Tangent of the Distance of the Perpendicular from the THEOREM VII. In any Jpberical Triangle ABC, it will be, as the Co-tangent of Half the Sum of the Angles at the Base, is to the Tangent of Half their Difference, so is the Tangent of Helf the vertical Angle, to the Tangent of the Angle which the perpendicular CD makes with the Line CF bisetting the vertical Angle. (See the preceding Figure.) Demonstration. It will be (by Corol. to Theor. 3.) Co-sine A : Cosine B :: Sine ACD : Sine BCDs and therefore, Cofine A + Co-sine B : Co-sine A-Co-fine B:: Sine ACD + Sine BCD : Sine ACD Sine BCD. But B +A B-A (by the Lemma) Co-tang. :Tang. (Co-fine A + Co-fine B : Co-sine A-Co-fine B:: Sine ACD + Sine BCD: Sine ACD - Sine BCD) Tang. ACF : Tang. DCF. 2. E. D. 2 2 B The Solution of the Cases of right-angled Spherical Triangles. Cala Angle A Given Sought Solution The Hyp. The oppo- As Radius : Sine Hyp. AC:: 1 AC and one site Leg Sine A : Sine BC (by the for BC mer Part of Theor. 1.). The Hyp. The adja- As Radius: Co-Gine of A :: 2 AC and one cent Leg Tang. AC:Tang. AB (by the Angle A AB latter Part of Theor. 1.) The Hyp. The other As Radius : Co-line of AC 3A AC and one Angle C 1: : Tang. A : Co-cang C (by Angle A Theor.5.) The Hyp. The other As Co-line A B : Radius :: 4 AC and one Leg BC Co-line AC : Co-line BC (by Theor. 2. The Hyp. The oppo- As Sine AC: Radius :: Sine 5 AC and one Gite Angle AB : Sine C (by the former Leg AB с Part of Theor, 1.) The Hyp. The adja- As Tang. AC : Tang. AB :: 6 AC and one cent Angle Radius: Co-sine A (by TheLeg AB A AB and the Leg BC gent A : Tangent B C ( by 7 adjacent Theor. 4.) Leg AB or. I. Case Theor. 3.) IO Given Sought Solution A B and the Gite Angle fine of AB : Co-line of C (by 8 adjacent C Angle A One Leg The Hyp. As Co-fine of A: Radius :: A B and the AC Tang. AB : Tang. A C (by Theor. 1.) opposite opposite C opposite Both Legs The Hyp. As Radius : Co-line AB :: 13 A Band BC AC Co-fine BC: Co-line AC (by Theor. 2.) Both Legs An Angle, As Sine AB : Radius:: Tang. 14 A Band B C suppose ABC:Tang. A (by Theor. 4.) II Theor. 3.) 12 Both Angles A Leg, As Sin. A : Co-line C:: Ra- AB 3:) Theor. 5.) Note, The poth, 11th and 12th Cases are ambiguous ; since it cannot be determined, by the Data whether AB, C, and AC, be greater or less than 90 Degrees each. D Tbe The Solution of the Cases of oblique Spherical Tri angles. Given | Care Sought Solution Two Sides AC, The Angle B As Sine BC: Sine A :: Sine AC : BC and an An opposite to Sine B (by Cor. 1. to Theor. 1.) Note, gle A opposite the other This Cale is ambiguous when B C is to one of them less than AC; fince it cannot be de termined from the Data whether B be acute or obtuse. Two Sides AC, The included Upon A B produced (if need be) let BC and an An Angle ACB fall the perpendicular CD: Then ( by gle A oppofite Theor. 5.) Rad. : Co-line AC :: to one of them Tang. A : Co.tang. ACD; but (by Cor. 2. to Theor. 1.) as Tang. BC: Tang. AC :: Co-sine ACD : Co-fine BCD, Whence ACB=ACD + BCD is known. Two Sides AC, gle opposite to 3 one of them The other As Rad. : Co-fine A :: Tang. AC :- 10 Theor. 2.) as Co-fin. AC: Colin. Two Sides AC, | The other AB and the in Side BC 4 cluded Angle A As Rad. : Co-fin. A :: Tang. AC: Tang. AD by Theor. 1.) whence BD is also known: Then (by Curol. to Theor. 2.) as Co-fine AD: Co fine I'wo Sides AC, Either of the As Rad. : Co-fine A : : Tang. AC: AB and the in- other Angles, Tång. AD (by Theor. 1.) whence BD cluded Angle A suppose B is known ; then (by Cor. 10 Thecr. 4.) as Sine BD: Sine AD:: Tang. A : Tang. B. Care Two Angles A, Either of the As. Rad. : Co-fine AC : : Tang. A: ACB and the other Sides, Co-tang. ACD ( by Theor. 5.) whence 7 Side A C be suppose BC BCD is also known: Then,as Co-line twixt them of BCD : Co-line ACD : : Tan, AC : Tang. BC ( by Cor. 2. to Tbeor. 1.) Two Angles A, The Side BC As Sine B : Sine AC :: Sine A : Sine 8 B and a Side opposite the BC (by Cor. 1. to Theor...) Two Angles A, The Side AB As Rad. : Co-fine A : : Tang. AC: B and a Side AC betwixt them Tang. AD (by Tbeor. 1.) and as Tan. 9 opposite to one B: Tan. A:: Sine AD: Sine BD of them (by Cor. to Tbeor. 4.) whence AB is also known. Two Angles A, The other As Rad. : Co-fine AC : : Tang. A : Band a Side AC Angle ACB Co-tang. ACD (by Theor. 5.) and as 10 opposite to one Co-sine A: Co-fine B :: Sine ACD of them : Sine BCD (by Cor. to Tbeor. 3.) whence ACB is also known. Note, In letting fall your Perpendicular, let it always be frein the End of a given Side and opposite to a given Angle, |