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A C, portions of the. Sides may
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 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
Sides Angles. cular be drawn from the Vertex
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.
4 and the
E LE M E N T S
Spherical Trigonometry .
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,
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.
a Right Line C D to the Centre thereof, the
the Circle A F B ; then, because in the Triangles
Interval of a Quadrant ; for, since the Angles
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.