5. The chord of an arc, is a right line drawn frow one extremity of the arc to the other: thus the line BE is the chord of the arc BAE or BDE, 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 BF is the sine of the arc AB or BD, Fig. 31. 7. The cosine of an arc, is that part of the diameter which is intercepted between the sine and the centre: thus CF is the cosine of the arc AB, and is equal to BI, 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 AB; and DF of DB, 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 AB, Fig. 31. 10. The secant of an arc, is the right line drawn from the centre and terminating the tangent; thus CG is the secant of AB, 31. 11. The cotangent of an arc, is the tangent of the complement of that arc; thus HK is the cotangent of AB, Fig. 31. 12. The cosecant of an arc, is the secant of the com. plement of that arc; thus CK is the cosecant of AB, 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 AB 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 arc BD, draw lines meeting BE, in 10, 20, 30, &c. and BE 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 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 off 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. OF THE TABLE OF LOGARITHMIC OR ARTIFICIAL SINES, TANGENTS, &e. THIS table contains the logarithms of the sine, tangent, &c. to every degree and minute of the quadrant, the radius being 10000000000, and consequently its logarithm 10. Let the radius CB, Fig. 32, be supposed to consist of 10000000000 equal parts as above, and let the quadrant DB be divided into 5400 equal arcs, each of these will therefore contain l'; and if from the several points of division in the quadrant, right lines be drawn perpendicular to CB, the sine of every minute of the quadrant, to the radius CB will be exhibited. The lengths of these lines being computed and arranged in a table, constitute what is usually termed a table of natural sines. The logarithms of those numbers taken from a table of logarithms and properly arranged form the table of logarithmic or artificial sines. In like manner the logarithmic tangents and secants are to be understood. The method by which the sines are computed is too abstruse to be explained in this work, but a familiar exposition of this subject as well as the construction of logarithms may be seen in Simpson's Trigonometry. To find, by the table, the sine, tangent, &c. of an arc or angle. If the degrees in the given angle be less than 45, look left hand column; then in the column marked at the top of the table, sine, tangent, &c. and against the minutes, is the sine, tangent, &c. required. If the degrees are more than 45, look for them at the bottom of the table, and for the minutes, in the right hand column; then in the column marked at the bottom of the table, sine, tangent, &c. and against the minutes, is the sine, tangent, &c. required. Note. The sine of an angle and of its supplement being the same, if the given number of degrees be above. 90. subtract them from 180°, and find the sine of the remainder. To find the degrees and minutes, corresponding to a given sine, tangent, &c. Find, in the table, the nearest logarithm to the given one, and the degrees answering to it will be found at the top of the table if the name be there, and the minutes on the left hand; but if the name be at the bottom of the table, the degrees must be found at the bottom, and the minutes at the right hand. EXAMPLES. 1. Required the degrees and minutes in the angle whose sine is 9.64390. Ans. 26° 8'. |