Proportion. The Angles being found by the 6th Cafe. S. Lb: Rad,:: ac: bc. S, 67°. 30': S. 90° :: 407.6:446. Operation. To the Ar.co, SL 667°30′ 0.034385 add Logar. ac, 407.6 2. 609659 Of Oblique-Angled Plane Triangles. Wherein there are 6 Cafes, and all rcsolved by the three following Axioms. In all Plane Triangles, the fides are proportional to the Sines of their opposite 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. ΑΧΙΟM IV. As the Pase, or longeft Side, is to the Sum of the other Sides; So is the Difference of those Sides, To the Difference of the Segments of the Bale. CASE I. Fig. 61. Two Sides, and an Angle opposite to one of them given; the Angle opposite to the other required. Example. Scd=139 cb=64 } 1 requir'd the 2<d=22° 301 SAcute L Proportion. cb: S,Ld::cd: S,L b 64: S, 22°, '30':: 1394 S, 56°15 per-Ax, 2.. Opera Operation. To the Ar. co, Logar. cb 64 8.193820 Sum-Radius, S. Lb, 56° 151 9.919802. By Ganters Scale. The extent from 64 2 on the Line of Num To 1395 bers. Reaches from S, 22° 30') on the Line of To S, 56-155 Sines. CASE II. Two Sides and an Angle oppofire to one of them given, the third Side required. Example. cd=139 Ld22°,30' Preparation. In the Abcd, given cb = 64 } requir'd b. By Cafe the first find the Angle b, and knowing the Angle d, you will find the Angle c to be 101° 151. Proportion, per Ax, 2. S, L b: cd:: S, Lcbd. Op 'ration. Ar.co.S, Lb 56° 15' 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 101° 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. S. 22°. 30': 64:: S, ICI 15: 164. Operation. To Ar. co, SL 22° 301 add the Log, cb 64 Sbd cd 0. 417160 9. 991574 1.806180 Sum-Radius= Log. bd. 164. 12. 214914 By Gunters Scale. The extent from the S, 22° 33' to the Sine 101° 15'on the Line of Sines, reaches from 64 to 164 on the Line of Numbers. Proportion. S. S, 6::cb:cd 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. bed, Sc=101, 1512 given, cb = 64 cd=139 Ld req. Preparation by the third Axiom. From 180°0" Substract 101, 15 Differ, cd-cb = 75. Remain 78, 45 to the 3 Angles of a Triangle. to the Sum of the opposite Angles. Remain, 39 22=to half the Sum of the opposite Angles. PROPORTION. cb + cd: cdcb:: T, LL: T, X4,4و 203:75:: T, 30°22′: T, 16'52'를 OPERATION. To Ar. co Log. cd+cb 203. cd--cb Adding T, Sum Sum--Rad T, Diff. 75 7. 692504 1.875061 39°.22′ 9.914172 16.52,+9.481737. |