,000290 4915 excess ,0000000050


sine 3° 0' 20002904915

4865 1'' rem. ,0526264415 line 3° 1' 50


4815 2° rem. ,0529169280 sine 3° 2' 50

2904815 4765 34 rem. ,0532074095 sine 3° 3' 50


4715 4h rem. ,0534978860 sine 3° 4' 50


4665 5th rem. ,0537883575 fine 3° 5 50

2904665 4615 6th rem. ,0540788240 fine 3° 6' 50

2904615 4565 7th rem. ,0543692855 sine 3° 7'

. 50


4515 geh rem. ,0546597420 sine 3° 8' 50


4465 9th rem. ,0549501935 fine 3° 9' 50


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


Again, as a second example, let it be required to find the lines of all the arches, to every minute, between 59° 15' and 60°00'; those of the two extremes being first found, by the preceding


method. In this case, the two extremes, being ,85940641 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. ,0000000730, nearly, for the first 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 90001486947; from whence the operation will be as follows:


9000 14 86947 excess ,85940641

sine 59° 15' ,0000003730 1“ prod. 0001486947

86217 1'' rem. ,8595551047 sine 59° 16' 730

1486217 85487 2d rem. ,8597037264 sine 59° 17' 730


84757 30 rem. ,8598522751 fine 59°18' 730


84027 4th rem. ,8600007508 sine 59° 19' 730


83297 5th rem. ,8601491535 sine 59° 20' 730


82567 6th rem. ,8602974832 sine 59° 21'


,8604457399 fine 59°22' &c.

&c. After the same manner the fines of all the intermediate arches between any other two proposed extremes may be derived, even up to 90 degrees ;

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

The reasons upon which the foregoing opera. tions 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,

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[ 23

Spherical Trigonometry.



Great-circle of a sphere is a section of the

sphere by a plane passing thro' 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

F 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 there. by form two spherical


D triangles ABC and FCE,

B the latter of the triangles so formed is said to be the complement of the former ; and vice versa.


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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 intersect 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 pass through, or coincide with the axis, which is perpendicular to it.


It follows moreover, that the periphery of a great-circle is every where 90 degrees distant from

its pole ; and that the B


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 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, Alihough a spherical angle is, properly, the inclination of two great-circles, yet it is commonly exprefed by the inclination of their jeripheries at the point where they interfect each Qiker.

4. Hence

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