Variation of the sine for values of the arc from zero to 360°,

Method from an arc> 90° to find one < 90° having the same sine,

Sines of negative arcs,

ib.

15

16

Definition of tangent,

ib.

Variations in the value of the tangent for arcs from zero up to 360°,

Definition of the secant,

Variation of the secant for variation of the arc from zero to 360°,

ib.

18

ib.

Expressions for the tangent, cotangent, secant, and cosecant, in terms of the

sine and cosine,

Relations of tangent and cotangent,

23

25

Exercises in the trigonometric functions,

Derivation of formulas for the solution of right angled plane triangles,

26

ib.

Method of finding the logarithm of a number between 100 and 10,000,

35

ib.

Method of finding the logarithm of a number beyond the limit of the

tables, .

of finding the number corresponding to any given logarithm,

Mode of finding the logarithms of a number by the tables of Callet, .

Examples in multiplication and division by logarithms,

Raising of powers and extraction of roots by logarithms,

Description of table of logarithms, sines, tangents, &c.,

47

Rules for finding the log. sin. tan., &c. of any given arc, expressed in

degrees and minutes,

48

Of an arc expressed in degrees, minutes, and seconds,

49

Rules for obtaining log. sec from log. cosine, and by log. cosec for log. sin

Logarithmic relations of tangents with cotangents,

ib.

ib.

Description and mode of using the trigonometric tables of Callet,

Method of finding the degrees, minutes, and seconds corresponding to any

given log. sine, tangent, &c.,

SOLUTION OF Right Angled TRIANGLES BY THE AID OF LOGARITHMS,
Definitions of trigonometric lines in terms of the ratios of sides of right
angled triangles and consequent rules of solution by two logarithms,
Rule for adding or rejecting 10 in the introduction of radius for homogeneity,
Examples in the solution of right-angled triangles,

Exercises in ditto,

Practical examples,

Practical exercises, .

Use of the arithmetical complement,

SOLUTION OF OBLIQUE ANGLED PLANE TRIANGLES,

Practical problems, .

Deduction of the formula for the cosine of an angle in terms of the three

sides of a triangle

Demonstration of formulas for the sine and cosine of the sum and difference

Derivation of formula for the sum of half an angle, in terms of the three

sides of a triangle,

Example under the above formula,

Blank form for the same,

Formulas for the sine of an angle, the cosine of half an angle, and the

tangent of half an angle in terms of the three sides,

Derivation of formulas for the sum and difference of the sines of two arcs

and their ratio,

Demonstration of formulas for the solution of a triangle when two sides

Demonstration of formula to be used in the last case, when only the

78

80

Miscellaneous exercises in Plane Trigonometry,

APPENDIX I.

Demonstration of formulas for the tangent of the sum and difference of two

Of the sine and cosine of 45°. Of the sine, &c., of 300 and 60°,

Extension of the demonstration for sine and cosine of (ab) to all the

Application of the formulas for computing a table of sines, &c.,

Formulas of verification in the computation of the tables,

Series for calculating sines, cosines, &c., .

Forms for finding the trigonometric lines and their logarithms for very small

arcs,

Derivation of series for calculating the logarithmic sines, cosines, &c., of

Forms for the sum and difference of the sines and cosines of two arcs,

Table of forms for the trigonometric lines in the different quadrants,

Forms for trigonometric lines of arcs greater than 360°,

Table of formulas expressing the most useful general relations of arcs,

Tables of formulas of less frequent occurrence,

An angle and the logarithms of the sides containing it being given, to solve

Demonstration of formula for solution of a triangle when two sides are

112

nearly equal,

Use of subsidiary angles,

Evasion of tabular errors for very large or very small angles,

Resolution of quadratic equations trigonometrically,

Determination of the increment of the sine, tangent, &c., from a small

Exercises in Plane Trigonometry and its applications,

114

ib.

113

115

ib.

116

Applications of right angled trigonometry to finding the sun's longitude

Demonstration of the formula for the cosine of an angle in terms of the

Of a formula for the cosine of half an angle in terms of the three sides,

Formulas for the sine and tangent of half an angle in terms of the three

130

sides of a spherical triangle, .

131

Blank form for the application of one of these formulas,

132

Derivation of the formulas for sine and cosine of a side in terms of the

New mode of deriving Napier's analogies from the Gauss equations,

Geographic example of the application of Napier's Analogies, .

Astronomic example of the same, viz. the determination of the latitude and

longitude of a heavenly body from its right ascension and decli-

nation, .

136

137

139

Mode of solution when two sides and the included angle are given, and the
unknown side only is required,

ib.

Mode of applying electro-magnetism to record the observation, .

Mode of solution when two angles and the included side, the unknown

Napier's rules for the solution of right angled triangles,

Astronomical example, from the sun's right ascension and declination, to

find his longitude, .

Exercises in right angled spherical trigonometry,

Astronomical Problem. Given the sun's declination, to find the time of his

Reduction of time of transit over any wire to time over imaginary middle

Page

140

141

143

144

145

ib.

147

148

149

150

ib.

ib.

152

wire,

ib.

Example of this reduction,

154

Demonstration of formula for determining the inclination of the axis,

ib.

Mode of computing the effect of this inclination on the time of meridian

transit, .

156

Demonstration of a formula for azimuth error,

157

Mode of computing its effect on the time of meridian transit,

160

Mode of computing the effect of error of collimation,

ib.

Example illustrating the application of all the corrections for instrumental

Conversion of siderial into mean solar time, and vice versâ,

164

Astronomical Problem. Given the latitude of a station and the declination

of a heavenly body, to find the time of its being on the six o'clock

hour circle, .

165

Solution of an oblique angled spherical triangle when two of the three given

parts are a side and its opposite angle,