center to every point of the plane is therefore greater than the radius, the plane must at every point be without the spherical surface.

PROP. VI.

If a perpendicular to a plane from the center of a sphere be shorter than the radius, the plane will cut the sphere, the section being a circle whose center is the foot of the perpendicular.

Let C be the center of the sphere, AB the plane, and CP the perpendicular to it. As CP is by supposition less than the radius, P must be within the surface of the sphere from P draw any line PD at right angles to CP,

E

P

and meeting the surface in D. Then a circle described in the plane AB, from P as a center, and with a radius equal to PD, will be the intersection of that plane with the sphere.

Draw in this circle any radius PE, and join CE, CD: the angle CPE is a right angle (Euc., Suppl. 2, Def. 1); and in the right-angled triangles CPE and CPD, as the sides PE and PD are equal, and CP is a common side, CE and CD are equal: but as D is a point in the surface of the sphere, CD is a radius; wherefore CE is also a radius, and the point E is in the surface. In like manner may it be shown that any other point in the circumference of the circle described about P as a center in the plane AB, as in the surface of the sphere: this

circumference therefore is the intersection of the plane with the surface of the sphere, and the circle itself is the intersection of the plane and sphere.

Cor. The nearer the cutting plane is to the center of the sphere, the larger will be the circular intersection. For the square of the radius of the circle added to the square of the perpendicular from the center of the sphere, is equal to the square of the radius of the sphere; thus, EP2+PC2=CE2: whence the radius of the circle must be greater, as the perpendicular from' the center is less.

The circle is greatest when its plane passes through the center of the sphere; in which case, as has been stated (Def. 7), it is called a great circle.

Every two great circles of a sphere bisect each other; for the center of the sphere is the center of each, and their common section is a diameter of each.

TWO SPHERES.

PROP. VII.

If the distance between the centers of two spheres be equal to the sum of their radii, the spheres will touch each other externally, the point of contact being in the line which joins their centers.

Let C, O, be the centers of the spheres, and let the

C

P

distance CO equal the sum of their radii, CP being equal to the radius of the sphere whose center is C, and OP consequently equal to the other radius. Then the point P must be in the surface of each sphere. And

Σ

this is the only point common to the spheres: for the lines drawn from any point to C and O, must at least be equal to PC and PO together, since CPO is the shortest distance between C and O: and if the point be within the surface of the sphere whose center is C, then as its distance from C is less than CP, its distance from O must be greater than OP, so that the point must be without the sphere whose center is O: again, if the point be in the surface of the sphere whose center is C, its distance from C equals CP, and its distance from O therefore either exceeds OP, so that the point is without the sphere whose center is O, or is equal to OP: but P is evidently the only point whose distance from C equals PC, and whose distance from O equals PO. The spheres therefore touch each other externally at P.

PROP. VIII.

If the distance between the centers of two spheres be greater than the sum of their radii, the spheres will be wholly exterior to each other.

For as the straight line that joins the centers is the shortest line between them, the distances from any point to these centers must together be greater than the sum of the radii of the spheres. And if any point be taken in one of the spheres, then as its distance from the center of this sphere is not greater than the radius, its distance from the center of the other sphere must be greater than the radius of that sphere, and the point must accordingly be without the surface.

PROP. IX.

If the distance between the centers of two spheres be equal to the difference of their radii, the smaller sphere will touch the other internally, the point of contact being in the straight line which passes through their centers. Let C, O, be the centers of two c spheres, OP being a radius of the smaller sphere, and COP a radius of the larger: the spheres will touch each other internally at the point P.

P

For as P is the extremity of a radius of each sphere, it is in the surface of each, and thus is common to the two spheres. And P is the only point of the smaller sphere that is not within the surface of the larger: for the distance of any point from the center C of the larger sphere, cannot exceed the distance of the same point from O, added to the line CO (Euc. 1, 20): and if the point be within the surface of the smaller sphere, then as its distance from O is less than OP, this distance added to CO is less than CP, wherefore the distance of the point from C is less than CP, which is a radius of the larger sphere, and the point is accordingly within the surface of this sphere. Again, if the point be in the surface of the smaller sphere, its distance from O equals OP, and this distance added to CO equals CP: the distance of the point from C is therefore less than CP the radius of the larger sphere, so that the point is within the surface of this sphere; or if the distance is not less, it is equal to CP: but evidently there is no other point than P, whose distances from C and O are equal to PC and PO. Since then the spheres meet at P, and this is the only point of the smaller sphere which is not within

the surface of larger, they touch each other internally at this point.

Σ

PROP. X.

If the distance between the centers of two spheres is less than the difference of their radii, the smaller sphere is wholly within the surface of the other.

Let C be the center of the larger and O the с center of the smaller of two spheres; CO the distance between the centers being less than the difference of the radii of the spheres: the smaller sphere is wholly within the other.

This is to be proved by showing that every point of the smaller sphere is within the surface of the larger, or what amounts to this, that the distance from any point whatever of the smaller sphere to the center of the larger, is less than the radius of the latter.

Now as the distance from any point of the smaller sphere to its center O, is not greater than the smaller radius, and the distance CO is less than the difference of the radii of the spheres, the sum of those two distances is less than the larger radius; but this sum is not less than the distance of the point from C the center of the larger sphere (Euc. 1, 20); wherefore this distance is less than the radius of the larger sphere; as was to be proved.

PROP. XI.

If the distance between the centers of two spheres be less than the sum but greater than the difference of their radii, the surfaces of the spheres will intersect in the circumference of a circle perpendicular to the line which joins the centers, and having its center in that line.