obLIQUE TRIG ONOME TRY. A X I O M II. N all Plane Triangles, the Sides are in direét Proportion to the Two Angles and one Side given, to find either of the other Legs. a Circle, it is plain that each Side becomes a Chord Now it has been shewn, that half the Chord is the Sine of the Arch subtended by that Chord ; therefore in the Triangle A B C, the Sines of the Angles will be as the Halves of their opposite Sides ; and since the Halves are as Wholes, it follows, that the Sines of the Angles are as their opposite Sides; that is, A C : C B :: Sine Ang. B : to Sine Ang. A, &c. 4. To * Extend from the Supplement of Angle D 78° 35' to Angle B 44°42' on the Line of Sines, that Extent will reach from the Side B C 76, to the Side D C 54,53 on the Line of Numbers.” . 2dly. ‘Extend from the Supplement of Angle D. 78° 35' to Angle C 33° 53' on the Line of Sines, that Extent will reach from the Side BC 76, to the Side B D 43.23 on the Line of Numbers." The Proportion by Axiom II. will be, . . . . To find D. C. 180,0 • The FXtent from 65 to 106 on the Line of Numbers, will reach from 31°49' to 59° 17' on the Line of Sines.” 2dly. “The Extent from 31°49' to 27°28' on the Line of Siness will reach from 65 to 56.88 on the Line of Numbers.” A X I O M III. In every Plane Triangle it will be, as the Sum of any two Sides is to their Difference, so is the Tangent of half the Sum of the Angles opposite these Sides to the Tangent of half their Difference. * Which half Difference, being added to half the Sum of the Angles, gives the greater ; but if subtracted, the Remainder will be the lesser Angle.” | Two Sides and their contained Angle given, to find either of the other given, to find the Angle B DC, or B C D, and the Side CD. opposite Angles, you take the sesser Angle; that is, if from Angle A B F £; take the Angle G B A, there will remain the Angle G B F = half the ifference of the opposite Angles: And so also if from C, E, # the Sum of the Legs, you take C B the lesser Leg, there will remain B E=# the Difference of the Legs. And since the Angle A B F is Right-angled, if B F be made Radius, A F will be the Tangent of the Angle A B F, (that is, the Tangent of ; the Difference of the opposite Angles ;) and in the little Triangle G B F, G, F will be the Tangent of the Angle G B F, (that is, the Tangent of halo, the Difference of the opposite Angles:) but the Segments of the Legs of any Triangle cut g Lines parallel to the Base being (by Eucl. 2. 6.) proportional as E C : C B :: FA : FG, that is, half the Sum of the Legs is to half their Difference, as the Tanent of halt the Sum of the opposite Angles is to the Tangent of half their Difference: but Wholes are as their Halves; wherefore the Sum of the Legs is to their Difference, as the Tangent of half the Sum of the opposite Angles, is to the Tangent of half their Difference, E 3 The *The Proportion by Axiom III. will be, - ... To find the Angles D and C, As the Sum of the Sides B C and B D-185 — 2.267.17 Is to their Difference - 3 - 1.5 1851 So is Tang. of ; the Sum of the Angles & and D 39° 15' 9.91224 - II.43075 2.267.17 To the Tan of # the Diff. of the Angles C and D 8° 17' 9.16358 To half the Sum of the Angles D and C * 39° 15' Subtracted, gives the lesser Angle C 3o 58 9.86786 To the Side D C required 144.8 ** - 2. 16076 “ The Extent from 185 to 33 on the Line of Numbers, will reach from 39° 15' to 8° 17' on the Line of Tangents.” . 2dly. “The Extent from Angle D 47° 32' to 78° 30' (the Sup. lement of Angle B) on the Line of Sines, will reach from the §. B C 109 to 144.8, the Side D C required, on the Line of Numbers,” t do A X I O M IV, In any Plane Triangle, as the Base or greatest Side is to the Sum of the other two Sides, so is the Difference of the Sides to the Difference of the Seguments of the Base, made by a Perpendicular let fall from the Angle opposite to the Base. ‘And if half the 'Difference of the Segments be added to half their Sum, it will give the greater Segment; but if subtracted, the Remainder will be the lessor Segment.’ - * The |