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A DV E R T IS E MENT

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THE FOURTH EDITION.

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This edition of the ensuing work has been carefully revised. All the astronomical examples which depend on the Nautical Almanac, are adapted to the year 1822, and they may be solved either with or without the Almanac; because the several articles necessary to be taken from that work are given at the end of the different examples. By these means the students will readily learn the use of the Nautical Almanac, and in a school where several are studying the same subject, their progress will not be retarded by waiting for the Almanac. The examples will be found sufficiently numerous and appropriate for the purpose of instruction. They might easily have been extended, with different dates, by a selection from various authors; but such examples are perfectly useless to a student, who has not in his possession a Nautical Almanac corresponding with each date.

No. 1. York-buildings, New-road, St. Mary-le-bone, London, October, 1820.

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1. To find the logarithm of any whole number, or mixed
decimal, consisting of one, two, three, or four figures
2. To find the logarithm of any whole number, or mixed

decimal, to five or six places of figures 3. To find the logarithm of a pure decimal

4. To find the logarithm of a vulgar fraction

5. To find the number answering to any logarithm to

four places of figures

6. To find the number answering to any logarithm, to

five or six places of figures

7. To find the product of two whole or mixed numbers

8. To divide one number by another

9. To involve a number to any power; that is, to square,

cube, a number, &c. -

10. To extract the square or cube root, &c. of any number
11. To find the value of a quantity having a vulgar frac-

tion for its exponent -

12. To find a fourth proportional to three given num-
bers; or to work a question in the rule of three by

logarithms

13. Promiscuous examples exercising all the propositions

CHAP. III. THE USE of THE TABLEs of sINES AND TAN

GENTS -

1. To find the natural sine or cosine of an arc, also the

logarithmical sine, tangent, secant, &c.

2. To find the logarithmical sine, cosine, &c. of an arc

to seconds -

3. To find the degrees, minutes; or degrees, minutes,
and seconds, corresponding to any given logarith-

mical sine, tangent, &c.

4. To find the matural or logarithmical versed sine of
an arc, by the help of a table of natural or logarith-

mical sines

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Page THE CONSTRUCTION AND USE OF THE PLAIN SCALE - - - • 16 2. Of the logarithmical lines, or GUNTER’s scale . 18 3. The construction of the logarithmical lines on GUNTER’s scale - - - . 20 4. GUNTER’s proportions for using the line of versed sines - - - . 22

5. Demonstration of GUNTER’s proportions (Note).22 and 23 6. The use of the logarithmical lines on GuntER's scale 23

GEOMETRICAL DEFINITIONS AND INTRODUC

TORY PROBLEMS - - • 24 1. Definitions, &c. of angles - - - 24 2. To erect a perpendicular from a given point in a

given line, or to make a right angle - . 26 3. To draw a straight line perpendicular to a given

straight line from a given point without it . 26

4. To make an angle of any proposed number of degrees upon a given straight line, by the scale of chords . 27

5. An angle being given, to find how many degrees it contains, by a scale of chords . . 27

6. Definitions and general properties of triangles . 27

BOOK II.

cHAPTER 1. DEFINITIONS OF PLANE TRIGoNOMETRY, &c. so

CHAP. II.

CHAP. III.

2. Investigation of general rules for calculating the sides
and angles of plane triangles - -

3. Formulae for the solutions of the different cases of
right-angled plane triangles - - . 40

4. Formulae for the solution of the different cases of
oblique-angled triangles - - -

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PRACTICAL RUILES FOR THE SOLUTION OF ALL
THE DIFFERENT CASES OF RIGHT-ANGLED
PLANE TRIANGLEs, witH THEIR APPLI-
CATION BY LOGARITHMS - 243 to 5

2. Practical rules for solving all the cases of oblique
triangles with their application by logarithms 52 to 6o

THE APPLICATION OF PLANE TRIGONoMETRY
TO THE MENSURATION OF DISTANCEs,
HEIGHTs, &c. - - . 61 to 82

The subject continued, and more minutely considered 83

CHAP. IV.

1. Observations on the admeasurement of a base line 83

2. Of the errors which occur in taking angles of elevation and depression with a theodolite - 84

Page 3. The nature of terrestrial refraction, and its effects on angles of elevation - - - . 87

4. Of the reduction of angles to the centre of the station 89 5. Of the reduction of angles from one plane to another 90

6. Of the dip, or depression of the horizon at sea . 94

7. Of the parallax of the celestial bodies - . 95

8. Of the admeasurement of altitudes by the barometer
and thermometer - - - . 96

CHAP. v. OF THE SIGNs of TRIGONoMETRICAL QUANTITIES, &c. - - - • 98

1. General properties of the sines, tangents, chords, &c.
of sINGLE ARcs, with a variety of useful formulae 102 to 106
2. General properties of thesines, tangents, &c.of double
ARcs and of HALF ARcs, with formulae, &c. 106 to 111
3. General properties of the sines, tangents, &c. of the
sums, and of the DIFFERENCEs of ARCs, including a
great variety of formulae - . . . 111 to 118
General properties of the sines, tangents, &c. of ARcs
in ARITHMETICAL PROGREssion - . 1 18 to 122
5. Of the sines, tangents, &c. of the MULTIPLEs of
ARCs - - - - . 122 to 124
6. Of the sines and cosines of the Powe Rs of
ARcs - - - - . 124 to 126
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. - - - 126 to 129
8. The construction of a table of sines, &c. . 129 to 132

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BOOK III.
CHAPTER I. DEFINITIONs, &c. OF SPHERICAL ANGLES,
ARCS, AND TRIANGLES - • 135

2. Spherical Geometry, or general properties of spheri-
cal angles, arcs, and triangles, &c. - 155 to 152

CHAP. II. THE STEREOGRAPHIC PROJECTION OF THE

SPHERE e - - • 152 1. Stereographical theorems - - 153 to 1.59 2. Stereographical problems 159 to 165,

CHAP. III. INVESTIGATION OF GENERAL RULES FOR CALCULATING THE SIDES AND ANGLES OF RIGHTANGLED SPHERICAL TRIANGLES, &c. 165 to 170

2. BARoN NAPIER’s universal rules for solving rightangled spherical triangles - - . 1 70

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