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Page Prob. 6. To reduce the declination of the moon, as given in the Nautical Almanac, to any other meridian, and to

any given time of the day - - . 27 1 7. To find the time of a star's culminating, or coming to the meridian of Greenwich - - , or 5 s. To find the time of the moon, or any planet's culminating - - - - - . .275 9. Given the observed altitude of a fixed star to find its true altitude - - • 276

10. Given the observed altitude of the sun's lower or upper limb, to find the true altitude of its centre . .277

11. Given the observed altitude of the moon's lower or
upper limb, to find the true altitude of its centre . 278

12. Given the sun's meridian altitude to find the latitude
of the place of observation - - . 280

CHAP. x1. THE APPLICATION OF RIGHT ANGLED spheRICAL TRIANGLES TO ASTRONOMICAL PROBLEMS - - - • 281

Prob. 1. Given the obliquity of the ecliptic and the sun's lon-
gitude, to find his right ascension and declination . 282
2. Given the latitude of the place, and the sun's declin-
ation, to find hisamplitude, ascensional difference, and
the time of his rising and setting - . 284
5. The latitude of the place, and the sun's (or a star’s)
declination being given, to find the altitude and azi-
muth, &c. at six o'clock - - . .289
4. The latitude of a place, and the declination of the sun
(or of a star) being given; to find the altitude, and
the times when it will be due east and west . 292
5. Given the latitude of the place, and the sun's altitude,
when on the equinoctial, to find his azimuth and the
hour of the day - - " - . 294
6. The difference of longitude between two places, both
in one parallel of latitude, being given, to find the dis-
tance between them, &c. - - - - . 296

CHAP. xII. THE APPLICATION of oblique-ANGLED sphe-
RICAL TRIANGLES TO ASTRONOMICAL PRO-

BLEMS • - - • 298 Prob. 7. Given the sun's declination, and the latitude of the place, to find the apparent time of day-break in the

morning, and the end of twilight in the evening . 298 8. Given the day of the mouth, the latitude of the place, the horizontal refraction, and the sun's horizontal parallax, to find the apparent time of his centre ap

pearing in the eastern or western part of the horizon 30

Prob. 9.

10.

1 1.

13. 14.

14.

15.

Page

Given the latitude of the place, the day of the month,
the moon’s horizontal parallax and refraction, to find
the time of her rising - - -

The latitude and longitude of a fixed star, or of a
planet, being given, to find its right ascension and

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declination, et contra - - - , 505

The right ascensions and declinations of two stars,
or the latitudes and longitudes of two stars being
given, to find their distance - - - -

. The places of two stars being given, and their dis

tances from a third star, to find the place of this

third star - - e ... • -
Given the latitude of the place, the sun’s declin-
ation and altitude, to find the azimuth - -

Given the latitude of the place, the sun's declin-
ation and altitude, to find the hour of the day e
Continued. The construction of the xv.1th of the
REQUISITE TABLEs, used in finding the latitude by
two altitudes of the sun (Note) - - -
Given the latitude of the place, the declination and

the altitude of a known fixed star, to find the hour

16.

16.

of the night when the observation was made ->
Given two altitudes of the sun and the time between
the observations, to find the latitude of the place .
Continued. A GENERAL RULE for finding the latitude
by two altitudes of the sun, the elapsed time and the

sun's declination being given - -

17.

17.

Given the apparent distance of the moon from the
sun, or from a star, and their apparent zenith dis-
tances, to find their true distance, as seen from the
earth’s centre - • - e -
Continued. Investigation of a GENERAL RULE for

determining the true distance of the moon from the

18.

CHAP. XIII.

Prop. 1.
2.

3.

sun, or from a fixed star - e -
The latitude of a place and its longitude by account,
the distance between the sun and the moon, or the
moon and a star in the NAUTICAL ALMANAc being

given, to find the correct longitude. - o
OF THE FLUXIONAL ANALOGIES OF SPHE •
RICAL TRIANGLES - - - - -
A preparatory proposition - - -

To find the fluxions of the several parts of a RIGHT-
ANGLED spherical triangle, when one of its oblique
angles is a constant quantity - -

To find the fluxions of the several parts of a RIGHT

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ANGLED spherical triangle, when one of its legs is a
constant quantity - - - -
To find the fluxions of the several parts of a Right-
ANGLED spherical triangle, when the hypothenuse is
a constant quantity - - -

. In any obliquE-ANGLED spherical triangle, supposing

an angle and its adjacent side to remain constant, it
is required to find the fluxions of the other parts
To find the fluxions of the several parts of an on-
Lique-ANGLED spherical triangle, when an angle and
its opposite side are constant quantities - -
To find the fluxions of the several parts of an on-
LiquE-ANGLED spherical triangle, when two of its
sides are constant quantities - - -
To find the fluxions of the several parts of an on-
LIQUE-ANGLED spherical triangle, when two of its
angles are constant quantities. - - -

USE OF THE FLUXIONAL ANALOGIEs.

To find when that part of the equation of time de-
pendent on the obliquity of the ecliptic is the greatest
possible - o -- - - *
Given the parallax in altitude of a planet, to find its
parallax in latitude and longitude - -
Given the altitude of the nonagesimal degree of the
ecliptic; the longitude of a planet from a nonage-
simal degree, and its horizontal parallax, to find its
parallax in latitude and longitude - -
To determine the correction for finding the time of
apparent noon, from equal altitudes of the sun

The error in taking the altitude of a star being given,
to find the corresponding error in the hour angle .
The error in the altitude of any tower, or other object,
is to the error committed in taking the angle of eleva-
tion; as double the height of the observed object, is
to the sine of double the angle of elevation .

CHAP. XIV. MISCELLANEOUS PROPOSITIONs, &c.

Prop. 1.

2.

3.

Of the FRENch division of the circle, and to turn
French degrees, minutes, &c. into English .
To turn English degrees, minutes, &c. into French
To find the distances of the observatories of Paris and
Pekin, by the French division of the circle . .
Ditto, by the English division of the circle -.
To find the surface of a spherical triangle

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. 360

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Page Prop. 4. To find the excess of the three angles of a spherical triangle, above two right angles - . 366

5. To reduce the angles of a spherical triangle (whose
sides are very small arcs) to those of a rectilineal tri-
angle, having its sides of equal length with the sides
of the spherical triangle - • • 368

6. Given two sides of a spherical triangle, and the angle
comprehended between them; to find the angle con-
tained between the chords of these sides, supposing
the chords not to differ materially from the arcs which
they subtend - - - - . 371

7. The angles of elevation of two distant objects being
given, together with the oblique angle contained be-
tween the objects, to find the horizontal angle • 573

1 BOOK IV. THE THEORY OF NAVIGATION. CHAPTER I. Definitions and Plane sailing - 376 to 380 CHAP. II. Parallel and Middle Latitude sailing . 381 to 383 çHAP. III. Mercator's sailing • - - 383 to 592 t TABLES. I. A Table of the Logarithms of numbers, from an unit to ten thousand . - - 393 to 408 II. A Table of NATURAL sines to every degree and minute of the quadrant - - • 409 to 417

III. A Table of Logan ITHMICAL SINEs and TANGENTs to every degree and minute of the quadrant 418 to 440 IV. A Table of the RErnaction in altitude of the heavenly

- bodies - - - • . 441
W. A Table of the depression or DIP of the horizon of
the sea - - e - . 441

VI. A Table of the sun’s PARALLAx in altitude . . 441 VII. A Table of the augmentation of the moon's semidiameter . . . . . . . . . . . . 441, VIII. A Table of the right ascensions and declinations of thirty-six principal fixed stars, corrected to the

. . . . . . beginning of the year 1822 • * * 442

Five copper-plates at the end of the book.

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