In the foregoing Examples it has been supposed that the altitudes of the objects were found by observation, it however sometimes happens in the night, that the distance between the Moon and a Star may be very correctly observed when the horizon is so obscure as to render the observed altitudes rather uncertain. Also in the practice of the Lunar Observation on shore, it is not always convenient to observe the altitude at the same time with the distance; in such cases it is necessary to find the Altitudes by calculation. For the computation of an altitude, it is necessary to have the following elements:

1. The Latitude of the place: and its Longitude by account.

2 The apparent time at that place when the altitude is required. 3. The declination of the object, and also its right ascension, together with that of the Sun, if the object, whose altitude is required, be the Moon or a Star.

In the following Rule, the right ascension and declination of the Sun or Moon are understood to be taken from the Nautical Almanac, and the right ascension and declination of a Star from Table I. of the Appendix, or from any other correct Catalogue. The places of all the Stars from which the Moon's distance is given in the Nautical Almanac, will be found in the Table containing the true apparent places of 24 of the principal fixed Stars, at the end of that work.

PROBLEM VI.

Given the Latitude of a place, and its Longitude by account, together with the Apparent Time, to find the True Altitude of a known Celestial Object.

RULE.

1. Find the horary distance of the object from the meridian. This, if the object be the Sun, is the interval between the given apparent time and noon; but if the object be the Moon or a Star, add the Sun's right ascension to the given apparent time, the sum rejecting 24 hours, if necessary, will be the right ascension of the meridian; the difference between this, and the right ascension of the given object, will be its horary distance from the meridian.

2. If the Latitude of the place, and the Declination of the given object, be both North, or both South, their difference will be the meridian Zenith distance of the object; but if one be North and the other South, their sum will be the meridian Zenith distance.

3. Add together the Logarithm of the horary angle of the object, Table XIII., the Logarithms of the Latitude and Declination used as Half Sums, in Table XII., and the Logarithm of the meridian Zenith distance, used as a Latitude, in Table XI; the sum of these 4 Logarithms, rejecting 10 from the Index, will be the Logarithm of an arch in time, in Table XIII.

4. Turn the above found arch into degrees, &c. and using it as a Latitude, take out its Logarithm from Table XI., which add to the Logarithm of the meridian Zenith distance, (before found) the sum will be the Logarithm of a Polar Distance, in Table XI., which will

EXAMPLE.

May 10th, 1824, by nautical time, at 10h. 39m. 25s. apparent time, P. M. in Latitude 37° 42′ N., and Longitude by account 67° 30′ W.: required the true altitude of Antares.

As the apparent altitudes are used in correcting a Lunar Distance, it is necessary to reduce the true altitudes, when found as above, to the apparent altitudes; this, when the object is the Sun or a Star, is done by taking the correction for the given altitude, from Table VI., and adding it to the true altitude, the Sum will be the apparent altitude. Thus the apparent altitude of a Star, when its true altitude is 17° 0', would be 17° 3′ 5′′, or the apparent altitude of the Sun, when the true altitude is 17° 0′, is 17° 2′ 56′′. But when the true altitude of the Moon is to be reduced to the apparent altitude, it will be necessary to proceed as follows:

With the Moon s true altitude, used as a Latitude, take out a Logarithm from Table XI. to this Log.; add the Proportional Logarithm of the Moon's horizontal parallax, the sum will be the Proportional Logarithm of the Moon's parallax in altitude, from which subtract the refraction in altitude, (the star's correction in altitude, Table VI. is the refraction in altitude, of any object;) the remainder being subtracted from the true altitude will leave the apparent altitude.

D

EXAMPLE.

Suppose the Moon's true altitude is 35° 23', when her horizontal What would be the apparent altitude of the

parallax is 59′ 42′′.

Moon?

I. If great accuracy were required, the operation of finding the Moon's correction in altitude ought to be repeated, using the Moon's apparent altitude, as found above, in place of the true altitude, and then subtracting the correction thus found from the true altitude; however, one operation is quite sufficient for the purpose of finding the Moon's apparent altitude, as required in the method of correcting the Lunar Distances, which is given in this work.

II. If an object be near the meridian, in bearing, or azimuth, when its altitude is to be computed, any probable error in the apparent time, will not cause a material error in the altitude; but any error in the Latitude will, in this case, cause nearly an equal error in the altitude.

III. If the object be near the prime vertical, that is near the east or west, at the time its altitude is to be found by calculation, any probable error in the Latitude of the place will cause very little error in the altitude; but an error in the apparent time will then greatly affect the altitude. In this case the error in the altitude arising from an error of 1 minute of time, will, in places near the Equator, be nearly 15 minutes of a degree; in high Latitudes the error is less.

IV. When the object is considerably distant, both from the meridian and prime vertical, its computed altitude is affected by an error either in the Latitude or apparent time; but the error of altitude, arising from an error in the Latitude, will not be so great as when the object is near the meridian, nor will the error, occasioned by an error in the apparent time be so great as when the object is near the prime vertical.

V. The apparent altitudes being found by computation, the true distance and time at Greenwich, are to be found in the same manner as before: but it very rarely happens at sea that the altitude of the Moon may not be observed with sufficient accuracy, for the purpose of clearing the distance of parallax and refraction, nor is it often necessary to calculate the altitude of a Star; however, as any given error in the altitude of the Star, will in general cause a greater error in the computed distance, than an equal error in the altitude of the Moon, it is proper, when the observed altitude of the Star is at all uncertain, to

In cases where there is not a sufficient number of observers to take the distance, and the altitudes of the objects, at the same time, it is necessary to observe the altitudes both before and after the time of taking the distance, and then reduce them, by the Rule of Proportion, to what they would be at the time the distance is observed. This may be done in the following manner :

Here the interval between the time of observing the first altitude of the Sun, and the mean of the times, when the distances were observed, is 3m. 35s.; and as the Sun's altitude decreasts 1° 19′ in the space of 7m. 34s. the change in 3m. 35s. will be 0° 37', which is to be subtracted from the first altitude, because the altitude is decreasing: hence the altitude of the Sun's lower limb, corresponding to the mean distance, is 26° 37'.

In the same manner the altitude of the Moon's upper limb, corresponding to the mean distance, is found to be 31° 36', we have therefore the following set of observation:

Time per watch of obs. Obs. dist & ) 's nrst. limbs. Obs. alt 's 1... Obs. alt. D's up.l. 3h. 27m. 51s. 26° 37 31° 36'

68° 35′ 40′′

From these, the Longitude is to be deduced in the same manner as before. It will be proper, however, to find the error of the watch by means of a set of altitudes, taken before or after the altitudes, to be employed in correcting the distance. It seldom happens but the altitudes of at least one of the objects, may be observed at the same time as the distances, in this case; it is generally proper to observe the altitudes of the Sun or Star along with the distances, and then deduce the altitude of the Moon, as in the foregoing Example.

d 2

This method of finding the Longitude depends on the same principle as the Lunar method, that is, on being able to find the respective times at two meridians, for the same instant of absolute time, when the difference of these times will give the difference of Longitude between the two meridians. For example,

Suppose a Chronometer that keeps mean time exactly, be set to mean time at Greenwich, and then taken to another meridian, where the mean time is found, by observation, to be 4 hours at the instant that the time by the Chronometer is only 2 hours; we know that the place of observation is 30° E. of Greenwich, because the time at that place is 2 hours farther advanced than the time at Greenwich: but if the time shewn by this Chronometer were 4 hours at the instant, the mean time found, by observation, is only 2 hours, then the Longitude of the place is 30° W. of Greenwich, because the time at Greenwich is 2 hours farther advanced, than the time at the place of observation.

A Chronometer generally deviates something from mean time in its rate of going; the portion of time which it gains or loses on mean ime, during 24 hours, is called its Daily Rate, or simply the Rate, and what a Chronometer is fast or slow, for mean time at a given neridian, is called its Error for that meridian. Those who reckon he Longitude from the meridian of Greenwich, should always have he errors of their Chronometers for that meridian. If the rate of a Chronometer and its error, for any particular time be known, the error for any other time is found by multiplying the rate by the number of days between the times. Thus, let the rate of a Chronometer be 5s. 4 gaining, and it is found to be fast for mean time at Greenwich, at noon, on the 5th of June, Oh. 11m. 31s.; the error on the 2d of July, at noon, would be Oh. 13m. 57s.; for here the number of days elapsed is 27, and 5s. 4. x 27 145s. 8. or 2m. 26s., and Oh. 11m. 31s. + 2m. 26s. Oh. 13m. 57s.

But it is better to set down the errors for the noon of each day at Greenwich. For example, Let there be two Chronometers, Nos. 185 and 230, No. 185 is slow for mean noon at Greenwich, on the 10th of June, 1824, Oh. Om. 37s., and is gaining on mean time 9s. 6 in