time as two or three seconds, a measurable quantity even with a small portable instrument (the sextant.)*

It is obvious, therefore, that if the distance of the moon's centre from any celestial body, in or near her path, be computed for any Greenwich time, and this distance be found the same as that given by actual observation at any place, then the difference between the time of observing the phenomenon and the time at Greenwich, when it was predicted to happen, will give the longitude of the place of observation. Now in the Nautical Almanac the distances of the moon from the sun, and from several of the fixed stars near her path, are given for every three hours of apparent Greenwich time, and for several years to come; and the Greenwich time, corresponding to any intermediate distance, is obtainable by simple proportion with all requisite accuracy; so that by means of the Nautical Almanac we may always determine the time at Greenwich when any distance observed at sea was taken.

The distances inserted in the Nautical Almanac are the true angular distances between the centres of the bodies, the observer being considered as at the centre of the earth, and to the true distance therefore every observed distance must be reduced; it is this reduction which constitutes the trigonometrical difficulties of the problem; and it consists in clearing the lunar distance from the effects of parallax and refraction; how to do this it is now our business to explain.

* The sextant is constructed upon the same principles as the quadrant. It is usually of brass, is made to hold in the hand, whereas the quadrant is suspended at the centre. It measures 1200°, having an arc equal to the circumference, from which unlike the quadrant it takes its name. The angular distance of two heavenly bodies apart is obtained by making the reflected image of one coincide with the other as seen directly.

refraction than it is depressed by parallax. Observation gives the apparent distance ms, and the apparent zenith distances zm, zs: by applying the proper corrections to these latter we also deduce the true zenith distances zм, zs, and with these data we are to determine the true distance, мs, by computation.

Put d for the apparent distance.*

Then in the triangle mzs, we have, (Art. 82,) R being 1,

hence, for the determination of D, we have this equation, viz.,

COS D-Sin a sin a'

COS A COS A'

cos d-sin a sin a'

cos a cos a'

* In observing d with the sextant, it is the nearest point of the limb of the moon which is made to coincide with the other heavenly body, and in observing a with the quadrant, it is the limb also which is made to coincide with the horizon; so that d and a must be corrected for the semidiater of the moon; similar remarks apply to the sun, if he be the other heavenly body.

† Observe that a and A' are the complements of zм and zs.

from which we immediately get

COS D= (cos d-sin a sin a')

COS A COS A'

cos a cos a'

+ sin a sin A'

But cos a cos a' sin a sin a' cos (a+a') Art. 69; transposing cos a cos a' and substituting the value of — sin a sin a' thus obtained, we have

COS D

cosd+cos(a+a')—cos a cos a'

cos a cos a'

-COS A COS A'+sin a sin a'

Dividing the last term of the numerator by the denominator, the quotient is -1; then observing that - COS A COS A' + sin a sin a'cos (A+A') and that cos d+cos (a+a')=2 cos (a+a+d) cos (a+ad) Art. 86, we have

A

2cos(a+a+d)cos(a+ad)cosacosa'

COSD=

cos a cos a'

-COS(A+A).(1)

EXAMPLE.

1. Suppose the apparent distance between the centres of the sun and moon to be 83° 57' 33", the apparent altitude of the moon's centre 27° 34′ 5′′, the apparent altitude of the sun's centre 48° 27' 32", the true altitude of the moon's centre 28° 20′ 48′′, and the true altitude of the sun's centre 48° 26′ 49′′; then we have

d=83° 57' 33", a 27° 34' 5", a' - 48° 27′ 32′′

A 28° 20′ 48′′, a′ = 48° 26' 49";

and the computation for D, by formula (1), is as follows:

(Reject 40 from index) 1.536926=log.344292+

A+A 76 47 37

nat. cos .228460

True distance 83° 20′ 54′′

nat. cos .115832

By glancing at the formula (1), we see that 30 must be rejected from the sum of the above column of logarithms, to wit, 20 for the two ar. comp. and 10 for R, which must be introduced into the denominator, in order to render the expression homogeneous, so that the logarithmic line resulting from the process is 9.536926. Now, as in the table of log. sines, log. cosines, &c., the radius is supposed to be 1010, of which the log. is 10, and in the table of natural sines, cosines, &c., the rad. is 1, of which the log. is 0; it follows that when we wish to find, by help of a table of the logarithms of numbers, the natural trigonometrical line corresponding to any logarithmic one, we must diminish this latter by 10, and enter the table with the remainder. Hence the sum of the foregoing column of logarithms must be diminished by 40, and the remainder will be truly the logarithm of the natural number represented by the first term in the second member of the equation (1). If this natural number be less than nat. cos (A+A'), which is to be subtracted from it, the remainder will be negative, in which case D will be obtuse.

Those who are desirous of entering more at large into the problem of the longitude, and of becoming acquainted with the best methods of shortening the computation by the aid of subsidiary tables, may advantageously consult, besides the works already referred to, the Quarto Tables of J. De Mendoza Rios, Lynn's Navigation Tables, Captain Kater's Treatise on Nautical Astronomy, in the Encyclopædia Metripolitana, Kerrigan's Navigator's Guide and Nautical Tables, and Dr. Myers's translation of Rossel on the Longitude.

Variation of the Compass.

114. We shall conclude this part of our subject by briefly considering the methods of finding the variation of the compass, or the quantity by which the north point, as shown by the compass, varies easterly or westerly from the north point of the horizon.

The solution of this problem merely requires that we find by computation, or by some means independent of the compass, the bearing of a celestial object, that we observe the bearing by the compass, and then take the difference of the two. The problem resolves itself, therefore, into two cases, the object whose bearing is sought being either in the horizon or above it: in the one case we have to compute its amplitude, and in the other its azimuth.