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Definition

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be

one of

Given Sought

Make, as
All the All the S, C:S, A :: AB:BC. And
Angles Sides S, C:S,B:AB:AC. Whence,
A,B,C. A B, if the Angles are given, the Pro-
2

A C, portions of the. Sides may
BC. \found, but not the Sides them-

lelves, unless one of them be first

known. The The AB: BC::S,C:S, A, which two Angles therefore may be found. When Sides A and A B, the Side opposite to C, the AB, B. given Angle is longer than C B, BC,

che Side opposite to the foughe and C,

Angle, the fought Angle is less 3 the An

than a Right one. But when it gle op

is shorter, because the Sine of an posite to

Angle, and that of its Comple

ment to two Right Angles, is the them.

fame, the Species of the Angle A must be firit known, or the Solu

tion will be ambiguous. The The Vid.Fig, to Prop. 1 3. BL=AB: two Angles BC-AB::T,A+C:T, A

A-C Sides

A and AB, C.

Whence is known the Difference BC,

of the Angles A and C, whose

Sum is given, and io (by Prop. inter

15.) the Angles themselves will jacent

be given. Angle

B.
All the The Vid. Fig. B. Letine Perpendi-

Sides Angles. cular be drawn from the Vertex
AB,

to the Bale, and find the SegBC,

ments of the Base by Prop. 14. A C.

viz, make as BC:AC+AB:: 5

AC-AB: DC-DB. And o BD, DC, are given from the

Analogy, and thence the Angles ABD, A CD, will be given by the Resolution of Right-angled Triangles.

THE

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THE

E LE M E N T S

OF

Spherical Trigonometry .

DEFINITIONS.

1.THE Poles of a Sphere are two Points in

the Superficies of the Sphere, that are the

Extremes of the Axis. II. The Pole of a Circle in a Sphere is a Point in

the Superficies of the Sphere, from which all Right Lines that are drawn to the Circumferenee of the

Circle are equal to one another. III. A great Circle in a Sphere is that whose

Plane passes through the Centre of the Sphere, and whose Centre is the same with that of the

Sphere. IV. A spherical Triangle is a Figure compre

bended under the Arcs of three great Circles in

a Sphere. V. A spherical Angle is that which, in the Su

perficies of the Sphere, is contained under two Arcs of great Circles; and this Angle is equal to the inclination of the Planes of ibe said Circles,

PRO

PROPOSITION I.
Great Circles ACB, AFB, mutually biseɛt each

otber.
OR, since the Circles have the same Centre,
their common Section thall be a Diameter of

each Circle, and so will cut them into two equal Paris, Coroll. Hence the Arcs of two great Circles in the

Superficies of the Sphere, being less than Semicircles, do not comprehend a Space; for they cannot, unless they meet each other in two opposite Points; that is, unless they are Semicircles.

F

1

PROPOSITION II.
If from the Pole C of any Circle A FB, be drawn

a Right Line C D to the Centre thereof, the
Jaid Line will be perpendicular to the Plane of
that Circle. Vid. Fig. to Prop. 1.
ET there be drawn any Diameters EF, GH, in

the Circle A F B ; then, because in the Triangles
CDF, CD E, the Sides CD, DF, are equal to the
Sides CD, DE, and the Base C F equal to the Base
CE (by Def. 2.); then (by 8 El. 1.) ihall the Angle
CDF=Angle CDE, and so each of them will be a
Righi Angle. After the same manner we demonftrate,
that the Angles CDG, C DH, are Right Angles ;
and so (by 4 El. 11.) C D shall be perpendicular to
the Plane of the Circle A FE. W. W.D.
Coroll, 1. A great Circle is distant from its Pole by the

Interval of a Quadrant ; for, since the Angles
CDO, CDF, are Right Angles, the Measures of

them, viz. the Arcs CĞ, CF, will be Quadrants. 2. Great Circles, that pass thro’ the Pole of some other

Circle, make Right Angles with it; and, contrariwise, if great Circles make Right Angles with fome other Circle, they shall pass thro' the Poles of that other Circle ; for they must neceffarily pass through the Right Line DC.

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