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(:: 2CG, or AC + BC, to AB). But (by Theor. 2.) EF: EI: tang. I (ACD) : radius; therefore, by compounding this proportion with the laft but one, we shall have, EI × EF : EI × CG:: tang. ACD x radius tang. Qx radius (by 11. 4.) and confequently EF: 2CG (AC + BC) :: tang. ACD : tang. Q. Whence the truth of the propofition is manifeft.

PROP. XIX.

In any plane triangle ABC, it will be, as the perpendicular is to the difference of the two fides, jo is the co-tangent of half the vertical angle, to the tangent of an angle; and, as radius is to the co-tangent of half this angle, fo is the difference of the fides to the bafe of the triangle.

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D

Let DP, DG, CF, &c. be as in the preceding propofition; alfo let PA and PC be drawn, and FI, parallel to PA, meeting AB

in I.

·

The right angled tri angles ADG and DPC, having DAG = DPC (by Cor. to 12. 3.) are fimilar

and therefore, AG2 : PC2 : : : AD2 : DP2 : : AE2 : AP (by Cor. to 11. 4.); whence, alternately, AG1 : AE2 :: PC2 (PF × PD) : AP2 (PE × PD) :: PF: PE:: AI: AE :: AIX AE: AE (by 7.4.); and confequently, AG' AIX AE = AE2 — EI × AE. Therefore, by Prop. 16. 4EI: AG :: radius: tangent of an angle (Q); and as radius: co-tang. Q:: AG: AE. But (by Theor. 2.) EF EI co-tang. EFI (ACD): radius ; which proportion being compounded with the laft

but

but one, &c. we shall have, EF: 2AG (AC-BC, fee Prop. 13.): co-tang. EFI : tang. Q; and as radius: co-tang. Q:: AG: AE::2AG (AC BC): 2AE (AB). Q, E. D.

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PROP. XX.

The bypotbenufe AC, and the fum, or difference, of the legs AB, BC, of a right-angled Spherical triangle ABC, being given, to determine the triangle.

Let AE be the fum, and AF the difference of the two legs. Because, radius: co-f. AB :: co-f. BC: co-f

AC (by Theor. 2.)

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rad. x co-f rad. x

therefore, co-f. AB x co-f. BC AC; but the former of these is

co-f. AE + co-f. AF (by Corol. 3. to Prop. 2.); therefore 2 × co-f. AC co-f. AE+co-f. AF. Whence it appears, that, if from twice the co-fine of the hypothenufe, the co-fine of the given fum, or difference, of the legs, be fubtracted, the remainder will be the co-fine of an arch which added to the faid fum, or difference, gives the double of the greater leg required.

COROLLAR Y.

Hence, if the two legs be fuppofed equal to each other (or the given difference=o), then will the co-fine of the double of each, be equal to twice the co-fine of the hypothenufe minus the radius.

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PROP. XXI.

One leg BC and the fum, or difference, of the bypothenufe and the other leg AB being given, to determine the hypothenufe (fee the last figure.)

Since rad. : co-fine BC:: co-fine AB: co-fine AC (by Theor. 2.), it will be (by comp. and div.) radius +co-fine BC: rad. — co-f. BC:; co-f. AB + co-f AÇ co-f. AB-co-f, AC, But the radius may be confidered as the fine of an arch of 90°, or the co-fine of o: and, therefore, fince (by the lemma in p. 30.) co-fine o + co-fine BC co-f. o

co-f

; and, co

fine AB+co-f. AC co-f. AB - co-f. AC :: com

BC+0

BC Q

BC :: co-tang.

: tang.

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BC

AC + AB

2 ACAB

equality, that co-tang. : tang. :: co-tang

2

: tang.

2

; that is, As the co

tang. of half the given leg, is to its tangent; fo is the co-tang. of half the fum of the hypothenufe and the other leg, to the tangent of half their difference,

BC

2

PROP. XXII.

The angle at the base and the fum, or difference, of the bypotbenufe and bafe, of a right-angled fpherical triangle being given, to determine the triangle.

First,

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с

B

whence, by arguing as in the last Prop. it will appear, that, co-tang. A tang. A:: rad. + co-f. A rad. —co-f. A (:: T. AC + T. AB: T. ACT, AB): S. AC + AB: S. AC AB (by Prop. 4.). Hence it appears, that, As the co-tangent of half the given angle, is to its tangent; fo is the fine of the fum of the hypothenuse and adjacent leg, to the fine of their difference.

PROP. XXIII.

The hypothenufe AC and the fum, or difference, of the two adjacent angles being given, to find the angles.

Let EC be perpendicular

to BC; and then it will be,
rad. co-f. AC:: T. A: E
T. ACE (by Theor. 5.)
From whence, by reasoning
ás above, we fhall, also,
have, co-tang. AC: tang.

A

B

AC:: S. A+ ACE: S. A-ACE; whereof the two laft terms, by fubftituting 90° - ACB for ACE, will become S. 90° +A — ACB (co-s. ACB

-A) and S. A+ACB-90° respectively. Whence it appears, that, As the co-tangent of half the bypothenufe, is to its tangent; fo is the co-fine of the difference of the angles at the hypothenufe, to the fine of the excess of their fum above a right-angle.

COROL

COROLLAR Y.

Hence, if the angles be fuppofed equal, then it will be, as radius: tang. AC :: tang. AC: fin. 2A

90°.

PROP. XXIV.

In two right-angled spherical triangles ABC, ADE, having one angle A common, let there be given the two perpendiculars BC, DE and the fum, or difference, of the hypothenufes AC, AE, to determine the triangles.

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gent of half the fum of the two perpendiculars, is to the tangent of half their difference; fo is the tangent of half the fun of the two hypothenuses, to the tangent of half their difference.

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tro right-angled spherical triangles ABC, ADE, le at the bafe, let there be given D and the fum or dif

ference

th. Bajes JH, AD, to determine the bafes

(jee the preceding figure.)

Since

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