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RULE II.

The Rectangle under the Radius, and the Sine of the middle Part, is equal to the Rectangle under the Cofine of the oppofite Parts.

Each of the Rules have three Cafes: For the middle Part may be the Complement of the Angle B, or C; or the Complement of the Hypothenufe BC; or one of the Legs, A B, A C.

Cafe 1. Let the Complement of the Angle C be the middle Part; then fhall AC, and the Complement of the Hypothenufe B C, be adjacent Extremes. By Prop. 28. the Cofine of the vertical Angle C is to Radius as the Tangent of C A is to the Tangent of the Hypothenufe BC. Then (by Alternation) we shall have Cof. C: T, CA::R:T, B C. But R: T, BC: Cot. BC: R (as has been before fhewn). Wherefore Cof. C: T, AC:: Cot. BC: R; whence R Cof. C=T, A CXCot. B C.

And the Complement of the Angle B, and AB, are oppofite Extremes to the fame middle Part, the Complement of the Angle C; and (by Prop. 25.) as the Cofine of the Angle C, to the Sine of the Angle CDF, fo is the Cofine of DF to Radius. But the Sine of CD FA E=Cof. B A, and Cof. DFS, EFS, Angle B. Whence it will be, as Cof. C : Cof. B AS, B: R. And R+Cof. C=Cof. BA xS, B; that is, Radius drawn into the Sine of the middle Part, is equal to the Rectangle under the Cofines of the oppofite Extremes.

Cafe 2. Let the Complement of the Hypothenufe BC be the middle Part; then the Complements of the Angles B and C will be adjacent Extremes. In the Triangle DCF (by Prop. 27.) it is, as S, CF : R :: T, DF: T,C. Whence (by Alternation) S, CF: T, DF:: (R: T, C: :) Cot. C: R. But S, CF Cof. B C and T, D F Cot. B. Wherefore R x Cof. B C Cot. CxCot. B; that is, Radius drawn into the Sine of the middle Part is equal to the Product of the Tangents of the adjacent extreme Parts.

X 3

And

And B A, A C, are the oppofite Extremes to the faid middle Part, viz. the Complement of BC; and (by Prop. 26.) Cof. BA: Cof. BC: R: Cof. A C. Wherefore we shall have RxCof, B C=Cọf. B A = Cof. A C.

Cafe 3. Laftly, let A B be the middle Part; and then the Complement of the Angle B, and A C, will be adjacent Extremes, and (by Prop. 27.) S, A B: R::T, CA: T, B. Whence S, A B: T, CA:: (R: T, B::) Cot. B: R. And fo R+S, A B=T, CA+Cot. B.

Moreover, the Complements of B C, and the Angle C, are oppofite Extremes to the fame middle Part A B; and in the Triangle GHD (by Prop. 25.) we have Cof. D : S, D G H : : Cos. G H : R. But Cof. D=Cof. A E=S, A B, and S, G=S, IF=S, BC. Alfo, Cof. G HES, HIS, C. Wherefore it will be, as S, ABS, BC:: S, C: R. And hence R x S, A B=S, B C×S, C.

And fo in every Cafe, the Rectangle under the Radius, and the Sine of the middle Part, fhall be equal to the Rectangle under the Cofines of the oppofite Extremes, and to the Rectangle under the Tangents of adjacent Extremes: And, confequently, if the aforefaid Equations be refolved into Analogies (by 16 El, 6.) the unknown Parts may be found by the Rule of Proportion. And if that Part fought be the middle one, then shall the first Term of the Analogy be Radius, and the fecond and third, the Tangents or Cofines of the extreme Parts. If one of the Extremes be fought, the Analogy muft begin with the other; and the Radius, and the Sine of the middle Part, must be put in the middle Places, that fo the Part fought may be in the fourth Place.

I N Oblique-angled Spherical Triangles (Fig. ta Prop. 31.) BCD, if a perpendicular Arc A C be let fall from the Angle C to the Base, continued, if need be, fo as to make two Right-angled Spherical Triangles BA C, DAC; then, by thofe Right-angled Triangles, may most of the Cafes of Oblique-angled ones be folved,

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PROPOSITION

XXXI.

The Cofine of the Angles B and D, at the Bafe BD, are proportional to the Sines of the vertical Angles B C A, DC A.

OR Cf. Angle B: S, BCA:: (Cof. CA: R : :) Cof. D: S, DCA (by 25 of this.)

PROPOSITION

XXXII.

The Cofines of the Sides B C, DC, are proportional to the Cofines of the Bafes B A, DA.

FOR

BA::
OR Cof. BC: Cof. B A :: (Cof. CA: R::)
Cof. DC: Col. D A (by 26 of this).

PROPOSITION XXXIII.

The Sines of the Bafes B A, D A, are in a reciprocal Proportion of the Tangents of the Angles B and D, at the Bafe 1s D.

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BEcaufe (by 27 of this) S, BAR:: T, AC: T, of the Angle B. And by the fame inverfly, R: S, DA:: T, of the Angle D: T, A C. Then will it be by the Equality of perturbate Ratio, according to Prop. 23. El. 5.) S, BA: S, DA:: T, Angle D: T, Angie B.

PROPOSITION XXXIV. The Tangents of the Sides BC, DC, are in a reciprocal Proportion of the Cofines of the vertical Angles B C A, DC A.

BE

Ecaufe, by alternating the 28th Propofition, we have T, BCR::T, CA: Cof. BCA, and by the fame, R: Cof. DCA:: T, DC: T, CA. Wherefore, by Equality of perturbate Proportion, T, BC: Cof. DCA:: T, DC: Cof. BCA.

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PROPOSITION XXXV.

The Sines of the Sides B C, DC, are proportional to the Sines of the oppofite Angles D and B.

BEcaufe (by the 29th of this) S, BC: R:: S, CA: S, of the Angle B; and, by the fame, inverting, R: S, DC:: S, Angle D: S, of CA. Whence, by Equality of perturbate Ratio, S, B C : S, DC : : S, D: S, B.

PROPOSITION XXXVI. In any Spherical Triangle A B C, the Rectangle CFXAE, or FM XA E, contained under the Sides of the Legs B C, BA, is to the Square of the Radius, as I L or I A-L A the Dif ference of the verfed Sines of the Bafe C A, and the Difference of the Legs A M, to G N, to the verfed Sine of the Angle B.

LET a great Circle PN be defcribed from the Pole

B; and let B P, B N, be Quadrants; and then PN is the Measure of the Angle B; alfo, describe from the fame Pole B a leffer Circle C F M thro' C; the Planes of thefe Circles fhall be perpendicular to the Plane BON (by the 2d of this.) And PG, CH. being perpendicular in the fame Plane, fall on the common Sections ON, FM; fuppofe in G, H. Again, draw H I perpendicular to A O; and then the Plane drawn thro' C H, HI, fhall be perpendicular to the Plane AO B. Whence A I, which is perpendicular to HI, will be perpendicular to the Right Line CI (by Def. 4. El. 11.); and fo AI is the verfed Sine of the Arc A C, and A L the verfed Sine of the Arc A M

BM-BA-BC-B A. The Ifofceles Triangles CFM, PON, are equiangular, fince MF, NO, as alfo CF, PO (by 16 El. 11.) are parallel. Wherefore, if Perpendiculars C H. P G, be drawn to the Sides FM, ON, the Triangles will be divided fimilarly, and we fhall have F M: ON:: MH:GN. Allo, because the Triangles A O E, DIH, DLM,

are

are equiangular, we shall have AE: AO::IL: M H.

But it has been proved, that FM:ON:: MH: GN. Wherefore it fhall be, as A EXF M: A OX ON:: ILXMH: MHXGN, or fo is IL to GN; that is, the Rectangle under the Sines of the Legs, is to the Square of Radius, as the Difference of the verled Sines of the Bafe, and the Difference of the Legs B C, BA, is to the verfed Sine of the Angle B. W. W.D.

PROPOSITION XXXVII. The Difference of the verfed Sines of two Arcs, drawn into half the Radius, is equal to the Rectangle under the Sine of half the Sum, and the Sine of half the Difference of those Arcs.

ET there be two Arcs, B E, BF, whofe Difference E F let be bifected in D; then fhall BD be the half Sum, and FD the half Difference of those Arcs G E IL is the Difference of the versed Sines of the Arcs BE, BF; alfo, FO is the Sine of the half Difference of the Ares. And because the Triangles CDK, FEG, are equiangular, we have DK: GE: (CD: FE:) ČD FE. Whence DKX FE, or DK×FO=G°£ × ¦ CD=I L × CD. W. W. D.

PROPOSITION

2

XXXVIII.

The verfed Sine of any Arc, drawn into half the Radius, is equal to the Square of the Sine of one half of the faid Arc.

THE

are Right Angles,

HE Triangles C BM, DE B, are equiangular, fince the Angles at M and E and the Angle at B is common. BD:: B M, B C.

And then

BMX BD; and EBX BC
BMq. W. W. D.

Wherefore EB: will E BxB C = BMX BD =

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