It has been my object to provide an introduction to Mr Todhunter's excellent Manual, and to write a simple book, which shall contain enough matter for a term's reading, and examples sufficiently easy for a boy to work out unaided, as evidence to himself that he understands what he has read. It gives me great pleasure to think that this little book bears a tribute to the memory of the late Canon Temple Chevallier, of Durham, who read it and gave me many very valuable suggestions. I have been also very much assisted in the idea and progress of the work by my Tutor and Friend, Mr J. F. Moulton, of Christ's College. J. C. P. A. Jan. 1st, 1874. 1 CHAPTER I. ANGLES AND THE MEASUREMENT OF ANGLES. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line. This is one of Euclid's Definitions, which describes the nature of the angles treated of in Plane Trigonometry. The inclination of one line to another, that is, the angle between the two lines, may be increased or diminished; it may also be doubled or halved, or any multiple may be taken of it. Also one angle may be said to be greater than another or less. Hence to distinguish and describe an angle it is natural to take a measure, for we see that angles differ in magnitude, and measurement can be applied to them as well as to lengths and other quantities. Now a measure or standard must be of the same kind as the thing to be measured, and must be of a constant size. Acres cannot be measured by pints, nor can we be sure that we have 30 yards of cloth when that quantity A. T. 1 2 has been measured to us unless all yard measures are of the same length. There is one well-known angle by which angles may be measured. DEF. When one straight line standing on another straight line makes the adjacent angles equal to one another, each of these angles is called a Right Angle. Now all right angles are equal to one another. So that any angle may be measured by them. 3 For example, of a right angle or 2 of a right 4 angle denote certain angles which we have expressed in terms of this measure or standard angle. We may refer to the Mariner's Compass as a use of the right angle as a measure; for when a pilot wishes the course of a ship altered, he speaks of turning it through two or three points; the right angle being divided into eight points this denotes that it is 1 3 turned through or of a right angle. 4 8 And as the ship might be turned round and round we might imagine it as turned through any number of points. Though we cannot draw an angle greater than two right angles, we see in this way that there is really no limit to its size. 3 It has been shewn that an angle may consist of any number of right angles or parts of a right angle, that is, an angle will be known if we know what fraction or multiple it is of a right angle. But it would be inconvenient to measure an angle only by its relation to a right angle, just as it would be |