11' from A, a line be drawn to any part or division of the quadrant, as G, at 60", then that line A G is the chord of that are, or of 60". And it' the line A a be drawn, it will be the chord of 90° ; and by setting one foot of the compasses in A, and extending the other to the several divisions, 10, 20, 30, 40. &e., they may be transferred from the circle to the line A a, which then be properly divided into a line of chords in 10, 20, 30,'10, &c,, to 90 as on the plane scale.
If' front any point, o, in the quadrant, you let fall a per peudieular, G I, to the radius A c, or G 11 to the 1'1111111S c a, the line c i is called the, sine of the arc A o ; and the 111133 r, 11i the sine of the are G 11, the complement of A G to 90". And if' all the divisions of the quadrant were transferred to c 13 by lines parallel to A n ; then the line C 13 will be divided as a line of sines in the points, 10, 20, 30, 40, &c., to 90".
By having a rule from the centre, c, to the several divisions of the 1pindrant. 10, 20, 30, S.r.e., it will cut the line A 11 111 1110 points 20, 30, which will be thereby divided into a line of tangents ; and here it must be observed, that the line of tangents, T, nn the sector, extends but to 45°, equal to C A or c n radius ; and that the. line of lesser tangents, t, are projected from a lesser radius, and begin from 45° at the distance of its radius from the centre of the sector.
By drawing the line c L through the division 60° to the line A 5, it makes A L the tangent of 60°, and c L the secant of 60°, or of the angle A C L. And if one foot of the com passes in c, the other is extended to the several divisions in the line A E, and transfer them to the line C F, then will the part a F be thus divided into a line of secants ; being placed at the distance of the radius c a, from the centre of the sector, and beginning at a, where the radius ends.
It may be of use in many cases to observe, that the chord 60°, A G, is equal to the radius C A or c G; that the sine 60°, G be5ects the radius A c in t , and therefore the sine G 11 of 30° is equal to half' the radius, or c t. Therefore the secant, C L, of 60° is equal to twice the radius, A c ; for C t is to c as c A to C L, and consequently the cosine is to radius as radius to the secant. Also, the tangent A L is to radius A C as radius
c is to the co-tangent a K, From what has been said, the reason appears why the line of lines (or equal parts L) terminates upon the sector at 10; the line of chords, c. at 60° ; the line of sines, s, at 90° ; the larger tangents, T, at 45° ; and that the lesser tangents, and also the secants are, of indefinite lengths.
From the nature of the sector, consisting of two pairs, or legs, movable upon a central joint, it is requisite that the lines should be laid on the sector by pairs, Piz., one of a sort on each leg, and all of them issuing from the centre-all of the same length, and every two containing the same angle. We shall now illustrate the nature of working problems by the sector, as lidlows, by the lines of lines, or equal parts L L.
Figure 2:2.-Let C L, C L, be the two lines of lines upon the sector, oitened to an angle, L C L ; join the divisions 4 and 4, 7 and 7, 10 and 10, by the dotted lines a b, c d, L L. Then by the nature of similar triangles, c L is to c bas L L to a b; and c L is to c d as L L to c therefore a b is the same part of L L as c b is of c L. Consequently, if L L be 10, then a b will be 4, and c d will be 7, of the same parts.
And hence, though the lateral scale c L be fixed, yet a parallel scale, L is obtainable at pleasure ; and therefore, though the lateral radius is of a determined length in the lines of chords, sines, tangents, and secants, yet the parollol radius may be had of any size required, by means of the sector, as tar as its length will admit ; and all the parallel sines, &e. peculiar to it ; as will be evident by the following etamples in each pair of lines.
Example i.-In the lines rf equal parts. Figure 22. Ilaving three numbers given, 4, 7, 16, to find out it fourth proportional To do this, take the lateral extent of 16 in the lice C L, and apply it parallel-wise, from 4 to 4, by a proper opening of the sector ; then take the parallel distance from 7 to 7 in the compasses; and applying one foot in c, the other will thll on 28 in the line of lines, c L, and is the number required ; tbr :Is 4 is to 7 so is 16 to 28.