night, on the fifth, or six hours before noon on the sixth, the semi-diameter must be taken out for midnight on the fifth, and noon on the sixth.

Suppose the moon's semi-diameter on the fifth

Then say, if twelve hours give 8", what will six hours give? Answer 4", which, added to 15' 14", gives 15' 18".* But if the semi-diameter had been decreasing, the 4" should have been subtracted, which would give 15' 10". In this manner, also, you must proceed with respect to the horizontal parallax.

From Table VIII. take out a number of seconds, answering to the moon's apparent altitude (which will be the augmentation of the moon's semi-diameter); which add to the moon's reduced semi-diameter, and their sum will be the moon's true semi-diameter.

In west longitude, when the sum of the longitude in time, and the time at ship, exceed twenty-four hours; twenty-four hours must be subtracted from it, and the remainder will be the reduced time, counted from noon the following day.

In east longitude, when the longitude in time is greater than the time at ship, add twenty-four hours to the time at ship, and subtract the longitude in time from it, and it will give the reduced time counted from noon of the preceding; and in every operation where one time is to be taken from another, add twenty-four hours to the time you subtract from, if the time which is to be taken from it be the greatest, and the remainder must be reckoned from noon the preceding day; and when one time is to be added to another, if the sum exceed twenty-four hours-take twenty-four hours from it, and the remainder must be reckoned from noon the following day.

On the sixth of the month, at two hours in the afternoon, by Nautical Almanac, in longitude by account 60° east, the reduced time be required.

The moon's semi-diameter and horizontal parallax may be proportioned to

reduced time, by Table XI. in this book.

preceding the sixth, which is twenty-two hours on the fifth by the Nautical Almanac. In this case the moon's semi-diameter and horizontal parallax must be taken out of the Nautical Almanac, for midnight, on the fifth, and noon on the sixth, and proportioned to the reduced time, by the preceding rules.

Again, on the sixth at four in the forenoon, civil account, which is sixteen hours on the fifth, by Nautical Almanac, in longitude 45° east.

In this case the moon's semi-diameter and horizontal parallax must be taken out of the Nautical Almanac, for midnight on the fifth, and noon on the sixth, and proportioned as abovementioned.

On the sixth, at four hours in the forenoon, civil account, which is sixteen hours on the fifth by Nautical Almanac, in longitude 120° east, the reduced time is required.

In this case the moon's semi. diameter is taken out of the Nautical Almanac for noon and midnight on the fifth, and proportioned to reduced time.

On the sixth at two hours in the afternoon, by Nautical Almanac, in longitude 60° west, reduced time is required.

In this case the moon's semi-diameter and horizontal parallax must be taken out of the Nautical Almanac, both for noon and midnight on the sixth, and reduced as aforesaid.

On the sixth, by Nautical Almanac, at eight hours in the afternoon, in longitude 909 west by account.

In this case the moon's semi-diameter and horizontal parallax, must be taken out for midnight and noon on the sixth, and reduced as before-mentioned.

On the sixth, at eight hours in the afternoon, by Nautical Almanac, longitude 60° west, the reduced time is required.

In this case the moon's semi-diameter and horizontal parallax, must be taken out of the Nautical Almanac, on the sixth, for midnight only.

In this case the moon's semi-diameter and horizontal parallax must be taken out of the Nautical Almanac, for midnight on the fifth, and noon on the sixth.

The moon's semi-diameter and horizontal parallax must be taken out, and for noon and midnight on the sixth.

On the sixth, at eight hours before noon, civil account, which is twenty hours on the fifth by Nautical Almanac, in longitude 60° west, required the reduced time.

EXAMPLE IX.

Time at ship,
Longitude 60° west in time,

Reduced time,

20 hours on the fifth, by N. A. 4 hours.

24

In this case the moon's diameter and horizontal parallax must be taken out, for noon on the sixth only.

On the sixth, at ten hours before noon, civil account, which is twenty-two hours on the fifth, by Nautical Almanac, in longitude 90° west, required the reduced time.

In this case the moon's semi-diameter and horizontal parallax must be taken out of page VII of the month in the Nautical Almanac, for noon and midnight on the sixth, and reduced as above.

The sun's semi-diameter is given in page III of each month, in the Nautical Almanac for every sixth day; no augmentation is applied to the sun's semi-diameter.

RULE III. To the moon's observed altitude, add 12', if the lower limb be taken, but if the upper limb be taken, subtract 20'; to the observed altitude of the sun's lower limb add 12', and from the star's observed altitude subtract 4', and you will have their apparent altitudes.

To the observed distance of the sun and moon add their semidiameters, and you will have their apparent distance.

To the observed distance of the moon and star, add the moon's semi-diameter, if the nearest limb be taken; but subtract it, if the farthest limb be taken their sum or difference will be the apparent distance.

It must be remembered, that, if there be an index error, that also, must be either added or subtracted, as the case may require, in order to obtain the apparent distance.

Having the true Distance, to find the Longitude.

AMONG the true distances of the moon's centre from the sun or fixed star, as given in pages VIII, IX, X and XI, of each month in the Nautical Almanac, find, on the given day, those two distances that are the next less and next greater than the true distance found by observation, which two distances place under the true distance found by observation. Take the difference between the true distance, and the first of the two dis

*

tances; also the difference between the two distances. Subtract the proportional logarithm of the second difference from the proportional logarithm of the first difference, and the remainder will be the proportional logarithm of a portion of time, which add to the time standing over the first of the two distances taken from Nautical Almanac, and the sum will be the apparent time at Greenwich. Take the difference between the apparent time at Greenwich, and the apparent time at ship, converted into degrees and minutes, at the rate of one hour to 15°, four minutes to one degree, and four seconds to one mile (or more briefly by Table XII.) and it will give the true longitude of the ship; which will be east, if the time at ship be greater than the time at Greenwich reckoned from the same noon, but west, if the time at ship be less than the time at Greenwich.

*

April 12th, 1817, the true distance of the sun and moon's centres, was 44 46 14. Required the apparent time at Greenwich.

N. B. There is no difference of cases, as you always take the difference between the true distance, and that standing first in the Nautical Almanac, and also the difference between the two distances found in the Nautical Almanac,

Examples for working Lunar Observations.

IN taking a lunar observation, two assistants are usual to observe the altitudes of the objects, while the principal takes the distance. If it be near noon when the observation is taken, the

Suppose a ship's time to be two hours on the sixth, by Nautical Almanac, and Greenwich time twenty-two hours, counted from the fifth, and the ship's time counted from the fifth, will give twenty-six hours. Difference between twenty-two and twenty-six is four hours, which gives longitude 60° east.