EXAMPLE I.

Suppose that on the 27th of June, 1820, sea account, in long. 800 W. from Greenwich, the meridian altitude of the moon's upper limb was observed to be 40° 0' bearing south; required the true latitude?

June 27th, sea account, is June 26th, by Nautical Almanac, on which day the moon passed the meridian of Greenwich at 12h. 45m. and the next day at 13h. 46m. the daily difference being 61m. In Table XXVIII. under 60 (which is the nearest number to 61 in the table) and opposite to the long. 800, stand 13m. which added to 12h. 45m. gives the time of passing the meridian, June 26d. 12h. 58m.

Suppose that on the 27th September, 1820, sea account, in long. 90° E. the meridian altitude of the moon's lower limb was observed 50° 0′, bearing south, required the true latitude?

September 27, sea account, is September 26, by the Nautical Almanac; on which day the moon passed the meridian of Greenwich at 16h. 17m. and the preceding day at 15h. 19m. differing 58m. in Table XXVIII. under 58' and opposite the long. 90° are 14m. which subtracted from 16h. 17m. leaves 16h. 3m. the time of passing the meridian of the place of observation.

Suppose that on the 29th November, 1820, sea account, in the longitude of 150 W. the meridian altitude of the moon's upper limb was observed 60° 26', bearing north, required the true latitude?

Nov. 29, sea account, is Nov. 23, by the Nautical Almanac, on which day the moon passed the meridian of Greenwich at 19h. 20m. and the next day at 19h. 59m. differing 39m. In Table XXVIII. under 39′ or 40′ and opposite the longitude of 1500 stand 17m. which added to 19h. 20m. gives 19h. 37m. the time of passing the meridian of the place of observation.

FROM page 4th of the month of the Nautical Almanac, take out the time of the planet's passing the meridian on the day nearest to that on which the observation was made; this will be nearly the time of passing the meri.. dian of the place of observation.*

Turn the ship's longitude into time, and add it to the time of passing the meridian, when in west longitude, but subtract in east, the sum or difference will be the time at Greenwich nearly. Take out the planet's declination, from the Nautical Almanac, for the times immediately preceding and following the day of observation, and note the difference of the declinations when they are of the same name, but their sum when of different names, and find the interval between these times marked in the Nautical Almanac ; take also the difference between the time first marked in the Nautical Almanac and the time of observation at Greenwich (remarking that this time is one day less than the sea account;) then as the former interval of time is to the latter, so is the sum, or difference of declinations, to the correction of the declination taken first from the Nautical Almanac, additive if that declination be increasing, but subtractive if decreasing; the sum or difference will be the declination of the planet at the time of observation. But you must observe that if the correction of declination be greater than the declination first marked in the Nautical Almanac, their difference will be the sought declination, which will be of a different name from the first declination.

From the observed altitude of the planet (taken by a fore observation) subtract the refraction and dip, the latter being in general about four minutes, and the remainder subtracted from 90° will give the correct zenith distance . nearly; with which, and the declination, the latitude may be found as by an observation of the sun.

If you wish to find the time of passing the meridian more accurately, you must take a proportional part of the difference of the times of coming to the meridian given in the Nautical Almanac, in the same manner as in finding the declination of the planet.

This time is also given by a chronometer, as in note page 124, or in the explanation prefixed to

the tables.

EXAMPLE.

Suppose that on the 23d October, 1820, in long. 650 W. Jupiter passed the meridian to the southward; his meridian altitude being observed was 45° 20′, and the dip 4'; required the true latitude?

9h. 26m.

October 23, sea account, is October 22, by the Nautical Almanac ; now on October 19, by the N. A. Jupiter passes the meridian at To this add the long. 650 W, in time

....

4 20

13 46

Then say, as 6 days (which is the interval between October 19 and October 25) is to 3 days 13 hours (which is the time elapsed between October 19th and October 224. 133 h.) so is 7 minutes to 4 minutes, which added to 7° 39′ S. gives 7° 43′ S. the true declination at the time of observation.

FORM I.-By double altitudes of the Sun.

WHEN by reason of clouds, or from other causes, a meridian altitude cannot be obtained, the latitude may be found by two altitudes of the sun, taken at any time of the day, the interval or elapsed time between the observations being measured by a good watch or chronometer, noticing the seconds, if possible, or estimating the times to a third or a quarter of a minute, if the watch is not furnished with a second-hand. The observed altitudes of the sun must be corrected, as usual, for the semi-diameter, dip, refraction and parallax, in the same manner as in finding the latitude by a meridian altitude. When great accuracy is required, the declination must be found at the time of each observation, using the third method of solution hereafter given, but when the sun's declination varies slowly, or the elapsed time is small, it will in general be sufficiently accurate to find the sun's declination for the middle time between two observations, and to consider it as invariable during the observations, computing the latitude by the first or second method.

This manner of finding the latitude is in general most to be depended upon where the sun's meridian zenith distance is great. If the sun passes the 'meridian near to the zenith, much greater care must be taken in measuring the altitudes and noting the times, than would be necessary under other circumstances. The nearer the sun is to the meridian at the time of one of the observations, the more correct the result will commonly be. In general the elapsed time ought to be as great or greater than the time of the nearest observation from noon. Similar remarks may be made upon every one of the following forms.

In all these observations it is supposed that the watch moves uniformly according to apparent time, measuring twenty-four hours from the time of the sun's passing the meridian on two successive days at the same place of observation. If the watch gain or lose on apparent time, supposing the observer

to be at rest, a correction must be applied for the gain or loss during the time elapsed between the observations, so as to obtain accurately the elapsed time or hour angle. It is not required that the watch should be regulated so as to give precisely the hour of observation; the only thing required is to find the lapsed time with all possible accuracy.

FORM II.-Double Altitudes of a Star.

Double altitudes of a fixed star may be used in finding the latitude, and the calculation is almost identical with that of double altitudes of the sun; the only difference consists in adding a small correction to the elapsed apparent time between the observations, on account of the daily acceleration of 3′ 56′′ in the time a star comes to the meridian on successive days. This correction is obtained to a sufficient degree of accuracy by adding one second for every six minutes of the elapsed time; the sums will be the corrected elapsed time or hour angle, to be used in the calculation, either by the first, second or third method. Thus if the elapsed time was 3h. or 180m. the correction would be 13° or 30′′, making the corrected elapsed time or hour angle 3h. 0′ 30′′. If great accuracy is required, find the correction in Table XXXI. in the column marked at top 3' 56", and at the side with the elapsed time. In the preceding example, this Table would give 29" for the correction, instead of 30".

In observations of a fixed star the altitudes are to be corrected for dip and refraction, as in finding the latitude by a meridian altitude. The declination of the star is to be found in Table VIII.*. With these altitudes, the declination and the hour angle, the calculation is to be made by either of the three methods hereafter given.

The chief difficulty in observations of this kind with a fixed star is the want of a good horizon in the night time. The method, however, might sometimes be used with success, soon after the dawn of day, or late in the evening twilight, at a time when the horizon is well defined, and the star sufficiently bright to bring its reflected image to the horizon. Sometimes a good horizon is produced by the aurora borealis, in which case a good observation might be made with stars in the northern horizon, but a single observation of the polar star will answer the same purpose, and be much more simple. FORM III.-Double Altitudes of a Planet.

Double altitudes of a planet (particularly Jupiter and Venus, on account of their great brightness) might sometimes be used with success. The observed altitudes must be corrected for dip and refraction. The parallax and semi-diameter being small may be neglected, except in cases where extreme accuracy is required. The declination of the planet is to be found in page IV. of the Nautical Almanac, for the supposed time at Greenwich. The daily variation of the time of coming to the meridian is also to be found in the same page where the hour is marked at intervals of 6 days, and thus the time elapsed between the passage of the planet over the meridian on two successive days is found; then the corrected elapsed time or hour angle is obtained by the following

RULE. As the interval of time between two successive passages of the object over the meridian is to 24 hours, so is the apparent elapsed time between the observations, to the corrected elapsed time or hour angle: Or more simply by means of Table XXXI. finding the daily variation in the time of coming to the meridian at the top and the elapsed time at the side, the corresponding correction is to be added to the elapsed time when the time of coming to the meridian is earlier on successive days, as is generally the case, but subtracted if later, the sum or difference will be the corrected elapsed time or hour angle nearly.

With this hour angle, the declination and corrected altitudes, the latitude may be found by either of the three following methods of calculation.

*Or more accurately in the Nautical Almanac, if any one of the twenty-four bright stars is observed, whose place is given in that work.

If the daily variation be less than 3 1-2 minutes, which is the smallest in the table, you may multiply the daily variation by 2 or 3, &c. and divide the result by the same number, and the correction will be stained. T

FORM IV.-Double Altitudes of the Moon.

Double altitudes of the moon may also be used in finding the latitude. These observations may be easily and very accurately made, but the calculation is much more complex than any of the preceding methods, on account of the great change in the moon's declination and right ascension, during the elapsed time between the observations. If, however, by the times of observation, and the longitude of the ship; or else by a chronometer, the time at Greenwich can be obtained within a few minutes; we may, from pages VI. VII. of the Nautical Almanac, find the corresponding declination, semidiameter and horizontal parallax of the moon for each of these observations. With the horizontal parallax and the moon's apparent altitude, find the corréction in Table XIX. which being subtracted from 59′ 42′′, leaves the correction of the moon's altitude for parallax and refraction, which is to be added to the corresponding observed altitude corrected for semi-diameter and dip, and in this way the moon's correct central altitude is to be obtained at each observation. Lastly, the time of the moon's passing the meridian on successive days in page VI. of the Nautical Almanac, gives the interval of time between two successive passages of the moon over the meridian,† and this time is to 24 hours as the elapsed time between the observations is to the corrected elapsed time or hour angle. With this hour angle, the correct central altitudes and the declinations, the latitude may be found by the third of the following methods of calculation, it being very rare that the two first methods can be used, on account of the great change in the moon's declination.

*

FORM V. By altitudes of two different objects, taken at the same time. The latitude may be obtained by observing, at the same moment of time, the altitudes of two heavenly bodies: as for example, (1) The sun and moon.‡ (2) The moon and a fixed star or planet. (3) A planet and a fixed star. (4) Two planets. (5) Two fixed stars. In these methods the altitudes are to be corrected as in the preceding Forms, for dip and refraction; also for parallax and semi-diameter when necessary, as is always the case in observations of the moon and sun. The declinations of the bodies are to be found for the supposed time of observation, reduced to the meridian of Greenwich, by means of the Nautical Almanac, or by Table VIII. for the fixed stars, as before taught. Then the difference of the right ascensions of the bodies (or that difference subtracted from 24 hours, if it exceed 12 hours) will be the hour angle which is to be used with these declinations and corrected altitudes in finding the latitude, by either of the two first methods if the declinations should be equal, or differ but one or two minutes, otherwise by the third method, which in fact may be considered as the only method to be used in this kind of observations, because, in almost all cases, the declinations of the objects differ considerably.

FORM VI.—By altitudes of two different objects, taken within a few minutes of each other, by one observer.

It may sometimes happen, for want of two good instruments, or from not having two observers, that the preceding Form V. cannot be employed. In this case the whole of the observations may be made by one person, noticing the interval between the observations, and making the calculation as in the following Form VII. But it is in general much better to make the observations as near to each other as possible, and then by a very simple process the calculation may be reduced to that of Form V. in which the observations are taken at the same moment. This is done by observing the first object twice, before and after observing the second object. For if the interval of time between these three observations are equal, as, for example, one minute, or

* When extreme accuracy is not required, we may find the correction for parallax and refraction from Table XXIX. which, if the altitudes are large, will not vary much from the truth.

†This time is given to minutes which in general is sufficient, because if the elapsed time is small, the effect of this correction would be only a few seconds. It might be obtained more accurately by means of the right ascensions of the sun and moon, using the second differences, as taught in the Appendix. * A particular case of this method occurs in taking a lunar observation, which will be treated of separately, because the distance of the two bodies being known, the calculation becomes more simple.