angle contained between that and the ship's course by compass, corrected for lee-way if she makes any, in the interval between the observations. With this angle as a course enter a traverse table, and the difference of latitude, answering to the distance run during the elapsed time, will be the reduction of altitude.

If the less altitude be observed in the forenoon, the reduction of altitude must be added to it, if the angle between the ship's course and the sun's bearing be less than eight points; but if that angle be greater than eight points, the reduction is to be subtracted from the less altitude. If the less altitude be observed in the afternoon, the reduction is to be subtracted from it, if the angle between the ship's course and the sun's bearing is less than eight points; but if greater, the reduction is to be added to the less altitude. With the corrected altitudes, the elapsed time, and the declination, the latitude at the time of the observation of the greatest altitude will be found, which may be reduced to noon by means of the dead reckoning.

1. Take half the interval between the observations, and call it the half elapsed time.

2. To the sine of the half elapsed time add the sine of the sun's polar distance, the sum, rejecting always ten in the index, will be arc first.

3. To the secant of arc first add the cosine of the polar distance, the sum will be the cosine of arc second, which will be of the same affection or character as the polar distance.

4. To the cosecant of arc first add the cosine of half the sum of the true altitudes, and the sine of half their difference, the sum will be the sine of arc third.

5. Add together the secant of arc first, the sine of half the sum of the true altitudes, the cosine of half their difference, and the secant of arc third, the sum will be the cosine of arc fourth.

6. The difference of arc second and arc fourth is arc fifth, when the zenith and the elevated pole are on the same side of the great circle, passing through the places of the sun at the times of observation, otherwise their sum is arc fifth.*

7. To the cosine of arc third add the cosine of arc fifth, and the sum will be the sine of the latitude.

Ex. 1.-On the 6th of June, 1828, in latitude 58° N., and longitude 48° W., by account at 10h 53m 20 A. м. per watch, the altitude of the sun's lower limb was 52° 20′, and at 1 17m 8s, the altitude of the same limb was 52° 54′, and the bearing per compass S. W. by W. The ship's course during the elapsed time was S., the wind E. S. E., and hourly rate of sailing 8 knots, and the ship making 14 pts. of lee-way. Required the true latitude at the time of observation of the greatest altitude, the height of the eye being 16 feet?

* Should there be any doubt whether the zenith and elevated pole are on the same side of the great circle, passing through the places of the sun, the latitude may be computed on both suppositions, which, being compared with that by account, the true latitude will, in general, be readily discovered with little additional trouble, for it is only arc fourth and its cosine that will require alteration.

Contained angle

31

Interval between the observations = 2h 23m 48-2.4
Distance run 2.4 x8 = 19.2 miles.

=

Now to course 34 points and distance 19′.2, the difference of latitude is 14'.84, and since the least altitude was observed in the afternoon, and the angle between the ship's course and sun's bearing is less than eight points, this reduction is subtractive.

In this example the computation is carried to seconds of arc, but such a degree of accuracy is unnecessary at sea.

2. On the 6th of March, 1827, in latitude 60° N. by account, and longitude 105° E., the altitude of the sun's lower limb was observed to be 19° 42′ at 40h 4m 20s in the forenoon, his centre bearing S. S. E. by compass, and at 1h 32m 36 afternoon it was 21° 8'. The ship's course during the elapsed time was N. W. by N., sailing at the rate of 9 knots per hour, and the height of the eye 16 feet. Required the ship's latitude at the time of taking the greater altitude? Ans.--60° 37' N.

3. August 31, 1827, in altitude 12° 40' S. by account and longitude 165° E. at 11h 13m 30s A. M., the altitude of the sun's lower limb was 66° 9′ 30′′, and at 1h 15m 12s P. м. it was 62° 0′ 15′′, bearing at the same time N. W. W. During the elapsed time the ship was sailing S. W. by W. at the rate of 4 knots per hour, and the height of the observer's eye was 28 feet. Required the latitude at the time of taking the first altitude?

Ans. 11° 37' S.

PROBLEM VI.

1

On finding the Longitude.

I. BY LUNARS.

Since the rotation of the earth about its axis is performed in a day, the sun appears to pass over 360° in 24 hours, and, consequently, over 15° in one hour; therefore it is obvious, that the difference of time between any two places will give the difference of longitude between those places.

A variety of methods have been proposed for determining the longitude of a place, but almost all of them depend upon one general principle, the comparison of the relative times under two different meridians; so that, if the time on two different meridians be known, the difference of these times turned into degrees, at the rate of 15° to an hour, will give the difference of longitude between these meridians.

As the sun apparently moves from the east towards the west, it is evident, that all places lying to the eastward of any meridian will have noon, or any other hour, sooner, or if westward, later, by the precise time the sun takes to pass from the meridian of the one place to that of the other. Hence, if the time on the meridian of Greenwich, the place from which our longitude is reckoned, and that of any other place at the same instant be known, the longitude of the latter place from Greenwich is also known, by turning the difference of time into degrees, at the rate of 15° to an hour.

Among the heavenly bodies which frequently present themselves for observation, there is none whose apparent velocity is so rapid with regard to the sun, planets, and fixed stars near the ecliptic, as that of the moon; the diurnal motion of that object being at a mean rate about 13° 11′. Hence, her distance from these bodies is continually changing in proportion to the time, and an error of 2" in the distance between the moon and any of these bodies will produce an error of about l' only of longitude. Of all the various modes, then, which have been proposed to determine the longitude at sea, it is probable the method by lunar observations will continue to be

the most practicable. It appears also from the numerous observations lately made by several of our most distinguished navigators, that a series of lunars taken at land with good instruments, will, when great nicety in the requisite observations and calculations is attended to, give the longitude with singular accuracy.

The instruments generally employed are a good chronometer for connecting observations taken at different times with one another, two good quadrants for obtaining the altitudes, and a sextant or reflecting circle for taking the distance. These instruments are all described in our usual treatises on navigation and nautical astronomy.

If the sun or star be at a sufficient distance from the meridian at the time of taking the distance, the true altitude of either of these objects will serve to compute the apparent time at the ship, and this compared with the Greenwich time, derived from the lunar distance, will give the longitude. The same thing may be obtained from the moon's altitude, but less readily, as her right ascension and declination must be very accurately computed by applying the equation of second difference. If neither of these objects be in a proper situation for determining the time, the error of the chronometer must be found when either of them or some other object is in a convenient position before or after taking the lunar distances. In correcting altitudes observed at sea, the dip should properly be first applied, and then the refraction computed for the altitudes thus corrected.

On the Minute Corrections of Lunar Distances.

In lunar observations the corrections for the spheroidal figure of the earth are sometimes made by diminishing the equatorial horizontal parallax by the reduction for the latitude only; but unless the latitude and altitudes are in like manner reduced, which leads to a complex calculation, the results are still inexact. The method here proposed is similar to that of Mendoza Rios,* requiring only a small table to facilitate its application. The table has been computed by my ingenious friend, Mr Thomas Henderson, for an ellip1 ticity of

300'

which seems to agree well with the latest measures, and to the mean horizontal parallax 57', which is sufficiently accurate for practical purposes, as the greatest error can hardly exceed 1", and at a mean not above half that quantity. This is within the limits of uncertainty arising from an error in the ellipticity, which

1

1

seems to vary between and even from the best measures, the 295 305

1

mean between which, has been here adopted. No doubt such

300❜

refinements are unnecessary in the usual sea-practice, and in that

See his Collection of Nautical Tables, published in 1801, for this method. Mendoza's Table XIII. (Edition 1805,) corresponds to our Table I., and his Table XIV, is merely the logs of twice the number in Table XIII., to save the constant log 0.30103. Consequently those who compute by Mendoza's Tables can easily employ his Tables XII., XIII., and XIV., by means of our rule.

9

case may be omitted; but as the lunar method, which is still capable of improvement, can be practised with great success at land, it was thought necessary to correct an erroneous rule, which I believe has been generally acted upon.

Rule. When computing the parallax in altitude; to the logarithm of the earth's radius (Table XXIII.) add the secant of the moon's apparent altitude, and the proportional logarithm of the moon's equatorial horizontal parallax, the sum of these will be the proportional logarithm of the moon's parallax in altitude to be employed in computing the true distance. Now from half the sum of the moon's polar distance, the sun's or star's polar distances, and the distance of the moon from the sun or star, subtract the moon's polar distance, and the distance from the sun or star respectively. Then to the constant logarithm 0.30103, add the cosecant of the moon's distance from the sun or star, the sines of the two remainders, and the logarithm of the number from the table (III.) here given; the sum of these is the logarithm of the number of seconds to be always subtracted from the computed distance, while the number from the table itself is always to be added to it to give the true distance on

1 300

the hypothesis of the earth being a spheroid of of ellipticity.

In general the correction of lunar distances for the earth's ellipticity is small, seldom amounting to 10" of distance or 5' of longitude, in most cases likely to occur in practice; and in any place within the tropics, the results on the spherical hypothesis may be considered almost perfectly correct. Within the polar circles, when the moon's equatorial parallax is diminished by the reduction for the la titude, the results will also be nearly correct.

On this subject Mr Henderson has remarked to me, that " the method prescribed by most authors, of allowing for the effects of the earth's spheroidal figure upon the lunar distances, by diminishing the equatorial parallax, is not altogether exact, but leaves an error uncorrected, which, at its maximum under any particular latitude, is nearly one-sixtieth of the reduction of latitude, or angle of the vertical with the radius. The greatest error therefore which can possibly happen in any part of the globe is under the parallel of 45°, where it may amount to 12". Under the equator and poles the error is nothing.

"If the equatorial parallax be employed in the computation of the true distance, the result is liable to a greater error. The maximum error under any particular latitude may be expressed by the hypothenuse of a right-angled plane triangle, in which one side is equal to the sixtieth part of the reduction of latitude, and the other to the correction of the equatorial parallax. Under the parallel of London, the maximum error is 14"."