Here again we may form a corresponding proposition of plane geometry by passing a plane through any three points of contact of the sphere and the tetrahedron. We shall then form the figure of a circle inscribed in a triangle. And just as in the case of the triangle we may have escribed circles by producing the sides, so in the case of the tetrahedron we may have escribed spheres by producing the planes indefinitely and proceeding in the same way as for the inscribed sphere. The figure is difficult to draw, but it is not difficult to understand, particularly if we construct the tetrahedron out of pasteboard. THEOREM. _A sphere may be circumscribed about any given tetrahedron._ By producing one of the faces indefinitely it will cut the sphere in a circle, and the resulting figure, on the plane, will be that of the analogous proposition of plane geometry, the circle circumscribed about a triangle. It is easily proved from the proposition that the four perpendiculars erected at the centers of the faces of a tetrahedron meet in a point (are concurrent), the analogue of the proposition about the perpendicular bisectors of the sides of a triangle. THEOREM. _The intersection of two spherical surfaces is a circle whose plane is perpendicular to the line joining the centers of the surfaces and whose center is in that line._ The figure suggests the case of two circles in plane geometry. In the case of two circles that do not intersect or touch, one not being within the other, there are four common tangents. If the circles touch, two close up into one. If one circle is wholly within the other, this last tangent disappears. The same thing exists in relation to two spheres, and the analogous cases are formed by revolving the circles and tangents about the line through their centers. In plane geometry it is easily proved that if two circles intersect, the tangents from any point on their common chord produced are equal. For if the common chord is _AB_ and the point _P_ is taken on _AB_ produced, then the square on any tangent from _P_ is equal to _PB_ x _PA_. The line _PBA_ is sometimes called the _radical axis_. Similarly in this proposition concerning spheres, if from any point in the plane of the circle formed by the intersection of the two spherical surfaces lines are drawn tangent to either sphere, these tangents are equal. For it is easily proved that all tangents to the same sphere from an external point are equal, and it can be proved as in plane geometry that two tangents to the two spheres are equal. Among the interesting analogies between plane and solid geometry is the one relating to the four common tangents to two circles. If the figure be revolved about the line of centers, the circles generate spheres and the tangents generate conical surfaces. To study this case for various sizes and positions of the two spheres is one of the most interesting generalizations of solid geometry.