Sidebilder
PDF
ePub

Spherical Trigonometry.

I.

"A

thereof.

DEFINITIONS.

Great-Circle of a Sphere is a Section of the
Sphere by a Plane paffing thro' the Center

2: The Axis of a Great-Circle is a Right-line paffing thro' the Center, perpendicular to the Plane of the Circle: And the two Points, where the Axis interfects the Surface of the Sphere, are call'd the Poles of the Circle.

3. A fpherical Angle is the Inclination of two Great-Circles.

4. A fpherical 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 DF and DA, standing at Right-angles, two other Great-Circles ACE and FCB be conceived to pass,and thereby form two fpherical

Triangles A B C and
FCE, the latter of the

F

E

D

B

Triangles fo formed is faid to be the Complement

of the former; and vice verfa.

[blocks in formation]

COROLLARIES.

1. It is manifeft (from Def. 1.) that the Section of two Great-Circles (as it paffes thro' the Center) will be a Diameter of the Sphere; and confequently, that their Peripheries will always interfect each other in two Points at the Distance of a Semicircle, or 180 Degrees.

2. It also appears (from Def. 2.) that all GreatCircles, paffing thro' the Pole of a given Circle, cut that Circle at Right-angles; because they pass through, or coincide with the Axis, which is perpendicular to it.

[blocks in formation]

3. It follows moreover, that the Periphery of a Great-Circle is every where 90 Degrees diftant from its Pole; and that the 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 Interfection of the Great Circles ADB and ACB (see Corol. 1.) and let the Plane, or Great-Circle, DEC be conceived perpendicular to that Diameter, interfecting the Surface of the Sphere in the Arch CD; then it is manifeft 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 propofed Circles.

*Note, Altho' a Spherical Angle is, properly, the Inclination of twb Great-Circles, yet it is commonly expreffed by the Inclination of their Peripheries at the Point where they interfect each ather.

4. Hence

[merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

In any right-angled Spherical Triangle it will be, as Radius is to the Sine of the Angle at the Bafe, fo is the Sine of the Hypothenufe to the Sine of the Perpendicular; and as Radius to the Co-fine of the Angle at the Bafe, fo is the Tangent of the Hypothenufe to the Tangent of the Bafe.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

Let ADL and AEL be two Great-Circles of the Sphere interfecting each other in the Diameter AL,

C 4

making

making an Angle DOE, meafúred by the Arch ED; the Plane DOE being fuppofed perpendicular to the Diameter AL, at the Center O.

Let AB be the Bafe of the propofed Triangle, BC the Perpendicular, AC the Hypothenufe, and BAC (or DAE DE DOE) the Angle at the = Bafe: Moreover, let CG be the Sine of the Hypothenufe, AK its Tangent, AI the Tangent of the Base, CH the Sine of the Perpendicular, and EF the Sine of the Angle at the Bafe; and let I, K and G, H be joined.

Because CH is perpendicular to the Plane of the Bafe (or Paper), it is evident, that the Plane GHC will be perpendicular to the Plane of the Bafe, and likewife perpendicular to the Diameter AL, because GC, being the Sine of AC, is perpendicular to AL. Moreover, fince both the Planes OIK and AIK are perpendicular to the Plane of the Bafe (or Paper), their Interfection IK will alfo be perpendicular to it, and confequently the Angle AIK a Right-angle. Therefore, feeing the Angles OFE, GHC and AIK are all Right-angles, and that the Planes of the three Triangles OFE, GHC and AIK are all perpendicular to the Diameter AL, we shall, by fimilar Triangles,

have

that is,

OE: EF:: GC: CH2
OE OF: AK: AI S

:

Radius Sine of EOF (or BAC) :: Sine
of AC: Sine of BC.

Radius: Co-fine of EOF (or BAC)::
Tang. AC: Tang. AB. 2, E. D.

COROLLARY I.

Hence it follows, that the Sines of the Angles of any oblique fpherical Triangle ADC are to one another, directly, as the Sines of the oppofite Sides.

For

B

D

For let BC be perpendicular to AD;

then Radius: Sin. A :: Sin. AC: Sin. BC

fince { Radius: Sin. D:: Sin. DC: Sin. BC by the } former Part of the Theorem; we shall have, Sin. Ax Sin. AC(= Radius x Sin. BC) = Sin. D x Sin. DC (by 3. 4.) and confequently Sin. A: Sin. D :: Sin. DC: Sin. AC; or Sin. A': Sin. DC :: Sin. D: Sin. AC.

COROLLARY 2.

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

For Rad.: Co-fin. ACB:: Tan. AC: Tan. BC fince Rad.: Co-fin. DCB :: Tan. DC: Tan. BC by the latter Part of the Theorem; we fhall (by arguing as above) have Co-fine ACB: Co-fine DCB:: Tang. DC: Tang. AC.

THEOREM II.

In any right-angled Spherical Triangle (ABC) it will be, as Radius is to the Co-fine of one Leg, fo is the Co-fine of the other Leg to the Co-fine of the Hypothenufe.

Demon

« ForrigeFortsett »