17 (d) Triangles which have equal altitudes are to one another as their bases, and triangles which have equal bases, as their altitudes; also any two triangles are to one another in the ratio, which is compounded of the ratios of their bases and altitudes 65, 66 (e) Triangles which have one angle of the one equal to one angle of the other are to another in the ratio, which is compounded of the ratios of the sides about the equal angles, or as the rectangles under those sides 66 (f) Triangles which have one angle of the one equal to one angle of the other, and the sides about the equal angles reciprocally proportional, are equal to one another; and, conversely, equal triangles, which have one angle of the one equal to one angle of the other, have the sides about the equal angles reciprocally proportional (g) Two triangles are similar, when they have 66 1. The three angles of the one equal to the three angles of the other, each to each 59 or 2. The three sides of the one proportional to the three sides of the other 59 or 3. One angle of the one equal to one angle of the other, and the sides about the equal angles proportionals 60 or 4. One angle of the one equal to one angle of the other, and the sides about two other angles proportionals, and the remaining angles of the same affection, or one of them a right angle 61 (h) Similar triangles are to one another in the duplicate ratio (or as the squares) of their homologous sides 67 ()Of all triangles having the same two sides, that one has the greatest area, in which the angle contained by the two sides is a right angle 103 (k) Of triangles which have equal bases, and equal areas, the isosceles has the least perimeter; and of triangles having equal bases and equal perime Now the area of the triangle ABC is equal to LF x AK, for it is equal to half the rectangle under the radius L F of the inscribed circle, and the sum 2 A K of the three sides (I. 26. cor.); also BFX BK is equal to LF x M K, because, LB M being a right angle, the right-angled triangles BFL, M KB are similar: but AKX AF: AKX FL :: AF FL, i. e.:: A K: KM, i. e. :: AK x FL: KM FL: therefore A Kx FL is a mean proportional between A K x AF and K M x F L or BFX BK, that is (if a, b, c represent the three sides opposite to the angles A, B, C respectively and S the half of (a+b+c) the area of the triangle is a mean proportional between Sx (Sa) and (S - b) x (S−c). PUBLISHED UNDER THE SUPERINTENDENCE OF THE SOCIETY LONDON: BALDWIN AND CRADOCK, PATERNOSTER ROW. MDCCCXXXIII. ELEMENTS OF TRIGONOMETRY. INTRODUCTION. THE term Trigonometry is derived from the two Greek words, Toiywvov, a triangle, and μerpéw, 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 ČA 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, subtending any angle ACP, this are cannot, A 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 7,, and the length of the semicircumference of the circle by II, |