rcle X CAM, r. 1. Prop. 2.) ; cle T MO, then › (by Cor. 1. Prop. rcle G H. In like

the Circle X BN, IN, the Arcs D N, (by Cor. 2. Prop. 3.) HD. And because will be the Pole of the premised,

and X B Quadrant (by K+X B, that is, NX Semicircle; and fo NX cK B, or of the Measure . Semicircle. In like manrant, and X A=Quadrant, at is, X M+AC=two Quaand, confequently, XM is Arc A C, which is the MeaGD. Likewife, fince MO, we fhall have MO+N T= cle: And therefore N M is the OT, or of the Measure of the micircle, W. W D.

DK, HT, are Quadrants, DK ID, are equal to two Quadrants, erefore K T, or the Measure of is the Supplement of the Side H D After the fame manner, it is demonthe Measure of the Angle X M N, t of the Side GH; and BA, the Angle X, is the Supplement of the W.D.

POSITION XV.

Spherical Triangles are alfo equilateral.

ir Supplementals (by 14 of this) are equiland therefore equiangular alfo; and fo themkewife equilateral (by Part 2. Prop. 14).

PRO

PROPOSITION

III.

If a great Circle ECF be defcribed about the Pole ; then the Arc C F, intercepted between AC, CF, is the Measure of the Angle C A F, or CD F. Vid. Fig. to Prop. 1.

THE Arcs A C, AF (by Cor. 1. Prop. 2.) are Quadrants; and, confequently, the Angles A DC, ADF, are Right Angles: Wherefore (by Def. 6. El. 11.) the Angle CDF (whofe Meafure is the Arc CF)is equal to the Inclination of the Planes A CB, A FB, and alfo equal to the Spherical Angle C A F, or CBF. W. W. D.

Coroll. 1.If the Arcs A C, A F, are Quadrants, then fhall A be the Pole of the Circle paffing thro' the Points C and F; for AD is at Right Angles to the Plane FDC (by 4 El. 11.)

2. The vertical Angles are equal: for each of them is equal to the Inclination of the Circles: alfo the adjoining Angles are equal to two Right Angles.

PROPOSITION IV.

Triangles fhall be equal and congruous, if they have two Sides equal to two Sides, and the Angles comprekended by the two Sides alfo equal.

PROPOSITION

V.

Alfo Triangles fhall be equal and congruous, if one Side, together with the adjacent Angles in one Triangle, be equal to one Side, and the adjacent Angles of the other Triangle.

PROPOSITION VI.

Triangles mutually equilateral, are alfo mutually equiangular.

PROPOSITION VII.

In Ifofceles Triangles, the Angles at the Bafe are

PROPOSITION

VIII.

And if the Angles at the Bafe be equal, then the Triangle fhall be Ifofceles.

THESE five laft Propofitions are demonftrated in the fame manner, as in plane Triangles.

PROPOSITION IX.

Any Two Sides of a Triangle are greater than the third.

FO

OR the Arc of a great Circle is the shortest Way, between any two Points in the Superficies of the Sphere.

PROPOSITION X.

A Side of a Spherical Triangle is less than a Semicircle.

LET AC, AB, the Sides of the Triangle ABC, be produced till they meet in D: Then fhall the Arc AC D, which is greater than the Arc AC, be a Semicircle.

PROPOSITION XI.

The three Sides of a Spherical Triangle are less than a whole Circle.

FOR BD+DC is greater than B C (by Prop. 9.); and, adding on each Side B A+A C; then D BA +DCA, that is a whole Circle, will be greater than BA+BC+A C, which are the three Sides of the Spherical Triangle A B C.

PROPOSITION XII. In a Spherical Triangle A B C, the greater Angle A is fubtended by the greater Side.

AKE the Angle B A D Angle B; then shall AD=BD (by 8 of this); and to BDC=DA+ DC, and thefe Arcs are greater than A C. Wherefore the Side B C, that fubtends the Angle B A C, is greater than the Side AC, that fubtends the Angle B.

PRO

PROPOSITION XIII. In any Spherical Triangle ABC, if the Sum of the Legs AB and BC be greater, equal, or less, than a Semicircle, then the internal Angle at the Bale BAC shall be greater, equal, or lefs, than the external and oppofite Angle BCD; and fo the Sum of the Angles A and A C B shall also be greater, equal, or less, than two Right Angles. FIRST, let A B+AC=Semicircle=AD; then BC BD, and the Angles B C D and D equal (by 8 of this); and therefore the Angle B C D fhall be Angle A.

Secondly, let A B+B C be greater than A BD; then fhall BC be greater than BD; and fo the Angle D (that is, the Angle A, by 12 of this) fhall be greater than the Angle BCD. In like manner we demonftrate, if A B+BC be together lefs than a Semicircie, that the Angle A will be lefs than the Angle BCD: And because the Angles BCD and BCA are two Right Angles, if the Angle A be greater than the Angle BCD, then fhall A and B C A be greater than two Right Angles; if the Angle ABCD, then fhall A and BCA be equal to two Right Angles, and if A be lefs than B CD, then will A and BCA be less than two Right Angles. W.W.D.

PROPOSITION XIV. In any Spherical Triangle G HD, the Poles of the Sides, being joined by great Circles, do conftitute another Triangle X M N, which is the Supplement of the Triangle G HD; viz. the Sides NX, XMN, and NM, fhall be Supplements of the Arcs that are the Measures of the Angles D, G, H, to the Semicircles; and the Arcs that are the Measures of the Angles M, X, N, will be the Supplements of the Sides GH, GD, and HD, to Semicircles.

FRO

ROM the Poles G, H, D, let the great Circles X CAM, TMNO, XKBN, be defcribed; then,

then, because G is the Pole of the C rcle X CAM, we shall have G M-Quadrant (Cor. 1. Prop. 2.); and fince H is the Pole of the Circle T M O, then will H M be also a Quadrant, and fo (by Cor. 1. Prop. 3.) M fhall be the Pole of the Circle G H. In like manner, because D is the Pole of the Circle X BN, and H the Pole of the Circle TMN, the Arcs DN, HN, will be Quadrants; and fo (by Cor. 2. Prop. 3.) N fhall be the Pole of the Circle HD. And because GX, D X, are Quadrants, X will be the Pole of the Circle GD. Thefe Things premised,

Because N K Quadrant, and X B=Quadrant (by Cor. 1. Prop. 2.); then will N K+X B, that is, N X +K B two Quadrants, or a Semicircle; and fo NX is the Supplement of the Arc K B, or of the Measure of the Angle HD G, to a Semicircle. In like manner, because M C Quadrant, and X A Quadrant, then will MC+X A, that is, X M+AC=two Quadrants, or a Semicircle; and, confequently, X M is the Supplement of the Arc A C, which is the Meafure of the Angle HGD. Likewise, fince MO, NT, are Quadrants, we fhall have MO+N T OT+NM Semicircle: And therefore N M is the Supplement of the Arç OT, or of the Measure of the Angle G H D to a Semicircle. W. W D.

Moreover, because DK, HT, are Quadrants, D K +HT, or KT+HD, are equal to two Quadrants, or a Semicircle; therefore K T, or the Measure of the Angle X N M, is the Supplement of the Side HD to a Semicircle. After the fame manner, it is demonftrated, that OC, the Measure of the Angle X M N, is the Supplement of the Side GH; and BA, the Measure of the Angle X, is the Supplement of the Side GD. W. W.D.

PROPOSITION XV.

Equiangular Spherical Triangles are also equilateral.

FOR

OR their Supplementals (by 14 of this) are equilateral, and therefore equiangular alfo; and fo themfelves are likewife equilateral (by Part 2. Prop. 14).

PRO