The first represents two truncated pyramids placed base to base. Here _e_ = 20, _f_ = 10, _v_ = 12, so that _e_ + 2 = _f_ + _v_. The second represents a crystal formed by replacing each edge of a cube by a plane, with the result that _e_ = 40, _f_ = 18, and _v_ = 24. The third represents a crystal formed by replacing each edge of an octahedron by a plane, it being easy to see that Euler's law still holds true. FOOTNOTES: [89] The actual construction of these solids is given by Pappus. See his "Mathematicae Collectiones," p. 48, Bologna, 1660. [90] The illustration is from Dupin, loc. cit. [91] For the historical bibliography consult G. Holzmueller, _Elemente der Stereometrie_, Vol. I, p. 181, Leipzig, 1900. CHAPTER XXI THE LEADING PROPOSITIONS OF BOOK VIII Book VIII treats of the sphere. Just as the circle may be defined either as a plane surface or as the bounding line which is the locus of a point in a plane at a given distance from a fixed point, so a sphere may be defined either as a solid or as the bounding surface which is the locus of a point in space at a given distance from a fixed point. In higher mathematics the circle is defined as the bounding line and the sphere as the bounding surface; that is, each is defined as a locus. This view of the circle as a line is becoming quite general in elementary geometry, it being the desire that students may not have to change definitions in passing from elementary to higher mathematics. The sphere is less frequently looked upon in geometry as a surface, and in popular usage it is always taken as a solid. Analogous to the postulate that a circle may be described with any given point as a center and any given line as a radius, is the postulate for constructing a sphere with any given center and any given radius. This postulate is not so essential, however, as the one about the circle, because we are not so concerned with constructions here as we are in plane geometry. A good opportunity is offered for illustrating several of the definitions connected with the study of the sphere, such as great circle, axis, small circle, and pole, by referring to geography. Indeed, the first three propositions usually given in Book VIII have a direct bearing upon the study of the earth. THEOREM. _A plane perpendicular to a radius at its extremity is tangent to the sphere._ The student should always have his attention called to the analogue in plane geometry, where there is one. If here we pass a plane through the radius in question, the figure formed on the plane will be that of a line tangent to a circle. If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane. THEOREM. _A sphere may be inscribed in any given tetrahedron._