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

PROP. XX.

The hypothenuse AC, and the Jum, or difference, of the legs AB, BC, of a right-angled spherical triengle ABC, being given, to determine the triangle.

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

B
AC (by Theor. 2.)
therefore, co-f. AB x co-f. BC = rad. X.co.f.
AC; but the former of these is = { rad. X
C0-1. AE + co-f. AF (by Corol. 3. to Prop. 2.);
therefore 2 x Co-f. AC = Co-f. AE + co-f. AF.
Whence it appears, that, if from twice the co-fine
of the hypothenuse, the co-fine of the given jum,
or difference, of the legs, be subtraited, the re-
mainder will be the co fine of an arch, wbich added
to the said fum, or difference, gives the double of the
greater leg required.

COROLLARY.

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Hence, if the two legs be supposed 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-line of the hypothenuse minus the radius,

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PROP. XXI.

One leg BC and the fun, or difference, of the bypothenuse and the other leg AB being given, to determine the hypothenuse (see the last figure.)

BC: : co-tang.

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Since rad.: co-fine BC:: co-fine AB: co-line AC (by Theor. 2.), it will be (by comp. and div.) radius + co-line BC:rad. Co-f. BC:: Co-f. AB + 'co-f. AC: co-f. AB CO-S. AC. But the radius may be considered as the fine of an arch of 90°, or the co-fine of o: and, therefore, since (by the lemma in p. 30.) co-fine o + co-fine BC: co-f. o-0-1, BC +o

BC

; and, co

2 fine AB + co-f. AC:co-f. AB - Co-f. AC::00

AC + AB AC-AB tang: tang.

j it follows, by

2 BC

BC equality, that co-tang. : tang:

:: Co-tang: AC + AB AC - AB

that is, As the cotang. of balf the given leg, is to its tangent; so is the co-tang. of balf the sum of the hypothenuse and the other leg, to the tangent of half their difference.

2

2

2

: tang

PROP. XXII.

The angle at the base and the sum, or difference, of the by'pothenuse and base, of a right-angled spherical triangle being given, to determine the triangle,

First,

First, it will be, rad. : co-f. A :: T. AC:T. AB (by Theor. 1.) and therefore rad. + co-fi A : rad. CO-S. A:: T. AC + T. AB : T. ACT. AB:

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. AC-T. 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; so is the line of the sum of the hypothenuse and adjacent leg, to the fine of their difference.

B

PROP. XXIII. The kypothenuse AG and the furr, 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: ET. ACE (by Theor. 5.) From whence, by reasoning as above, we shall also have co-tang. {AC : tang.

AC::S. A + ACE:S. A---ACE; whereof the two last terms, by fubftituting 90° — ACB for ACE, will become S. 90°+ A- ACB (co-f. ACB -A) and S. A+ACB-90° respectively. Whence it appears, that, As the co-tangent of half the bypothenuse, is to its tangent; so is the co-line of the difference of the angles at the hypothenuse, to the fine of the excess of their sum above a right-angle.

COROL

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COROLLARY. Hence, if the angles be supposed equal, then it will be, as radius : tang. {AC :: tang. JAC : fin. 2A-900.

PROP. XXIV.

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

It is evident (from Tbeor. 1.) that S. DE: S. BC :: S. AE. S. AC; therefore S. DE + S. BC : S.DE-S. BC::S. AE

+ S, AC:S. AE S. B

AC : whence (by the

lemma in p. 30. and equaDE + BC

DE-BC
lity) tang
AE + AC AE- AC
: tang.

: that is, As the tan-
2
gent of balf the sum of the two perpendiculars, is to
the tangent of half their difference; so is the tangent
of balf the sum of the two hypot benufeș, to the tan-
gent of half their difference.

D

: tang

: : tang

2

2

PROP. XXV.

In two right-angled spherical triangles ABC, ADE, having the fame angle A at the base, let there be given the two perpendiculars BC, DE and the sum or difference of the bases AB, AD, to determine the bases (see the preceding figure.)

Since

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Since T. DE : T. BC:: S. AD: S. AB (by Theor. 4. and equality); therefore is T. DE + T. BC:T. DE- T.BC :: S. AD +S. AB:S. AD S. AB; whence, (by Prop. 4. and the lemma in P. 30.) it will be, S. DE + BC:S. DE-BC::T. AD + AB AD AB

; that is, As the fine of the fum of the two perpendiculars, is to the fine of their difference ; so is the tangent of half the sum of the two bases, to the tangent of half their difference.

: T.

2

2

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PROP. XXVI:

The produ&t of the square of radius and the co-fine of the base of any spherical triangle ABC, is equal to the product of the lines of the two sides and the co-fine of the vertical angle, together with the product of radius and the co-fines of the fame sides.

For let AD be perpendi-
cular to BC; then, since co-f.
S.CBXS.CD+co-f.CBXCO-SCD

rad.
(by Cor. 1. to Prop. 2.) it is
evident, that co-f. BD : co-f.
CD::

А
S.CBXS.CD+co-f.CB x co-f.CD

rad.

S. CB x S. CD : co-f. CD::

+ co-f.CB: rad. (by Co-f. CD

S.CD mult, each term by

T.CD, Co-S.CD CO-T.CD rad.

S. CB x T.CD therefore our proportion will be

rad.

rad.

. But

+

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