only, consists in observation of the altitude of the pole star at any time, whether in the meridian or not. Tables for this purpose, with explanation, are given in the Nautical Almanac, and are hence unnecessary here.

To determine the Meridian.

I. Bisect the horizontal angle formed by two positions of a star or of the sun when arriving at the same altitude in the east and in the west, and bring this direction down vertically by plumb-line or otherwise to the ground, and mark it firmly. At night a light showing through a hole in a board is convenient for this purpose.

2. The mean between the eastern and western elongations of a circumpolar star is the meridional direction. 3. At the moment when a truly vertical circle cuts both the pole star and the star Alioth (ɛ Ursæ Majoris) this direction is the meridian.

4. If it be preferred to make use of some easily fixed and found position for an instrument, and also of some conspicuous object, such as a church-spire, for reference at any subsequent time, the azimuth of this latter may be obtained by means of the mean of the angles made with it by the extreme positions of stars explained in methods Nos. 1 and 2.

5. If the latitude is known, and the sun or a star of known declination be observed, a single altitude will suffice, and the azimuth may be calculated by the same process, as time is found from a single altitude, the required angle in this case being Z instead, of P (see the figure); for PZS is the supplement of the azimuth.

6. The variation of the compass being simply the variation of the magnetic from the true meridian, any of the above methods for obtaining either the true meridian or a correct azimuth of a fixed object enable it to be obtained from them; but at sea it is usually found by observing the bearing of the sun at sunrise and at sunset, or when it is a semi-diameter above the horizon, as this allows roughly for refraction.

7. If the true azimuth of the sun be required at any time at sea, it can be calculated from the altitude (a), the polar distance (p), and the latitude (/), in the same triangle as the hour-angle (see figure 40) by the formula 2 cos (p+1+a), cos (p-l- a) sin2 =

The Longitude.

Ist. The difference of longitude is most easily and, if the places are not very distant, also most exactly determined by means of good chronometers that gain or lose steadily and equably. If a chronometer giving true local mean time at one place be conveyed to another place whose local mean time is accurately observed and compared with that of the transferred chronometer, the difference between these two local times is the difference of longitude between the two places expressed in time, after making allowance for the gaining or losing of the transferred chronometer.

The longitude in time is reduced to longitude in arc by multiplying severally the hours, minutes, and seconds of time by fifteen, and thus converting them into degrees, minutes, and seconds of arc.

Comparing timepieces is a matter that requires a little practice, more especially when the number of beats to a second is different in each of them. The mode is to listen to the beats of both timepieces while watching them, and counting those of one of them up to the exactly coincident beat, when the time by the first timepiece is immediately recorded, and the counting of the beats of the other commenced and kept up, till by seeing its hand at some conspicuous division, the minutes and hours of the latter can be noted by eye. The exact moment of coincidence can then be fully recorded.

2nd. When chronometers fail or require checking, and when from any cause a telescope cannot be adjusted to the meridian, as at sea, the method of lunar distances may be employed; this, if all the observations and the rather long calculation be very accurately performed,