2. From this exeefs let the first product be continually fubtracted; that is, firft, from the excess itself; then from the remainder; then from the last remainder, and fo on 44 times.

3°. To the leffer extreme add the forementioned excefs; and, to the fum, add the first remainder; to this fum add the next remainder, and fo on continually then the several fums thus arifing will refpectively exhibit the fines of all the intermediate arches, to every single minute, exclusive of the laft; which, if the work be right, will agree with the greater extreme itself, and therefore will be of ufe in proving the operation.

But to illuftrate the matter more clearly, let it be propofed to find the fines of all the intermediate arches between 3° 00′ and 3° 45′ to every fingle minute, thofe of the extremes being given, from the foregoing method, equal to 05233595 and ,06540312 refpectively. Here, the fum of the fines of the extremes being multiplied by,oooco00423, the first product will be ,00000000498, &c. or,0000000050, nearly (which is fufficiently exact for the prefent purpose); and this, again, multiplied by 22, gives 00000011 for a 2d product; which added to ,0002903815, part of the difference of the two given extremes, will be,co02904915, the excefs of the fine of 3° or above that of 3° 00'. From whence, by proceeding according to the 2d and 3d rules, the fines of all the other intermediate arches are had, by addition and fubtraction only. See the operation.

,000290 4915 excess

,0000000050

,05233595
,0002904915

fine 3° 0

4865 1 rem. 0526264415 fine 3° 1′

4815 2 rem. ,0529169280 fine 3° 2′

4765 3 rem. 0532074095 fine 3° 3′

4715 4th rem, 0534978860 fine 3° 4′

4665 5th rem. 0537883575 fine 3°5′

4615 6th rem. 0540788240 fine 3°6′

4565 7th rem. 0543692855 fine 3° 7′

4515 8th rem. 0546597420 fine 3° 8'

4465 9th rem. 0549501935. fine 3° 9′

50

2904465

4415 10th rem. 0552406400 fine 3o 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° 00'; thofe of the two extremes being first found, by the preceding

method.

method. In this cafe, the two extremes, being 85940641 and ,86602540, their fum will be 172543, &c. and their difference =,00661899; whereof the former, multiplied by ,0000000423 (fee the rule) gives ,00000007298, &c. or ,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 ,0001486947; from whence the operation will be as follows:

I

0001486947 excess ,0000000730 1 prod.

3

86217 1 rem.

730

8595551047 fine 59° 16′ 1486217

85487 2 rem. 8597037264 fine 59° 17′

84757 3 rem. ,8598522751 fine 59° 18′

84027 4th rem. 8600007508 fine 59° 19′

83297 5th rem. 8601491535 fine 59° 20′

82567 6th rem. 86029748 32 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 best found from thofe below, as has been fhewn elsewhere.

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

Spherical Trigonometry.

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 fphere, 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 surface of the sphere 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

E

D

B

fo formed is faid to be the complement of the

former; and vice versa.