Cos AD + cos BD : cos AD ~ cos BD:: co-tang # (AD+DB): tang # (AD ~ DB) (R.112.) ‘." Co-tang # (AC+Bc): tang (Ac-Bc)::co-tang # (AD+ DB) : tang # (AD ~ DB.) But # (AD+DB)=AE, and 4 (AD ~ DB)=DE, hence co-tang # (Ac--Bc): tang # (Ac-Bc)::co-tang AE : tang DE. But when the perpendicular falls without the triangle, # (AD ~ DB)= AE and # (AD+DB)=DE. "." Co-tang # (Ac + BC) : tang # (Ac-Bc)::co-tang DE : tang AE; and by inversion, tang # (Ac-Bc); co-tang # (Ac--Bc) :: tang AE : co-tang DE. Q. E. D. (R) CoRoLLARY I. The distance of a perpendicular from the middle of the base, or as some writers call it the altern, or alternate base, is always equal to half the difference of the segments of the base, when the perpendicular falls within the triangle; or equal to half the sum of the segments, when a perpendicular falls without the triangle. Either of the above rules will bring the same conclusion, whether the perpendicular falls within or without the triangle; only in the first case, the fourth number will be less than half the base, and in the second case, it will be greater. That the rules are both the same may be shewn thus, When the perpendicular falls within the triangle. Co-tang # (Ac-HBC) : tang # (AC - BC)::co-tang AE : tang DE; and inversely, tang # (AC - BC): co-tang # (AC-H BC):: tang DE : co-tang AE. But the tangents are reciprocally as their co-tangents. "." Tang # (Ac-Bc) : co-tang # (AC + BC) :: tang AE : cotang DE; the same conclusion as when the perpendicular falls without the triangle. (S) CoRoLLARY II. The tangent of half the base, Is to the tangent of half the sum of the sides; As the tangent of half the difference of the sides, Is to the tangent of the distance of a perpendicular from the middle of the base. Or, The tangent of half the sum of the sides, is to the co-tangent of half their difference; as the tangent of half the base, is to the co-tangent of the distance of a perpendicular from the middle of the base. And, According as this distance is less or greater than half the base, the perpendicular falls within, or without the triangle. For we have already shewn, Co-tangent}(Ac-H BC): tang?! (Ac- BC): : co-tang AE : tang DE; and tango (Ac- Bc) : co-tang # (Ac--Bc):: tang AE : cotang DE. But, since the tangents are reciprocally as the co-tangents, Tang AE : tang # (AC+Bc):: tang # (AC - BC): tang DE; and tang # (AC+BC): co-tang 3 (AC - BC):: tang AE : co-tang I).E. ' (T) CoRollary III. If the triangle be isosceles, or equilateral, the perpendicular will fall on the middle of the base, except the sides be quadrants, and then it may fall in any part of the base. (U) CoRollary IV. In a right-angled triangle, the rectangle of the tangents of half the sum and half the difference, of the hypothenuse and one leg, is equal to the square of the tangent of half the other leg. For in this case B and D will coincide, and DE will be equal to # AB, or equal AE. PROPosition xxvii. (See the Fig. to PROP. xxvi.) (W) In any spherical triangle, I. If the perpendicular fall within the triangle. The co-tangent of half the sum of the angles at the base, Is to the tangent of half their difference ; As the tangent of half the vertical angle, Is to the tangent of the excess of the greater of the two vertical angles, made by a perpendicular, above half the aforesaid vertical angle. II. Or, If the perpendicular fall without the triangle. The tangent of half the difference between the base angles, Is to co-tangent of half their sum , As the co-tangent of half the vertical angle, *Is to the co-tangent of the excess of the greater of the two vertical angles, formed by a perpendicular upon the base, above half the aforesaid vertical angle. DEMONSTRATION. Cos A+ cos B : cos A-cos B::sine ACD + sine BCD : sine AcD ~ sine BCD. (B.175.). Cos A+cos B : cos A-cos B::cot # (A + B) : tang # (A - B) (R. 112.) Sine ACD +sine BCD : sine ACD ~sine BCD :: tangă(ACD. +BCD): tang # (ACD ~ BCD) (P. 111.) Therefore, Cot # (A+B) : tang # (A - B):: tang # (ACD + BCD) : tang # (ACD ~ BCD.) But # (ACD+BCD)=# ACB, and 4 (ACD ~BCD)=ECD, hence cot # (A+B) : tang # (A - B):: tang # ACB : tang Ecd. But when the perpendicular falls without the triangle, # (ACD-BCD)=# ACB, and EcD=# (ACD + BCD.) Cot # (A - B) : tang # (A + B):: tang # (ACD – BCD) : tang # (ACD + BCD); by inversion, tang # (A+B): cot # (A - B):: tang RCD : tang # ACB. But the tangents and co-tangents are reciprocally proportional. Tang # (A - B): cot # (A+B)::cot # ACB : cot. EcD. Q.E.D. (X) CoRoLLARY I. The excess of the greater of the two vertical angles (formed by a perpendicular) above half the vertical angle, is equal to half the difference of those vertical angles, when the perpendicular falls within the triangle; or half their sum, when it falls without. Either of the above rules will bring the same conclusion, whether the perpendicular falls within or without the triangle; only in the former case, the fourth number will be less than half the vertical angle, and in the latter it will be greater. That the rules are the same may be shewn thus, When the perpendicular falls without the triangle. Tang # (A - B) : co-tang # (A + B)::cot 4 ACB : cot ECD ; by inversion, cot # (A + B): tang # (A - B)::cot ECD : cot # ACB. But the tangents are reciprocally as the co-tangents. '-' Cot # (A+B) : tang # (A - B):: tang # ACB : tang ECD, the same expression as when the perpendicular falls within the triangle. (Y) CoRoLLARY II. The co-tangent of half the sum of the angles at the base, is to the tangent of half their difference; as the tangent of half the vertical angle, is to tangent of half the difference between the two vertical angles, formed by a perpendicular, or to tangent of half their sum, according as the perpendicular falls within or without the triangle. CHAP. W. INVESTIGATION OF GENERAL RULES FOR, CALCULATING THE SIDES AND ANGLEs of OBLIQUE-ANGLED spher ICAL TRIANGLES WITHOUT MAKING USE OF A PERPENDICULAR. PRoPosition xxvii.1.” (Z) If the cosine of any side of a spherical triangle be multiplied by the radius, and the rectangle of the cosines of the other two sides be deducted from the product ; the remainder divided by the rectangle of the sines of these two sides, will be equal to the cosine of the included angle divided by the radius, (cos AB x rad)–(cos Acx cos Bc) viz. in any spherical triangle ABC, sine Acx sine BC COS Z. C T rad * Legendre's Geometry, 6th Edition, page 386, et seq. DEMONSTRATION. Let ABC be the triangle proposed, and leto be the centre of the sphere; join Ao, Bo, and co. Take any point D in oc, and in the planes AOC, Boc, draw DE and DF each at right angles to oc, and join EF. Then because the Z. EDF is the measure of the inclination of the planes Aoc, Boc, it is also the measure of the spherical ZACB. (D. 133.) 2 *— 2 COS Epo ED* + DF*— EF (N. 92.) rad 2DE x DF also in the pl iangl cos Eof Eoo--of-EF" also in the plane triangle Eof, rad TTT2Eox of From the second of these equations EF-Eo°+of"— 2EO × OF x cos EOF In the plane triangle EDF, , which substituted in the first equation gives COSEDF consequently cos EDF = sine b. sine a These are the formulae from which Lagrange and Legendre" begin their investigations, and of the four quantities involved, any three being given the fourth may be found. They are applicable to every species of spherical triangles, whether rightangled, quadrantal, or oblique-angled, but the formulae for right-angled and quadrantal triangles have already been given. (O. 169, and P. 169.) (B) If A, B, C represent the three angles of a spherical triangle, the opposite sides may be represented by 180°-A, 180°–B, 180°-c; and if a, b, c represent the three sides, their opposite angles may be represented by 180°–a, 180°–b, 180°–e (U. 137.) Hence ) rad” . cos (180°–A)—rad . cos (180°–b) . cos (180°-c Cos (180°-a)= ( sine ão - #. ...}. But cos (180°–a) = — cos a; cos (180°–A) = — cos A, &c. (N. 101.) consequently (rad”. cos A) + (rad . cos B. cos c) - sine B. sine C In the same manner the cosines of the other sides may be determined. (C) It has been shewn (G. 176.) that the sines of the sides of any spherical triangle have the same ratio to each other as the sines of their opposite angles; hence by using the notation of Legendre, we shall have sine a . sine B_sine a . sine c COS a I. Sine A- - -- - (D) The general expressions for the cosines, which have been obtained by this proposition, may be arranged thus: , * The former in the Journal de L’Ecole Polytechnique, and the latter in his Eléments de Géométrie. |