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 Bafe AC shall be greater, equal, or less, than the external and oppofite Angle BCD, and fo the Sum of the Angles A and ACB fhall also be greater, equal, or lefs, than two Right Angles.

=

IRST, let AB+ BC Semicircle AD, then fhall BCBD, and the Angles BCD and D equal, (by 8 of this) and therefore the Angle BCD fhall be Angle A.

Secondly, Let AB+BC be greater than ABD; then shall 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 AB+BC be together lefs than a Semicircle, 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 BCA, be greater than two Right Angles; if the Angie ABCD, then fhall A and BCA be equal to two Right Angles. And if A be lefs than BCD, then will A and BCA be lefs than two Right Angles. W. W.D.

PROPOSITION XIV.

In any Spherical Triangle GHD, the Poles of the Sides being joined by great Circles, do conftitute another Triangle X MN, which is the Supplement of the Triangle GHD, viz. the Sides NX, XM, 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.

ROM the Poles G, H, D, let the great Circles
XCAM, TMNO, XKBN, be described; then

because

because G is the Pole of the Circle X CAM, we shall have GMQuadrant (by Cor. 1. Prop. 2.) and fince H is the Pole of the Circle TMO, then will HM be alfo a Quadrant; and fo (by Cor. 1. Prop. 3.) M fhall be the Pole of the Circle GH. In like Manner. becaufe 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. 1. Prop. 3.) N fhall be the Pole of the Circle HD. And because GX, DX, are Quadrants, X will be the Pole of the Circle GD. These things premised.

Because NKQuadrant (by Cor. 1. Prop. 2.) then will NK+XB, that is, NX+KBtwo Quadrants, or a Semicircle; and fo NX is the Supplement of the Arc KB, or of the Measure of the Angle HDG to a Semicircle. In like manner, because MC Quadrant, and XA Quadrant, then will MC+XA; that is, XM+AC= two Quadrants, or Semicircle; and confequently, XM is the Supplement of the Arc AC, which is the Measure of the Angle HGD. Likewife, fince MO, NT, are Quadrants, we fhall have MO+NT=OT+NM

=

Semicircle. And therefore NM is the Supplement of the Arc OT, or of the Measure of the Angle GHD, to a Semicircle. W. W. D.

Moreover, because DK, HT, are Quadrants, DK+HT, or KT+HD, are equal to two Quadrants, or a Semicircle. Therefore KT, or the Meafure of the Angle XNM, is the Supplement of the Side HD to a Semicircle. After the fame manner it is demonftrated, that OC, the Measure, of the Angle XMN, 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 fpherical Triangles are also equilateral.

FOR

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

RG

PROPOSITION XVI.

The three Angles of a spherical Triangle, are greater than two Right Angles, and less than fix.

FOR

OR the three Measures of the Angles G, H, D, together with the three Sides of the Triangle XNM, make three Semicircles (by 14 of this) but the three Sides of the Triangle X N M, are less than two Semicircles (by 11 of this.) Wherefore the three Measures of the Angles G, H, D; are greater than a Semicircle; and fo the Angles G, H, D, are greater than two Right Angles.

The fecond Part of the Propofition is manifeft; for in every spherical Triangle, the external and internal Angles together, only make fix Right Angles; wherefore the internal Angles are lefs than fix Right Angles.

PROPOSITION XVII.

If from the Point R, not being the Pole of the Circle AFBE, there fall the Arcs RA, RB, RG, RV, of great Circles to the Circumference of that Circle; then the greatest of thofe Arcs is RA, which paffes thro' the Pole C thereof; and the Remainder of it is the least; and those that are more diftant from the greatest are less than thofe which are nearer to it, and they make an obtufe Angle with the former Circle AFB, on the Side next to the greatest Arc. Vid. Fig, to Prop. I.

Ecaufe C is the Pole of the Circle AFB, then

fhall CD and RS, which is parallel thereto, be perpendicular to the Plane AF B. And if SA, SG, SV, be drawn, then fhail SA (by 7. El. 3.) be greater than SG, and SG greater than SV. Whence in the Right-angled plain Triangles RSA, RSG, RSV, we fhall have RSq+SAq, or RA q, greater than RSq+SGq, or R&q; and fo RA will be greater than RG, and the Arc RA greater than the Arc RG. In like manner, RSq+SGq, or RGq fhall be greater

than

than R Sq+SVq, or R Vq; and fo R G fhall be greater than RV, and the Arc R G greater than the Arc RV.

zdly. The Angle RGA is greater than the Angle CGA, which is a Right Angle, (by Cor. Prop. 3.) and the Angle RVA is greater than the Angle CVA, which alfo is a Right Angle. Therefore the Angles R GA, RV A, are obtufe Angles.

PROPOSITION XVIII.

In a Spherical Triangle right-angled at A, the Legs containing the Right Angle, are of the fame Affection with the oppofite Angles; that is, if the Legs be greater or less than Quadrants, then accordingly will the Angles oppofite to them be greater or less than Right Angles. Vid. Fig. to Prop.1.

FOR if AC be a Quadrant, then will C be the Pole

of the Circle AFB, and the Angles AG C, or AVC, will be Right Angles. If the Leg AR be greater than a Quadrant, then fhall the Angle AGR be greater than a Right Angle, (by 17. of this ;) and if the Leg AX be lefs than a Quadrant, the Angle AGX fhall be less than a Right Angle.

PROPOSITION

XIX.

If two Legs of a right-angled Spherical Triangle be of the fame Affection; (and confequently the Angles,) that is, if they are both less or both greater than a Quadrant, then will the Hypothenuse be lefs than a Quadrant. Vid. Fig. to Prop. 1. IN the Triangle ARV, or BRV, let F be the Pole

Ν

of the Leg AR, then will RF be a Quadrant, which is greater than RV, (by 17. of this.)

PROPOSITION XX.

If they be of a different Affection, then fhall the
Hypothenufe be greater than a Quadrant. Vid.
Fig. to Prop. 1.

OR in the Triangle ARG, the Hypothenufe
RG is greater than RF, which is a Quadrant.
PRO-

PROPOSITION XXI.

If the Hypothenufe be greater than a Quadrant, then the Legs of the Right Angle, and fo the Angles oppofite to them, are of a different Affection; but if leffer, of the fame Affection. Vid. Fig. to Prop.1. THIS Propofition, being the Converse of the for

mer ones, eáfily follow from them.

PROPOSITION XXII.

In any Spherical Triangle ABC, if the Angles at the Bafe B and C, be of the fame Affection, then the Perpendicular falls within the Triangle ; and if they be of a different Affection, the Perpendicular falls without the Triangle.

N the first Cafe, if the Perpendicular does not fall within, let it fall without the Triangle, (as in Fig. 2.) then in the Triangle ABP, the Side AP is of the fame Affection with the Angle B. And in like manner, in the Triangle ACP, AP is of the fame Affection with the Angle ACP. Therefore fince ABC, and ACP, are of the fame Affection, the Angles ABC, ACB, fhall be of a different Affection; which is contrary to the Hypothefis.

In the fecond cafe, if the Perpendicular does not fall without, let it fall within, (as in Fig. 1.) Then in the Triangle A B P, the Angle B is of the fame Affection with the Leg AP. So likewife, in the Triangle A CP, the Angle C is of the fame Affection with AP; and therefore the Angles B and C are of the fame Affection; which is contrary to the Hypothefis.

PROPOSITION XXIII. In fpherical Triangles BAC, BHE, right-angled at A and H, if the fame acute Angle B be at the Bafe BA,or BH,then the Sines of the Hypothenuses shall beproportional to theSines of the perpendicular Arcs. OR the Right Lines CD, EF, being perpendicular to the fame Plane, are parallel. Alfo FR, DP, perpendicular to the Radius OB, are likewise

FOR

paral