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8. The geometrical proofs (Art. 138) of the propositions that in any spherical right triangle:

I. If the sides including the right angle are in the same quadrant, the hypotenuse is 90°; if they are in different quadrants, the hypotenuse is > 90°.

II. An angle is in the same quadrant as its opposite side.

9. The discussion of the properties of spherical right triangles before those of spherical oblique triangles; see Chapters XI. and XII.

10. The reduction of the number of cases in the complete demonstration of the fundamental formulæ for spherical right triangles, to three, by application of the results proved geometrically in Art. 138; see Art. 143.

11. The discussions of the ambiguous cases in the solution of spherical oblique triangles (Arts. 171 and 172); especially the rules given on pages 130 and 132 for determining the number of solutions.

At the end of the book will be found a collection of formulæ in form for convenient reference.

Teachers who desire a briefer course are recommended to omit Chapter IV., which may be done without interrupting the logical completeness of the rest of the work. Chapter VI. may also be omitted by those who have taken the subject of Logarithms in their course in Algebra. The course might be still further abridged, if desired, by the omission of the exercises at the end of Chapter V.


BOSTON, 1887.








1. Trigonometry is that branch of mathematics in which algebraic processes are used to treat of the properties and measurement of geometrical figures.

In Plane Trigonometry we consider plane figures only.

2. An angle is measured by finding its ratio to another angle adopted arbitrarily as the unit of measurement.

3. The usual unit of measurement for angles is the degree, or an angle equal to the ninetieth part of a right angle.

To express fractional parts of the unit, the degree is divided into sixty equal parts, called minutes, and the minute into sixty equal parts, called seconds.

Degrees, minutes, and seconds are denoted by the symbols,,", respectively; thus, 43° 22' 37" denotes an angle of 43 degrees, 22 minutes, and 37 seconds.


4. Another method of measuring angles, and one of great importance, is known as the circular method, in which the unit of measurement is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

5. Let AOB be any angle, and AOC the unit of circular

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That is, the circular measure of an angle is equal to the ratio of its subtending arc to the radius of the circle.

For example, the circular measure of a right angle is equal to the ratio of one-fourth the circumference to the radius. But the circumference of a circle is equal to the radius multiplied by 2π, where 3.14159265...

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Hence, if R denotes the radius, we have

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6. Since the circular measure of 90° is the circular


measure of 180° is π; of 60°, ; of 45°,



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That is, an angle expressed in degrees may be reduced to circular measure by finding its ratio to 180°, and multiplying the result by T.

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