HBM, HKM; therefore HB will be equal to BM, and the tri-, angle DBG equiangular to BHK. For the fame reason, MCL will be bifected by CK, and MC equal to CL.

Now, because BF, FC, are equal to BH, CL; AH, AL, are equal to the fum of the fides AB, BC, AC, and AH equal to half the fum of the fides. And, because the triangles DBG, BHK, are equiangular, GD is to DB as BH is to HK f; and f 4. 6. the rectangle under DG, HK, equal to the rectangle DBH8, 8 16. 6. that is, to BFC; but the triangles ADG, AHK, are equiangular; therefore AD is to DG as AH is to HK, and the rectangle under DG, HK, equal to the rectangle HAD; therefore the rectangle HAD is to the rectangle DBH, or BFC, as the fquare of AD is to the fquare of DG; but AD is the excess h 22. 6. of AH above the base BC, and BF, FC the right lines by which AH exceeds the fides AB, AC; and, if AD is taken rad. then DG is the tangent of half the angle BAC. Wherefore, &c.

IN every plain triangle, the bafe is to the fum of the fides as the difference of the fides is to the fum or difference of the fegments of the bafe, as the greater or lesser fide of the triangle is taken for the bafe.

Let ABC be a triangle; from the vertex A let fall the perpendicular AD; then the bafe is to the fum of the fides as the difference of the fides is to the fum or difference of the fegments CD, BD, according as the base BC is the leffer or greater fide. From the vertex A, let fall the perpendicular AD upon the base BC; with the center A, and distance AC, the greater of the other two fides, defcribe the circle CEF, and produce AB both ways to F and E, and CB to G; then, becaufe the right lines FE, GC, cut one another in B, the rectangle FBE is equal to the rectangle GBC; but CB is to BF asa 35. §. BE is to GB; that is, when the base BC is the greateft, theb 16. 6. base to the fum of the fides, as the difference of the fides to the difference of the fegments of the bafe; but, when BC is the leaft, GB is the fum of the fegments of the bafe. Wherefore,

&c.

PROP, VI.

THE fum and difference of any two quantities being given to find thefe quantities.

U

Let

Let AB, BC, be the two quantities; place them in the fame right line, as AC; and bife&t AC in E; and cut off AD equal to BC; then DB is the difference of the two quantities, and EB half their difference; therefore, if to AE, half their fum, EB, half their difference, be added, the fum is equal to AB, the greater quantity; and if from AE, half the fum, ED, half their difference, be taken, gives AD equal to BC, the leffer quantity. Wherefore, &c,

THE

THE

ELEMENTS

O F

SPHERICAL

TRIGONOMETRY,

T

DEFINITIONS.

I.

HE poles of a fphere are two points in the fuperficies of the sphere that are the extremes of the axis.

II.

The pole of a circle in a sphere, is a point in the fuperficies of the fphere from which all right lines, drawn to the circumference of the circle, are equal to one another.

III.

A great circle in a sphere is that whose plain paffes through the center of the sphere, and whofe center is the fame with that of the fphere, or whose plain bifects the sphere.

IV:

A spherical triangle, is a figure comprehended under the arches of three great circles of a sphere.

V.

A spherical angle is that which is contained under two arches of greater circles in the superficies of the sphere.

GRE

RE AT circles in a sphere mutually bifect each other.

Let the two great circles be ACB, AFB, they will mutually bifect each other; for their common fection AB is the diameter of both circles.

PROP:

a def. 2.

b 8. I.

IC 4. II.

d 19. 11.

PROP. II.

IF from the pole of any circle, to its center, a right line be drawn, it will be perpendicular to the plain of that circle.

Let the circle be AFB, and its pole C; from which draw CD to the center, then CD will be perpendicular to the plain of that circle.

For, in it draw any diameters EF, GH, and join CG, CH, CE, CF; then, in the triangles CDF, CDE, the two fides CD, DE, are equal to the two fides CD, DF, and their bases CF, CE, are equal; therefore the angle CDF is equal to the angle CDE; therefore CD is perpendicular to the plain of the circle AFB. Wherefore, &c.

COR. I. Hence, if this circle be a great circle, the distance upon the fuperficies of the fphere betwixt the pole and great circle is a quadrant, for the plain of it bifects the sphere.

II. Great circles, that pass through the pole of some other circle, make right angles with it; for the right line CD is the common fection of fuch plains 4.

a cor. 2.

Fa great circle is defcribed about the pole of a sphere, and from that pole two right lines be drawn to the circle, the arch of that circle contained by the two right lines is the measure of the angle at the pole.

Let A be the pole of a fphere, and ECF the great circle defcribed about it, and let the right lines AC, AF, be drawn to the great circle; then the arch CF is the measure of the angle at A.

For, let D be the center of the sphere, then the angles ADC, ADF, are right angles "; and the angle CDF is the inclination of the plains ACB, AFB, and equal to the spherical triangle b def. 6. CAF, or CBF b.

II.

C 4. II.

COR. I. If the arches AC, AF, are quadrants, then A is the pole of the circle paffing through the points C, F; for AD is at right angles to the plain FDC.

II. The vertical angles are equal, for each is equal to the inclination of the circles; alfo, the adjacent angles are equal to two right angles.

I

PROP. IV.

F two fpherical triangles have two fides of the one, equal to two fides of the other, and the angle contained by the two fides of the one equal to the correspondent angles of the other, the two triangles will be equal.

For, if the two arches containing the angles are equal, their chords or fubtenfes are likewise equal, and contain equal angles; a 29. 3 therefore their bases are equal, and remaining angles of the one, equal to the remaining angles of the other, each to each; and the right lined triangles equal; but equal right lines cut off equal b 4 8. circumferences; wherefore the spherical triangles are equal to € 28. 3. one another.

COR. I. Hence triangles will be equal and congruous, if two angles of the one be equal to two angles of the other, each to each, and a fide of the one equal to a fide of the other, either the fide that lies betwixt the equal angles, or fubtending one of them ".

d 29. and

24. 3. and 26. I.

II. Equilateral triangles are likewife equiangular. *. III. In ifofceles triangles, the angles at the bafes are equal; e29. and and, if the angles at the bafes are equal, the triangles are celes f.

ifof-24. 3. and

IV. Any two fides of a triangle are greater than the third; for any two of their chords or fubtenfes, are greater than the third

ANY

PROP. V.

NY fide of a fpherical triangle is less than a femicircle.

Let AC, AB, the fides of the triangle ABC, be produced till they meet in D, then the femicircle ACD is greater than the arch AC.

I. I.

f 29 and 24. 3. 5.

and 6, 1.

g 20, I.

THE

HE three fides of a spherical triangle are less than a whole
circle.

For