enables the longitude to be obtained with certainty to within twenty miles.

be

The direct observation of lunar distance may true to 30 seconds with tolerable certainty; taking this amount of error as a guide to results, it will occasion a difference of one minute in time, or of fifteen angular minutes in longitude, while the errors inherent in the corrections for refraction will certainly produce more than five minutes' more error, or 20 minutes in all.

In the event of calculated altitudes being used instead of observed ones, there is then the liability to error in juxta-meridional cases from the assumed latitude on which they depend, or, in other cases, from error in time.

It has been asserted that by this method the longitude may be obtained from a set of such observations to within ten miles, on the supposition that the errors will compensate each other in the mean; this cannot, however, be justified, as means of sets simply have the result of diminishing the effect of the worst observations, while they also diminish that of the best. It is hence waste of labour to attempt by this method to get results nearer than the limit above mentioned.

In making use of the moon's motion to determine longitude under the above circumstances, its position must be determined at the moment of observation by its angular distance from some known heavenly body measured with a sextant or reflecting circle, while the true altitudes, both of the moon and of the star, sun or planet observed, are also necessary for the same moment. The Nautical Almanac gives for every three hours of Greenwich mean time the distances of the moon from the sun, the planets, and certain principal stars; hence

the observer can find the Greenwich mean time corresponding to the ascertained lunar distance, while the difference between this and the true local mean time is the longitude estimated from Greenwich expressed in time, which can then be reduced to angular measurement, as before explained.

The most convenient mode of conducting this observation at sea is to have three observers with three instruments, so that the two altitudes and the lunar distance may be perfectly simultaneous; the lunar distance requires correction for semi-diameter and for refraction, while the observed altitudes must be corrected as before explained in the paragraph on latitude. The lunar distance itself requires most precision, as one second of error in observation causes about two minutes of error in longitude. The calculation necessary is best illustrated by example, for which refer to the collection of examples at the end of this chapter.

When one observer alone is available, the lunar distance is observed, and the two altitudes may be calculated; the latitude in this case must be known, the right ascension and declination of the celestial object, and the sidereal time of mean noon, if the object is other than the sun; so also by previous observation the local time. Having these, the two zenith distances must be calculated for the same moment, and corrected for refraction and parallax to get the apparent zenith distances. This calculation is the converse of the problem before explained for determining the hour angle. (See figure 40.) In this case ZS is one true zenith distance required for the given time, ZP is the given meridian zenith distance for the same object obtained from the latitude and declination, and ZPS is the hour-angle; then ZP, PS, and the

hour angle P are given, and ZS is the side required. The formulæ1 required and used in the Example are

and the required altitude=90—z.

The other method for the single observer involves his observing the altitudes also; in this case he must take several sets of observations, noting the times of each, and using the means of the observed times and lunar distances. Each set of observations will then include, first, an observed altitude of the star used; second, an observed altitude of the moon; third, the lunar distance itself, or several reduced to a mean distance for a mean time; fourthly, the altitude of the moon; and fifthly the altitude of the same star or celestial body. The required altitude of either moon or star corresponding to the time of the mean lunar distance can be obtained by interpolation from the differences of times and differences of altitude. The time intervening between the two observations of altitude of the moon should be as short as possible, on account of the rapid change of declination.

If it is required also to fix the time simultaneously, it is advantageous to have either the moon or the other celestial object near the prime vertical; and if several lunar distances are observed, it is also better to choose celestial objects that are about the same distance from the moon, but on opposite sides of it, thus neutralising the effect of certain errors in those pairs of cases.

1D is a mere intermediate angle in the calculation.

The lunar distance itself is measured between the moon's bright limb and the nearest limb of the other celestial body, and corrected for the semi-diameters of both. When the altitudes of the moon and the star are very unequal, the effect of refraction on each semidiameter will be different, and the correction for semidiameter does not admit of accurate determination even with the aid of tables. It is usual to avoid such cases when possible. Thus, if in the attached figure 41, RV is the difference of altitudes of M and V,

representing the projection of moon and the planet or star, if the angle VMR is greater than 30°, or if either altitude AM or A'V is less than 15°, the case is unsuited to very accurate calculation.

The calculation for lunar distances may be made with the aid of special Lunar Tables, which give many pages of special tables for this purpose; but these are generally very intricate and confusing. The more natural plan is to make use of ordinary tables of logarithms,

Y

and a more self-evident mathematical method, as follows. Let a and ẞ be the apparent altitudes, and a' B' the true altitudes of the star and moon respectively, and y the apparent lunar distance, and σ=1(a+B+y).

whence y, the true lunar distance, is obtained.

;

It is hardly necessary to add, that at sea an indistinct horizon would prevent the adoption of observed altitudes and render calculated altitudes necessary as before explained.

In this class of calculation, the involved computation of time is often unavoidably tedious; but as regards the clearing of the lunar distance itself, far too many difficulties have been made. The problem consists mathematically in the reduction of an arc that undergoes change due to the corrections in the two altitudes. The corrections are due to two causes, refraction and parallax. First as to refraction; the barometric and thermometric readings enable this to be calculated (see table of altitude-corrections in the Pocket Logarithms) within moderate limits of accuracy; beyond these, it must be noticed that such readings are true for the actual position of the observer only, while the visual ray, affected by refraction, passes through atmosphere under other conditions, hence the estimated corrections for refraction in altitude cannot be absolutely correct. It would, however, be an error to adopt mean refraction in preference, and thus arrive further from the truth; while it would be equally unwise to make use of such Lunar tables as 1 D is a mere intermediate angle in the calculation.