334 MATHEMATICS. 



We proceed to treat first of Trigonometry ; then of Coordinates, 

 and their immediate applications ; and lastly of the Conic Sections. 



1. Plane Trigonometry, has for its object the solution of pro- 

 blems concerning plane triangles ; the sides of which are always straight 

 lines. It is subdivided, in reference to the different kinds of triangles, 

 into Right Angled and Oblique. In any right angled triangle, ABC, 

 Plate VII. Fig. 13, if, from the vertex, A, of one of the acute angles, 

 as a centre, and with the hypothenuse. AC, for a radius, we describe 

 an arc of a circle, the side, BC, opposite to the vertex used, becomes 

 the sine, and the adjacent side, AB, becomes the cosine, of the angle, 

 A, in question. The cosine, prolonged, becomes another radius of 

 the same arc ; and the prolonged part, BD, beyond the triangle, is 

 called the versed sine of the arc or angle in question. If now we 

 apply a scale, on which the hypothenuse or radius shall be equal to 

 unity or 1, the sine and cosine will be expressed by decimals, which 

 are called the natural sine, and cosine, of the angle in question. 

 But if we take the radius equal to 10,000,000,000, (whose logarithm 

 is 10), and then find the logarithms of the corresponding lengths of 

 the sine and cosine, we shall have the logarithmic sine, and cosine, 

 of the same angle. Thus, angles may be designated by their sines, 

 or cosines. 



Again, if from the same vertex, A, as a centre, and with the base, 

 AB, as a radius, we describe an arc, then the other leg, BC, is 

 called the tangent, and the hypothenuse, AC, is called the secant 

 of the same angle. The tangent and secant of the complement of an 

 angle, or what it wants of 90, are called the co-tangent and co-secant 

 of the angle itself. It is chiefly by means of Tables of the sines and 

 co-sines, tangents and co-tangents of angles, that all problems of Tri- 

 gonometry are resolved. In every plane triangle, we must have given 

 at least three parts, sides and angles, one of which at least must be a 

 side, in order to find the other parts. 



Thus, in a right angled triangle, if we have given the base, and 

 angle at the base, the right angle being of course known, then, the 

 base is to the perpendicular, or other leg, as the cosine of the angle 

 at the base, is to the sine of the same angle ; and the base is to the 

 hypothenuse, as the cosine of the angle at the base, is to radius, or 

 the sine of 90. In an oblique angled triangle, the sides are pro- 

 portional to the sines of the opposite angles : also, the sum of any 

 two sides is to their difference, as the tangent of the half sum of the 

 two opposite angles, is to the tangent of their half difference : and 

 finally, the sine of half of either angle, is equal to radius multiplied 

 by the half sum of the three sides minus one of the adjacent sides, 

 this multiplied by the same half sum minus the other adjacent side, 

 and the whole divided by four times the product of the two adjacent 

 sides, adjacent to the angle sought. 



2. Spherical Trigonometry, has for its object the resolution of 

 spherical triangles, formed by arcs of great circles on the surface of a 

 sphere. The angles of such a triangle, (Plate VII. Fig. 14), are 

 those formed by the planes of its sides, with each other ; and its sides 

 are measured as arcs, by the number of degrees which they subtend, 

 at the centre of the sphere. In spherical triangles, the sines of the 



