CHAP. XII, OF TRIGONOMETRY. 1. Definitions. TRIGONOMETRY is the art of finding all the fides and angles of a triangle, from having any three of these, one of which, at least, must be a fide. Or, to find the ratio of the fides, when the angles are given; and converfely the ratio of the angles, when the fides are given. And it is founded on the mutual proportion which fubfifts between the fides and angles of triangles, which proportions are known by finding the relations between the radius of the circle and certain lines drawn in and about the circle, called chords, fines, tangents, fecants, &c. For this purpose, the circumference of a circle is divided. into 360 parts, called degrees; and every degree fubdivided into 60 other parts, called minutes, and every minute into 60 feconds, and every fecond into 60 thirds, &c. and any angle is faid to confift of fo many degrees, minutes, and feconds, as are contained in the arc, that measures the angle; or that is intercepted between the legs or fides of the angle; the point of the faid angle being at the centre of the circle. 2. The 2. The complement of an arc is the difference between the arc and a quadrant. 3. The Supplement of an arc is the difference between the arc and a femicircle. 4. The right fine of an arc, commonly called the fine, is a perpendicular falling from one end of the arc, to the radius, drawn through the other end of the fame arc, as DE (fig. 1, plate 9) is the fine of the arc D B, and it is always equal to half the fubtenfe of double the arc. Thus, DE is equal to chord of the arc DO; half of DO, which is the fubtense or and the arc DO is double the arc DB. Hence the fign of an arc of 30 degrees is equal to one half the radius, because the chord of 60 degrees is equal to the radius. 5. The fine complement of an arc is that part of the radius intercepted between the centre and the right fine, as CE, and is also the fine of the complement of the arc to a qua drant; for CE is equal to F D, which is the fine of the arc DH. 6. The cofine of an arc is the fame as the fine complement. 7. The verfed fine is that part of the radius intercepted between the right fine and the circumference of the circle, as E B. 8. The tangent of an arc is a perpendicular drawn from the extremity of the radius to the fecant, as B G, which is the tangent of the arc D B. 9. The fecant of an arc is a line drawn from the centre of the circle, through one end of the arc, till it meets the tangent; as C G. 10. The cofecant and cotangent of an arc is the fecant and tangent of that arc, which is the complement of the former arc. And the chord of an arc, and the chord of its complement to a circle, is the fame; fo likewise the fine, tangent, and fecant of an arc are the fame as the fine, tangent, and fecant of its fupplement or complement to a femicircle. Thus, the fine ED, the tangent BG, and fecant CG, is the the fine, tangent, and fecant of the arc, D A, which is the fupplement of the former arc D B. 11. The finus totus is the greatest fine, being the fine of an arc of 90 degrees, or one quarter of a circle, and is equal to the radius of the circle. Thus, the fines always increase from B, at which place they are nothing, till they come to the radius C H, which is the greateft, being the fine totus. From hence they decrease all the way along the fecond quadrant from H to A, and at length vanish at the point A; whereby we fee that the fine of the femicircle BH A, is nothing. After this, the fines are negative, as they proceed along the next femicircle A O B, being drawn on the oppofite fide, or downwards, from the diameter A B. As D E is the cofine of DH: the fine, cofine, and radius of any arc form a right-angled triangle; as, CDE, or CDF; of which, the radius C D is the hypothenufe: and therefore the fquare of the radius is equal to the fur of the fquares of the fine and cofine of any arc. The fines, cofines, tangents, &c. of every degree and minute, in a quadrant, are calculated to a radius of 1, and ranged in tables for ufe. But to fhorten the operation in calculations in trigonometry, we ufe the logarithms of them, instead of the natural numbers, which are called the artificial fines, tangents, &c. and these numbers fo ranged in tables, form the trigonometrical canon; and contain every species of right-angled triangles; fo that no triangle can be propofed, but one fimilar to it can be found there by comparison, with which the propofed one may be computed by analogy of proportion. Lastly, fometimes the proportion is not expreffed in numbers; but the feveral fines, tangents, &c. are actually laid down upon lines of fcales; from whence the line of fines, of tangents, &c. on the plane fcale, the construction and use of which follow : The plane fcale is a mathematical instrument of most extenfive use, commonly two feet in length. The lines ufually drawn VOL. II. N drawn upon it, are the following:-1. Lines of equal parts. 2. Of Chords.-3. Rhumbs.-4. Sines.-5. Tangents.6. Secants.-7. Semi-tangents.-8. Longitude.-and Latude. (Fig. 2.) 1. The lines of equal parts are of two kinds: viz.-fim. ply divided, and diagonally divided. The firft of these are formed by drawing three parallel lines, and dividing them into any number of equal parts, by fhort lines drawn across them; and in like manner fubdividing the first divifion into ten other equal smaller parts, by which numbers or dimen. fions of two figures may be taken off. Upon fome rules feveral of these scales of equal parts are ranged parallel to each other, with figures fet to them, to fhew into how many equal parts they divide the inch; as 20, 25, 30, &c. 2. The diagonal divifions are formed by drawing eleven long parallel lines, equidiftant from each other, which are divided into equal parts, and croffed by other fhort lines, as the former; then the first of these equal parts have the two outermost of the eleven parallel lines divided into ten equal parts, and the points of divifion connected by diagonal lines, as fhewn in Menfuration. The whole fcale is thus divided into dimen fions of three places of figures. The other lines upon the scale are commonly used in Trigonometry, Navigation, Aftronomy, Dialling, &c. &c. and are all constructed from the divifions of a circle, as follows:→→→ 2. Describe a circle* with any convenient radius, and di vide it in four equal parts, by two diameters, drawn at right angles to each other, (fig. 2.) Continue one diameter C D towards F, and draw the tangent line E A, parallel thereto; then draw the chords D A and D B. 3. To conftruct the line of chords, divide the quadrant A D into 90 equal parts: then on A, as a centre, with the * Only half the circle is drawn in the figure for want of room; but in general a complete circle is formed. compaffes, |