SECTION ELEVENTH.

ASTRONOMICAL PROBLEMS.

PROBLEM I.

To convert Time into degrees, minutes, &c.

RULE.

As one hour is to 15 degrees, so is the time given to the answer.

1. How many degrees are equal to 8 hours, 20 minutes, and 30 seconds?

2d. The Sun passes the meridian of Detroit 1 hour, 19 minutes after 12 o'clock, noon at Boston, how far are those places asunder ?

PROBLEM II.

To convert degrees, minutes, &c. into Time.

RULE.

As 15 degrees are to an hour, so are the number of degrees given to the time.

P

EXAMPLES.

1. The apparent distance of Venus from the Sun, can never be above 50 degrees, and when at that distance, how long does she rise before the Sun, or set after him?

2. The greatest elongation of Mercury is said to be 28 degrees, 20 minutes and 19 seconds, how long can he set after the Sun, when an evening star?

PROBLEM III.

The diurnal arc of the Sun, or of any planet being given, to find the time of the rising or setting of the Sun.

RULE.

Bring the diurnal arc into time by Problem 2d. Divide this time by two, and the quotient will be the time at which the Sun sets. Take this time from 12 hours, and the remainder will be the time at which the Sun rises.

EXAMPLES.

1. Suppose the Sun's diurnal arc be 174 degrees and thirty minutes, at what time does he rise and set. Ans. 5 hours 49 minutes, the time of the Sun's setting, and he rises at 6 hours and 11 minutes.

2. The diurnal arc of Venus' is found to be 96 degrees and 44 minutes, at what hours does the Sun rise,and when does he set ?

3. The diurnal arc of Mars, is 198 degrees, 14 minutes and 50 seconds.

The diurnal arc of Jupiter, is 201 degrees, 33 minutes and 16 seconds.

The diurnal arc of Saturn, is 196 degrees, and 14 minutes: and the diurnal arc of Herschel is 213 degrees, 41 minutes, and 58 seconds; when, according to the above mentioned numbers, does the Sun rise and set?

PROBLEM IV.

The time which the Sun, or any planet remains above the horizon being given, to find the length of his diurnal, or nocturnal arc.

RULE.

Divide the given time by two, and the quotient will be the time of the Sun's setting. Take this time from 12 hours, and the remainder will be the time of his ri sing. Multiply the given time by 15 degrees, and the product will give the Sun's, or planet's diurnal arc ;— this subtracted from 360 degrees, will leave the nocturnal arc.

EXAMPLES.

1. On the fourth of July, the Sun rose at 43 minutes past 5 o'clock; at what time did he set on that day, and what was the length of his diurnal arc ?

2. September 7th, 1825, the Sun rose at 5 o'clock and 52 minutes, at what time did he set, and what are the dimensions of both arcs ?

PROBLEM V.

To find the time which elapses between two conjunctions, or two oppositions, or between one conjunction, and one opposition of any two planets.

RULE.

Find the difference between the given daily motions of the two given planets, as given in the following table of the daily motions, then say, as the difference of their daily motions, is to one day, so is 360 degrees, to the difference in the times of the two conjunctions, or oppositions required. But for one conjunction, and one opposition, or, for a superior and an inferior conjunction; say as the difference of their daily motions is to one day, so is 180 degrees to the time, which elapses between a conjunction, and an opposition of the two given planets.

EXAMPLES.

1. How many days elapse between a conjunction, and an opposition of Mercury and Venus.

Thus Mercury's daily motion,

4,0928 degrees Less the daily motion of Venus, 1,6021-2,4907 Degrees,then as 2,4907 d: 1 day :: 180 d: 72,25 days, the time required.

2. How many days is Venus a morning and an evening star, alternately to the earth?

3. How many days is Jupiter a morning and evening star, alternately to the earth?

4. How many days is Mercury east, and how many west of the Sun to us?

PROBLEM VI.

The heliocentric longitude of any two planets being given, to find when they will be in heliocentric conjunction.

RULE.

Subtract the given longitude of the planet nearest the Sun, from that of the planet farthest from him, if practicable, but if not, add to the latter 360 degrees, and then subtract, say, as the difference of the daily motions of the given planets is to one day, so is the difference of their longitudes, to the time when the given planets will be in conjunction.