ELEMENTS

SPHERICAL TRIGONOMETRY

A

DEFINITIONS..

I.

Great circle of the sphere is any circle on the fuperficies of the fphere of which the plane paffes through the centre of the fphere, and of which the centre therefore is the fame with the centre of the sphere.

II.

The pole of a great circle of the sphere is a point in the fu perficies of the sphere, from which all ftraight lines drawn to the circumference of the circle are equal.

III.

A fpherical angle is that which on the fuperficies of a sphere is contained by two arches of great circles, and is the fame with the inclination of the planes of these great circles.

IV.

A fpherical triangle is a figure upon the fuperficies of a sphere comprehended by three arches of three great circles, each of which is lefs than a femicircle.

PROP.

PRO P... I.

NY two great circles of a fphere bifect one another.

For, as they have the fame centre, their common fection is a diameter of both, and therefore bifects both.

TH

PROP. II.

HE arch of a great circle between the pole and the circumference of another great circle is a quadrant.

Let ABC be a great circle, and D its pole; if a great circle CD país through D, and meet ABC in C, the arch DC is a quadrant.

Let the great circle CD meet ABC again in A, and let AC be the common fection of the great circles, which will pafs through E the centre of the fphere: Join DE, DA, DC; By def. 2. DA, DC are equal, and AE, EC are alfo equal, and DE is common; therefore (8. 1.) the angles DEA, DEC are equal; wherefore the arches

DA, DC are equal, and confequently each of them is a quadrant. Q. E. D.

COR. The circle ABC has two poles, one on each fide of its plane, which are the extremities of a diameter of the fphere perpendicular to the plane ABC; and no other point but these two can be a pole of the circle ABC.

PROP.

IF

PRO P. III.

F the pole of a great circle be the fame with the interfection of other two great circles; the arch of the first mentioned circle intercepted between the other two, is the meafure of the spherical angle which the fame two circles make with one another.

Let the great circles BA, CA on the superficies of a sphere, of which the centre is D, interfect one another in A, and let BC be an arch of another great circle, of which the pole is A; BC is the measure of the spherical angle BAC.

Join AD, DB, DC; fince A is the pole of BC, AB, AC are quadrants, and the angles ADB,... ADC are right angles; therefore (4. def. 7.) the angle CDB is the inclination of the planes of the circles AB, AC, and is (def. 3. Sp.. T.) equal to the spherical angle BAC. Q.E. D.

COR. 1. If through the point A, two quadrants AB, AC, be drawn, the point A will be the pole of B the great circle BC, paffing through

their extremities B, C.

A

E

C

Join AC, and draw AE, a ftraight line to any other point E in BC; join DE: Since AC, AB are quadrants, the angles ADB, ADC are right angles, and AD will be perpendicu lar to the plane of BC: Therefore the angle ADE is a right angle, and AD, DC are equal to AD, DE each to each; therefore AE, AC are equal, and A is the pole of BC, by

def. 2.

PROP, IV.

F there be two great circles of a fphere, of which the firft paffes through the poles of the fecond, the fecond alfo paffes through the poles of the first.

Let ACBD and AEBF be two circles, the one of which ACBD paffes through C and D, the poles of the other AEBF, the circle AEBF paffes through the poles of the circle ACBD.

Let G be the centre of the fphere; join CG, GD, which will be in the same straight line: Draw AGB the diameter, which is the common fee

tion of the circles ACBD, AEBF; and in the plane of the circle AEBF draw from the point G the ftraight line EGF perpendicular to AB.

Then,becanfe Cand D are the poles of the circle AEBF, the arch intercepted between either of them and any point in the circumference AEBF, is a quadrant (Prop. 3.), and therefore

the straight line CD is perpendicular to the plane of the circle AEBF; wherefore alfo the plane of the circle ACBD, which paffes through CD, is perpendicular (17.7.) to the plane AEBF. And fince GE in the plane AEBF, is perpendicular to AB, the common fection of the two planes, it is perpendicular to the plane (2. def. 7.) ACBD. The arch of a great circle, therefore, intercepted between the point E and any point of the circumference ACBD is a quadrant, and therefore, (Prop. 3.) the point E is the pole of the circle ACBD. For the fame reafon, F is the other pole of the circle ACBD; and E and F are in the circumference of the circle AEBF. Q. E. D.

Let ABC be a spherical triangle, having the fide AB equal to the fide AC; the spherical angles ABC and ACB are equal. Let D be the centre of the fphere; join DB, DC, DA, and from A on the straight lines DB, DC, draw the perpendiculars AE, AF; and from the points E and F draw in the plane DBC the ftraight lines EG, FG perpendicular to DB and DC, meeting one another in G: Join AG.

A

Because DE is at right angles to each of the ftraight lines AE, EG, it is at right angles to the plane AEG, which paffes, through AE, EG, (4. 7.); and therefore, every plane that paffes through DE is at right angles to the plane AEG (17.7.); wherefore, the plane DBC is at right angles to the plane AEG. For the fame reafon, the plane DBC is at right angles to the plane AFG, D and therefore AG, the common fection of the planes

G

E B

AFG, AEG is at right angles (18.7.) to the plane DBC, and the angles AGE, AGF are confequently right angles.

But, fince the arch AB is equal to the arch AC, the angle ADB is equal to the angle ADC. Therefore the triangles ADE, ADF, have the angles EDA, FDA equal, as also the angles AED, AFD, which are right angles; and they have the fide AD common, therefore the other fides are equal, viz. AE to AF, (26. 1.) and DE to DF. Again, because the angles AGE, AGF are right angles, the fquares on AG and GE are equal to the fquare of AE; and the fquares of AG and GF to the fquare of AF. But the fquares of AE and AF are equal, therefore the fquares of AG and GE are equal to the fquares of AG and GF, and taking away the common

fquare