TO FIND THE TIME AT SEA, AND REGU LATE A WATCH, BY THE SUN'S ALTITUDE.

We have already noticed the difference between the civil, astronomical, and nautical computation of time; but as it is a subject of great importance, it may not be unnecessary again to repeat, that a civil day is reckoned from midnight to midnight, and is divided into 24 hours; the first 12 hours are marked A. M., the latter 12 hours P. M. being reckoned from midnight in numeral succession from 1 to 12, then beginning again at 1 and ending at 12. Astronomers begin their computation at the noon of the civil day, and count the hours in numeral succession from 1 to 24, so that the morning hours are reckoned from 12 to 24. Navigators begin their computation at noon, 12 hours before the commencement of the civil day, (and 24 hours before the commencement of the astronomical day,) marking their hours from 1 to 12 P. M. and A. M., as in the civil computation.

There are two kinds of time, mean and apparent. Mean time is that shown by a chronometer, which is always regulated to mean solar time. Apparent time is that shown by the sun, estimating the apparent noon to commence at the passage of his centre over the meridian of any place. There is sometimes a difference of a quarter of an hour between mean and apparent time, owing to the unequal motion of the earth in its orbit, and the inclination of its axis. This difference is called the equation of time, which is given in Table IV., A., or more accurately in the Nautical Almanac. It is always necessary to take notice of the equation of time when regulating a chronometer to mean solar time, by means of an altitude or transit of the sun.

We may obtain the apparent time at sea, when the ship makes no way through the water, by observing an altitude of the sun in the morning, and again in the afternoon when at the same altitude, and noting the times by a chronometer; for the middle time between these two observations will be nearly the apparent time of the sun's passage by the meridian; hence the error of the chronometer may be found. A small correction is necessary for the variation of the sun's declination during the interval between the observations, and the method of calculating this correction will be given in this work, but this method cannot often be made use of at sea, by reason of the motion of the vessel.

The best method of obtaining the apparent time at sea, is by observing, by a fore observation, the altitude of the sun's lower limb when rising or falling fastest, or when bearing nearly E. or W.; to this altitude we must add the semidiameter and parallax, and subtract the dip, (or, instead of these three corrections, add 12',† which will answer very well for an observation taken on the deck of a common-sized vessel;; subtract also the refraction, taken from Table XII., and the remainder will be the correct altitude. The ship's latitude must be found at the time of observation by carrying the reckoning forward to that time. The declination must be taken from Table IV., or from the Nautical Almanac, and corrected for the ship's longitude and the time

*There is, as we have already observed in a note on page 147, another method of computing the time, made use of by astronomers, called Sideral time, in which the interval between two successive transits of a fixed star over the meridian is estimated at 24 hours, commencing the day at the time the first point of Aries is on the meridian, so that the hour in sideral time is the same as the right ascension of the meridian.

The semidiameter is, in general, about 16', the parallax never exceeds 9', and the dip is about 4'; and as the two former corrections are additive, and the latter subtractive, the effect of all three corrections will not differ materially from 12' additive.

This must be carefully attended to, because, when the ship is sailing in a northerly or southerly direction, the latitude at the time of regulating the chronometer, may vary considerably from the observed latitude at noon.

The declination may also be found very easily, by taking it out for the time at Greenwich, as

of the sun from the meridian by Table V. Then, if the latitude and declination be both north or both south, subtract the declination from 90°, and you will have the polar distance; but if one be north and the other south, add the declination to 90°, and you will have the polar distance.

Having thus found the correct altitude, latitude, and polar distance, the apparent time of observation may be found by either of the three following methods, of which the first is the most simple, since it does not require the table of natural sines, all the logarithms being found in Table XXVII. This method is abridged by means of the table of hours affixed to the table of log. sines; in using which you must observe, that, if the sine or cosine of the logarithm sought is marked at the top of the table, the title "Hour A. M." or "Hour P. M." is also to be found at the top, and the contrary if the sine or cosine is marked at the bottom.

FIRST METHOD.

Add together the correct altitude of the sun's centre, the latitude and the polar distance; from the half-sim subtract the sun's altitude, and note the remainder. Then add together the log. secant of the latitude, (this and all the other logs. being found in Table XXVII.) the log.cosecant of the polar distance, (rejecting 10 in each index,) the log. cosine of the halfsum, and the log. sine of the remainder; half the sum of these four logarithms, being sought for in the column of log. sines, will correspond to the apparent time of the day in one of the hour columns. To this apparent time we must apply the equation of time, taken from Table IV., A., or from the Nautical Almanac, and we shall obtain the mean time of the observation.

EXAMPLE I.

Suppose that, on the 10th of October, 1864, sea account, at 8h 21m, A. M., per watch, in the latitude 51° 30′ N., and longitude 130° E. from Greenwich, by account, the altitude of the sun's lower limb by a fore observation, was 13° 32', the cor rection for semidiameter, parallax, and dip, 12′. Required the apparent time of observation.

By Example III., page 157, the declination was 6° 40' S.; this added to 90° gives the polar distance 96° 40′. To the sun's observed altitude 13° 32′, I add 12 minutes. and subtract the refraction 4'; the remainder is the correct altitude, 13° 40.

Suppose that, on the 10th of May, 1836, sea account, at 5h 30m P. M., per watch, in latitude 39° 54' N., longitude by account 35° 45′ E. from Greenwich, the altitude of the sun's lower limb, by a fore observation, was 15° 45', the correction for dip, parallax, and semidiameter, being 12 minutes, consequently the correct altitude 15° 54' Required the apparent time of observation.

15° 54' 39 54

By Example IV., page 157, the sun's declination was 17° 29′ N., which, being subtracted from 90°, leaves the polar distance 72° 31'.

's altitude

Latitude

Secant...... 0.11511

Sine..... 9.82388 corresponding to which,

in the column P. M., is.... 5h 34m 28, the apparent time at the place of observation. Equation of time, sub...

3 49

Mean time of observation 5 30 39
Time per watch....... 5 30 00

Watch too slow...... 0.00 39

EXAMPLES TO EXERCISE THE LEARNER.

1. In latitude 36° 39′ S.,'s declination 9° 27' N., the altitude of the 's lower limb in the morning was observed 10° 33';* required the apparent time. Answer, 7h 23m 51'. 2. In latitude 36° 21′ S., O's declination 8° 44′ N., altitude's lower limb in the morning 10° 48';* required the apparent time. Answer, 7h 22m 11'. 3. In latitude 29° 25′ N., 's declination 23° 20′ N., observed altitude of 's lower limb in the afternoon 14° 58;* required the apparent time. 4. In latitude 3° 31′ S., ☺'s declination 20° 3′ S., observed 38° 41' in the afternoon; required the apparent time. 5. In latitude 13° 17′ N., 's declination 22° 10′ S., in the of 's lower limb 36° 26'; required the apparent time. 6. In latitu 21° 36′ S., ☺'s declination 3° 37′ S., in the of 's lower limb 35° 48'; required the apparent time.

SECOND METHOD.

Answer, 5h 41m. altitude's lower limb Answer, 3h 18m 47'. morning observed altitude Answer, 9h 17m 8'. morning observed altitude Answer, 8h 29m 50'.

Find, as in the former method, the sun's correct altitude, the ship's latitude, and the polar distance; thence the sun's correct zenith distance, and the complement of latitude; then add together the zenith distance, co-latitude, and polar distance; from half their sum subtract the zenith distance, and note the remainder. Add together the log. cosecant of the co-latitude, (this and all the other logs. being found in Table XXVII.,) the log. cosecant of the polar distance, (rejecting 10 in each index,) the sine of the half-sum and the sine of the remainder; half the sum of these four logarithms, being found among the log. cosines, will correspond in one of the adjoined columns to the apparent time of day. This may be reduced to mean time, by applying the equation of time found in Table ÏV., Å.,

or in the Nautical Almanac.

The preceding examples I. and II. are thus worked by this method:

:

Cosine.. 9.94170 corresponding to which in the column A. M is 87m 46, the apparent time of day, wnich agrees nearly with the other method.

The correction for dip and semidiameter being 12 additive, the correction for refraction is also to

Cosine.. 9.87241 corresponding to which, in the column P. M., is 5h 34m 25", the apparent time of day, which agrees nearly with the first method.

By the preceding method you may find the beginning or ending of the twilight, by calculating the hour when the sun's zenith distance is 108°, (or when the sun is 18° below the horizon ;) for by observation it has been found that the twilight begins or ends when the sun is at that distance from the zenith.

EXAMPLE III.

Required the time of beginning and ending of the twilight, in the latitude of 42° 23′ N., when the declination is 23° 27′ N.

corresponds to 2h 6m 20 A. M., and 9h 53m 40 P. M. Therefore, the first appearance of the twilight in the morning was at 2 6m 20', and the end of it in the evening at 9h 53 40, apparent time.

THIRD METHOD.

If the sun's declination, and the latitude, be both north or both south, take their difference, but if one be north and the other south, take their sum, and from the natural cosine of this difference, or sum, subtract the natural sine of the true altitude, (both being found in Table XXIV.;) find the log. of their difference, (in Table XXVI. add thereto the log. secant of the latitude (from Table XXVII.) and the log. secant of the sun's declination, (from the same table,) rejecting 10 in each index; the sum of these three logarithms being found in the column of rising (Table XXIII.) the hours, minutes, and seconds, corresponding, will be the apparent time from noon; and by applying the equation of time to the apparent time, we get the mean time.

The preceding examples 1. and II. are thus worked by this third method:

sponding to which, in column rising, is apparent time 5h 34m 30

Equation of time....

Mean time of observation...

Time per watch...........

.sub. 3 49

5 30 41
5 30 00

Watch too slow.....

41 agreeing nearly

with the other methods. The differences between the results of the different methods arise chiefly from not taking notice of the seconds in the angles, and sometimes from not having the natural sines and cosines to 6 or 7 places of decimals; and we remark generally, that it is always best to retain the seconds in the calculation. This is easily done, in the first and second methods, by means of the columns A, B of proportional parts in Table XXVII.; and by retaining the seconds, we are sure to obtain a more correct result in the calculation.