99 28 (c) Triangles which stand upon the same ters, the isosceles has the greatest area 17 (a) To describe a triangle, when there 1. Two sides and the included angle. triangles which have equal bases, as 2. Two sides, and an angle opposite their altitudes; also any two triangles to one of them. are to one another in the ratio, which 3. Two angles and the interjacent is compounded of the ratios of their side. bases and altitudes 65, 66 4. Two angles and a side opposite (e) Triangles which have one angle of to one of them. the one equal to one angle of the 5. The three sides 27 other are to another in the ratio, which (6) To describe a triangle, when there is compounded of the ratios of the are given the sides about the equal angles, or as the 1. Vertical angle, base, and sum (or rectangles under those sides 66 difference) of the two sides. (f) Triangles which have one angle of 2. Vertical angle, base, and area. the one equal to one angle of the other, 3. Vertical angle, sum (or differand the sides about the equal angles ence) of sides and area. reciprocally proportional, are equal to 4. Base, sum (or difference) of sides one another; and, conversely, equal and area 121 triangles, which have one angle of the (c) To describe a triangle one equal to one angle of the other, 1. Which shall be equal to a given have the sides about the equal angles rectilineal figure, and have a side reciprocally proportional 66 and adjoining angle the same (9) Two triangles are similar, when with a given side, and adjoining they have angle of the figure 1. The three angles of the one 2. Which shall be equiangular with equal to the three angles of the a given triangle, and have a given other, each to each 59 28 or 2. The three sides of the one pro 3. Which shall have for two of its portional to the three sides of the sides the parts which are cut off other 59 by the third side from two straight or 3. One angle of the one equal to lines given in position, and the one angle of the other, and the third passing through a given sides about the equal angles pro point 77 portionals 60 (d) To describe a right-angled triangle or 4. One angle of the one equal to which shall have its three sides proone angle of the other, and the portionals 74 sides about two other angles Şee « Circle.” Triplicate, one ratio said to be of another 34 one of them a right angle 61 Trisection of an arc or angle, under what (h) Similar triangles are to one another form the problem has been put 113 in the duplicate ratio (or as the squares) of their homologous sides 67 Undecagon. See “ Hendecagon.” (i) Of all triangles having the same Ungula. See “ Sphere." two sides, that one has the greatest Unit of length, or linear unit, is any arbitrary area, in which the angle contained by straight line, as an inch, a foot. the two sides is a right angle 103 -of surface, is the square of the linear unit sch. 18 (k) Oftriangles which have equal bases, and equal areas, the isosceles has the --of content, is the cube of the linear unit least perimeter; and of triangles sch. 142 having- equal bases and equal perime- Variation, a short form of expressing certain proportions sch. 62 Now the area of the triangle ABC is equal to Vertex of a triangle 2, polyhedron 126, LF X A K, for it is equal to half the rectangle pyramid 127, cone 267. under the radius L F of the inscribed circle, and of the diameter of a conic section 220 the sum 2 A K of the three sides (I. 26. cor.); also BF XBK is equal to LF X M K, because, LBM -a term in perspective projection 208 being a right angle, the right.angled triangles BFL, -of a conic section 215 M K B are similar : but AK XAF : AK X FL Vertical plane, in perspective projection 208 : : AF : FL, 1. e. : : AK: KM, i.e. : : AKX Vertical angle, of a triangle. See" Triangle.” FL: KMX FL: therefore A KX F L is a mean proportional between A K x A F and K M F L or Volume: See “ Content.” BF x B K, that is (if a, b, c represent the three sides opposite to the angles A, B, C respectively and Wedge, spherical. See “ Sphere." S the half of (a+b+c) the area of the triangle is a mean proportional between Sx (S –a) and ($ - b) X (S-c). Zone, spherical. See " Sphere." . . . sor LIBRARY OF USEFUL KNOWLEDGE. E L E M E N TS OF TRIGONOMETRY. BY W. HOPKINS, ESQ., M.A. 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, Tpiywvov, a triangle, and perpéw, 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 CA ; 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 Tin and the length of the semicircumference of the circle by II, P |