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method. In this cafe, the two extremes, being 85940641 and ,86602540, their fum will be 1,72543, &c. and their difference =,00661899; whereof the former, multiplied by ,0000000423* (See the rule) gives ,00000007298, &c. or ,0000000730, nearly, for the firft product (which is exact enough for our purpose); therefore the 2d product, or ,0000000730 X 22, will be ,0000016060; which, added to of the difference, gives ,0001486947; from whence the operation will be as follows:

1 75

,0001486947 excefs 0000000730 1 prod.

,85940641 fine 59° 15′ 0001486947

$6217 1 rem.

730

8595551047 fine 59° 16′ 1486217

85487 2 rem. 8597037264 fine 59° 17′

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84757 3 rem. 8598522751 fine 59° 18′.

730

1484757

84027 4th rem. 8600007508 fine 59° 19′

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83297 5th rem. 8601491535 fine 59° 20′

730

1483297

82567 6th rem. 8602974832 fine 59° 21′

&c.

1482567

,8604457399 fine 59° 22′

&c.

After the fame manner the fines of all the intermediate arches between any other two propofed extremes may be derived, even up to 90 degrees;

C 3

but

but those of above 60° are beft found from those below, as has been fhewn elsewhere.

The reafons upon which the foregoing operations are founded, depend upon principles too foreign from the main defign of this treatise, to be explained here, (even would room permit); however, as to the correctnefs and utility of the method itself, I will venture to affirm, that, whoever has the inclination, either to calculate new tables, or to examine thofe already extant, will not find one quarter of the trouble, this way, as he unavoidably muft according to the common methods.

Spherical Trigonometry.

1.

A

DEFINITION S.

Great-circle of a sphere is a section of the sphere by a plane paffing rhro' the center. 2. The axis of a great-circle is a right-line paffing through the center, perpendicular to the plane of the circle: and the two points, where the axis interfects the furface of the sphere, are called 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 furface of the fphere included by the arches of three great-circles; which arches are called the fides of the triangle.

5. If thro' the poles A and F of two greatcircles DF and DA, ftanding at right-angles, two other great-circles ACE and FCB be conceived to pass, and thereby form two fpherical triangles ABC and FCE, the latter of the triangles

F

E

C

B

fo formed is faid to be the complement of the

former; and vice verfa.

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COROLLARIES.

1. It is manifeft (from Def. 1.) that the fection of two great-circles (as it paffes through 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 femicircle, or 180 degrees.

2. It also appears (from Def. 2.) that all greatcircles, paffing through the pole of a given circle, cut that circle at right-angles; because they pafs through, or coincide with the axis, which is perpendicular to it.

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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 whofe pole is the angular point A. For let the diameter AB be the interfection of the great-circles ADB and ACB (fee 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° (Corol. 1.) and that CD is the measure of the angle DEC (or CAD) the inclination of the two propofed circles.

* Note, Although a spherical angle is, properly, the inclination of two great circles, yet it is commonly expreffed by the inclination of their peripheries at the point where they interfect each

other.

4. Hence

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In any right-angled spherical triangle it will be, as radius is to the fine of the angle at the bafe, fo is the fine of the bypothenufe to the fine of the perpendicular; and as radius to the co-fine of the angle at the bafe, fo is the tangent of the hypothenuse to the tangent of the base.

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Let ADL and AEL be two great-circles of the sphere interfecting each other in the diameter AL,

making

K

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