'circles perpendicular to each. [The student will readily perceive how this is done.] The intersection of these perpendiculars, o, will be the pole of the small circle required (?). Then from o, as a pole, with an arc oB draw the circumference of a small circle: it will pass through A, B, and C (?), and hence is the circumference required.

OF TANGENT PLANES.

564. A Tangent Plane to a curved surface at a given point is the plane of two lines respectively tangent to two plane sections through the point.

565. Theorem.-A tangent plane to a sphere is perpendicular to the radius at the point of tangency.

S

༤

M

N

U

DEM.-Let P be any point in the surface of a sphere; pass two great circles, as PaA, etc., and PmAR, through P, and draw ST tangent to the arc mP, and UV tangent to the arc aP; then is the plane SVTU a tangent plane at P, and perpendicular to the radius OP. For, a tangent (as ST) to the arc mP is perpendicular to the radius of the circle, i. e., to OP, and also a tangent (as VU) to the arc aP is perpendicular to the radius of this circle, i. e., to OP. Hence, OP is perpendicular to two lines of the plane SVTU, and consequently to the plane of these lines (?). Q. E. D.

566. COR. 1.--Every point in a tangent plane to a sphere, except the point of tangency, is without the sphere.

For, OP, the perpendicular, is shorter than any line which can be drawn from O to any other point in the plane (?), hence any other point in the plane than P lies farther from the centre of the sphere than the length of the radius, and is, therefore, without the sphere.

567. COR. 2.-A tangent through P to ANY circle of the sphere passing through this point, lies in the tangent plane.

DEM. Thus MN, tangent to the small circle PnRb through P, lies in the tangent plane. For, conceive the plane of the small circle extended till it intersects the tangent plane. This intersection is tangent to the small circle, since it touches it at one point, but cannot cut it; otherwise the tangent plane would have another point than P common with the surface of the sphere. But there can be only one tangent to a circle at a given point. Hence this intersection is MN, which is consequently in the tangent plane.

OF SPHERICAL TRIANGLES.

568. A Spherical Triangle is a portion of the surface of a sphere bounded by three arcs of great circles. In the present treatise these arcs will be considered as each less than a semicircum fer

ence.

The terms scalene, isosceles, equilateral, right angled, and oblique angled, are applied to spherical triangles in the same manner as to plane triangles.

PROPOSITION XI.

569. Theorem.-The sum of any two sides of a spherical triangle is greater than the third side, and their difference is less than the third side.

DEM.-Let ABC be any spherical triangle; then is BCBA + AC, and BC - AC <BA; and the same is true of the sides in any order. For, join the vertices A, B, and C, with the centre of the sphere, by drawing AO, BO, and CO. There is thus formed a triedral O-ABC, whose facial angles are measured by the sides of the triangle (208). Now, angle BOC <BOA + AOC (434), whence BCBA + AC: and subtracting AC from both members, we have BC AC < BA.

-

B

FIG. 328.

PROPOSITION XII.

570. Theorem.-The sum of the sides of a spherical triangle may be anything between 0 and a circumference.

DEM.-The sides of a spherical triangle are measures of the facial angles of a triedral whose vertex is at the centre of the sphere. Hence their sum may be anything between 0 and the measure of 4 right angles, as these are the limits of the sum of the facial angles of a triedral (436).

571. SCH.-As the sides of a spherical triangle are arcs, they can be measured in degrees. Hence, we speak of the side of a spherical triangle as 30°, 57, 115° 10′, etc. In accordance with this, we say that the limit of the sum of the sides of a spherical triangle is 360°.

PROPOSITION XIII.

572. Theorem.-The sum of the angles of a spherical triangle may be anything between two and six right angles.

DEM.-The sum of the angles of a spherical triangle is the same as the sum of the measures of the diedrals of a triedral having its vertex at the centre of the sphere, as in (569). Now the limits of the sum of the measures of these diedrals are 2 and 6 right angles (439). Hence the sum of the angles of any spherical triangle may be anything between 2 and 6 right angles. Q. E. D.

573. SCH.-It will be observed, that the sum of the angles of a spherical triangle is not constant, as is the sum of the angles of a plane triangle. Thus, the sum of the angles of a spherical triangle may be 200°, 290°, 350°, 500°, anything between 180° and 540°.

574. DEF.-Spherical Excess is the amount by which the sum of the angles of a spherical triangle exceeds the sum of the angles of a plane triangle; i. e., it is the sum of the spherical angles —180°, or π.

ILL. It is not difficult to observe the occasion of this excess in the case of the cquilateral spherical triangle. Thus,let ABC be such a triangle. Conceive the plane

B

FIG. 329.

triangle formed by the chords AB, AC, and CB. The sum of the angles of this plane triangle is 180°. But each angle of the spherical triangle is larger than the corresponding angle of the plane triangle. Thus, the spherical angle BAC is the same as the plane angle C'AB', included be tween the tangents C'A and B'A, which are perpendicular to the edge of the diedral C-AO-B, and include its measuring angle. Now, CA and BA

being different lines from C'A and B'A are oblique to the edge AO, and include an angle less than its measure, and consequently less than CAB. For a like reason the plane angle ACB < the spherical angle ACB, and plane angle ABC spherical angle ABC. Moreover, it is easy to see that the inequality between any plane angle and the corresponding spherical angle increases as the chords BA and CA deviate more from the tangents. Whence we see why the sum of the angles of the spherical triangle is not a fixed quantity.

575. COR.—A spherical triangle may have one, two, or even three right angles; and, in fact, it may have one, two, or three obtuse angles; since, in the latter case, the sum of the angles will not necessarily be greater than 540°.

576. DEF.-A Trirectangular Spherical Triangle is a spherical triangle which has three right angles.

PROPOSITION XIV.

577. Theorem.-The trirectangular triangle is one-eighth of the surface of a sphere.

DEM.-Pass three planes through the centre of a sphere, respectively perpendicular to each other. They will divide the surface into 8 trirectangular triangles, any one of which may be applied to any other. Thus, let ABA'B', ACA'C', and CBC'B' be the great circles formed by the three planes, mutually perpendicular to each other. The planes being perpendicular to each other the diedrals, as A-CO-B, C-BO-A, C-AO-B, etc., are right, and hence the angles of the 8 triangles formed are all right. Also, as AOB is a right angle, AB is a quadrant; as BOC is a right angle, CB is a quadrant, etc. Hence, each side of every triangle is a quadrant. Now any one triangle may be applied to any other. [Let the student make the application.] Hence the trirectangular triangle is one-eighth of the surface of a sphere. Q. E. D.

B

A'

FIG. 330.

578. COR.-The trirectangular triangle is equilateral and its sides are quadrants.

PROPOSITION XV.

579. Theorem.-In an isosceles spherical triangle the angles opposite the equal sides are equal; and, conversely, If two angles of a spherical triangle are equal, the triangle is isosceles.

A

DEM.-Let ABC be an isosceles spherical triangle in which AB = AC; ther angle ABC ACB. For, draw the radii AO, CO, ari BO, forming the edges of the triedral O-ABC. Now, since AB AC, the facial angles AOC and AOB are equal, and the triedral is isosceles. Hence the dicdrals A-OB-C and A-OC-B are equal (442), and consequently the spherical angles ABC and ACB are equal (558). Again, if angle ABC = angle_ACB, side AC side AB. For in the triedral O-ABC, the diedrals A-OB-C and A -OC-B are equal, whence the facial angles AOB and AOC are equal (443), and conse

FIG. 331.

quently the sides AB and AC which measure these angles.

580. COR.-An equilateral spherical triangle is also equiangular; and, conversely, If the angles of a spherical triangle are equal the triangle is equilateral.

PROPOSITION XVI.

581. Theorem.-On the same or on equal spheres two isosceles triangles having two sides and the included angle of the one equal to two sides and the included angle of the other, each to each, can be superimposed, and are consequently equal.

AB

=

AC'; and let AB'C' ; then

DEM.-In the triangles ABC and AB'C', let AB = AC, AB' : AB', BC = B'C', and angle ABC can the triangle AB'C' be superimposed upon ABC. For, since the triangles are isosceles, we have angle ABC ACB, AB'C' = AC'B', and, as by hypothesis ABC = AB'C', these four angles are equal each to each. For a like reason AB AC AB' AC'. Now, applying AC' to its equal AB, the extremity A at A and C' at B, with the angle B' on the same side of AB as C, the convexities of the arcs AC' and AB being the same, and in the same direction, the arcs will coincide. Then, as angle AC'B' ABC, C'B' will take the direction BC, and since these arcs are equal by hypothesis, B' will fall at C. Hence B'A will fall in CA, as only one arc of a great circle can pass between C and A, and the triangle AB'C' is superimposed upon ABC; wherefore they are equal. [Let the student give the application when other parts are assumed equal.]

FIG. 332.

582. Symmetrical Spherical Triangles are such as have the parts (sides and angles) of the one respectively equal to the parts of the other, but arranged in a different order, so that the triangles are not capable of superposition.