99 they have (© Triangles which stand upon the same ters, the isosceles has the greatest area or upon equal bases, and between the 17 (a) To describe a triangle, when there 1. Two sides and the included angle. 2. Two sides, and an angle opposite to one of them. 3. Two angles and the interjacent side. 65, 66 4. Two angles and a side opposite to one of them. 5. The three sides 27 are given the 1. Vertical angle, base, and sum (or. difference) of the two sides. 2. Vertical angle, base, and area. 3. Vertical angle, sum (or differ- ence) of sides and area. 4. Base, sum (or difference) of sides and area 121 1. Which shall be equal to a given rectilineal figure, and have a side and adjoining angle the same with a given side, and adjoining 28 2. Which shall be equiangular with a given triangle, and have a given 59 28 3. Which shall have for two of its sides the parts which are cut off 59 by the third side from two straight lines given in position, and the third passing through a given point 77 60 (d) To describe a right-angled triangle 74 See « Circle.": Triplicate, one ratio said to be of another 34 Trisection of an arc or angle, under what form the problem has been put 113 Undecagon. See “ Hendecagon." Ungula. See “ Sphere.” Unit of length, or linear unit, is any arbitrary straight line, as an inch, a foot. -of surface, is the square of the linear unit sch. 18 --of content, is the cube of the linear unit sch. 142 proportions sch. 62 Vertex of a triangle 2, polyhedron 126, of the diameter of a conic section 220 -a term in perspective projection 208 215 Vertical anglé, of a triangle. See“ Triangle.” Volume, See" Content." Wedge, spherical. See “ Sphere." Zone, spherical. See “ Sphere." . 66 PUBLISHED UNDER THE SUPERINTENDENCE OF THE SOCIETY FOR THE DIFFUSION OF USEFUL KNOWLEDGE. LONDON: MDCCCXXXIII. ELEMENTS OF TRIGONOMETRY. INTRODUCTION. The term Trigonometry is derived from the two Greek words, Tplywvov, a triangle, and uerpéw, I measure, and originally signified simply the science by which those relations are determined which the sides and angles of a triangle bear to each other, being called plane or spherical trigonometry according as the triangle was described on a plane or spherical surface. At present, however, the term has a much more extensive signification, as the science now embraces all the theorems expressing the relations between angles and those functions of them to be hereinafter described ; the terms plane and spherical still denoting those branches of the science immediately connected respectively with plane and spherical triangles. The present treatise will be found to contain the fundamental theorems of the science, with their applications to plane trigonometry and to the construction of trigonometrical tables. A knowledge of these theorems should be acquired by the student before he proceeds to the Differential Calculus; the remainder of the subject, as a branch (and a most important one) of pure analysis, he may, in many respects, read more advantageously when some knowledge of the Calculus shall have prepared him to enter on a wider field of analytical investigation. The ancients, it is well known, cultivated astronomy with considerable assiduity and success; but as little advance could be made in it without a knowledge of trigonometry, their cultivation of this branch of mathematics was necessarily co-existent with that of astronomical science. Little, however, of what was written by them on this subject is come down to us. During the dark ages which overshadowed the nations of Europe, trigonometry, in common with other sciences, seems to have made some little progress among the Arabians, from whom it was derived by the Europeans after the revival of literature and science among them, about the beginning of the fifteenth century. After this period, the attention of scientific men, in imitation of the ancients, was principally directed to astronomy; and those to whom that science was most indebted for its progress were those to whom trigonometry also was equally indebted. Among the first of these may be mentioned George Purbach, professor of mathematics and astronomy at the University of Vienna, and John Müller, his pupil and successor, sometimes called Regiomontanus, from Mons Regius, or Koningsberg, a small town in Franconia, the place of his nativity. The former was born in 1423, and died in 1462 ; the latter was born in 1436, and died in 1476. Copernicus also, the celebrated astronomer, wrote a treatise on trigonometry about the year 1500. Several others might also be mentioned, the greater part of whom were natives of Germany, the progress of astronomy and trigonometry, for a considerable period after their revival in Europe, having been due very principally to the philosophers of that country. Vieta, a native of France, was born in 1540, and was one of the first mathematicians of his time. He gave improved methods of calculating trigonometrical tables, and enriched the science with a variety of theorems. He appears to have been the first who made any considerable application of algebra to this subject. The following authors also wrote on trigonometry: George Joachim Rheticus, professor of mathematics in the University of Wittemburg, who died in 1576; Pitiscus; Valentine Otho, mathematician to the electoral Prince Palatine; and Christopher Clavius, a German Jesuit. These authors lived during the latter part of the sixteenth century. At the period we have just mentioned, all the fundamental formulæ of trigonometry, and their applications to the calculation of tables and the sides and angles of triangles, were well known; but the immense progress of modern analysis has since opened a wide field for the applications of trigonometry, beyond those primary objects of the science to which the first part of this treatise will be devoted. Further historical notices will be reserved, to be made in immediate connexion with such parts of the subject as they may tend to elucidate, or render more interesting to the student. SECTION I. Trigonometrical definition of an Angle-Complements and Supplements of Angles, and of the arcs subtending them—Numerical measure of Angular space-Sexagesimal and Centesimal divisions of the Circle. (1.) Def. An angle, in geometry, denotes the inclination of one straight line to another, and, in this simple acceptation, must be less than two right angles. In trigonometry, the term has a more extended meaning. (See Geom. III. $ 2. Prop. 13. Schol.) Let Č A be a fixed line, and C a given point in it; and suppose CP to revolve in one plane about C, coinciding at first with C A ; then is the whole angular space described by CP in its revolution about C, called an angle, which may, therefore, in this case, be of any magnitude whatever; also, if with the centre C and any radius we describe a circular arc, sub A tending any angle A CP, this arc cannot, according to the geometrical definition of an angle, be greater than the semi-circumference of the circle; but, according to the trigonometrical definition of an angle, the subtending arc may be of any magnitude, consisting of any number of circumferences, or any part of a circumference. (2.) If we denote the angle ACP by A, the subtending arc by a, the sum of two right angles by Ty, and the length of the semicircumference of the circle by II, |