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,000290 4915 excess

,0000000050

,05233595
,0002904915

fine 3° 0′

4865 ft rem. ,0526264415 fine 3° 1′

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4815 2a rem. ,0529169280 fine 3° 2′

[blocks in formation]

4765 3a rem. 0532074095 fine 3° 3′

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4715 4th rem. 0534978860 fine 3° 4′

[blocks in formation]

4665 5th rem. 0537883575 fine 3° 5′

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4615 6th rem. ,0540788240 fine 3o 6′

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4565 7th rem. 0543692855 fine 3° 7′

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4515 8th rem. 0546597420 fine 3° 8' 50

2904515

4465 9th rem. 0549501935 fine 3° 9′

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4415 10th rem. ,0552406400 fine 3°10′ &c.

&c.

Again, as a fecond example, let it be required to find the fines of all the arches, to every minute, between 59° 15′ and 60° oo'; those of the two extremes being firft found, by the preceding

method.

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 first product (which is exact enough for our purpofe); therefore the 2d duct, or ,0000000730 x 22, will be ,0000016060; which, added to of the difference, gives ,0001486947; from whence the operation will be as follows:

73

,0001486947 excess 0000000730 1st prod.

pro

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86217 1' rem.

730

,8595551047 fine 59°16′ 1486217

59°17′

85487 2 rem. ,8597037264 fine 59° 17′

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84757 3 rem. ,8598522751 fine 59°18′

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84027 4th rem. 8600007508 fine 59°19′

730

1484027

83297 5th rem. ,8601491535 fine 59° 20′

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82567 6th rem. ,8602974832 fine 59°21′

&c.

1482567

,8604457399 fine 59° 22' 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;

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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 treatife, 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 those already extant, will not find one quarter of the trouble, this way, as he unavoidably muft according to the common methods.

Spherical

Spherical Trigonometry.

I.

A

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 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 pafs, and thereby form two spherical triangles ABC and FCE, the latter of the triangles

F

C

D

B

fo formed is faid to be the complement of the

former; and vice versa.

COROL

C 4

COROLLARIES.

1. It is manifeft (from Def. 1.) that the section 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.

B

E

D

A

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 furface 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 proposed 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 ather.

4. Hence

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