I must not omit to express my thanks to the friends who have given me assistance and advice from time to time during the progress of the work, and in particular to Mr Walton, of Trinity College, for his very kind interest and supervision. T. P. H. TRINITY COLLEGE, CAMBRIDGE, The Changes in Sign and Magnitude of the Trigonometrical Functions. Another mode of defining the Trigonome- trical Functions. The Groups of Angles corresponding Trigonometrical Functions of the Sum and Difference of Determination, à priori, of the number of Values which any assigned Trigonometrical Function may have when Trigonometrical Functions of Three Angles and relations A Collection of Examples in illustration of the preceding Examples illustrating Chapter I. Examples illustrating Chapter II. Examples illustrating Chapter III. Examples illustrating Chapter IV. Examples illustrating Chapter VI. Examples illustrating Chapter VIII. Examples illustrating Chapter IX.. DEFINITIONS. CHAPTER I. UNITS AND MEASUREMENT OF ANGULAR MAGNITUDES. 1. THE HE object of that branch of mathematical science, which is called Trigonometry, is the investigation of all geometrical properties and relations in which angular magnitude is concerned. In the earlier stages of its progress it was, as its name implies*, applied exclusively to the measurement of triangles, and to the establishment of propositions connected immediately with them. Its methods, however, have now received an extension and a generality which render it a most valuable analytical instrument in the higher departments of mathematics. Of all the elementary branches of mathematical science it is perhaps the one of which the practical utility is most distinctly apparent. The student will, for instance, without difficulty foresee how indispensable such methods of calculation are to the surveyor, the navigator, and the astronomer. 2. Extension of the definition of an angle. Euclid defines a plane rectilineal angle to be the inclination of two straight lines to one another, which meet together but are not in the same straight line. He does not in his definition take into account the direction in which this inclination is supposed to be estimated, and, moreover, necessarily limits the signification of the word to angular magnitudes which are less than two right angles. In Trigonometry, however, we regard an angle as capable of being of any magnitude whatever, and consequently Tρlywvov, a triangle, and perpéw, I measure. H. T. * B must have proper regard to the direction in which we estimate the inclination or opening between the two straight lines which contain the angle; i. e. to the direction in which one of the straight lines must be supposed to revolve from coincidence with the other in order to pass over the angular space in question. For instance, the straight lines AB, AC, according to Euclid, would only bound one right angle, but in accordance with the more extended definition of an angle, they may also be considered as containing an angle whose magnitude is three right angles, the line AB in this case B being supposed to revolve from right to left in order to move into coincidence with AC. A trigonometrical angle then must be regarded not merely as the opening between two straight lines, but as the angular space swept over by a revolving line, which starts from coincidence with one of the bounding lines of the angle and moves into coincidence with the other. Moreover, in order to effect this, the revolving line may be supposed to have made any number of complete revolutions, so that under this supposition we can have angular space of any magnitude whatever. For instance, the minute hand of a watch at a quarter past four o'clock will since twelve o'clock have revolved through an angle the magnitude of which is 17 right angles. 3. Angular units. In order to measure angles some particular angle must be chosen as a standard or unit. This selection is of course quite arbitrary, and is influenced only by considerations of convenience. 4. Degrees, minutes, seconds. Sexagesimal division of the right angle. The ninetieth part of a right angle is called a degree, the sixtieth part of a degree a minute, and the sixtieth |