Example 11.-ha the line of chords. Figure 23. Sup pose it required to lay off an angle, A C a, equal to 35° ; then with any convenient opening of the sector, take the extent from 60 to 60 and with it, as radius, on the point C describe the arc A D indefinitely ; then in the same opening of the sector take the parallel distance from 35° to 35°, and set it from A to 11 in the arc A D, and draw A snake the angle at c as required.
Erumple Ill.-1n the lines of sines. Figure 21. The lines of sines, tangents, and secants, are used in conjunetion with the lines of lines in the solution of all the cases of plain trigonometry ; thus let there be given in the triangle A 13 c, the side A a = 230, and the angle A 11 C = 36° 30', to find the side A c. 'Here the angle at cis 35° 30' ; then take the lateral distance 230 from the line of lines, and make it a parallel from 53° 30' to 53° 30' in the line of sines ; then the parallel distance between 36" 30' in the same lines, will reach laterally from the centre to 170.19 in the line of lines for the side A c required.
Example the lines of tangents. If, instead of making the side c radius, as before, A B be made radius; then A c, which before was a sine, will be the tangent of the angle B; and, therefore. to find it, the lines of tangents must be used thus :—Take the lateral distance, 230, from the line of lines, and make it a parallel distance on the tangent radius, viz., from 45° to then the parallel tangent from 36° 30' to 36° 30', will measure laterally on the line of lines, 170.19, as before, for the side A c.
Example V.—In the lines of secants. In the same tri angle, in the base A B, and the angles at B and c given, as before, to find the side or hvpothenuse B c. Ilere B C is the secant of the angle B. Take the lateral distance 230 on the line of lines, make it a parallel distance at the radius, or beginning of the lines of secants; then the parallel secant of 60° 30' will measure laterally on the line of lines 287.12, for the length of B c, as required.
_Example V1.—In the lines of sines and tangents conjointly. Figure 25. In the solution of spherical triangles, use the line of sines and tangents only, as in the following example. In the spherical triangle A B C, right-angled at A, there is given the side A B = 36° 15' and the adjacent angle a = 42° 34' to find the side A c. The analogy is, as radius is to sine of A a, so is tangent of a to tangent of A C ; therefore ktake the lateral sign of 36° 15' a parallel at radius, or between 90 and 90; then the parallel tangent of 42° 34' will give the lateral tangent of 28° 30' for the side A C.
Example V11.—In the lines of polygons. Figure 24. It has been observed, that the chord of 60° is equal to radius, and 60' is the sixth part of 360°; therefore, such a chord is the side of a hexagon, inscribed in a circle. So that in the line of polygons, if the parallel distance between 6 and 6 be made the radius of a circle, as A C, and the parallel distance between 5 and 5 be taken and placed from A to 13, the line A 13 will be the side of a pentagon A B D E F, inscribed in the circle. In the same manner may any other polygon, from 4 to 12 sides, be inscribed in a circle, or upon any given line A B.
Of Gunter's lines.—Having thus shown the use of all that are properly called sectoral lines, or that are to be used secto• wise, it only remains w describe another set of lines usually put upon the sector, which will in a more ready and simple manner give the answers to the questions in the above exam ples; these are called artificial lines of numbers, sines, and tangents, because they are only the logarithms of the natural numbers. sines, and tangents, laid upon lines of scales; this method was first invented by Mr. Edmund Gunter, and the Imes are called Gunter 's lines, or the Gunter. Logarithms are a set of numbers, so contrived, that, by addition and sub traction, the answers to all questions in multiplication, divi sion, proportion, and the analogies of plain and spherical trigonometry, are found. Therefore, in the the extent, or ratio, between the first and second terms will always he equal to the extent, or ratio, between the third and fourth terms; consequently, if with the extent between the first and second terms, one foot of the compasses be placed on the third term, the other foot, on turning the com p:is:es about, will fall on the fourth term sought.
Thus. in Example I., of the three given numbers 4, 7, and 16, take the extent from 4 to 7 in the compasses, then place one foot in 16, and the other will fall on 28, the answer, in the line of numbers, marked a.
Again, the artificial line of numbers and sines are used together iu plain trigonometry, as in Example III., where the two angles a and c, and the side A B, are given ; for here, if the extent of the two angles 53° 30' and 36' 30' be taken in the line of sines, marked s, and one be placed upon 230 in the line of numbers, n, the other will reach to 170.19, the answer.
Also, the line of numbers and tangents are used conjointly, as in Example IV.; thus, take in the line of tangents, t. the extent from 45°, radius, to 36° 30', and it will reach from 230 to 170.19, the answer, as before.
Lastly, the artificial lines of sines and tangents are used together in the solution of spherical triangles.
Thus, Example VI. is solved by taking in the line of sines, s, the extent from 90°, radius, to 36° 15', and in the line of tangents, it will reach from 42° 34' to 28° 30', the answer required.
It may he farther observed, that each pair of sectoral lines contains the same angle, viz., six degrees in the common six inch sector ; therefore, to open these lima, to any given angle, as 35", for instance, take 35° laterally from the line of chords, and apply it parallel-wise from 60° to 60° in the same lines, and they will all be opened to the given angle 35°.
It' to the angle 35° be added the angle 6^, which they con tain, the sum is 41°; then take 41° laterally from the line of chords, and apply it parallel from 60 to 60, then will the sides or edges of the sector contain the same angle, 35°. In this case, the sector becomes a general recipe-angle, which is an instrument for taking the quantity of any angle contained between two inclining