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For, let AB and K AC be the two proposed arches, and

I ler BG and CH be their fines, and OG

S and OH their cofines : imoreover,

B В let the arch BC be equally divided in D, fo that CD may

1 11 GA be half the dif. ference, and AD half the sum, of AB and AC: let the radii OD and OC be drawn, and also the chord CB, meeting OE in E and OA (produced) in P; draw ES parallel to AO, meeting CH in S, and EF and OK perpendicular to AO, and let the latter meet EC (produced) in I; lastly, draw QDK perpendicular to OD, meeting OA,OC and Ol (produced) inQ, Land K.

Because CD=BD, it is manifest that OD is not only perpendicular to the chord BC, but bisects it in E; whence, also, EF bisects HG; and therefore CH+BG = 2EF, and CH - BG = 2CS; also OG + OH = 20F, and OG - OH = 2HF : but 2EF (CH+BG): 2CS (CH-BG) :: EF : CS :: EP:EC (by 14. 4.) :: DQ (the tangent

of AD): DL (the tangent of DC, by 20. 4.) And 20F (OG+OH): 2HF (OG-OH)::EI: EC :: DK (the co-tang. of AD): DL (the tang. DC). 2: E. D

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THEOREM VI. In any spherical triangle ABC it will be, as the to-tangent of balf the sum of the two fides is to the tangent of half their difference, so is the co-tangent of balf the base to the tangent of the distance (DE) of the perpendicular from the middle of the base.



Since co-fine AC : co-fine BC :: co-fine AD': co-fine BD (by Cor. to Tbeor. 2.) there

fore, by compofition and B

division, co-fine AC + E F D

Co-sine BC:co-fine AC - Co-sine BC :: co-fine AD + co-fine BD:co. fine AD - Co-sine BD, But (by the preceding lemma) co-fine AC +co-sine BC : co-fine ACco-fine BC :: co-tang.

AC+BC. AC-BC : tang


2 and co-fine AD + co-fine BD : co-fine AD

AD+BD Con-fine BD:: Co-tang.of AE





BD); whence; by equality, co-tang.



AC-BC : tang

:: Co-tang. AE: tang. 2 DE.


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Since the last proportion by permutation, be-

AC+BC comes co-tang. : Co-tang. AE :: tang. AC-BC.

tang. DE, and it is proved, in p. 13. that the tangents of any two arches are, inversely, as their; co-tangents; it follows, therefore, that

AC - BC tang. AE: tang.



AC+BC: ; tang.



: tang. DE; or, that the tangent of half the base, is to the tangent of half the sum of the sides, as the tangent of half the difference of the sides, to the


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tangent of the distance of the perpendicular from the middle of the base.

THEOREM VII. In any spherical triangle ABC, it will be, as the co-tangent of half the sum of the angles at the base, is to the tangent of half their difference, so is the tangent of half the vertical angle, to the tangent of the angle which the perpendicular CD makes with the line CF biseating the vertical angle. (See the preceding figure.)


It will be (by Corol. to Theor. 3.) co-line A :cofine B :: sine ACD: fine BCD; and therefore, cofine A + co fine B: co-fine A - Co-fine B :: sine ACD+fine BCD: sine ACD-sine BCD. But

B+A B-A r by the lemma) co-tang.

: tang. (co-line A+co-fine B : co-line A - co-line B: : fine ACD + sine BCD : sine ACD-fine BCD) tang. ACF : tang. DCF. 2. E. D.



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The folution of the cases of right-angled spherical



Given Soüglit


The hyp 1The oppo-As radius : fine hyp. AC:: IAC and one fite legfine A : fine BC: (by the forangle A


nier part of Theor. 1.) The hyp. The adja. As radius : co-iine of A:: 2 AC and one cert leg tang. AC; tang. AB (by the angle A AB

latter part of Theor. 1.) The hyp. The other As radius : co-fine of AC 3 AC and one angle C :: tang. A : co-tang. C (by angle A

Theor. 5.) The hyp. | The other As co-line AB : radius : : 4 4C and one leg BC co-line AC : co-fine BC (by 1 leg AB

Theor. 2.) The hyp. The oppo-jAs fine AC : radius :: sine 5 AC and one fite angle AB : sine C.(by the former

C part of Theor. 1.) The hyp. The adj.- As tang. AC : tang. AB :: 6 AC and onc cent a. gladius : co-fine A (by

A Theor. 1.)
One leg The other sAs radius : fine AB :: tan-
JAB and the leg BC Igent A : tangent BC (by

angle A

leg AB

leg AB

Theor. 4.)






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One leg

One leg The oppo-As radius : fine A : : coAB and the site angle line of AB : co-line of C (by 8

adjacent с
angle A
One leg | The hyp. As co-line of A : radius : :

AC Itang. AB : tang. AC (by

Theor. 1.) angle A

The other jAs tang. A : tang. BC :: raBC and the leg AB dius : line AB (by Theor. 4.) 10

opposite angle A

The adja- As co-fine BC': radius : :
BC and the cent angle co-line of A : fine C (by

opposite С Theor. 3.)
angle A

The hyp: As sin. A : fin. BC :: radius
BC and the

AC : sine AC (by Theor. 1.)
angle A

Both legs | The hyp. As radius : co-fine AB : : 13 AB and BC

AC co-fine BC : co-fine AC

(by Theor. 2.) Both legs An angle, As fine AB : radius : : tang. AB and BC suppose A BC : tang. A (by Theor. 4.)

One leg

One leg


15! A and C

Both angles A leg, As fin. A: co fine C :: ra

suppose Jdius i co-line AB (by Tbeor. AB

13.) Both angles. The hyp. As tang. A: co-tang. C:: 16) A and C

AC radius: co-fine AC (by

Theor. 5.)

Note, The 10th, 11th, and 12th cases are ambiguous ; fence it cannot be determined, by the data, whether AB, C, and AC, be greater or less than 90 degrees each.


D 2


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