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Spb. Trig.

PRO P. XXIII.
N any spherical triangle, as the rectangle contained

by the fines of two sides, is to the rectangle contained by the fines of half the sum and half the difference of the base and the excess of the fides, fo is the square of the radius to the square of the fine of half the angle oppolite to the base.

Let ABC be a spherical triangle, of which the fide AC is greater than AB; a d make AD equal to AC, and AE to AB; the rectangle contained by the fines of BA, AC is to the rectangle contained by the fine of half the sum of CB, BD, and the fine of half the difference of the same CB, BD, as the square of the radius to the square of the fine of half the angle BAC.

Describe through C, D and B, E the great circles CFD, BGE, and bileet the angle BAC by the great circle AGF, and join BC, BD, DC, BE ; and let O be the centre of the sphere,

and join OF, OG, meeting CD, BE in H, K, and join HK, and a 11. 1: draw a BL perpendicular to CD. And because in the spherical

triangles CAF, DAF, the two fides CA, AF, are equal to the two ĎA, AF, and the angle CAF

is equal to DAF; therefore the b 6. S. T. bale CF is equal to DF 6, and

G the angle CFA to AFD; there

fore the plane AGF is perpenc 3. Def. dicular c to the plane CFD: and

CD is perpendicular to their
common section OF, because OF,

which bisects the arch CFD, dCor.39.3. bisects CD at right angles d; e4. Def... therefore CD is perpendicu iar

to the plane AGF, and DHK is
therefore a right angle: For the

same reason, BE is perpendicular to the plane AGF; therefore f 6.11.

BK is parallel to DH': and BL, HK are parallel, because BLH,

LHK are right angles; therefore BL HK is a parallelogram, ḥ 34. I. and BK is equal to LH 6: and because the sides CB, BD of the

triangle CBD are placed in equal circles CB, ABD, the rect

angle CH, HL, or CH, BK, is equal to the rectangle contained k LEM. by half the sum and half the difference of the arches CB, BDk;

but in the right angled spherical triangle AGB, the fine of AB I 20. S. T. is to BK the line of BG, as the radius to the fine of BAG ! half of BAC; and in the right angled triangle ACF, the fine

of

KI

S. T.

OUI

S. T.

of AC is to CH the fine of CF, as the radius to the fine of FAC Spb. Trig.
half of BAC ; that is, as the fine of AB to BK; therefore
the rectangle CH, BK is fimilar to the rectangle contained by the
fines of AB, AC: Wherefore, as the rectangle contained by the
fines of BA, AC is to the rectangk CH, BK contained by the
fines of half the sum and half the difference of the arches CB,
BD, so is the square of the radius to the square of the line of
half the angle BACH,

m 22. 6.

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PROP. XXIV.
IN any spherical triangle, the rectangle contained

by the fines of two fides, is to the rectangle con-
tained by the fine of half the sum of the three sides,
and the line of its excess above the base, as the square
of the radius to the square of the cofine of half the
angle opposite to the base.

Let ABC be a spherical triangle; the rectangle contained by
the fines of two fides AB, AC is to the rectangle contained by
the fine of half the sum of the three sides, and the fine of the
excess of the said half fum above the base BC, as the square of
the radius to the square of the cofine of half the angle BAC.

Produce BA, CA to D, E, and make AD equal to AC, and
AE to AB; and describe the great circles CFD, BGE; and let
the great circle GAF bisect the angle CAD, and meet the great
circles CFD, BGE in F, G; and join EB, BC, CD, DB; and
draw BL perpendicu-
lar to DC ; and let O
be the centre of the
sphere, and join OF,
OG, meeting CD, BE,
in H, K; and join
HK.
. It
may

be demon-
strated, as in the pre-
ceding proposition, that C
HL is equal to BK ;
and that the rectangle
contained by the fines

K
of BA, AC is to the

В
rectangle BK, CH, or
CH, HL, as the square of the radius to the square of the fine
of half the angle BAE or CAD : but, because CD is bisected

in

A А

G

Sph, Trig. in H, the rectangle CH, HL is equal to the rectangle contained

by the fines of half the sum, and half the difference of the arches BD, BC, and the sum of BD, BC is the sum of the three fides, and the excess of the half of BD above half of BC is the excess of half the sum of the three fides above BC; therefore the rectangle CH, HL is equal to the rectangle contained by the fines of half the sum of the three sides, and of its excess above BC: also, because the angles CAB, CAD are equal to two right angles, the half of CAB and the half of CAD are together equal to a right angle; therefore the fine of half CAD is the cofine of half CAB: But the rectangle contained by the fines of BA, AC, is to the rectangle CH, HL as the square of the radius to the square of the fine of half the angle CAD; therefore the rectangle contained by the fines of BA, AC, is to the rectangle contained by the fines of half the sum of the three sides, and of its excess above the base BC, as the square of the radius to the square of the cofine of half the angle BAC

Cor. Because the rectangle contained by the fines of half the sum of the three fides, and of its excels above BC, is to the rectangle contained by the fines of BA, AC as the square of the cotine of half BAG to that of the radius ; and that the rectangle contained by the fines of BA, AC, is to the rectangle contained by the fines of half the sum, and half the difference of the base

and the excess of the sides, as the square of the radius to the $ 22. 5. square of the fine of half BAC; therefore, by equality, the

rectangle contained by the fines of half the sum of the three sides, and of its excess above the base, is to the rectangle contained by the lines of half the sum, and half the difference of the base, and the excess of the fides, as the square of the cofine of half the angle BAC to the square of its fine; that is, as the

square of the radius to the square of the targent of half the b 2. Cor. angle BACO. Def. 10.

P. T.

SOLUTION of the Cases of OBLIQUE-ANGLED

SPHERICAL TRIANGLES.

GENERAL PROPOSITION.
IN any spherical triangle, of the three sides and

three angles, any three being given, the other three may be found.

The

The several cases of this proposition may be resolved by the spb. Trig. help of the three general proportions, together with the 22d, and any one of the subsequent propositions ; as in the following

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Table, in which ABC is any spherical triangle; and the perpen-
dicular AD either falls within the triangle, or meets the base
BC produced beyond C.

The cases referred to, are those of the preceding Table.

I

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Cases. Given. 1 Songht.

Solution.
AB, AC and the angle|Sin. AC : fin. AB : : fin B: fin.

B oppofite opposite to C. If the sum of BA, AC
to AC.
AB.

be less than 180°, and AB less
than AC; the angle at C is
acute: Or, if the sum of BA,
AC be greater than 180°, and
AB greater than AC; ACB
is obtuse. In other cases,

ACB is ambiguous.
AB, AC and BC the third R: cos. B :: tan. AB : tan. BD

B opposite side. (case 2.) and cos. AB : cos.
2 to AC.

AC: : cos. BD : cos. DC.
When ABC is acute, DC,
CA are of the same affection,
otherwise they are of con-

trary affection.
If CD be not less than DB,

their fum is CB; if CD be
less than DB, but their sum
not less than 180°, their dif
ference is CB. In other cases,
CB is ambiguous.

Cases,

Sph. Trig. Cafes.

Given. | Souq:
AB, AC and A the angle Ricos. AB :: tan. B : cot. BAD,

B oppofite contained (case 3.), and tan. AC: tan.
3
to AC.
by the sides. AB :: cos. BAD: cos. DAC.

If В be acute, DAC and AC are of the same affection, otherwise they are of different affection. If DAC be not less than BAD, their sum is BAC: if DAC be less than BAD, but their fum not less than 180°, their difference is BAC. In other cases BAC

is ambiguous. B, C and AB, AC the fide Sin. C : fin. B :: fin. AB : fin.

two angles opposite to AC. If the sum of B and C 4 and the side B.

be less than 180°, and B less opposite to

than C, AC is acute : or if the one of them

fum of B and C be greater C.

than 180°, and B greater than C, AC is obtufe. In other

cafes, AC is ambiguous. B, C and AB, A the third R: cos. AB::tan. B: cot. BAD,

two angles angle. (case 3 ), and cos. B : cos. C:: 5 and the side

lìn. BAD: fin. DAC, (p. 3.), oppofite to

which is less than BAD, if one of them

B, C be of diff. affection, or C.

less than the supplement of BAD, if B and C be of the same affection : In other cases it is ambiguous. When B and C are of the same affec. tion, BAСis the sum of BAD, DAC, otherwise it is their difference.

Cases.

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