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corrections for second, third, and fourth differences, having regard to the signs of all the quantities, will be the longitude or latitude required.
Note 2. When great precision is required in the moon's parallax and semidiameter, the corrections for second differences should be applied in the same manner as for the longitude or latitude.
Taking the time, as in the first example, let the moon's longitude be required, as corrected for third and fourth differences.
To find the approximate Time of New or Full Moon, for a given Year and Month.
For New Moon.
Take from table XVIII., the mean new moon in January, for the given year, and the arguments I., II., III., and IV. Take from table XIX., as many lunations, and the corresponding arguments I., II., III., and IV., as the number* of the given month exceeds a unit, and add these quantities to the former, rejecting the ten thousands in the first two arguments, and the hundreds in the other two. Take the number of days corresponding to the given month, from the second or third column of table XX., according as the given year is a common or bissextile year, and subtract it from
* The numbers for the months are, for January, 1; for February, 2; for March, 3; &c.
the sum, in the coluinn of mean new moon; the remainder will be the tabular time of mean new moon in the given month. If the number of days, taken from table XX., is greater than the sum of the days in the column of mean new moon, as will sometimes be the case, one lunation more than is directed above, with the corresponding arguments, must be added.
With the arguments I., II., III., and IV., take the corresponding equations from table XXI., and add them to the time of mean new moon; the sum will be the approximate time of new moon, expressed in mean time at Greenwich.
For Full Moon.
When the time of mean new moon in January of the given year is on or after the 16th, subtract from it, and the arguments I., II., III., and IV., a half lunation, with the corresponding arguments, taken from table XIX., increasing when necessary, either or both of the first two of the former by 10,000, and of the two latter by 100; but add them, when the time is before the 16th. The result will be the tabular time of mean full moon in January, and the corresponding arguments. Then proceed to find the approximate time of full moon, in the same manner as directed for the new moon.*
EXAM. 1. Required the mean time of new moon in August, 1821, expressed in approximate time at Greenwich.
August, 27 3 17 Approximate time.
* When the half lunation and arguments are to be added, the addition may be
left till the proper number of lunations, with their corresponding arguments, are placed under, and thus make one addition serve.
2. Required the approximate time of full moon in July, 1823, expressed in mean Greenwich time.
3. Required the approximate times of new and full moon in February, 1822, expressed in mean time at Greenwich.
To determine what Eclipses may be expected to occur in any given year, and the times nearly at which they will take place.
For the Eclipses of the Sun.
Take, for the given year, from table XVIII., the time of mean new moon in January, the arguments and the number N.* If the number N differs less than 53, from 0,500, or 1000, an eclipse of the sun may be expected at that new moon. If the difference is less than 37, there must be one. When the difference is between 37 and 53, there is a doubt, which can only be removed by calculation.
If an eclipse may or must occur in January, calculate the approximate time of new moon by problem XII., and it will be the time, nearly, at which the eclipse will take place, expressed in mean time at Greenwich. This time may be reduced to the meridian of any other place by problem V.
Look in column N of table XIX., and, excluding the number belonging to the half lunation, seek the first number that, added to the number
* The number N, in this table, designates the sun's mean distance from the moon's ascending node, expressed in thousandth parts of the circle.
N of the given year, will make the sum come within 53, of 0,500, or 1000. Take the corresponding lunations and arguments, and this number N, and add them to the similar quantities for the given year. Take from the second or third column of table XX., according as the given year is common or bissextile, the number of days next less than the sum of the days in the column of mean new moon, and subtract it from the time in that column; the remainder will be the tabular time of mean new moon in the month corresponding to the days, taken from table XX. At this new moon an eclipse of the sun may be expected; and if the sum of the numbers N, differs less than 37 from the numbers mentioned above, there must be one. Find the time nearly, of the eclipse, by calculating the approximate time of new moon as directed above.
If there are any other numbers in the column N, of table XIX., that, when added to the number N of the given year, will make the sum come within the limit 53, proceed in a similar manner to find the time of the eclipses.
Note. When the time at which an eclipse of the sun will take place is thus found, nearly, and reduced to the meridian of a given place in north latitude, if it comes during the day time, and if the sum of the numbers N, or the number N itself when the eclipse is in January, is a little above 0, or a little less than 500, there is a probability that the eclipse will be visible at the given place. When the number N in January, or the sum of the numbers N in other months, is more than 500, the eclipse will seldom be visible in northern latitudes, except near the equator.
For the Eclipses of the Moon.
When the time of new moon in January of the given year is on or after the 16th, subtract from it, from the arguments, and the number N, a half lunation, the corresponding arguments, and the number N; but when it is before the 16th, add them. The results will be the time of mean full moon in January, and the corresponding arguments, and number N. Proceed to find the times at which eclipses of the moon may or must occur, exactly as directed for the sun, except that the limits 35 and 25, must be used instead of 53 and 37.
Note. In an eclipse of the moon, when the time is found nearly, and reduced to the meridian of a given place, if it comes in the night, it will be visible at that place.
EXAM. 1. Required the eclipses that may be expected in the year 1822, and the times nearly, at which they will take place.