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To find the Sun's Right Ascension :

Pro. part

Sun's right ascension at noon, August 20, 1824, per Nautical
Almanac,

8.50 0:48: Variation in 24 hours = 3:52".

to 12: 0 and 3! 0 = 1:30" 0 0"."
Do, to , 0.47 and 3. 0

and 3. 0 = 0. 5.52.30
Do. to 12. 0 and 0.50 = 0.25, 0. 0
Do. to 0.47 and 0.50 = 0. 1.37.55
Do. to 12. 0 and 0. 2 = 0. l. 0. 0
Do. to 0.47 and 0.2 0. 0. 3. 55

Pro. part

to 12:47" and 3.52" is 2. 3. 34. 20 = +2:3534"

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17:44:41:

Pro. part

Sun's declination at noon, August 2d, 1824, per Nautical

Almanac,
north, decreasing, and var. in 24 = 15:36:

to 12! 0 and 15: 0 = 7:30("' 0"."
Do. to 0.47 and 15. 0 = 0.29. 22.30
Do. to 12. 0 and 0.30 = 0. 15. 0. 0
Do. to 0.47 and 0. 30 =0.0.58. 45
Do. to 12. 0 and 0.6 = 0. 3. 0. 0
Do. to' 0.47 and 0.6 = 0. 0. 11.45

Pro. part

to 12:47 and 15:36" is 8. 18.33. 0=

8:19

Sun's declination, as required

17:36:22

554:24:

To find the Equation of Time:-
Equation of time at noon, August 2d, 1824, per Nautical

Almanac,
decreasing, and variation in 24 hours = 430"!

to 12! 0"and 4" 0% = 25. 0"' 0"."
Do. to 0.47 and 4. O = 0. 7.50
Do. to 12. 0 and 0.30 = 0. 15. 0
Do. to 0.47 and 0.30 = 0. 0.58

Pro. part

Pro. part

to 12.47" and 4"30" is 2.23.45 =

2:24"

Equation of time, as required

5:52: 0!

Remark.-Should the proportional part corresponding to the daily variation of the sun's longitude and any given time be required, it may be taken from the first page of the Table, by esteeming the seconds of variation, in that page, as minutes, and then raising the signs of the corresponding proportional parts one grade higher than what are marked at the top of the said page : the seconds of variation will, of course, be taken out after the usual manner. Thus,

Suppose that the daily variation of the sun's longitude be 57. 40", and the Greenwich time 9 hours 50 minutes, to find the corresponding equation, or proportional part.

Pro. part

Do.
Do.
Do.
Do.

to 9 0 and 50: 0 = 18:45 05 0"."
to 9. 0 and 7.0 2.37.30. 0
to 0.50 and 50. O 1.44. 10. 0
to 0.50 and 7.0 0.14.35. 0
to 9. 0 and 0.40 0. 15. 0. 0
to 0.50 and 0.40 0. 1. 23. 20

Do.

Pro: part

to 9:50 and 57:40" is 23.37.38.20 = 23.38 +

Note. It is easy to perceive that the foregoing operations might have been much contracted, by taking out two or more of the proportional parts at once; but, lest doing so should appear anywise ambiguous to such as are not well acquainted with the method of taking out tabular numbers, it was deemed prudent to arrange the said operations according to their present extended form, so as to render them perfectly intelligible to every capacity.

The present Table was computed agreeably to the established principles of the rule of proportion ; viz., As one day, or 24 hours, is to the variation of the sun's right ascension, declination, &c. &c., in that time, so is any other portion of time to the corresponding proportional part of such variation.

Table XVI,

To reduce the Moon's Longitude, Latitude, Right Ascension, Declination,

Semidiameter, and Horizontal Parallax, as given in the Nautical Almanac, to any given Meridian, and to any given Time under that Meridian.

This Table is arranged in a manner so nearly similar to the preceding, that any explanation of its use may be considered almost unnecessary; the only difference being, that the proportional parts are computed to variation in 12 hours, instead of 24. By means of the present Table, the proportional part corresponding to any variation of the moon's longitude, latitude, right ascension, &c. &c. &c., may be easily obtained, to the greatest degree of accuracy, as follows; viz.

Turn the longitude of the ship or place into time (by Table I.), and add it to the apparent time at such ship or place, if it be west ; but subtract it if east: and the sum, or difference, will be the corresponding tiine at Greenwich.

Take from pages V., VI., and VII. of the month, in the Nautical Almanac, the moon's longitude, latitude, right ascension, declination, semidiameter, and horizontal parallax, (or any one of these elements, according to circumstances) for the noon and midnight immediately preceding and following the Greenwich time, and find their difference; which difference will express the variation of those elements in 12 hours.

Enter the Table with the variation, thus found, at top, and the Greenwich time in the left-hand column; in the angle of meeting will be found the corresponding equation, or proportional part, which is always to be added to the moon's longitude and right ascension at the preceding noon or midnight, but to be applied by addition, or subtraction, to the moon's latitude, declination, semidiameter, and horizontal parallax, according as they are increasing or decreasing. And, since the Greenwich time and the variation in 12 hours will be very seldom found to correspond exactly; it is the sum, therefore, of the several equations making up those terms, that will, in general, express the required proportional part,

Example. Required the moon's longitude, latitude, right ascension, declination, semidiameter, and horizontal parallax, August 2d, 1824, at 3.10., in longitude 60:30:, west of the meridian of Greenwich? Apparent time at ship or place

3?10 Longitude 60: 30! west, in time =

4. 2

.

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Greenwich time

7:12

To find the Moon's Longitude :

Do.

Moon's longitude at noon, August 22, 1824, per Nautical
Almanac,

· · 7

7:17:16:27"
Variation in 12 = 6:31:59".
Propor, part to 70 and 6: 0: 0". = 3:30. 05. 0

to 0.12 and 6. 0. 0 = 0. 6. 0. 0
Do. to 7. 0 and 0.30. 0 = 0.17: 30. O
Do. to 0.12 and 0.30. 0 = 0. 0.30. 0
Do. to 7: 0 and 0. 1. 0 = 0. 0.35. 0
Do. to 0.12 and 0. l. 0 = 0. 0. 1. 0
Do.

to 7.0 and 0. 0.50 0. 0. 29. 10
Do. to 0.12 and 0. 0.50 = 0. 0. 0.50

to 7. 0 and 0. 0. 9 = 0. 0. 5. 15 Do. to 0.12 and 0. 0. 9 = 0. 0. 0. 9

Do.

Propor. part to 7?12" and 6:31:59" is 3.55. 11. 24=+3:55.11!!

Moon's longitude, as required.

7:21:11:38:

.

To find the Moon's Latitude:

406:59

Moon's latitude at noon, August 2d, 1824, per Nautical

Almanac,

south, decreasing, and var. in 12 hours = 23:35"
Proportional part to 7? 05 and 20: 0". = 11:40. 0".

Do. to 0.12 and 20. 0 0. 20.0
Do. to 7. 0 and 3. 0 1.45. 0
Do. to 0.12 and 3. 0. 0. 3.0
Do. to 7. 0 and 0.30 0.17.30
Do. to 0.12 and 0.30 0. 0.30
Do. to 7. 0 and 0. 5 0. 2.55
Do. to 0.12 and 0.5 0. 0. 5

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Proportional part to 7"12" and 23:35” is 14. 9. 0 = - 14' 9".
Moon's latitude, as required

3:52:50" Note.-- In consequence of the unequal motion of the moon in 12 hours, (when her place is to be determined with astronomical precision, the proportional part of the variation of her longitude and latitude, found as above, must be corrected by the equation of second difference contained in Table XVII.; and the same may be observed of her right ascension and declination,

To find the Moon's Right Ascension :

.

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Moon's right ascension at noon, August 2d, 1824, per
Nautical Almanac,

223:33:36"
Var. in 12. = 6:51! 49.
Propor. part to 7? 07 and 6: 0: 0" = 3:30: 0. 0).
Do. to 0.12 and 6, 0.0

= 0. 6. 0. 0
Do. to 7.0 and 0.50. O 0. 29. 10. 0
Do..

to 0.12 and 0.50. 0 = 0. 0.50. O
Do. to 7. 0 and 0. l. 0 = 0. 0.35. 0
Do. to 0.12 and 0. l. 0 = 0. 0..1. 0
Do. to 7. 0 and 0. 0.40 = 0. 0.23. 20
Do. to 0.12 and 0. 0.40 = 0. 0. 0.40
Do. to 7.0 and 0. 0. 9 = 0, 0. 5. 15
Do.
to 0.12

and 0. 0. 9 = 0. 0. 0. 9

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Propor, part to 7.12. and 6:51:49" is 4. 7. 5. 24=+4: 7' 5"

Moon's right ascension, as required .

227:40:41"*

To find the Moon's Declination :

20:57! 7!

Moon's declination at noon, August 2d, 1824, per Nautical

Almanac,

south, increasing, and var. in 12 ho.=1:23:43 Propor. part to 7° 0 and 1: 0: 0 = 35! 00!!

Do. to 0.12 and 1. 0. O .1. 0. 0
Do. to 7. 0 and 0. 20. O = 11.40. 0
Do. to 0.12 and 0. 20. 0 0. 20.0
Do. to 7. 0 and 0. 3. 0 1.45. 0
Do. to 0.12 and 0. 3. 0 0. 3. 0
Do. to 7.0 and 0. 0.40 0. 23. 20
Do. to 0.12 and 0. 0.40 0. 0.40
Do.

to 7. 0 and 0. 0. 3 0. 1.45
Do. to 0.12 and 0. 0. 3 0. 0. 3

Propor. part to 7.12" and 1:23:43" is 50. 13.48 =

+ 50:14

Moon's declination, as required

21:47:21"*

* When accuracy is required, the moon's right ascension and declination must be corrected by the equation of second difference, on account of the irregularities of ber motion in 12 hours.

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