Tangent of the Distance of the Perpendicular from the
Middle of the Base.

THEOREM VII.

In any Jpberical 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 Helf the vertical Angle, to the Tangent of the Angle which the perpendicular CD makes with the Line CF bisetting the vertical Angle. (See the

preceding Figure.)

Demonstration. It will be (by Corol. to Theor. 3.) Co-sine A : Cosine B :: Sine ACD : Sine BCDs and therefore, Cofine A + Co-sine B : Co-sine A-Co-fine B:: Sine ACD + Sine BCD : Sine ACD Sine BCD. But

B +A

B-A (by the Lemma) Co-tang. :Tang. (Co-fine A + Co-fine B : Co-sine A-Co-fine B:: Sine ACD + Sine BCD: Sine ACD - Sine BCD) Tang. ACF : Tang. DCF. 2. E. D.

2

2

B

The Solution of the Cases of right-angled Spherical

Triangles.

Cala

Angle A

Given Sought

Solution The Hyp. The oppo- As Radius : Sine Hyp. AC:: 1 AC and one site Leg Sine A : Sine BC (by the for

BC mer Part of Theor. 1.). The Hyp. The adja- As Radius: Co-Gine of A :: 2 AC and one cent Leg Tang. AC:Tang. AB (by the Angle A

AB latter Part of Theor. 1.) The Hyp. The other As Radius : Co-line of AC 3A AC and one Angle C 1: : Tang. A : Co-cang C (by Angle A

Theor.5.) The Hyp. The other As Co-line A B : Radius :: 4 AC and one Leg BC Co-line AC : Co-line BC (by

Theor. 2. The Hyp. The oppo- As Sine AC: Radius :: Sine 5 AC and one Gite Angle AB : Sine C (by the former Leg AB

с Part of Theor, 1.) The Hyp. The adja- As Tang. AC : Tang. AB :: 6 AC and one cent Angle Radius: Co-sine A (by TheLeg AB

A
One Leg The other As Radius : Sine AB :: Tan-

AB and the Leg BC gent A : Tangent B C ( by 7 adjacent

Theor. 4.)
Angle A

Leg AB

or. I.

Case

Theor. 3.)

IO

Given Sought

Solution
One Leg The oppo. As Radius : Sine A:: Co-

A B and the Gite Angle fine of AB : Co-line of C (by 8

adjacent C Angle A One Leg The Hyp. As Co-fine of A: Radius ::

A B and the AC Tang. AB : Tang. A C (by
9
adjacent

Theor. 1.)
Angle A
One Leg The other As Tang. A : Tang. BC::
B C and the Leg AB Radius : Sine AB (by Theor. 4.)

opposite
Angle A
One Leg The adja- As Co-line BC: Radius :
BC and the cent Angle Co-sine of A : Sin. C (by

opposite C
Angle A
One Leg The Hyp. As Sin. A : Sin. BC:: Radius
BC and the AC Sin. AC (by Theor. 1.)

opposite
Angle A

Both Legs The Hyp. As Radius : Co-line AB :: 13 A Band BC

AC

Co-fine BC: Co-line AC (by

Theor. 2.) Both Legs An Angle, As Sine AB : Radius:: Tang. 14 A Band B C suppose ABC:Tang. A (by Theor. 4.)

II

Theor. 3.)

12

Both Angles A Leg, As Sin. A : Co-line C:: Ra-
A and C suppose dius : Co-line AB (by Theor.

AB 3:)
Both Angles The Hyp. As Cang. A : Co-tang. C::
A and C AC Radius : Co-fine AC ( by

Theor. 5.)

Note, The poth, 11th and 12th Cases are ambiguous ; since it cannot be determined, by the Data whether AB, C, and AC, be greater or less than 90 Degrees each.

D

Tbe

The Solution of the Cases of oblique Spherical Tri

angles.

Given

| Care

Sought

Solution

Two Sides AC, The Angle B As Sine BC: Sine A :: Sine AC : BC and an An opposite to Sine B (by Cor. 1. to Theor. 1.) Note, gle A opposite the other This Cale is ambiguous when B C is to one of them

less than AC; fince it cannot be de

termined from the Data whether B be acute or obtuse.

Two Sides AC, The included Upon A B produced (if need be) let BC and an An Angle ACB fall the perpendicular CD: Then ( by gle A oppofite

Theor. 5.) Rad. : Co-line AC :: to one of them

Tang. A : Co.tang. ACD; but (by Cor. 2. to Theor. 1.) as Tang. BC: Tang. AC :: Co-sine ACD : Co-fine BCD, Whence ACB=ACD + BCD is known.

Two Sides AC,
BC and an An-

gle opposite to 3

one of them

The other As Rad. : Co-fine A :: Tang. AC :-
Side AB Tang. AD (by Tbeor. i.) and (by Cor.

10 Theor. 2.) as Co-fin. AC: Colin.
BC :: Co-fin. A D: Co-fin. B D.
Note, This and the last Care are both
ambiguous when the first is so.

Two Sides AC, | The other

AB and the in Side BC 4 cluded Angle A

As Rad. : Co-fin. A :: Tang. AC: Tang. AD by Theor. 1.) whence BD is also known: Then (by Curol. to

Theor. 2.) as Co-fine AD: Co fine
BD :: Co-sine AC : Co-sine BC.

I'wo Sides AC, Either of the As Rad. : Co-fine A : : Tang. AC: AB and the in- other Angles, Tång. AD (by Theor. 1.) whence BD cluded Angle A suppose B is known ; then (by Cor. 10 Thecr. 4.)

as Sine BD: Sine AD:: Tang. A : Tang. B.

Care

Two Angles A, Either of the As. Rad. : Co-fine AC : : Tang. A:

ACB and the other Sides, Co-tang. ACD ( by Theor. 5.) whence 7 Side A C be suppose BC BCD is also known: Then,as Co-line twixt them

of BCD : Co-line ACD : : Tan, AC : Tang. BC ( by Cor. 2. to Tbeor. 1.)

Two Angles A, The Side BC As Sine B : Sine AC :: Sine A : Sine 8

B and a Side opposite the BC (by Cor. 1. to Theor...)
AC opposite to other
one of them

Two Angles A, The Side AB As Rad. : Co-fine A : : Tang. AC:

B and a Side AC betwixt them Tang. AD (by Tbeor. 1.) and as Tan. 9 opposite to one

B: Tan. A:: Sine AD: Sine BD of them

(by Cor. to Tbeor. 4.) whence AB is also known.

Two Angles A,

The other As Rad. : Co-fine AC : : Tang. A : Band a Side AC Angle ACB Co-tang. ACD (by Theor. 5.) and as 10 opposite to one

Co-sine A: Co-fine B :: Sine ACD of them

: Sine BCD (by Cor. to Tbeor. 3.) whence ACB is also known.

Note, In letting fall your Perpendicular, let it always be frein the End of a given Side and opposite to a given Angle,