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

Let AB be the base of the propofed triangle, BC the perpendicular, AC the hypothenuse, and BAC (or DAE DE DOE) the angle at the base: moreover, let CG be the fine of the hypothenufe, AK its tangent, AI the tangent of the base, CH the fine of the perpendicular, and EF the fine of the angle at the bafe; 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 alfo 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 shall, by fimilar triangles,

JOE: EF:: GC: CH2
OE OF: AK: AI (

have {OE

that is,

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.

COROLLARY I.

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

For

D

BC } by

the

For let BC be perpendicular to AD; then radius: fine A:: fine AC: fine BC s finceradius: fine D; : fine DC : fine BC former part of the theorem; we fhall have, fine Ax fine AC (radius ×fine BC) fine D×fine DC (by 10. 4.) and confequently fine A: fine D:: fine DC: fine AC; or fine A: fine DC:: fine D : fine AC.

COROLLAR Y 2.

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

For radius: co-fine ACB:: tan. AC: tan. { BC? fince radius: co-fine DCB :: tan. DC: tan. BCS by the latter part of the theorem; we fhall (by arguing 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 bypothenufe.

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.J

B

COROLLARY.

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

the co-fines of their bafes.
For

rad: co-fin. BC:: co-fin. AB: co-fineAC, 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 fpherical 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

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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. 2. E. D.

COROLLARY.

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

For

radius: fine BCA:: co-fine CB: co-fine A, fince radius: 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 spherical 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

E

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B

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of the bafes will be to

B

each other, inversely, as the tangents of the angles

at the bases:

:

For radius fine AB:: tang. A: tang. BC fince radius: fine DB:: tang. D: tang. BC we fhall (by reafoning as in Cor. 1. Theor. 1.) have Sine AB: fine DB:: tang. D: tang. A.

THEOREM V.

In any right-angled spherical triangle it will be, as radius is to the co-finé of the bypothenufe, fo is the tangent of either angle to the co-tangent of the other angle.

For (CEF being as in the laft) it will be, as radius fine CE: tang. C: tang. EF (by Theorem 4.) that is, radius: co-fine AC:: tang. C: cotang. A. Q. E. D.

LEMMA.

As the fum of the fines of two unequal arches is to their difference, fo is the tangent of half the fum of thofe arches to the tangent of half their difference: and, as the fum of the co-fines is to their difference, fo is the co-tangent of half the fum of the arches to the tangent of half the difference of the Jame arches.

For,

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