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than RSq+S V q, or R Vq; and so R G shall be greater than RV, and the Arc RG greater than the Arc R V.

2dly, The Angle RG A 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 also is a Right Angle. Therefore the Angles RGA, RV A, are obiule Angles.

PROPOSITION XVIII.
In Spherical Triangles A GR, A GX, Right-

angled at C, the Legs containing the Right An-
gle are of the fame Affe Etion with the opposite
Angles ; that is, if ihe Legs be greater or less
than Quadrants, iben, accordingly, will the
Angles opposite to them be greater, or less than

Righi 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 AGC, A V C, will be Righe Angles. If the Leg A R be greater than a Quadrant, then shall the Angle AGR be greater than a Right Angle (by 17 of this); and if the Leg A X be leis than a Quadrant, the Angle A Ġ X Thall be less than a Right Angle.

P. R O POSITION XIX.
If two Legs of a Right-angled Spherical Triangle

be of the same Ajfe Elion (and consequently the

Angles), that is, if they are borb less, or both greater, than a Quadrant, then will the Hypothenuse be lejs than a Quadrant. Vid. Fig. to Prop. I. N the Triangle ARV, or BRV, let F be the

Pole of the Leg AR: then will RF be a Qua-, drant, which is greater than R V (by 17 of this).

PROPOSITION XX.
If they be of a different Affection, then fsall the
H puthenuje be greater than a Quadrant. Vid.

Prus. I.
in the Tr wigle ARG, the Hypothenuse RG
acer than RF, which is a Quadrant.

PRO.

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PROPOSITION XXÍ.
If tbe Hypotbenuse be greater than a Quadrant,

then the Legs of the Right Angle, and so the
Angles opposite 10 them, are of a different Af.
fection ; but if lefjer, of ibe Jame Affection.

Vid. Fig. to Prop. 1.
THIS Proposition being the Converfe of the former

ones, casily follows from them.

PROPOSITION XXII.
1112 any Spherical Triangle A B C, if tbe Angles at

the Base B and C be of the same Affection, then
the Perpendicular falls within the Triangle ;
and if they be of a different Affection, the Per-
pendicular 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 A BP, the Side AP is of the same Affection with the Angle B. And, in like manner, in the Triangle ACP, A Pis of the fame Affection with the Angle ACP; therefore, fince ABC and ACP are of the same Affection, the Angles ABC, ACB, shall be of a different Affection; which is contrary to the Hypothesis.

In the second Case, if the Perpendicular does not fall without, let it fall within (as in Fig. 1.) Then, in the Triangle ABP, the Angle B is of the fame Affection with the Leg A P. So likewise in the Trio angle ACP, the Angle C is of the fame Affection with AP; and therefore the Angles B and Care of the same Affection ; which is contrary to the Hypothisis.

PROPOSITION XXIII. .
In Spherical Triangles B AC, B H E, Right-en-

gied at A und H, if the same accute Angle B be
at the Base B A, or BH, iben the Sines of the
Hypothenuses shall be proportional to the Sines

of ibe perpendicular Arcs. FO OR the Sight Lines C D, EF, being perpendicular to the same Plane, are parallel. Allo, FR,

DP

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DP, perpendicular to the Radius O B, are likewise parallel : Wherefore the Planes of the Triangles E FR, CDP, are also parallel (by 15 El. 11.) Wherefore CPER, the common Sections of those Planes, with the Plane pafling thro' BE, CO, will be parallel (by 16 El. 11. Wherefore, the Triangles CDP, EFR, Thall be equiangular. Wherefore CP, the Sine of the Hypothénule B C, is to CD, the Sine of the perpendicular Arc C A, as ER, the Sine of the Hypothenuse BE, to EF, the Sine of the perpendicular Arc EH. W. W. D.

PROPOSITION XXIV. The same Things being supposed, AQ, HK, the

Sines of the Bases are proportional to IA, GH,

the Tangents of the perpendicular Arcs. FOR, after the same manner as in the laft Propofi

tion, we demonstrate, that the Triangles AI, KHG, are equiangular ; whence QA:AI::KH: HG.

PROPOSITION XXV. In a Spherical Triangle ABC, Right-angled at A,

as the Cofine of the Angle B, at the Base BA, is to the Sine of the verlical Angle ACB, so is the Cofine of the Perpendicular to the Radius.

PREPARATION. LET the Sides A B, BC, CA, be produced, so

that BE, BF, CI,CH, be Quadrants; and from the Poles B and C draw the great Circles E FDG, IHG; then will the Angles at E, F, I, H, be Right Angles : and fo D is the Pole of B A E (by Cor. 2. Prop. 2. of this), and G the Pole of IFCB: Allo, A E will be = Complement of the Arc B A, and F E the Measure of the Angle B= GD, and D F their Com. plement: Allo, BC shall be =FI = Measure of the Angle G, and CF their Complement: Likewise, C AHD, and D C their Complement. These Things premised, in the Triangles HIC, DCF, Rightangled at I and F, and having the fame acute Angle Cance B A is less than a Quadrant, it will be, ass,

DF:

DF:S, Hi::S, DC:S, HC; that is, the Cofine of the Angle B is to the Sine of the vertical Angle BCA, as the Cofine of CA is to Radius. W. W.D.

PROPOSITION XXVI. The Cofine of the Base: Celine of the Hypotbenuje

::R:Cof. of the Perpendicular, FOR in the Triangles A E D, C F D, Right

angled at E, F, having the same acute Angle D; because A E is less than a Quadrant, we have, S, EA: S, CF::S, DA:S, DC. W. W. D.

PROPOSITION XXVII.
S, of the Base: R::T, of the Perpendicular :

T, of the Angle at the Base.
OR in the Triangles B A C, B E F, Right-angled

at A and E, and having the same acute Angle B: because A C is lets than a Quadrant, we have S, BA: S, BE::T,AC:T, EF. W. W, D.

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PROPOSITION XXVI!I. Cof. of the Vertical Angle : R::T, of the Per

pendicular : T, of the Hypothenuse. N the Triangles GIF, GHD, Right-angled at

I and H, and having the same acute Angle G, because H D is less than HC, or a Quadrant, it is, as S, GH:S, GT::T, HD: TIF.

PROPOSITION XXIX. S, of the Hypotbenuse : R::S, of the Perpendi

cular : S, of ike Angle at the Base. IN N the aforesaid Triangles we have S, IF: S, GF

::S, HD:S, GD.

PRO

PROPOSITION XXX.
R: Cof. of the Hypothenuse : : T, of the vertical

Angle: Cot. of the Angle at the Base.
IN the

Triangles HIC, D F C, Right-angled at I and F, and having the same acute Angle C, because D F is less than a Quadrant, we have S, CI: S, CF::T, HI:T, DF; that is, R: Cof. BC ::T, C: Cot. B.

The last fix Propositions are sufficient for solving all the sixteen Cases of Right-angled Spherical Triangles. These fixteen Cases, with their Analogies deduced from the said Propofitions, are as follow :

AC and

Given Sought\Vid. Fig. tu Prop. 25, 26, 27, 28,
besides

29, 30.
the
Right
Angle.

B. R; Cor. CA :: S, C : Col. B, By the C. of the same Kind with CA. Inverse

of Prop.

25. AC and C. Cos. CA:R :: Cor. B:S, C; By Prop. B. this is ambiguous.

25 Band AC. S, C : Cof. B::R: Cof. C A, By Prop

of the same Kind with the An. 25, and

3) C.

gle B.

18.

AB,
СА.

4

BC. R: Cos. BA :: Cof. AC:Cor.BC.By Prop'

If BA, A C, be of the fame Af-26, 19,
fection, and not Quadrants, then and 20.
BC will be less than a Quadrant.
If they be of a different Affection,
B C shall be a greater than a Qua.
Jdrant.

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