PREFACE.

In the present work I have had chiefly in view the requirements of the Previous Examination as laid down in the new Scheme which is to come into operation for the first time in December of the present year.

I have also aimed at rendering the work a suitable Manual for the Junior Candidates at the various University Local Examinations, and to this end I have devoted a Chapter to the Numerical Solution of Triangles, and worked out some Examples in detail. The subjects embraced in the volume are likewise co-extensive with those required for the First Three Days of the Mathematical Tripos. The Chapters and Articles which I consider may safely be omitted by Candidates for Honours at the Previous Examination, have an asterisk (*) prefixed.

The Examples have been mostly selected from the College and University Examination Papers, and they have been all carefully worked out and the results given at the end of the volume.

The MECHANICS and HYDROSTATICS are in preparation, and will shortly appear.

My best thanks are due to J. H. Smith, Esq., M.A., Caius College, and to various friends, for their suggestions and advice respecting the work during its progress through the

press.

I shall be thankful for any corrections or suggestions from teachers or students.

PEMBROKE COLLEGE,

CAMBRIDGE,

Nov. 1st, 1866.

CONTENTS.

CHAPTER I.

Convention respecting the signs + and Circumference and Area of a Circle. Modes of measuring Angles

PAGE

1

CHAPTER II.

Definitions and Relations of Trigonometrical Ratios. Changes in the Sign and Magnitude of the Trigonometrical Ratios.

*CHAPTER III.

Construction of Angles with assigned Trigonometrical Ratios. General formulae for Angles with given Trigonometrical Ratios.

CHAPTER IV.

Trigonometrical Functions of two Angles.

*CHAPTER V.

Trigonometrical Ratios of known Angles. Formulae of Verification and

CORRECTIONS AND ALTERATIONS.

Page 1, Art. 2, line 2, after "intersecting" insert at right angles. Page 5, line 15, for "measure of D" read measure of A.

Pages 5 and 6, for "D" substitute 7.

Page 13, line 2, for "PM" read P'M'.

Page 27, line 11, for "prove them" read prove those in Art. 27. Page 30, line 19, omit "smaller."

PLANE TRIGONOMETRY.

CHAPTER I.

CONVENTION RESPECTING THE SIGNS + AND -. CIRCUMFERENCE AND AREA OF A CIRCLE. MODES OF MEASURING ANGLES.

1. The following Elementary Treatise on Plane Trigonometry contains "the modes of measuring angles, trigonometrical ratios, functions of two angles, and the properties of triangles," besides a Chapter on the solution of Triangles.

2. Let X'X, Y'Y be two indefinite straight lines intersecting in and dividing the plane of the paper into the four quadrants XOY, YOX', X'OY' and Y'OX, which are called respectively the first, second, third, and fourth quadrants.

If OX and OY be considered the positive directions, then OX' and OY' will be the negative directions.

Thus, if PM and PM' be perpendicular to X'X and P,N, and P,N' to Y'Y, then OM or PN is positive, and also ON or PM; whilst OM' or P,N' and ON' or PM' are negative.

In other words, perpendicular distances from XX and above it are positive, below it, negative. Similarly perpen

B