fine of 52" 44" 03" 45""" is found to be .00025566346186138 Pl. Trig. 32219. And because the cofine of an arch is to its fine, as the

:

radius to the tangent of the fame arch°; therefore the tangent of o 2. Cor. 52" 44" "03" 45" is .co025566347, which is nearly equal to Def. P. T. the fine of it but the arch is greater than the fine, and less than the tangent; therefore the arch is more nearly equal to the fine. Confequently, in fmall arches, the fines are to one another as the arches. Wherefore, as 52" 44" 03" 45"""" to 1', fo is .00025566 34618, the fine of the first, to .00029088820866572, the fine of 1'; and if the fquare of this fine be fubtracted from 1, the fquare-root of the remainder .999999957692025328 is the cofine of minute.

Again, because the triangles ADE, ABG, have the angle at A common, and AED, AGB right angles, they are equiangular P; P 32. I. therefore AD is to DE, as AB to BG: and if AF be an arch of f 4. 6. 1', AB is an arch of 2'; therefore, as the radius to the cofine of I', fo is twice the fine of 1' to the fine of two minutes. The fines of all the other arches may be found by the following.

L

PROPOSITION

b.

ET AB, AC, AD be three arches, fuch that BC is equal to CD; as the radius to the cofine of DC the common difference, fo is twice the fine of AC the middle arch to the fum of the fines of AB, AD the extreme arches.

c

Make the arch DE equal to AB: and let F be the centre, and join FC, FD, AE, BD; and draw a AK, EL perpendicular to FD, and EM parallel to it; therefore MK is equal to EL the fine of DE or AB: and AK is the fine of ADd; the refore AM is the fum of the fines of AB, AD: and becaufe ED is equal to AB, and DC to CB, the whole EC is

equal to CA; and therefore

a 12. I.

b 31. I.

E

C

Τι

C 34. L.

B

d 3. Def.

N

H

P. T.

FC bisects AE, BD at right angles ; and AH, DG are the fines eCor.30.3. of AC, CD; and FG is the cofine of DC: and because the angle

f

FDG is equal to FNA, that is, to MEA; and FGD, EMA f 29. 1. are right angles; the triangles FGD, AME are equiangular ; 8 32. 1. and therefore, as DF to FG, fo is EA to AMh; that is, as the h 4. 6. radius to the cofine of DC, fo is twice AH the fine of AC to the fum of the fines of AB, AD.

244

Pl. Trig.

PLANE TRIGONOMETRY.

Let DC, CB be arches of 1': then, if AC be 2', R is to cos, 1', as twice fin, 2' to the fum of the fines of 3' and 1′: If, therefore, a fourth proportional be found to R, cos. 1' and twice fin. 2', the excefs of this fourth proportional above fin. 1' is the fine of 3'. In the fame manner, if a fourth proportional be found to R, cos. 1' and twice fin. 3', its excefs above the fin. 2′ is the fine of 4'; and so on.

In this manner, the table of fines may be computed: and from it the table of tangents may be calculated; for the cofine of any k 2. Cor. arch is to its fine, as the radius to the tangent of the fame arch *. Def. P. T. As also the table of fecants, because the secant of any arch is a 13. Cor. third proportional to its cofine and the radius 1.

Def. P. T.

To find the Length of the Circumference. Because the arch of 52" 44" 03" 45""" is the 24576 the part of the circumference; and that the fine is lefs than the arch; if the fine of 52" 44" 03" 45" be multiplied by 24576, the product 6.28318524 is lefs than the circumference of the circle, of which the radius is : and because the tangent is greater than the arch; if the tangent of 52" 44" 03" 45" be multiplied by 24576, the product 6.28318544 is greater than the circumference. Confequently the circumference is nearly 6.28318531 when the radius is 1; or 3.14159265 when the diameter is 1. Hence the diameter of a circle is to its circumference, nearly as 1 to 3.14159265; or as 7 to 22; or as 113 to 355.

THE

THE

ELEMENTS

OF

SPHERICAL TRIGONOMETRY.

DEFINITIONS.

A of

I.

GREAT circle of the sphere is the common section of the Sph. Trig. sphere, with a plane passing through its centre.

II.

The pole of a great circle of the sphere is a point on the superficies of the sphere, from which all straight lines drawn to the circumference of the circle are equal.

A spherical angle is that made by two arches of great circles, and is the fame with the inclination of the planes of these circles.

IV.

A fpherical triangle is a figure on the fuperficies of the sphere, contained by three arches of great circles, each of them less than a femicircle.

PROP. I.

IF

F a sphere be cut by a plane paffing through its centre, the common fection is a circle.

For every ftraight line drawn from the centre to the fuperficies of the sphere, is equal to the radius of the femicircle, by which the sphere is described; therefore all straight lines drawn from the centre of the sphere to the common section of its fuper

ficies

Sph. Trig. ficies with the cutting plane, are equal: and therefore the common fection of the plane and the fphere is a circle.

COR. 1. The centre of a great circle is the fame with the centre of the sphere.

COR. 2. All great circles are equal to one another; for their radii are equal to the radius of the circle, by which the sphere is defcribed.

COR. 3. Great circles bifect one another. For having the fame centre, their common fection is a diameter of each of them.

THE

PROP. II.

HE arch of a great circle between the pole and the circumference of another great circle, is the fourth part of the circumference.

Let ABC be a great circle, and D its pole; and let the great circle ADC pass through D, and meet the circumference ABC, in C: the arch DC is the fourth part of the circumference. Let AC be the common fection of

the circles ABC, ADC; therefore

a 3. Cor. 1. ADC is a femicircle

: join DA, S. T. DC and becaufe D is the pole of b 2. Def. the great circle ABC, DA is equal to DC; therefore the arch AD is

S. T.

C

B

c 28. 3. equal to the arch DC; and the arch DC is the half of ADC: but A the arch ADC is half of the circumference; therefore the arch DC is the fourth part of the cir-cumference.

A

PROP. III.

SPHERICAL angle made at the pole of a great circle, is measured by the arch of that great circle, intercepted between the circles, which contain the angle.

Let AB, AC be arches of great circles, which pafs through A the pole of the great circle BC, the arch BC is the measure of the spherical angle BAC.

Let the planes of the great circles cut one another in the ftraight lines AD, DB, DC: and because the great circles AB, a. Def. AC pafs through the centre of the sphere, their common

S. T.

fection

A

b 2. S. T.

fection AD paffes through it. For the fame reafon, BD paffes Sph. Trig. through it; therefore D is the centre of the fphere: and because A is the pole of the great circle BC, AB is the fourth part of the circumference b; therefore ADB is a right angle. For the same reason, ADC is a right angle; therefore the angle BDC is the inclination of the planes of the circles AB, AC £; it is therefore the fame with the spherical angle BAC : and the arch BC is the meafure of the angle BDC; therefore the arch BC is the measure of the spherical angle BAC.

D

B

C

COR. 1. The straight line drawn from the pole A of a great circle BC to the centre of the sphere D, perpendicular to the plane of that great circle.

is

COR. 2. The poles of a great circle are the extremities of the diameter of the fphere, which is perpendicular to the plane of that great circle.

c6.Def.11.

d 3. Def.

S. T. e 1. Def.

ET.

IF

PROP. IV.

F two arches of different great circles be drawn from the fame point, and each of them be the fourth part of the circumference; that point is the pole of the great circle, which paffes through the extremities of the arches.

Let AB, AC be arches of great circles not in the fame plane, and each of them the fourth part of its circumference: The point A, in which they meet, is the pole of the great circle BCE which paffes through their extremities.

D

A

Let the planes of the great circles cut one another in AD, DB, DC; therefore D is the centre of the fphere. In the circumference BC take any point E; and join DE, AE, AC: and because each of the arches AB, AC is the fourth part of the circumference, the angles ADB, ADC are right angles; therefore AD is perpendicular to the plane BDC ; and therefore ADE is a right angle and AD, DE are equal to AD, DC, and ADE to ADC; therefore AE is equal to AC ; that is, any ftraight line drawn from b 4. 1. A to the circumference BC, is equal to AC; therefore A is the pole of BC.

:

E

B

C

COR.

a 4. 1%.

C 2. Def.

S. T.