Spherical Trigonometry.


I. Great-Circle of a Sphere is a Sečtion of the Sphere by a Plane passing thro’ the Center thereof. 2: The Axis of a Great-Circle is a Right-line passing thro’ the Center, perpendicular to the Plane of the Circle: And the two Points, where the Axis intersects the Surface of the Sphere, are call'd the Poles of the Circle. 3. A spherical Angle is the Inclination of two Great-Circles. 4. A spherical Triangle is a Part of the Surface of the Sphere included by the Arches of three

Great-Circles: Which Arches are called the Sides

of the Triangle. 5. If thro’ the Poles A and F of two GreatCircles D F and DA, standing atRight-angles, two other Great-Circles . ACE and FCB be conceived to pass,and thereby form two spherical Triangles A B C. and FCE, the latter of the Triangles so formed is said to be the Complement of the former; and vice versa.

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B F. its Pole; and that the

.4 ** Measure of a spherical Angle CAD “ is an Arch of a Great-Circle intercepted by the two Circles ACB, ADB forming that Angle, and whose Pole is the angular Point A. For let the Diameter AB be the Interse&tion of the Great Circles ADB and ACB (see Corol. 1.) and let the Plane, or Great-Circle, DEC be conceived perpendicular to that Diameter, interseóting the Surface of the Sphere in the Arch CD; then it is manifest that AD = BD = 90°, and AC = BC = 90° (Cor. 1.) and that CD is the Measure of the Angle DEC (or CAD) the Inclination of the two proposed Circles.

* Note, Altho' a spherical Angle is, properly, the Inclination of two Great-Circles, yet it is commonly expressed by the Inclination of their Peripheries at the Point where they inters & each other.

4. Hence

4. Hence it is also manifest, that the Angles B and E, of the ComplementalTriangles ABC and FCE, are both Right-angles; and that CE is the Complement | of AC, CF of BC, B D . (or the Angle F) of AB, B and EF of ED (or the Angle A).

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Let ADL and AEL be two Great-Circles of the Sphere intersecting each other in the Diameter AL, C 4 making

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making an Angle DOE, measured by the Arch
ED; the Plane DOE being supposed perpendicular
to the Diameter AL, at the Center O.
Let A B be the Base of the proposed Triangle,
BC the Perpendicular, AC the W. and
BAC (or DAE = DE = DOE) the Angle at the
Base: Moreover, let CG be the Sine of the Hypo-
thenuse, AK its Tangent, AI the Tangent of the
Base, CH the Sme of the Perpendicular, and EF
the Sine of the Angle at the Base; and let I, Kand
G, H be joined.
Because CH is perpendicular to the Plane of the
Base (or Paper), it is evident, that the Plane GHC
will be perpendicular to the Plane of the Base, and
likewise perpendicular to the Diameter AL, because
GC, being the Sine of AC, is perpendicular to AL.
Moreover, since both the Planes OIK and AIK are

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Hence it follows, that the Sines of the Angles of any oblique spherical Triangle A DC are to one another, directly, as the Sines of the opposite


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It follows, moreover, that, in right-angled spherical Triangles ABC, DBC, having one Leg BC common, the Tangents of the Hypothenuses are to each other, inversely, as the Co-sines of the

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