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

THEOREM VII.

In any Ipberical Triangle ABC, it will be, as the Co-tangent of Half the Sum of the Angles at the Bafe, is to the Tangent of Half their Difference, fo is the Tangent of Half the vertical Angle, to the Tangent of the Angle which the Perpendicular CD makes with the Line CF bijecting the vertical Angle. (See the preceding Figure.)

Demonftration.

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

(by the Lemma) Co-tang.

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(Co-fine A+ Co-fine B: Co-fine A-Co-fine B:: Sine ACD+Sine BCD: Sine ACD Sine BCD) Tang. ACF: Tang. DCF. 2. E. D.

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The

B

The Solution of the Cafes of right-angled Spherical

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Triangles.

Sought

Given

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Solution

As Radius: Sine Hyp. AC::
Sine A: Sine BC (by the for-
mer Part of Theor. 1.)

The adja-As Radius: Co-fine of A::
Tang. AC: Tang. AB (by the
latter Part of Theor. 1.)

cent Leg

AB

The other

As Radius: Co-fine of AC

Angle C:: Tang. A: Co-tang C (by
Theor.5.)

The other

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Leg BC

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BC and the

oppofite
Angle A

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One Leg The adja-As Co-fine BC: Radius::
BC and the cent Angle Co-fine of A: Sin. C (by
oppofite
C Theor. 3.)
Angle A

One Leg The Hyp. As Sin. A: Sin. BC:: Radius
:Sin. AC (by Theor. 1.)

BC and the

oppofite

Angle A

AC

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

Both Legs 14 A Band BC

AC

Co-fine BC: Co-fine AC (by
Theor. 2.)

An Angle, As Sine AB : Radius:: Tang.
fuppofe ABC: Tang. A (by Theor. 4.)

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

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

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