As the circumference of every circle is commonly divided into 360 equal parts, a quadrant, which corresponds to a right angle, contains 90 of those parts, that is, 90 degrees. Degrees, minutes, seconds, and thirds, are denoted O by these marks, °,!," """. Thus 27°, 30', 40", 56"", sig nifies 27 degrees, 30 minutes, 40 seconds, 56 thirds. Angles smaller than seconds are generally expressed in decimals. In most of the modern continental works, the decimal division of angles is adopted, of which we shall say something in a subsequent part of this work DEFINITIONS. 1. The complement of an arc is its defect from a quadrant, or what it wants of 90°; and its supplement is its defect from half the circumference, or what it wants of 180°. 2. The sine of an arc is a perpendicular let fall from one of its extremities upon a diameter passing through the other extremity. 3. The versed sine of an arc is that part of the diameter between its sine and the circumference. 4. The chord of an arc is a straight line, less than the diameter, joining the extremities of the arc. 5. The tangent of an arc is a perpendicular drawn at one extremity of a diameter, and produced to meet another diameter passing through the other extremity. 6. The secant of an arc is a right line drawn from the centre through its extremity, and produced to meet the tangent. the other two can least being given. of equiangular tr obviously follow, be discovered fr the considerati As the port. angle rests, iangles, in cale though angles indifferently minutes, ami measureme: inclinations any tangible crease or com they ther the calculate parison of e angles step a measure The is to in those r which acc fav. 60° is the side of a hexigon inscribed in a circle, it mal to radius also, (Prop. 15, 4.) And if the angle OB, measured by the arc AB, be 45°, the angle 'O is 45°, (32.1); therefore (6.1) AO=AP; that the tangent of 45° is equal to the radius: hence the *** of 90°, the chord of 60°, the tangent of 45°, and us, are all equal. 2. As the diameter which bisects an arc, bisects also → chord of that are at right angles, it necessarily follows it half the chord of an arc is the sine of half that are. or, as Fk is equal to kE, the angles FOX, E0x e equal (27.3), and FO being equal to OE, and common, Fr is equal to Er, and the angle F equal to OxE (4.1): hence, Fr or Er is the ne of the arc Fk or kE; that is, of half the are kE. Let ABFE be a circle, of which the diameters AF and CE are at right angles; having taken any arc AB, erect the perpendicular AP, meeting OB produced; also draw CG and DB perpendicular to OC, and BS perpendicular to OA. Then CB is the complement of the assumed arc AB; its supplement is BCF; BS is the sine; BD or OS is the cosine; SA is the versed sine; DC the coversed sine; the supplementary versed sine is FS; the tangent is AP, and cotangent CG; the secant of AB is OP; cosecant OG; and the chord of the arc FkE is EF. In naming the sine, tangent, or secant of the complement of an arc, we generally use the abbreviated terms, cos. cot. co-sec.; we also use the letter R for radius. From these definitions numerous obvious consequences flow: 1. The sine of a quadrant, or 90°, is equal to the radius, OC being the sine of ABC; and as the chord of 60° is the side of a hexigon inscribed in a circle, it is equal to radius also, (Prop. 15, 4.) And if the angle AOB, measured by the arc AB, be 45°, the angle APO is 45°, (32.1); therefore (6.1) AO=AP; that is, the tangent of 45° is equal to the radius: hence the sine of 90°, the chord of 60°, the tangent of 45°, and radius, are all equal. 2. As the diameter which bisects an arc, bisects also the chord of that arc at right angles, it necessarily follows that half the chord of an arc is the sine of half that are. For, as Fk is equal to kE, the angles FOx, EOx are equal (27.3), and FO being equal to OE, and Ox common, Fx is equal to Ex, and the angle Ox F equal to OxE (4.1): hence, Fx or Ex is the sine of the arc Fk or kE; that is, of half the are FkE. |