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5. Of the sines, tangents, &c. of the MULTIPLES OF
ARCS.
Page
116 to 118
6. Of the sines and cosines of the POWERS of
118 to 120
7. The determination of the value of the sine and of
the cosine, &c. of any arc, in terms of that arc, by
infinite series, &c.
120 to 123
8. The construction of a table of sines, &c.
9. Formulæ for the solutions of the different cases of
right-angled plane triangles
123 to 126
126
10. Formulæ for the solutions of the different cases of
oblique-angled plane triangles
127
BOOK III.
CHAPTER I. DEFINITIONS, &c. OF SPHERICAL ANGLES,
CALCULATING THE SIDES AND ANGLES OF
RIGHT-ANGLED SPHERICAL TRIANGLES, &c. 161
2. BARON NAPIER's universal rule for solving right-
angled spherical triangles
165
3. General rules for the solutions of all the different
cases of right-angled spherical triangles
168
4. Of the different species or affections of right-angled
spherical triangles
171
5. Practical examples, exercising the rules in right-
angled spherical triangles, solved by construction and
by calculation
172 to 187
CHAP. IV.
GENERAL RULES FOR SOLVING THE DIFFER-
ENT CASES OF RECTILATERAL, OR QUA-
DRANTAL, SPHERICAL TRIANGLES
2. Practical examples exercising the rules of rectilateral
187
triangles,
CHAP. V. INVESTIGATION OF GENERAL RULES FOR SOLV-
ING THE DIFFERENT CASES OF OBLIQUE-
ANGLED SPHERICAL TRIANGLES, WITH A
CHAP. VI.
CHAP. VII.
PERPENDICULAR
The manner of applying BARON NAPIER'S rule to
oblique spherical triangles
191
191 to 200
2. Rules for solving all the different cases of oblique-
angled spherical triangles, with a perpendicular
3. Practical examples, exercising the rules in oblique-
angled spherical triangles, where a perpendicular is
used
200
201 to 211
INVESTIGATION OF GENERAL RULES FOR CAL-
CULATING THE SIDES AND ANGLES OF
OBLIQUE-ANGLED SPHERICAL TRIANGLES,
WITHOUT A PERPENDICULAR
211 to 223
2. Rules for solving all the different cases of oblique-an-
gled spherical triangles without a perpendicular 223 to 228
3. Practical examples, exercising the rules of oblique-
angled spherical triangles, where a perpendicular is not
used, solved by construction and by calculation 229 to 240
ASTRONOMICAL DEFINITIONS, &C. WITH THE
APPLICATION OF RIGHT-ANGLED SPHERICS
2. Given the obliquity of the ecliptic and the sun's
longitude, to find his right ascension and declination 246
3. To turn degrees, or parts of the equator into time,
et contra (Note)
247
4. Given the latitude of the place, and the sun's decli-
nation, to find his amplitude, ascensional difference,
and the time of his rising and setting, &c.
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 6 o'clock
249
6. The latitude of a place, and the declination of the
253
บ
sun
sun (or of a star) being given; to find the altitude,
and the times when it will be due east and west
7. Given the latitude of the place, and the sun's alti-
tude, when on the equinoctial, to find his azimuth
and the hour of the day
256
259
8. The difference of longitude between two places,
both in one parallel of latitude, being given, to find
the distance between them, &c.
260
PROBLEMS
262 to 308
262
1. To find the beginning, end, and duration of twi-
light
2. Given the day of the month, the latitude of the place,
the horizontal refraction, and the sun's horizontal
parallax, to find the apparent time of his upper limb
appearing in the eastern or western part of the ho- rizon
3. To reduce the sun's declination, as given in the
Nautical Almanac, to any other meridian, and to
any given time of the day (Note)
265
267
4. 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 and setting
268
5. To find the time of the moon, or any planet's cul-
minating (Note)
6. The latitude and longitude of a fixed star, or of a
planet, being given, to find its right ascension and
declination, et contra
270
7. The right ascensions and declinations of two stars,
or the latitudes and longitudes of two stars being
given, to find their distance
273
8. The places of two stars being given, and their dis-
tances from a third star, to find the place of the third
9. Given the latitude of the place, the sun's declination
and altitude, to find the azimuth
10. Given the latitude of the place, the sun's declina-
tion and altitude, to find the hour of the day
278
281
11. The
11. The construction of the XVIth of the REQUISITE
TABLES, used in finding the latitude by two alti-
tudes of the sun (Note)
12. Given the latitude of the place, the declination and
the altitude of a known fixed star, to find the hour of
the night when the observation was made
13. Given two altitudes of the sun and the time between
the observations, to find the latitude of the place
14. A GENERAL RULE for finding the latitude by two
altitudes of the sun, the elapsed time, and the sun's
declination being given
15. 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 .
16. Investigation of a GENERAL RULE for determining
the true distance of the moon from the sun, or from
a fixed star
17. 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
282
284
288
291
297
298
302
SHAP. IX.
OF THE FLUXIONAL ANALOGIES OF SPHE- RICAL TRIANGLES
308 to 323
308
1. A preparatory proposition
2. To find the fluxions of the several parts of a RIGHT-
ANGLED spherical triangle, when one of its oblique
angles is a constant quantity
3. To find the fluxions of the several parts of a RIGHT-
ANGLED spherical triangle, when one of its legs is a
constant quantity
4. To find the fluxions of the several parts of a RIGHT-
ANGLED spherical triangle, when the hypothenuse
is a constant quantity.
5. In any OBLIQUE-ANGLED spherical triangle, sup-
posing an angle and its adjacent side to remain con-
stant, it is required to find the fluxions of the other
parts
6. To find the fluxions of the several parts of an OB-
309
311
312
313
LIQUE-
LIQUE-ANGLED spherical triangle, when an angle and its opposite side are constant quantities
7. To find the fluxions of the several parts of an OB-
LIQUE-ANGLED spherical triangle, when two of its
sides are constant quantities
8. To find the fluxions of the several parts of an OB-
angles are constant quantities .
THE USE OF THE FLUXIONAL ANALOGIES.
9. To find when that part of the equation of time de-
pendent on the obliquity of the ecliptic is the greatest
possible
315
317
318
319
320
10. Given the parallax in altitude of a planet, to find
its parallax in latitude and longitude
11. Given the altitude of the nonagesimal degree of the
ecliptic; the longitude of a planet from the nonage-
simal degree, and its horizontal parallax, to find its
parallax in latitude and longitude
321
12. To determine the correction for finding the time of
apparent noon, from equal altitudes of the sun
13. The error in taking the altitude of a star being given,
to find the corresponding error in the hour angle
322
323
CHAP. X.
MISCELLANEOUS PROPOSITIONS, &c.
1. Of the FRENCH division of the circle
324
2. To turn French degrees, minutes, &c. into English 324
3. To turn English degrees, minutes, &c. into French 325
4. 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
326
327
5. To find the surface of a spherical triangle
328
6. To find the excess, of the three angles of a spherical
triangle, above two right angles
329
7. 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.
8. 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
331
the