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DEMONSTRATION.

I. It is already proved; That,

The Sine of the Sum of the Angles B and E,

Is to the Sine of the Difference of those Angles B and E;

As the Tangent of half the Base B Е,

Is to the Tangent of half BC.

And multiplying the latter part of this Proportion, by the Tan

gent of half the Base BE: It is,

As the Sine of the Sum of the Angles Band E,

Is to the Sine of the Difference of those Angles BandE;

So is the Square of the Tangent of half BE,

To the Rectangle made of the Tangent of half BE, and the Tan-
gent of half B С.

But, the Rectangle made of the Tangent of half BE, and the
Tangent of half BC, is equal to the Rectangle made of the
Tangent of the half Sum, and half Difference, of the Sides B A
and AE: Therefore,

As the Sine of the Sum of the Angles B and E,

Is to the Sine of the Difference of the Angles B and E;

So is the Square of the Tangent of half BE,

To the Rectangle made of the Tangent of the balf Sum, and half
Difference of the Sides B A and A Ε..

And the former Part of this Proportion, being multiplied by the
Sine of the Sum of the Angles B and E: It is,

As the Square of the Sine of the Sum of the Angles B and E,
Is to the Rectangles made of the Sines of the Sum, and Difference

of those Angles;

So is the Square of the Tangent of half BE,

To the Rectangle made of the Tangents of the half Sum, and half
Difference of the Sides BA and A É.

But, the Rectangle made of the Sines of the Sum and Difference
of the Angles, is equal to the Rectangle made of the Sum and
Difference of the Sines.

And therefore,

As the Square of the Sine of the Sum of the Angles B and E,

Fig. XXVIII.

Is

Fig. Is to the Rectangle made of the Sum and Difference of the Sines;
XXVIII. So is the Square of the Tangent of half B E,

To the Rectangle made of the Tangent of the half Sum, and half
Difference of the Sides B A and A E.

II. As the Difference of the Sines of B and E,

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Is to the Sum of the Sines of the Angles B and E;
So is the Tangent of the half Difference of the Sides,
To the Tangent of the half Sum of the Sides BA and A E.

And multiplying the former Part of this Proportion, by the Dif-
ference of the Sines of the Angles; and the latter Part thereof,
by the Tangent of half the Difference of the Sides A B and
A E: It will be,

As the Square of the Difference of the Sines of B and E,

Is to the Rectangle made of the Sum and Difference of the Sines

of Band E;

So is the Square of the Tangent of the half Difference of B A
and A E,

To the Rectangle made of the Tangents of the half Sum, and balf
Difference of the Sides.

But, by the First Section of this Demonstration it is proved, That,

As the Square of the Sine of the Sum of B and E,

Is to the Rectangle made of the Sum and Difference of the Sines;

So is the Square of the Tangent of half B E,

To the Rectangle made of the Tangents of the half Sum, and half

Difference of the Sides.

Therefore,

As the Square of the Sine of the Sum of B and E,

Is to the Square of the Difference of the Sine of Band E;

So is the Square of the Tangent of half B E,

To the Square of the Tangent of half the Difference of the Sides.

And also,

As the Sine of the Sum of the Angles B and E,

Is to the Difference of their Sines;

So is the Tangent of half the Bafe B E,

To the Tangent of half the Difference of the Sides B A and A E.

But,

P

But,

As the Sine of the half Sum of Band E,

Is to the Sine of their half Difference

So is the Sine of the Sum,

To the Difference of the Sines.

And therefore,

As the Sine of the half Sum of the Angles B and E,

Is to the Sine of the half Difference of B and E;

So is the Tangent of half B E,

To the Tangent of half the Difference of the Sides B A and A E,

which is the first Part of the Propofition.

III. Having already proved,-That the Sum of the Sines of the Angles B and E, is to the Difference of the Sines of those Angles; as the Tangent of the half Sum of the Sides, is to the Tangent: of their half Difference. Therefore,

If you multiply the former Part of this Proportion, by the Sum of the Sines of the Angles of Brand E; and the latter Part thereof by the Tangent of the half Sum of the Sides of AB and AE: Then it will be,

As the Square of the Sum of the Sines of B and E,

Is to the Rectangle made of the Sum and Difference of the Sines;
So is the Square of the Tangent of half the Sum of the Sides;
To the Rectangle made of the Tangent of the half Sum, and half
Difference of the Sides.

Bút,

As the Square of the Sine of the Sum of B and E,

Is to the Rectangle made of the Sum and Difference of the Sines;

So is the Square of the Tangent of half B E,

To the Rectangle made of the Tangent of the half Sum, and half Dif

ference of the Sides B A and AE.

Therefore,

As the Square of the Sine of the Sum of B and E,

Is to the Square of the Sum of the Sines of those Angles;

So is the Square of the Tangent of half BE,

To the Square of the Tangent of the half Sum of the Sides BA

and A E.

And,

Lig XXVIII. Fig. XXVIII.

Fig.

۱

And,

As the Sine of the Sum of the Angles B and E,

Is to the Sum of the Sines of the Angles B and É;

So is the Tangent of half the Bafe B E,

To the Tangent of the half Sum of the Sides A B and A E.

But,

As the Co-fine of the Sum of B and E,

Is to the Co-fine of the Difference of the Angles B and E;

So is the Sine of the Sum of the Angles B and E,

To the Sum of the Sines of the Said Angles.

And therefore,

As the Co-fine of the Sum of the Angles B and E,

Is to the Co-fine of the Difference of the Angles B and E;

So is the Tangent of half the Base B E,

To the Tangent of the half Sum of the Sides A B and AE.

Which was to be Demonstrated.

THEOREM III.

In all Spherical Triangles:

As the Difference of the Verfed Sines, of the Sum and Difference

of any Two Sides (including an Angle,)

Is to the Diameter;

So is the Difference between the Verfed Side of the Third Side, and the Verfed Sine of the Difference of the other Two Sides, To the Verfed Sine of the Angle comprehended by the faid Two Sides.

DEMONSTRATION.

Let the Sides of the Triangle ZSP be known, and let the VerXXIX. tical Angle be SZP: Then shall Z S, the one Side, be equal to ZR, and PR equal to their Sum; and PB the Versed Sine of PR, and PC, is the Difference of the Sides ZS and ZP, and the Versed Sine of PC, is PM.

Now then, MB is the Difference between B P, the Versed Sine of PR. the Sum of the Sides, and PM the Versed Sine of PC, the Difference of the Sides.

MH is the Difference between PH the Versed Sine of PS, and PM the Versed Sine of PC the Difference of the Sides: QV

is

L

is the Diameter, and OV the Versed Sine of PZS the AngleFig. Sought: And the Right Lines NC, KL, and RG, being Paral- XXIX. lel, by the Work, their Inter-segments MB and RC, and alfo

MH and SC, are proportional.

And therefore,

MB:MH::RC:SC;
RC:SC::QV:OV;

Therefore, MB:MH::QV: OV,

Or,

MB:MH:: half QV : half O V.

Which was to be Demonstrated.

THEOREM IV.

In all Spherical Triangles;

As the Rectangle of the Sines of the Sides containing the Angle enquired,

Is to the Square of the Radius;

So is the Difference between the Versed Sine of the Base, and the
Verfed Sine of the Difference of the other Two Sides,

To the Verfed Sine of the Angle Sought.

DEMONSTRATION.

Let the Sides of Triangle A EK be given, and let the Angle at A be enquired; and from O, let fall the Perpendicular O B.

Now then, OE being the Difference of the Sides A E and AK, equal to AO; the Right Line OQ is the Right Sine thereof, and EQ the Versed Sine. In like manner, SM's the Right Sine, and EM the Versed Sine of E S; that is, of the Base EK; and MQ or OB, is the Difference of those Versed Sines.

OK, is the Versed Sine of the Angle OAK, in the Measure of the Parallel O F; and DX is the Versed Sine of the fame Angle in the measure of a Great Circle, whose Diameter is HD. Now then, because of their like Arches, it shall be

As DC:DX::OL:OK:

And because N E and OK are Parallel, as also EC and OB, the Angles BOK and CEN are equal: And the Triangles E CN and OKB like; and therefore the Sides NE, EC, OB, MQ and OK, are Proportional: And it will be

M

As

Fig. XXX.

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