3. Making BC the Radius. T. C: AB:: Sec. C: AC. i. e. As the Tangent of C to AC, 47° 40 10.04048 190 2.27875 47°. 401 10.17170 12.45045 As the Tangent of C 47°. 401 10.04048 is to AB, So is Radius, to BC, 190 90° 2.27875 10.00000 12.27875 Or having found one of the required Sides, the other may be obtained, by one or the other of the Cors. to Theo. 14. Sect. 1. By Gunter's Scale. 1. When AC is made the Radius. Extend from 47° 40', to 90°, on the Line of Sines; Sines; that Distance will reach from 190 to 257, on the Line of Numbers, for AC. 2. When AB is made the Radius, the firft Stating is thus performed. Extend from 45° on the Tangents, (for the Tangent of 45° is equal to the Radius, or to the Sine of go, as before) to 42°. 20'; that Extent will reach from 190, on the Line of Numbers, to 173, for BC. 3. When BC is made the Radius, the second Stating is thus performed. Extend from 47°. 40', on the Line of Tangents, to 45°, or Radius; that Extent will reach from 190 to 173, on the Line of Numbers, for BC; for the Tangent of 47°. 40', is more than the Radius; therefore the fourth Number must be lefs than the fecond, as before. The two firft Statings of this Cafe, answer the Question without a Secant. CASE III. The Angles and Perpendicular given; to find the Bafe and Hypothenufe. Plate V. Fig. 6. In the Triangle ABC, there is the Angle A 40", and confequently the Angle C 50°, with BC 172, given to find AC and AB. Geo Geometrically. Make an Angle CAB of 40° in blank Lines; (by Prob. 16. Sect. 1.) with BC 172, from a Line of equal Parts, draw the popped Lines EF parallel to AB (by Prob. 8. Sect. 1.) the lower Line of the Angle, and from the Point where it cuts the other Line in C, let fall a Perpendicular BC (by Prob. 7. Sect. 1.) and the Triangle is conftructed: The Measures of AC, and AB, from the fame Scale that BC was taken, will answer the Question. What has been faid in the two foregoing Cafes, is fufficient to render the Operations in this, both by Calculation and Gunter's Scale, so obvious, that it is needless to infert them; however, for the fake of the Learner, we give for Answer, AC 264. 5, and AB 202. 6. CASE IV. The Bafe and Hypothenufe given; to find the Angles and Perpendicular. Plate V. Fig. 7. In the Triangle ABC, there is given, AB 300, and AC 500: The Angles A and C, and the Perpendicular BC, are required. Geometrically. From a Scale of equal Parts, lay goo from A, to B; on B erect an infinite blank perpendicular Line, with AC 500, from the fame Scale, and one Foot of the Compass in A, crofs the perpendicular Line in C; and the Triangle is conftructed. By By Prob. 17. Sect. 1. Measure the Angle A, and let BC be measured from the fame Scale of equal Parts that AC and AB were taken from; and you have the Answer. By Cor. 2. Theo. 5. 90-36°. 52'=53°. 081 the Angle A. 300 10.00000 2.47712 So is the Tangent of A, 53°.08 10.12499 to BC, 400 2.60211 Or BC may be found from Cor. 1. Theo. 14Sect. I. By Gunter's Scale. 1. Making AC the Radius, Extend from 500 to 300, on the Line of Numbers; that Extent will reach from 90°. on the Line of Sines to 36°. 52', for the Angle C. Again. Extend from 90° to 53°. 08', on the Line of Sines, that Extent will reach from 500 to 400, on the Line of Numbers, for BC. 2. Making AC the Radius the second Stating is thus performed. Extend from Radius, or the Tangent of 45°, to 53°. 08', that Extent will reach from 300 to 400, for BC. |