34 and will be 100th parts. If H E be tens, EB will be units, B C will be 10th parts. If H E be hundreds, B E will be tens, and B C units. And so on, each set of divisions being tenth parts of the former ones. For example, suppose it were required to take off 242 from the scale. Extend the dividers from E to 2 towards H; and with one leg fixed in the point 2, extend the other till it reaches 4 in the line E B; move one leg of the dividers along the line 2, 7, and the other along the line 4, till they come to the line marked 2, in the line B C, and that will give the extent required. PLANE TRIGONOMETRY. DEFINITIONS. 1. PLANE TRIGONOMETRY is the art by which, when any three parts of a plane triangle, except the three angles, are given, the others are determined. 2. The periphery of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds, &c. 3. The measure of an angle is the arc of a circle, contained between the two lines that form the angle, the angular point being the centre; thus the angle ABC, Fig. 30, is measured by the arc D E, and contains the same number of degrees that the arc does. The measure of a right angle is therefore 90 degrees; for D H, Fig. 31, which measures the right angle D C H is one fourth part of the circumference, or 90 degrees. Note. The degrees, minutes, seconds, &c. contained in any arc, or angle, are written in this manner, 50° 18′ 35" which signifies that the given arc or angle contains 50 degrees, 18 minutes and 35 seconds. 4. The complement of an arc, or of an angle, is what it wants of 90°; and the supplement of an arc, or of an angle, is what it wants of 180°. 5. The chord of an arc, is a right line drawn from one extremity of the arc to the other: thus the line BE is the chord of the arc B A E or B D E, Fig. 31. 6. The sine of an arc, is a right line drawn from one extremity of the arc, perpendicular to the diameter which passes through the other extremity: thus B F is the sine of the arc A B or B D, Fig. 31. 7. The cosine of an arc, is that part of the diameter which is intercepted between the sine and the centre: thus C F is the cosine of the arc A B, and is equal to B I, the sine of its complement HB, Fig. 31. 8. The versed sine of an arc, is that part of the diameter which is intercepted between the sine and the arc: thus AF is the versed sine of A B; and D F of D B, Fig 31. 9. The tangent of an arc, is a right line touching the circle in one end of the arc, being perpendicular to the diameter which passes through that end, and is terminated by a right line drawn from the centre through the other end: thus AG is the tangent of the arc A B, Fig 31. 10. The secant of an arc, is the right line drawn from the centre and terminating the tangent; thus C G is the secant of A B. 31. 11. The cotangent of an arc, is the tangent of the complement of that arc; thus H K is the cotangent of A B. Fig. 31. 12. The cosecant of an arc, is the secant of the complement of that arc; thus C K is the cosecant of A B. 13. The sine, cosine, &c. of an angle is the same as the sine, cosine, &c. of the arc that measures the angle. PROBLEM I. To construct the lines of chords, sines, tangents, and secants, to any radius. Fig. 32. Describe a semicircle with any convenient radius CB; from the centre C draw CD perpendicular to A B and produce it to F; draw BE parallel to CF and join AD. Divide the arc AD into nine equal parts as A, 10; 10, 20; &c. and with one foot of the dividers in A, transfer the distances A, 10; A, 20; &c. to the right line AD; then will AD be a line of chords constructed to every ten degrees. Divide BD into nine equal parts, and from the points of division 10, 20, 30, &c. draw lines parallel to CB, and meeting CD in 10, 20, 30, &c. and CD will be a line of sines. From the centre C, through the divisions of the are BD, draw lines meeting BE, in 10, 20, 30, &c. andBE will be a line of tangents. With one foot of the dividers in C transfer the distances from C to 10, 20, &c. in the line BE, to the line CF which will then be a line of secants. By dividing the arcs AD and BD each into 90 equal parts, and proceeding as above, the lines of chords, sines, &c. may be constructed to every degree of the quadrant. PROBLEM II. ་ At a given point A, in a given right line AB, to make an angle of any proposed number of degrees, suppose 38 degrees. Fig. 33. With the centre A, and a radius equal to 60 degrees, taken from a scale of chords, describe an arc, cutting AB in m; from the same scale of chords, take 38 degrees and apply it to the arc from m to n, and from A through n draw the line AC, then will the angle A contain 38 degrees. Note. -Angles of more than 90 degrees are usually taken of at twice. PROBLEM III. To measure a given angle A. Fig. 34. Describe the arc mn with the chord of 60 degrees, as in the last problem. Take the arc mn between the dividers, and that extent applied to the scale of chords, will show the degrees in the given angle. Note. When the distance mn exceeds 90 degrees, it must be taken off at twice, as before. |