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IN

PROP. XXI.

N right angled fpherical triangles, the co-fine of an angle is to the radius, as the tangent of the fide adjacent to that angle is to the tangent of the hypotenufe.

The fame construction remaining: In the triangle CEF, fin EF: R :: tan CE: tan CFE (18.); but fin EF = cof ABC; tan CE cot BC, and tan CFE cot AB, therefore cof ABC: R:: cot BC: cot AB. Now, because (Cor. 1. def. 9. Pl. Tr.) cot BC: R::R: tan BC, and cot AB: R::R: tan AB, by equality inversely, cot BC: cot AB :: tan AB : tan BC; therefore (11. 5.) cof ABC: R:: tan AB : tan BC. Therefore, &c. Q. E. D.

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COR. I. From the demonftration it is manifeft, that the tangents of any two arches AB, BC are reciprocally proportional to their co-tangents.

COR. 2. Becaufe cof ABC: R :: tan AB : tan BC, and R: cot BC: tan BC: R, by equality, cof ABC : cot BC:: tan AB: R. That is, the co-fine of any of the oblique angles is to the cotangent of the hypotenufe, as the tangent of the fide adjacent to the angle is to the radius.

PROP.

IN

PROP. XXII.

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N right angled fpherical triangles, the co-fine of either of the fides is to the radius, as the cofine of the hypotenufe is to the co-fine of the other fide.

The fame conftruction remaining: In the triangle CEF, fin CFR :: fin CE: fin CFE, (19.); but fin CF

cof CA, fin CE cof BC, and fin CFE cof AB; therefore, cof CA: R:: cof BC; cof AB. Q. E. D.

IN

PROP. XXIII.

right angled fpherical triangles, the co-fine of either of the fides is to the radius, as the co-fine of the angle oppofite to that fide is to the fine of the other angle.

The fame construction remaining: In the triangle CEF, fin CF R :: fin EF : fin ECF, (19.); but fin CF cof CA, fin EF cof ABC, and fin ECF fin BCA; therefore, cof CA: R:: cof ABC : fin BCA. Q. E. D.

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IN

PROP. XXIV.

N spherical triangles, whether right angled or oblique angled, the fines of the fides are proportional to the fines of the angles oppofite to them.

Firft, Let ABC be a right angled triangle, having a right angle at A; therefore, (19.) the fine of the hypotenuse BC is to the radius, or the fine of the right angle at A), as the fine of the fide AC

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Secondly, Let ABC be an oblique angled triangle, the fine of of the fides BC, will be to the fine of any of the other two AC, as the fine of the angle A oppofite to BC, is to the fine of the angle B oppofite to AC. Through the point C, let there be drawn an arch of a great circle CD perpendicular to AB; and in the right angled triangle BCD,

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fin BC: R:: fin CD: fin B, (19.); and in the triangle ADC, fin AC: R:: fin CD: fin A; wherefore, by equali ty inverfely, fin BC: fin AC :: fin A: fin B. In the fame manner, it will be proved that fin BC: fin AB :: fin A : fin C, &c. Therefore, &c. Q. E. D.

PROP

IN

PROP. XXV.

'N oblique angled spherical triangles, a perpendicu lar arch being drawn from any of the angles upon the oppofite fide, the co-fines of the angles at the bafe are proportional to the fines of the fegments of the vertical angle.

Let ABC be a triangle, and the arch CD perpendicular to the base BA; the co-fine of the angle B will be to the co-fine of the angle A, as the fine of the angle BCD to the fine of the angle ACD.

For having drawn CD perpendicular to AB, in the right angled triangle RCD, (23.) cof CD: R:: cof B: fin DCB; and in the right angled triangle ACD, cof CD: R:: cof A: fin ACD; therefore (11. 5.) cof B: fin DCB :: cof A: fin ACD, and alternately, cof B: cof A:: fin BCD: fin ACD. Q. E. D.

TH

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HE fame things remaining, the co-fines of the fides BC, CA, are proportional to the co-fines of BD, DA, the fegments of the bafe.

For in the triangle BCD, (22.), cof BC: cof BD :: cof DC: R, and in the triangle ACD, cof AC: cof AD: : cof DC: R; therefore (11. 5.) cof BC : cof BD : : cof AC : cof AD, and alternately, cof BC: cof AC:: cof BD: cof AD. Q. E. D..

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TH

HE fame conftruction remaining, the fines of BD, DA, the fegments of the base are reciprocally proportional to the tangents of B and A, the angles at the bafe.

In the triangle BCD, (18.) fin BD: R:: tan DC : tan B; and in the triangle ACD, fin AD: R:: tan DC : tan A; therefore, by equality inversely, fin BD : fin AD: : tan A: tan B. QE. D.

(See Figure on p. 360.)

PRO P. XXVIII.

THE fame conftruction remaining, the co-fines of the fegments of the vertical angle are reciprocally proportional to the tangents of the fides.

Because (21.) cof BCD: R:: tan CD: tan BC, and also, cof ACDR:: tan CD: tan AC, by equality inverfely, cof BCD : cof ACD :: tan AC: tan BC. Q. E. D.

PROP.

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