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PROBLEM I.

638. To inscribe a sphere in a given tetrahedron.

A

B

CONSTRUCTION. Through any edge and any point from which perpendiculars to its two faces are equal, pass a plane.

Do the same with two other edges in the same face as the first, and let the three planes so determined intersect at O. O shall be the center of the required sphere.

(602. The locus of all points from which the perpendiculars on the same sides of two planes are equal, is a plane determined by one such point and the intersection line of the given planes.)

639. COROLLARY. In the same way, four spheres may be escribed, each touching a face of the tetrahedron externally, and the other three faces produced.

640. The solid bounded by a sphere is called a Globe.

641. A globe-segment is a portion of a globe included between two parallel planes.

The sections of the globe made by the parallel planes are the bases of the segment.

Any sect perpendicular to, and terminated by, the bases, is the altitude of the segment.

642. A figure with two points fixed can still be moved by revolving it about the line determined by the two points.

This revolution can be performed in either of two senses, and continued until the figure returns to its original position. The fixed line is called the Axis of Revolution. If the axis of revolution is any line passing through the center, a sphere slides upon its trace. This is because every section of a sphere

by a plane is a circle.

Any figure has central symmetry if it has a center which bisects all sects through it terminated by the surface. The sphere has central symmetry, and coincides with its trace throughout any motion during which the center remains fixed. Thus any figure drawn on a sphere may be moved about on the sphere without deformation. But, unlike planes, all spheres are not congruent. Only those with equal radii will coincide.

In general, a figure drawn upon one sphere will not fit upon another. So we cannot apply the test of superposition, except on the same sphere or spheres whose radii are equal. Again, if we wish the angles of a figure on a sphere to remain the same while the sides increase, we must magnify the whole sphere: on the same sphere similar figures cannot exist.

643. Two points are symmetrical with respect to a plane when this plane bisects at right angles the sect joining them. Two figures are symmetrical with respect to a plane when every point of one figure has its symmetrical point in the other. Any figure has planar symmetry if it can be divided by a plane into two figures symmetrical with respect to that plane. The sphere is symmetrical with respect to every plane through its Any two spheres are symmetrical with respect to every plane through their line of centers.

center.

Every such plane cuts the spheres in two great circles; and the five relations between the center-sect, radii, and relative position of these circles given in 410, with their inverses, hold for the two spheres.

Any three spheres are symmetrical with respect to the plane determined by their centers.

THEOREM VI.

644. If a sphere be tangent to the parallel planes containing opposite edges of a tetrahedron, and sections made in the globe and tetrahedron by one plane parallel to these are equivalent, sections made by any parallel plane are equivalent.

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HYPOTHESIS. Let KJ be the sect the edges EF and GH in the tangent planes.

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Let sections made by plane 1 KJ at R and 1 DT at I, where KR=DI, be equivalent; that is, MO = © PQ. □ O Draw any parallel plane ACBLSN. CONCLUSION. LN = AB.

PROOF.

Since ▲ LEU ~A MEW, and ▲ LHV ~ ▲ MHZ, :. MW : LU :: EM : EL :: JR : JS;

MZ : LV :: HM : HL :: KR : KS;

(593. If lines be cut by three parallel planes, the corresponding sects are propor

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(542. Mutually equiangular parallelograms have to one another the ratio which is

compounded of the ratios of their sides.)

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© PQ: 0 AB :: PI2 : AC2 :: TI. ID : TC. CD. (2)

(546. Similar figures are to each other as the squares on their corresponding sects.)

(522. If from any point in a circle a perpendicular be dropped upon a diameter, it will be a mean proportional between the segments of the diameter.)

By hypothesis and construction, in proportions (1) and (2), the first, third, and fourth terms are respectively equal,

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645. The sects joining its pole to points on any circle of a sphere are equal.

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P'

PROOF. (600. If through the center of a circle a line be passed perpendicular to its plane, the sects from any point of this line to points. on the circle are equal.)

646. COROLLARY. Since chord PA equals chord PB, therefore the arc subtended by chord PA in the great circle PA equals the arc subtended by chord PB in the great circle PB.

Hence the great-circle-arcs joining a pole to points on its circle are equal.

So, if an arc of a great circle be revolved in a sphere about one of its extremities, its other extremity will describe a circle of the sphere.

647. One-fourth a great circle is called a Quadrant.

648. The great-circle-arc joining any point in a great circle with its pole is a quadrant.

P

649. If a point P be a quadrant from two points, A, B, which are not opposite, it is the pole of the great circle through A, B; for each of the angles POA, POB, is right, and therefore PO is perpendicular to the plane OAB.

650. The angle between two intersecting curves is the angle between their tangents, at the point of intersection.

A

E

CB

When the curves are arcs of great circles of the same sphere, the angle is called a Spherical Angle.

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