Of the Solution of Right-angled Spherical Triangles, by the five circular Parts.

HE Lord Napier, (the noble Inventor of Logarithms,) by a due Confideration of the Analogies, by which right-angled spherical Triangles are folv❜d, found out two Rules, easy to be remembred, by means of which, all the fixteen Cafes may be folv'd'; for fince in these Triangles, befides the right Angle, there are three Sides and two Angles; the two Sides comprehending the right Angle, and the Complements of the Hypothenufe, and the two other Angles, were called by Napier, Circular Parts. And then there are given any two of the faid Parts, and a third is fought; one of these three which is called the Middle Part, either lies between the other two Parts, which are called Adjacent Extremes, or is feparated from them, and then are called Oppofite Extremes; fo if the Complement of the Angle B (Fig. to Prop. 25.) be supposed the middle Part, then the Leg AB, and the Complement of the Hypothenufe B C, are adjacent extreme Parts; but the Complement of the Angle C, and the Side A C, are oppofite Extremes. Alfo if the Complement of the Hypothenufe B C, be supposed the middle Part, then the Complements of the Angles B and C are adjacent Extremes, and the Legs A B, A C, are oppofite Extremes. In like manner, fuppofing the Leg AB the middle Part, the Complement of the Angle B and A C, are adjacent Extremes; for the right Angle A does not interrupt the Adjacence, because it is not a circular Part. But the Complement of the Angle C, and the Complement of the Hypothenufe BC, are oppofite Extremes to the faid middle Part. Thefe Things premised.

RULE I.

In any right-angled Spherical Triangle, the Rectan gle under the Radius, and the Sine of the middle Part, is equal to the Rectangle under the Tangents of the adjacent Parts.

RULE

The Rectangle under the Radius, and the Sine of the middle Part, is equal to the Rectangle under the Cofines 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, AB, A C.

Cafe 1. Let the Complement of the Angle C be the middle Part. Then fhall A C, and the Comple ment 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 CA, is to the Tangent of the Hypothenufe BC. Then (by Alteration) we fhall have Cof. C: T, CA:: R:T, BC. But R: T, BC:: Cot. BC: R. (as has been before fhewn.) Wherefore Cof. C: T, AC:: Cot. BC: R; whence, Rx Cof. C=T, ACx Cot. BC.

And the Complement of the Angle B, and A B, 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 CDF, AE= Cof. BA, and Cof. DF= S, EF S, Angle B. Whence it will be as Cof. C: Cof. BA:: S, B: R. And Rx 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 Hypothenuse 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, DF Cot. B. Wherefore Rx Cof. B C Cot. Cx Cot. 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.

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

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

Moreover, the Complement of BC, and the Angle C, are oppofite Extremes to the fame middle Part AB; and in the Triangles GHD (by Prop. 25.) we have Cof. D: S, DGH:: Cof. GH: R. But Cof. D-Cof. AES, A B, and S, G=S, IF=S, BC. Alfo Cof. GHS, HIS, C. Wherefore it will be as S, AB: S, BC:: S, C: R. And hence Rx S, ABS, BCXS, 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 the 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 the Part fought be the middle one; then shall the firft 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, muft be put in the middle Places, that fo the Part fought may be in the fourth Place.

IN oblique-angled spherical Triangles (Fig. to Prop 31.) BCD, if a perpendicular Arc AC be let fall from the Angle C to the Bafe, continued, if need be, fo as to make two Right-angled fpherical Triangles BAC, DAC; then by thofe Right-angled Triangles may moft of the Cafes of oblique-angled ones be foly'd.

PROPOSITION XXXI.

The Cofines of the Angles B and D, at the Bafe BD, are proportional to the Sines of the Vertical Angles BCA, DCA.

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

PROPOSITION XXXII.

The Cofines of the Sides BC, DC, are proportional to the Cofines of the Bafes BA, DA.

FOR

OR Cof. BC: Cof. BA:: (Cof. CA: R::)
Cof. DC: Cof. DA. (by 26. of this.)

PROPOSITION XXXIII.

The Sines of the Bafes BA, DA, are in a reciprocal Proportion of the Tangents of the Angles B and D at the Bafe BD.

Ecause (by 27. of this) S, BA: R::T, AC: T, of the Angle B. And by the fame inversely, 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,Angle B.

PROPOSITION XXXIV. The Tangents of the Sides BC, DC, are in a reciprocal Proportion of the Cofines of the Vertical Angles BCA, DCĂ.

BEcaufe by alternating the 28th Proposition, we

have

T, BC: R:: T, CA: Cof. B C A, 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.

PROPOSITION XXXV.

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

B

Ecause 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 fpherical Triangle ABC, the Rectangle CF × AE, or FM x.AE, contained under the Sines of the Legs, BC, BA, is to the Square of the Radius, as IL or IA-LA the Difference of the verf ed Sines of the Base CA, and the Difference of the Legs AM, to GN, the verfed Sine of the Angle B.

ET a great Circle PN be described from the

Pole B; and let BP, BN be Quadrants, and then PN is the Measure of the Angle B; also describe from the fame Pole B a leffer Circle CFM 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 HI, perpendicular to AO, and then the Plane draw thro' CH, 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 AL the verfed Sine of the Arc AM-BM-BA-BC-BA. The Ifofceles Triangles CFM, PON, are equiangular, fince MF, NO, as alfo CF, PO (by 16. El. 11.) are parallel. Wherefore, if Perpendiculars CH, PG be drawn to the Sides FM, ON, the Triangles will be divided fimilarly, and we shall have FM:ON:: MH: GN. Alfo, because the Triangles AOE, DIH, DL M, are equiangular, we fhall have AE: AO:: IL: MH.

But