(: : 2CG, or AC + BC, to AB). But (by Theor. 2.) EF:El :: tang. I (ACD): radius ; therefore, by compounding this proportion with the last but one, we shall have, EIX EF: EI X CG:: tang. ACD X radius : tang. Qx radius (by 11. 4.) and consequently EF : 2CG (AC + BC) :: tang. ACD : tang. Q. W bence the truth of the proposition is manifeft. PROP. XIX In any plane triangle ABC, it will be, as the perpendicular is to the difference of the two sides, so is the co-tangent of half the vertical angle, to the tangent of an angle ; and, as radius is to the co-tangent of balf this angle, so is the difference of the sides to the base of the triangle. P E B Let DP, DG, CF, &c. be as in the preceding proposition ; also let PA and PC be drawn, and FI, parallel to PA, meeting AB in I. The right • angled triangles ADG and DPC, having DAG = DPC (by Cor. to 12. 3.) are similar ; and therefore, AGʻ: PC? ::: AD2: DP2 :: AE? : AP (by Cor. to 11. 4.); whence, alternately, AG” : AE? :: PC (PF XPD) : AP (PE XPD) :: PF: PE:: AI: AE :: AI X AE: AE(by 7. 4.); and consequently, AGʻ = AI X AE = AE - EI <AE. Therefore, by Prop. 16. EI: AG :: radius : tangent of an angle (Q), and as radius : co-tang. Q:: AG: AE. But (by Theor. 2.) EF : EI :: co-tang. EFI (ACD) : radius; which proportion being compounded with the last but but one, &c. we shall have, EF: 2AG (AC - BC, fee Prop. 13.) :: co-tang. EFI : tang. Q; and as radius : co-tang. Q:: AG:AE:: 2AG (AC BC): 2AE (AB). 2: E. D. PROP. XX. The hypothenuse AC, and the sum, or difference, of the legs AB, BC, of a right-angled Spherical triangle ABC, being given, to determine the triangle. Let AE be the с sum, and AF the difference of the two legs. Because, radius : co-f. AB E :: co-f. BC: co-f. I B AC (by Theor. 2.) therefore, co-f. AB X co-f. BC = rad. X co-f. AC; but the former of these is = rad. x co-f. AE + co-f. AF (by Corol. 3. to Prop. 2.); therefore 2 x co-f. AC = co-f. AE + co-f. AF. Whence it appears, that, if from twice the co-line of the hypothenuse, the co-fine of the given sum, or difference, of the legs, be subtracted, the remainder will be the co-fine of an arch which added to the said sum, or difference, gives the double of the greater leg required. COROLLARY. Hence, if the two legs be supposed equal to each other (or the given difference = o), then will the co-fine of the double of each, be equal to twice the co-line of the hypothenuse minus the radius. 1 PROP. XXI. One leg BC and the fum, or difference, of the bypothenuse and the other leg AB being given, to determine the bypothenuse (see the last figure.) 2 2 Since rad. : Co-sine BC :: co-fine AB : co-fine AC (by Theor. 2.), it will be (hy comp. and div.) radius + co-fine BC: rad. — Co-f. BC :: Co-s. AB + co-f. AÇ: co-f. AB — Co-f, AC. But the radius may be considered as the sine of an arch of 90°, or the co-fine of o: and, therefore, since (by the lemma in P. 30.) co-sine o + co-line BC : co-f.o - Co-f, BC+0 BC Q BC :: co-tang. : tang: ; and, cofine AB + Co-f. AC : 00-f. AB -- co-f. AC :: co AC + AB AC - АВ tang : tang į it follows, by BC BC equality, that co-tang, : tang :: CO-tang AC + AB AC - AB : tang. ; that is, As the co2 tang. of half the given leg, is to its tangent ; fo is the co-tang. of half the sum of the hypothenuse and the other leg, to the tangent of half their difference, 2 2 2 2 2 The angle at the base and the fum, or difference, of the hypothenuse and base, of a right-angled Spherical triangle being given, to determine the triangle, First, First, it will be, rad. : co-f. A::T. AC: T. AB (by Theor. 1.) and therefore rad. + co-f. A: rad. co-f. A:: T. AC + T. AB: T. AC -- T. AB: B whence, by arguing as in the last Prop. it will appear, that, co-tang. {A : tang. A :: rad. + Co-f. A : rad. — co-f. A (:: T. AC + T. AB:T. AC — T, AB) :: S. AC + AB: S. AC - AB (by Prop. 4.). Hence it appears, that, As the co-tangent of half the given angle, is to its tangent ; fois the fine of the sum of the hypothenuse and adjacent leg, to the fine of their difference. PROP. XXIII. The hypothenuse AC and the sum, or difference, of ibe two adjacent angles being given, to find the ongles. Let EC be perpendicular to BC; and then it will be, rad. : co-f. AC :: T. A: E B as above, we shall, also, A have, co-tang. AC : tang. AC::S. A + ACE:S. A - ACE ; whereof the two last terms, by substituting 90° - ACB for ACE, will become S. 90° +A -ACB (co-f. ACB -A) and S. A + ACB-90° respectively. Whence it appears, that, As the co-tangent of half the hypothenuse, is to its tangent ; so is the co-fine of the difference of the angles at the hypothenuse, to the fine of the excess of their sum above a right-angle. COROL COROLLAR Y. Hence, if the angles be supposed equal, then it will be, as radius : tang. AC :: tang. AC : fin. 2A - 90°. PROP. XXIV. In two right-angled Spherical triangles ABC, ADE, baving one angle A common, let there be given the two perpendiculars BC, DE and the fum, or difference, of the bypotbenuses AC, AE, to determine the triangles. It is evident (from Theor. 1.) that S. DE: S. BC :: S. AE. S. AC; therefore S. DE + S. BC : S.DE-S. BC::S. AE + S. AC: S. AE — S. D lemma in p. 30. and equa- : tang. :: tang 2 2 AE+AC AE - AC : that is, As the tan : tang 2 2 gent of b:!f the sum of the two perpendiculars, is to the tangent of half their difference ; so is the tangent of half the fun of the two hypotbenuses, to the tangent of half their difference. T PROP. XXV. " the base, let there be given D: and ibe fum or differol 7es AD, to determine the bases (jee the preocurig figure.) Since Di |