6. The Sine of an arch, is a right line drawn from one extremity of that arch, perpendicularly to a diameter passing through the other extremity. 7. The Versed-Sine of an arch is that part of the diameter, intercepted between the sine and the periphery. 8. The Co-Sine of an arch is that part of the diameter, between the sine and the centre and is equal to the sine of the complement. 9. The Tangent of an arch is a right line, touching the arch in one extremity, and produced from thence till it meets a right line drawn from the centre, through the other extremity. 10. The Secant of an arch is that line drawn from the centre to the extremity of the tangent. 11. The Co-Tangent, and Co-Secant, are the tangent, and secant of the complement of the arch. 12. Every sine, co-sine, tangent, co-tangent, &c. corresponds to two arches, which are supplements to each other the one being as much greater than 90°, as the other is less. Thus the sine, &c. of 1°, is also the sine of 179°; the sine of 10°, is the sine, &c. of 170°; and the sine, &c. of 60° is also that of 120°, Thus, let BH, or AH, be any arch; then is A Table of Logarithm sines, tangents, secants, &c. is the Logarithms of the lengths of those lines, calculated to a given radius, for every degree, and minute of the quadrant; by the help of which, and the following Theorems, any three parts of a triangle, except the three angles, being given, the rest may be found. RECTANGULAR TRIGONOMETRY. fig. 61. B 1. In every right-angled plane triangle ABC, if the hypothenuse AC be made the radius, and with it a circle, or an arc of one, be described from each end; it is plain (from def. 5.) that BC is the sine of the angle A, and AB is the sine of the angle C; that is, the legs are the sines of their opposite angles. Because the sine, tangent or secant, of any given arc in one circle is to the sine, tangent, or secant of a like arc (or to one of the like number of degrees) in ano the other; therefore the sine, tangent, or secant of any arc is proportional to the sine, tangent, or secant of a like arc, as the radius of the given arc is to 10.00000, the radius from whence the logarithmick sines, tangents, and secants, in most tables are calculated, i. e. If AC be made the radius, the sines of the angle A and C, described by the radius AC, will be proportional to the sines of the like arcs, or angles in the circle, that the tables now mentioned were calculated for. So if BC was required, having the angles and AB given it will be fig. 61. As S.C: AB::S.A: BC. i. e. As the sine of the angle C in the tables, is to the length of AB; (or sine of the angle C, in a circle whose radius is AC ;) so is the sine of the angle A in the tables, to the length of BC, (or sine of the same angle, in the circle, whose radius is AC.) In like manner, the tangents and secants represented by making either leg the radius, will be proportion al to the tangents and secants of a like arc, as the radius of the given arc is to 10.00000, the radius of the tables aforesaid. Hence it is plain, that if the name of each side of the triangle be placed thereon, a proportion will arise to answer the same end as before; thus if AC be made the radius, let the word radius be written thereon; and as BC and AB, are the sines of their opposite angles; upon the first let S.A, or sine of the angle A, and on the other let S.C, or sine of the angle C, be wrote: then,. When a side is required, it may be obtained by this proportion, viz. As the name of the side given is to the side given, So is the name of the side required to the side required. Thus if the angles A and C, and the hypothenuse AC were given, to find the legs; the proportions will be That is, as radius is to AC, so is the sine of the angle A. to BC. And, That is, as radius is to AC, so is the sine of the angle C, to AB. When an angle is required, we use this proportion, viz. As the side that is made the radius," is to radius, So is the other given side, to its name. Thus if the legs were given to find the angle A, and if AB be made the radius, it will be |