Another instrument frequently used in trigonometrical constructions, is THE SECTOR. This consists of two equal scales, moveable about a point as a centre. The lines which are drawn on it are of two kinds; some being parallel to the sides of the instrument, and others diverging from the central point, like the radii of a circle. The latter are called the double lines, as each is repeated upon the two scales. The single lines are of the same nature, and have the same use, as those which are put upon the common scale; as the lines of equal parts, of chords, of latitude, &c. on one face; and the logarithmic lines of numbers, of sines, and of tangents, on the other. The double lines are A line of Lines, or equal parts, marked A line of natural Sines, A line of natural Tangents to 45°, A line of tangents above 45°, A line of Polygons, Lin. or L. Cho. or C. Sin. or S. Tan. or T. Tan. or T. Sec. or S. Pol. or P. The double lines of chords, of sines, and of tangents to 45°, are all of the same radius; beginning at the central point, and terminating near the other extremity of each scale; the chords at 60°, the sines at 90°, and the tangents at 45°. (See art. 95.) The line of lines is also of the same length, containing ten equal parts which are numbered, and which are again subdivided. The radius of the lines of secants, and of tangents above 45°, is about one fourth of the length of the other lines. From the end of the radius, which for the secants is at 0, and for the tangents at 45°, these lines extend to between 70° and 80°. The line of polygons is numbered 4, 5, 6, &c. from the extremity of each scale, towards the centre. The simple principle on which the utility of these several pairs of lines depends is this, that the sides of similar triangles are proportional. (Euc. 4. 6.) So that sines, tangents, &c. are furnished to any radius, within the extent of the opening of the two scales. Let AC and AC' (Fig. 40.) be any pair of lines on the sector, and AB and AB' equal por tions of these lines. As AC and AC' are equal, the triangle ACC' is isosceles, and similar to ABB'. Therefore, AB: AC:: BB': CC'. Distances measured from the centre on either scale, as AB and AC, are called lateral distances. And the distances between corresponding points of the two scales, as BB' and CC', are called transverse distances. Let AC and CC' be radii of two circles. Then if AB be the chord, sine, tangent, or secant, of any number of degrees in one; BB' will be the chord, sine, tangent, or secant, of the same number of degrees in the other. (Art. 119.) Thus, to find the chord of 30°, to a radius of four inches, open the sector so as to make the transverse distance from 60 to 60, on the lines of chords, four inches; and the distance from 30 to 30, on the same lines, will be the chord required. To find the sine of 28°, make the distance from 90 to 90, on the lines of sines, equal to radius; and the distance from 28 to 28 will be the sine. To find the tangent of 37′, make the distance from 45 to 45, on the lines of tangents, equal to radius; and the distance from 37 to 37 will be the tangent. In finding secants, the distance from 0 to 0 must be made radius. (Art. 201.) To lay down an angle of 34°, describe a circle, of any convenient radius, open the sector so that the distance from 60 to 60 on the lines of chords shall be equal to this radius, and to the circle apply a chord equal to the distance from 34 to 34. (Art. 161.) For an angle above 60°, the chord of half the number of degrees may be taken, and applied twice on the arc, as in art. 161. The line of polygons contains the chords of arcs of a circle which is divided into equal portions. Thus the distances from the centre of the sector to 4, 5, 6, and 7, are the chords of,,, and of a circle. The distance 6 is the radius. (Art. 95.) This line is used to make a regular polygon, or to inscribe one in a given circle. Thus, to make a pentagon, with the transverse distance from 6 to 6 for radius, describe a circle, and the distance from 5 to 5 will be the length of one of the sides of a pentagon inscribed in that circle. The line of lines is used to divide a line into equal or proportional parts, to find fourth proportionals, &c. Thus, to divide a line into 7 equal parts, make the length of the given line the transverse distance from 7 to 7, and the distance from 1 to 1 will be one of the parts. To find of a line, make the transverse distance from 5 to 5 equal to the given line; and the distance from 3 to 3 will be of it. In working the proportions in trigonometry on the sector, the lengths of the sides of triangles are taken from the line of lines, and the degrees and minutes from the lines of sines, Thus in art. 135, ex. 1, tangents, or secants. 35: R:: 26: Sin 48°. To find the fourth term of this proportion by the sector, make the lateral distance 35 on the line of lines, a transverse distance from 90 to 90 on the lines of sines; then the lateral distance 26 on the line of lines, will be the transverse distance from 48 to 48 on the lines of sines. For a more particular account of the construction and uses of the Sector, see Stone's edition of Bion on Mathematical Instruments, Hutton's Dictionary, and Robertson's Treatise on Mathematical Instruments. Multiplying both the numerator and denominator by cot a cot b, dividing by R2, and proceeding in the same manner, for cot(a-b) we have, NOTE L. p. 109. The errour in supposing that arcs less than 1 minute are proportional to their sines, can not affect the first ten places of decimals. Let AB and AB' (Fig. 41.) each equal 1 minute. The tangents of these arcs BT and B'T are equal, as are also the sines BS and B'S. The arc BAB' is greater than BS+B'S, but less than BT+BT. Therefore BA is greater than BS, but less than BT: that is, the difference between the sine and the arc is less than the difference between the sine and the tangent. Now the sine of 1 minute is 0.000290888216 And the tangent of 1 minute is 0.000290888204 The difference between the sine and the arc of 1 minute is less than this; and the errour in supposing that the sines of 1', and of 0' 52'′ 44′′'' 3''"' 45''''' are proportional to their arcs, as in art. 223, is still less. NOTE M. p. 110. There are various ways in which sines and cosines may be more expeditiously calculated, than by the method which is given here. But as we are already supplied with accurate trigonometrical tables, the computation of the canon is, to the great body of our students, a subject of speculation, rather than of practical utility. Those who wish to enter into a minute examination of it, will of course consult the treatises in which it is particularly considered. There are also numerous formulæ of verification, which are used to detect the errours with which any part of the calculation is liable to be affected. For these, see Legendre's and Woodhouse's Trigonometry, Lacroix's Differential Calculus, and particularly Euler's Analysis of Infinites. A TABLE OF NATURAL SINES AND TANGENTS; TO EVERY TEN MINUTES OF A DEGREE. IF the given angle is less than 45°, look for the title of the column, at the top of the page; and for the degrees and minutes, on the left. But if the angle is between 45° and 90°, look for the title of the column, at the bottom; and for the degrees and minutes, on the right. |