Example. In the ▲ a b c, given ac, 407. ab,168. Proportion. 61. } req. b c The Angles being found by the 6th Cafe. S, 67. 30': S. 90° :: 407. 6: 446. Operation. To the Ar.co, SL b 67°. 30′ add Logar. ac, 407.6 Sum = Logar. bc 446. By Gunters Scale. 0. 034385 2. 609659 2.644044. The Extent from S, 67° 30' on the Line of To the Sine Reaches from • To 90,00 $ Sines. 487:6 on the Line 446. o of Numbers. Of Oblique-Angled Plane Triangles. Wherein there are 6 Cafes, and all rcfolved by the three following Axioms. AXIOM II. In al! Plane Triangles, the fides are proportional to the Sines of their oppofite Angles. AXIOM III. In all Plane Triangles, As the Sum of the Legs of any Angle, is to their Difference; So is the Tangent of half the Sum of their oppofite Angles, To the Tangent of half their Difference. AXIOM IV. As the Pale, or longeft Side, is to the Sum of the other Sides; So is the Difference of thofe Sides To the Difference of the Segments of the Bale. CASE T Fig. 61. Two Sides, and an Angle oppofite to one of them given; the Angle oppofite to the other required. Example. In the Abcd, Scd=139 requir'd the cb=64 given, 2<d=22° 30° Acute L Proportion cb: S, 4 d::cd: S,4 b per-Ax, 2. 64: S, 22°, 30′ :: 1394 S, 56°152 Opera. Operation. To the Ar. co, Logar. cb 64 add the {S, Ld 22°. 30 = 8.193820 2. 143142 9. 582840 Sum-Radius, S. L b, 56° 151 9.919 802. By Gunters Scale. The extent from 64 on the Line of Num To 139 bers. Reaches from S, 22° 30' on the Line of To S, 56-15 S Sines. CASE II. Two Sides and an Angle oppofire to one of them given, the third S de required. Example. cd=139 In the abcd, g`ven cb=64} requir'd b. Ld22°,30' Preparation. By Cafe the firft find the Angle b, and knowing the Angle d, you will find the Angle c to be ICI 151. Proportion, per Ax, 2. S, 4 b: cd:: S, L c: b d. Ar.co. S, Lb 56° 15' S. Lc 101. IS 0.080554 2. 143142 9.991594. Sum-Radius=Lg. bd, 164. 12.214870. By Gurters Scale. The extent from the Sine of 56° 15 to IOI 15' on the Line of Sines, reaches from 139 to 164 on the Lines of Numbers... Note, That the Sine of 101' 15, is the Sine of the Complement to 180. degrees Sine of 71° 45'. CASE III. Two Argles and a Side given, the other Side required. In Obl. bcd given, Example. Sle=101°.151? 4 d = 22.30` requ. Za b, S, Ld:cb::SLc:bd. S. 22°. 30': 64: S, ICI 15: 164. Operation. To Ar. co, SL d 22° 301 add the Log, cb 64 Sbd 0. 417160 9. 991574 1.806180 Sum-Radius Lɔg. bd. 164. 12. 214914 By Gunters Scale. The extent from the S, 22° 33' to the Sine 101° 151 on the Line of Sines, reaches from 64 to 164 on the Line of Numbers. S, 22. 30: S,56. 15:64:139. CASE CASE IV. Two Sides and an Argle comprehended by them, given; the other Angles required. Example. In obl. A bcd, Sc=1010, 1517 given, cb=64 cd 139. L d req.4.b Preparation by the third Axiom. cd =139 cb 64 Differ, cd-cb= 75. to the 3 Angles of a Triangle. From 180° Subftract 101, 15 Remain 78, 45 Remain, 3922 Angles. to the Sum of the oppofite Angles. to half the Sum of the oppofite PROPORTION. cb – cả : cả -cb :: T, 1 x LL : T, = X L,L, . 203: 75: T, 30'22':T, 16′ 52′ 1⁄2 OPERATION. 2230 C 20 |