Ans. N. M. 29th day III o'clock 52 m. 28 sec. and Sun's dist. from node 11 S. 29° 56' 57". (Sun eclipsed.*)

3. Required the true time of New Moon, and at that time the Sun's distance from the ascending node, in April, A. D. 1819, N. S. Ans. New Moon 24th day at 46 m. 21 sec. past XI. A. M. and Sun's dist. from node 12° 10' 9". (Sun eclipsed, visible.)

4. Required to find the true time of opposition of the Sun and Moon, and the simultaneous distance of the Sun from the node,. in October, A. D. 1819, N. S.

Ans. True Full Moon 3d day, at 7m. 48 sec. past III. in the evening, and Sun's dist. from descending node 50' 5 (Moon eclipsed, total.)

5. Required the true time of conjunction of the Sun and Moon, September, A. D. 1820, N. S.

Ans. On the 7th day, at 16 m. 55 sec. past II. P. M. (Sun. eclipsed.)

6. Required the true time of Full Moon at Boston, Long. 70 ° 37' 15" W. in May, A. D. 1826, N. S.

Ans. 21st day, at 29 m. 58 sec. past X in the morning. (Moon eclipsed.)

7. Let it be required to find the true time of New Moon in July, 1980, O. S. and how far short the Sun will be at that time from the Moon's ascending node.

Ans. 29th day, at 52 m. 28 sec. past III. in the morning. And the Sun will be only 3', 3", short of the Moon's node, N. ascending. (Consequently, the Sun must suffer a total eclipse!)

To calculate the true Place of the Sun for any given Moment of

Time.

Precept 1. In Table XII. find the next lesser year in number to that in which the Sun's place is sought, and write out his mean longitude and anomaly answering thereto to which add his mean motion and anomaly for the complete residue of years, months, days, hours, minutes, and seconds, down to the given time, and this will be the Sun's mean place and anomaly

*Note. When the Sun is within 17 degrees of either of the Moon's nodes at the time of New Moon, he will be eclipsed at that time and when he is within 12 degrees of either of the nodes at the time of Full Moon, the Moon will be eclipsed at that time. See the method of calculating Eclipses.

J

at that time, in the old style;* provided the said time be in any year after the Christian Era.

2. Enter Table XIII. with the Sun's mean anomaly; and making proportions for tlfe odd minutes and seconds thereof, take out the equation of the Sun's centre: which being applied to his mean place, as the title Add or Subtract directs, will give his true place or longitude from the vernal equinox, at the time for which it was required.

EXAMPLE I.

Required the Sun's true place, July 13th 1748, Old Style, at 23 hours 19 minutes 58 seconds past noon? In common reckoning, July 14th, at 19 minutes 58 seconds past XI. in the fore

*N. B. Although this Table is constructed according to the Old Style, yet it will serve, with equal exactness, for the New, by diminishing the day of the month in this Table by 12, for the present age. Thus, suppose the required time to be on the 28th day of May N. S. Instead of the numbers answering to that day, write out those for the 16th day, &c. But if the required time be within the limits of the 18th century, subtract 11 days from the given time. See Example II, on the next page.

EXAMPLE II.

Required the Sun's true place, March 31st, 1764, New Style, at 22 hours 30 minutes 25 seconds, past the noon of that day?

Sun's true place at the same time. 0 12 10 7 or 12 10

of Aries.

EX. III. Required the Suns true place and anomaly, July 28th, 15h. 52m. 26 sec. past noon, in the year 1980, . S.

Ans. 4 S. 18° 35'8 "from the vernal equinox, and Sun's anomaly, 1 S. 7 ° 18' 9 ".

To find the Sun's Distance from the Moon's Ascending Node, at the time of any given New or Full Moon; and consequently to know whether there is an Eclipse at that Time, or not.

The Sun's distance from the Moon's ascending node, is the argument for finding the Moon's fourth equation in the syzy gies, and therefore it is taken into all the foregoing examples in finding the times of these phenomina.

Thus, at the mean time of New Moon in July, 1748, the Sun's mean distance from the ascending node is 5 S. 25° 30' 1". See Example I. page 175.

The descending node is opposite to the ascending one, and they are, therefore, just six signs distant from each other.

When the Sun is within 17 degrees of either of the nodes at the time of New Moon, he will be eclipsed at that time: and when he is within 12 degrees of either of the nodes at the time

of Full Moon, the Moon will be eclipsed.* Thus we find there will be an eclipse of the Sun af the time of New Moon in July, 1748.

But the true time of that New Moon comes out by the equations to be 6 minutes 10 seconds later than the mean time thereof, by comparing these times in the above example: and therefore, (in this, and all similar cases) we must add the Sun's motion from the node during that interval to the above mean distance, 5 S. 25° 30' 1", which motion is found in Table XII for 6 minutes, 10 seconds, to be 14". And to this we must apply the equation of the Sun's mean distance from the node, in Table XV. found by the Sun's anomaly, which at the mean time of New Moon in example I. we estimated at 25 °, and then we shall have the Sun's true distance from the node, at the true time of New Moon, as follows:

At the mean time of New Moon
in July, 1748.

Sun's motion from the 6 minutes
node for

Sun's mean distance from node

at true New Moon

Equation of mean distance from
node, subtract

Sun's true distance from

the ascending node; that is,

{

Sun from Node.

}

the descending node; which being far within the above limit of 17 degrees, shows that the Sun must then be eclipsed.

And now we shall shew how to project this, or any other eclipse, either of the Sun or Moon.

*Note. This admits of some variation: for in apogeal eclipses, the solar limit is but 16 degrees; and in perigeal eclipses, it is 18. When the Full Moon is in her apogee, she will be eclip sed if she be within 10 degrees of the node; and when she is full in her perigee, she will be eclipsed if she be within 12 degrees of the node.

TO PROJECT AN ECLIPSE OF THE SUN.

In order to this, we must find the ten following Elements by means of the Tables.

1. The true time of conjunction of the Sun and Moon; and at that time,

2. The semidiameter of the Earth's disk* as seen from the Moon, which is always equal to the Moon's horizontal parallax.

3. The Sun's distance from the solstitial colure to which he is then nearest.

4. The Sun's declination.

5. The angle of the Moon's visible path with the ecliptick. 6. The Moon's latitude.

7. The Moon's true horary motion from the Sun.

8. The Sun's semidiameter.

9. The Moon's semidiameter.

10. The semidiameter of the penumbra.

We shall now proceed to find these elements for the Sun's Eclipse in July, 1748. O. S.

1. To find the true time of New Moon. This, by example I. page 175, is found to be on the 14th day of the said month, at 19 minutes 58 seconds past XI. in the morning.

2. To find the Moon's horizontal parallax, or semidiameter of the Earth's disk, as seen from the Moon. Enter Table XVII. with the signs and degrees of the Moon's anomaly, (making proportions, because the anomaly is in the table only to every 6th degree,) and thereby take out the Moon's horizontal parallax; which, for the above time answering to the anomaly 10 ° 56'56", is 54' 33".

3. To find the Sun's distance from the nearest solstice, viz. the beginning of Cancer, which is 3 signs, or 90 degrees fromthe beginning of Aries.

It appears by the example on page 187 (where the Sun's place is calculated to the above time of New Moon) that the

*Note. The body, or face of the Sun, or Moon, as it appears to a spectator on the Earth; or of the Earth, as it would appear to a spectator at the Sun, or Moon, is called its DISK.