Spherical trigonometry

AE Tangent of half the Angle D; the Right Line E G Tangent of half C; and AG = CoTangent of half B. Alfo AP = Tangent of half the Sum of the Angles D, C ; and AO

of half their Difference.

Tangent

6. Thus the Points O, G, P, are in a Semicircle defcribed from its Pole E, thro' G, whofe Center fuppofe projected into the Point n.

7. Then take BI BC= BL; and draw the Diameter Bm Ab; and draw D K parallel to B m, and to d H. Hence B D = db; and m K = Hm; and FIDL.

8. Draw I w perpendicular to Dd; then the Triangle Idy is equiangular to D Ly; by Euclid Lib. 3. Prop. 21. Hence DL K is equiangular to Id w.

9. Wherefore we have thofe Analogies Ly dy :: D L : d I. And as D L:di::LK: dw; and therefore Ld is parallel to K w.

10. But the Points d, I, H, w, are in a Circle whofe Diameter is d I; therefore the Angle 1 wH = IdH= LdD=Kwy. Therefore the Angle Kw H =ywa Right Angle; hence m H = m K = mw. Hence alfo 1 K: LK:: (w I: w R) Tangent of the Angle Dd I, the Tangent of the Angle D d L. That is,

11. As the Sine of the Sum of the Sides, BD + BCDI BL,

Is to the Sine of the Difference of the Sides, BCBD = DL,

So is the Tangent of half the Suin of the Sides,
To the Tangent of half the Difference of the Sides.
Therefore

12. As the Sine of the Sum of the Angles D+ C,
Is to the Sine of their Difference
D-C;
So is the Tangent of half the Sum of the Angles,
To the Tangent of half their Difference.

But

13. As the Sine of the Sum of the Sides, Is to the Sine of their Difference,

(So is the Sine of the Sum of the Angles, to the Sine of their Difference ;)

So is the Tangent of half the Sum of the Angles, To the Tangent of half their Difference.

14. That is, IK: IH:: AP: AO. Therefore IK+IHIK—IH AP+40 A P — AO

mw:: Ann G.

the Angle G An;

15. Moreover the Angle m Iw therefore the Triangles Im w and An G are equiargular, (by Euclid, Lib. 6, Prop. 7;) as are alfo the Triangles of the Sum of the Angles, IH w, and AO G; and alfo the Triangles IKw and AP G, are equiangular, Therefore

16. We have Iw:IH::AG: AO, and Iw:IK::AG:AP. But we have DI: Iw :: DL (: L K =): I H. And alfo d I: Iw::d L:LH=I K.

Therefore

17. As DI: DL:: (Iw: IH::) AG: AO. And dI: dL::(Iw: IK::) AG: AP. Which in Words afford these following Theorems or Propofitions, viz.

PROPOSITION I.

As the Sine of half the Sum of the Sides,

is to the Sine of half their Difference;

fo is the Co-tangent of half their contained Angle, to the Tangent of half the Difference of the other Angles.

PROPOSITION II.

As the Co-fine of half the Sum of the Sides, is to the Co-fine of half their Difference;

fo is the Co-tangent of half the contained Angle, to the Tangent of half the Sum of the other two Angles.

PROPOSITION III.

As the Sine of half the Sum of the two Angles, is to the Sine of half the Difference ;

fo is the Tangent of half the contained Side,

to the Tangent of half the Difference of the other two Sides.

PROPOSITION IV.

As the Co-fine of half the Sum of the Angles, is to the Co-fine of half the Difference;

fo is the Tangent of half the included Side, to the Tangent of half the Sum of the other two Sides.

PROPOSITION V.

As the Tangent of half the Bafe,

is to the Tangent of half the Sum of the other two Sides; fo is the Tangent of half the Difference of the Sides, to the Tangent of half the Difference of the Segment of the Bafe.

That is, AG: AP :: AO: AT.

Thefe Propofitions, together with the Theorems 22, 35, and 39, of Spherical Geometry, are fufficient to folve all the Cafes of Oblique Spherical Triangles without a Perpendicular; or any kind of Ambiguity; as appears by the following Examples. And on this Account, this Method is, in fome refpects, preferable to that by a Perpendicular; efpecially to fuch as understand not the Manner of projecting them on the Plane, which will be taught in the next Chapter.

Cafe