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In any right-angled Spherical triangle it will be, as radius is to the fine of the angle at the bafe, fo is the fine of the hypothenufe to the fine of the perpendicular; and as radius to the co-fine of the angle at the bafe, fo is the tangent of the bypothenufe to the tangent of the bafe.

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Let ADL and AEL be two great-circles of the sphere interfecting each other in the diameter AL,

making

making an angle DOE, meafured by the arch ED; the plane DOE being fuppofed perpendicular to the diameter AL, at the center O.

Let AB be the bafe of the proposed triangle, BC the perpendicular, AC the hypothenufe, and BAC (or DAE = DE = DOE) the angle at the base: moreover, let CG be the fine of the hypothenufe, AK its tangent, AF the tangent of the bafe, CH the fine of the perpendicular, and EF the fine of the angle at the base; and let I, K and G, H be joined.

Because CH is perpendicular to the plane of the base (or paper), it is evident, that the plane GHC will be perpendicular to the plane of the bafe, and likewife perpendicular to the diameter AL, because GC, being the fine of AC, is perpendicular to AL. Moreover, fince both the planes OIK and AIK are perpendicular to the plane of the base (or paper), their interfection IK will also be perpendicular to it, and confequently the angle AIK a right-angle. Therefore, feeing the angles OFE, GHC and AIK are all right-angles, and that the planes of the three triangles OFE, GHC and AIK are all perpendicular to the diameter AL, we fhall, by fimilar triangles,

have

that is,

OE: EF:: GC: CH
OE: OF :: AK: AI

Radius: fine of EOF (or BAC):: fine
of AC fine of BC.

Radius co-fine of EOF (or BAC) :: tang. AC tang, AB. 2. E. D.

COROLLAR Y I.

Hence it follows, that the fines of the angles of any oblique spherical triangles ADC are to one another, directly, as the fines of the oppofite fides,

For

D

For let BC be perpendicular to AD; then S radius: fine A:: fine AC : fine BC 2 fince radius: fine D.:: fine DC; fine BC by the former part of the theorem; we fhall have, fine Ax fine AC (radius x fine BC)= fine D x fine, DC (by 8. 4.) and confequently fine A: fine D:: fine DC: fine AC; or fine A: fine DC :: fine D: fine AC.

COROLLARY 2.

It follows, moreover, that, in right-angled fpherical triangles ABC, DBC, having one leg BC common, the tangents of the hypothenufes are to each, other, inverfely, as the co-fines of the adjacent angles.

For radius: co-fine ACB.:: tan. AC: tan, BC2 finceradius: co-fine DCB:: tan. DC: tan, BC by the latter part of the theorem, we shall (by ar guing as above) have co-fine ACB: co-fine DCB:: tang. DC: tang. AC.

THEOREM II.

In any right-angled fpherical triangle (ABC) it. will be, as radius is to the co-fine of one leg, fo is the co-fine of the other leg to the co-fine of the bypotbenufe.

DEMON

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Radius fine F :: fine CF : fine CE; that is, Radius co-fine BA :: co-fine CB: co-fine AC (See Cor. 4. p. 25.)

2. E. D.

COROLLAR Y.

Hence, if two rightangled spherical triangles ABC, CBD have the fame perpendicular DBC, the co-fines of their hypothenufes will be to each other, directly, as

the co-fines of their bases. For rad: co-fin. BC:: co-fin. AB: co-fine AC, fincerad: co-fin. BC:: co-fin. DB: co-fine DC, therefore, by equality and permutation, co-fine AB: co-fine DB:: co-fine AC: co-fine DC.

THEOREM III.

In any right-angled Spherical triangle (ABC) it will be, as radius is to the fine of either angle, fo is the co-fine of the adjacent leg to the co-fine of the oppofite angle.

DEMON

DEMONSTRATION.

Let CEF be as in the preceding propofition; then, by Theor. 1. Cafe 1. it will be, radius: fine C: fine CF: fine EF; that is, radius: fine C :: co-fine BC co-fine A. Q, E. D.

COROLLAR Y.

Hence, in right-angled spherical triangles ABC, CBD, having the fame perpendicular BC (See the laft figure), the co-fines of the angles at the base will be to each other, directly, as the fines of the vertical angles :

For S radius: fine BCA :: co-fine CB : co-fine A, finceradius: fine BCD :: co-fine CB : co-fine D, therefore, by equality and permutation,

Co-fine A co-fine D:: fine BCA : fine BCD.

:

THEOREM IV.

In any right-angled fpherical triangle (ABC) it will be, as radius is to the fine of the bafe, fo is the tangent of the angle at the bafe to the tangent of the perpendicular.

For, fuppofing CEF as before,

it will be, as radius: co-fine of F:: tang. CF: tang. FE (by the latter part of Theor. 1.) that is, radius : fine AB: co-tang. BC: co-tang, A tang. A: tang. BC (by Corol. 5. P. 13.) 2. E. D.

F

B

E

COROL

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