(9) The angles in one regular polygon are twice as many as in another polygon; and an angle of the former is to an angle of the latter as 3: 2; find the number of sides.

(10) The angles of a quadrilateral are in increasing geometrical progression, and the difference between the third angle and the fourth part of the first is 90°; find the angles. Let A, Ar, Ar2, Ar3, be the angles;

.. 4 (1 + r + 9 +23) = 360°,

PREFACE TO THE THIRD EDITION.

IN submitting this Rudimentary Treatise on Plane Trigonometry, after its having been carefully revised and corrected, the author would also respectfully invite the reader's attention to the collection of Mathematical Tables, forming part of this Series of Scientific Treatises; they will be found amply sufficient for all the practical purposes of Trigonometry, and will therefore furnish all the aid necessary in computations connected with the present subject.

The author has given, to illustrate the principles, a great number of examples fully worked out, and which will be of service to those who have not the aid of a teacher.

"In compiling the work, the best authors, whether French or English, have been consulted. Amongst others, those of Bonnycastle, Cape, De Morgan, Gaskin, Hall, Hind, Hymers, Snowball, Woodhouse, and Gregory; with Davies's edition of Hutton's Course.

"The problems have been principally taken from the Ladies and Gentleman's Diaries, the Cambridge Problems, and Laybourn's Repository.

"The demonstration of Demoivre's Theorem is taken from an able French work on Trigonometry, by Lefebure De Fourcy."

J. HANN.

CONTENTS.

CHAPTER I.

The object of Trigonometry. French and English Measures

Angular measures

Trigonometrical ratios

Tracing the various trigonometrical functions.

Values of sine, cosine, tangent, cotangent, secant and cosecant of

30°, 45°, and 60°

Page

1

2, 3

8

12, 13

nverse trigonometrical functions

32, 33

alues of the sine and cosine of 15°, 18°, 36o, 54°, 74°.

34, 35

Ambiguous case

51, 52, 53

Area of a triangle in terms of the radius of the inscribed circle

Area of the triangle in terms of the radius of the circumscribed

circle, &c.

`ractical remarks on the solution of triangles

75

TRIGONOMETRY.

CHAPTER I.

1. TRIGONOMETRY was originally considered to be the doctrine of triangles, but in its present improved state it has a much more extensive signification, which we shall hereafter shew even in this rudimentary treatise.

2. In estimating angular measures, we suppose the right angle to be the primary one, and to be divided into 90 equal parts, each of which is called a degree; each degree is supposed to be divided into 60 equal parts, each of which is called a minute; each minute is supposed to be divided into 60 equal parts, each of which is called a second, and so on to thirds, fourths, &c. Here one degree is considered as the angular unit.

3. Modern French writers, instead of using the sexagesimal division, use the centesimal; and it is to be regretted that the latter is not universally used, from the great ease with which all calculations are made in that division.

We shall, however, shew how to reduce French into English measures, and vice versâ.

If E and Frepresent the number of English degrees and French grades in the same angle,

4. The circumference of a circle is known to be about 3.14159 times its diameter, or, in other words, the ratio of the circumference to the diameter is represented by 3.14159; for this number writers generally put the Greek letter π.

B