THEOREM XXXIII. The Sines of the Bafes B A, D A, are in a reciprocal Proportion to the Tangents of the Angles B and D, at the Bafe B D. Demonftration. By Theorem 27, we have sBA: RtAC: tB; and by the fame inverfely RsDA::t D:t AC. Then it will be, by Equality of perturbate Ratio, SBA:SDA::t D:t B. Q. E. D. See Euclid 5. Prop. 23. THEOREM XXXIV. The Tangents of the Sides BC, DC, are in a reciprocal Proportion to the Co-fines of the Vertical Angles BCA, DC A. Demonftration, By Alternation of Theorem 28, tBC: R::tCA: cs BÇA. And by the fame, we have R:cs DCA :: tDC : t CA. Wherefore, by Equality of Perturbate Proportion, tBC:cs DCA::tDC:cs BC A 2. E. D. THEOREM XXXV. The Sines of the Sides BC, DC, are proportional to the Sines of the oppofite Angles D, and B. Demonftration. By Theorem 29, we have s BC RSCA: sB. And by Inverfion of the fame, R: $DCsD: SCA VOL. II. N 2 Where Wherefore, by equality of Perturbate Proportion SBC:SDC:: SD:s B. 2. E. D. THEOREM XXXVI. In any Spherical Triangle ABC, the Rectangle CFX AE, or FMX AE, contained under the Sines of the Legs BC, B A, is to the Square of Radius, as IL, or IA-LA, the Difference of the versed Sines of the Base C A, and the Difference of the Legs AM, to G N, is verfed Sine of the Angle B. Let a great Circle Demonftration. be described on the Pole and let be Quadrants and then shall be the Measure of the Angle Alfo from the fame Pole defcribe a leffer Circle Then fhall the Planes of those Circles be perpendicular to the Plane and the Perpendiculars fall on the common Sections fuppofe in the Points Again draw perpendicular to the Radius and then the Planes drawn through fhall be perpendicular to the Plane Whence the Line (which is perpendicular to will be perpendicular to the Line (by Euclid 11. Def. 4.) and fo is the versed Sine of the Arch Alfo the Right Line PN, BP, and B N, P N, B. B, CFM; P N, CM, AO; С Н, НІ, AO B ; HI) CI, AI, AC: A L, is the versed Sine of the Arch AM BM-BA The Ifofceles Triangles are equiangular, fince =BC-BA. CFM, PON, MF, NO, and C F, P 0, are parallel (by Euclid 11. Prop. 16. Wherefore if Perpendiculars be drawn to the Sides ) CH, PG, FM, ON, the Triangles will be divided fimilarly; and we shall have FM:ON:: MH: GN. Also because the Triangles AO E, DIH, D L M, are equiangular, we hall have A E AO :: IL: MH; But it has been proved that FM:ON:: MH:G N. Wherefore it shall be, as A E× FM: A OXON :: ILX MH: MHXGN. AEXFM: Rq :: I L : G N. 2. E.D. That is, as THEOREM XXXVII. The Difference of the verfed Sines of two Arches drawn into half the Radius, is equal to the Rectangle under the Sine of half the Sum, and Sine of half the Difference of those Arches. is the Difference of the verfed Sines Now the Sine of Sum of the Arches is and the Sine of their Difference is and because the Triangles B E, B F, E F D: BD, FD, GEIL, I B, LB: DK, FO; CD K, FEG, are are Equiangular; we have DK: GE::CD: FE:: The Verfed Sine of any Arch drawn into half the Radius, is equal to the Square of the Sine of one half of the faid Arch. and confequently EBX BC= BMX BD = BMq. 2, ED. THEOREM XXXIX. In any Spherical Triangle ABC, the Rectangle under the Sines of the Sides BC, AB, comprehending the Angle B, is to the Square of Radius, as the Rectangle of the Sines of the Half Sum, and Half Difference, of the Bafe AC, and Difference of the Sides or Legs A M, is to the Square of the Sine of one Half of the Angle B. Demonftration. By Theorem 36, we have AEX FM: Rq::IL: G N. Alfo it will be AE XFM: Rq :: IL × GNXIR. ILX R And And fince (by Theorem 37) ILX R, is equal to the Rectangle of the Sines of the Half Sum of the Sides and the Half Difference of the fame And because (by Theorem 38.) AC+AM AEX FM: Rq:: S X S 2 2 In any Spherical Right-angled Triangle, according to the Lord Napier's Invention; the Radius is to the Tangent of one of the Extremes Conjunct, :: as the Tangent of the other Extream Conjunct, is to the Sine of the Middle Part. Demonftration. Cafe 1. Let the Complement of the Angle : C, AC, BC, be |
