Tangent of the Distance of the Perpendicular from the Middle of the Bafe. THEOREM VII. In any Ipberical Triangle ABC, it will be, as the Co-tangent of Half the Sum of the Angles at the Bafe, is to the Tangent of Half their Difference, fo is the Tangent of Half the vertical Angle, to the Tangent of the Angle which the Perpendicular CD makes with the Line CF bijecting the vertical Angle. (See the preceding Figure.) Demonftration. It will be (by Corol. to Theor. 3.) Co-fine A: Cofine B:: Sine ACD: Sine BCD, and therefore, Cofine A+ Co-fine B: Co-fine A-Co-fine B:: Sine ACD+ Sine BCD: Sine ACD Sine BCD. But (by the Lemma) Co-tang. (Co-fine A+ Co-fine B: Co-fine A-Co-fine B:: Sine ACD+Sine BCD: Sine ACD Sine BCD) Tang. ACF: Tang. DCF. 2. E. D. The B The Solution of the Cafes of right-angled Spherical Triangles. Sought Given Solution As Radius: Sine Hyp. AC:: The adja-As Radius: Co-fine of A:: cent Leg AB The other As Radius: Co-fine of AC Angle C:: Tang. A: Co-tang C (by The other Leg BC BC and the oppofite One Leg The adja-As Co-fine BC: Radius:: One Leg The Hyp. As Sin. A: Sin. BC:: Radius BC and the oppofite Angle A AC Both Legs The Hyp. As Radius: Co-fine AB:: 13 A Band BC Both Legs 14 A Band BC AC Co-fine BC: Co-fine AC (by An Angle, As Sine AB : Radius:: Tang. Note, The oth, 11th and 12th Cafes are ambiguous; fince it cannot be determined, by the Data,whether AB, C, and AC, be greater or less than 90 Degrees each. |