method. : In this case, the two extremes, being
385940641 and ,86602540, their sum 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 purpose); therefore the 2d pro-
duct, or ,0000000730 X 22, will be ,0000016060;
which, added to it's of the difference, gives
,0001486947; from whence the operation will
be as follows:
,0001486947 excess ,85940641 fine 59° 15'
g0000000730 14 prod. 0001486947

$6217 Ift rem. ,8595551047 sine 59° 16


85487 24 rem. ,8597037264 sine 59° 17'


84757 34 rem. ,8598522751 fine 59° 18':

84027 4th rem. ,8600007508 fine 59° 19'

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


82567 6th rem. ,86029748 32 sine 59° 21'



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,8604457399 fine 59° 22"


&c. After the same manner the lines of all the intermediate arches between any other two proposed extremes may be derived, even up to go degrees ; C 3


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

The reasons upon which the foregoing operations are founded, depend upon principles too foreign from the main design of this treatise, to be explained here, (even would room permit); however, as to the correctness 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 must according to the common methods.

Spherical Trigonometry.



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 passing through the center, perpendicular to the plane of the circle: and the two points, where the axis interfects the surface of the sphere, are called 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 DF and DA, ftanding at right-angles, two other great-circles ACE and FCB be conceived to pass, and thereby form two spherical triangles ABC 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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COROLLARIES. 1. It is manifest (from Def. 1.) that the section of two great-circles (as it passes through the center) will be a diameter of the sphere, and consequently, 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, passing 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.


3. It follows more1.

over, that the periphery of a great-circle is every where go

degrees distant from E

its pole; and that the А

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 intersection of the great-circles ADB and ACB (see Corol. 1.) and let the plane, or great-circle, DEC be conceived perpendicular to that diameter, intersecting the furface of the sphere in the arch CD; then it is manifest that AD-BD=90°, and AC=BC=90° (Corol. 1.) and that CD is the meafure of the angle DEC (or CAD) the inclination of the two proposed circles.



Note, Although a spherical angle is, properly, the inclination of too great circles, get it is commonly expressed by the inclination of their peripheries at the point where they interfe& cach otber.

A. Hence


4: Hence it is also manifest, that the angles B and E, of the complemental triangles ABC and FCE, are both right angles; and that CE is the complement

A of AC, CF of BC, BD for the angle F) of AB and EF of ED (or the angle A).


THEOREM I. In any right-angled spherical triangle it will be, as radius is to the fine of the angle at the base, so is the fine of the bypothenuse to the fine of the perpendicular; and as radius to the co-fine of the angle at the base, so 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 intersecting each other in the diameter AL,


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