Here _O_ is the center of arc _BC_, _M_ of arc _AB_, and _N_ of arc _CD_. Since _XY_ is perpendicular to _BM_ and _BO_, it is tangent to arcs _AB_ and _BC_, so there is no abrupt turning at _B_, and similarly for _C_.[90] THEOREM. _The volume of a circular cone is equal to one third the product of its base by its altitude._ It is easy to prove this for noncircular cones as well, but since they are not met commonly in practice, they may be omitted in elementary geometry. The important formula at this time is _v_ = 1/3[pi]_r_^2_h_. As already stated, this proposition was discovered by Eudoxus of Cnidus (born _ca._ 407 B.C., died _ca._ 354 B.C.), a man who, as already stated, was born poor, but who became one of the most illustrious and most highly esteemed of all the Greeks of his time. THEOREM. _The lateral area of a frustum of a cone of revolution is equal to half the sum of the circumferences of its bases multiplied by the slant height._ An interesting case for a class to notice is that in which the upper base becomes zero and the frustum becomes a cone, the proposition being still true. If the upper base is equal to the lower base, the frustum becomes a cylinder, and still the proposition remains true. The proposition thus offers an excellent illustration of the elementary Principle of Continuity. Then follows, in most textbooks, a theorem relating to the volume of a frustum. In the case of a cone of revolution _v_ = (1/3)[pi]_h_(_r_^2 + _r'_^2 + _rr'_). Here if _r'_ = 0, we have _v_ = (1/3)[pi]_r_^2_h_, the volume of a cone. If _r'_ = _r_, we have _v_ = (1/3)[pi]_h_(_r_^2 + _r_^2 + _r_^2) = [pi]_hr_^2, the volume of a cylinder. If one needs examples in mensuration beyond those given in a first-class textbook, they are easily found. The monument to Sir Christopher Wren, the professor of geometry in Cambridge University, who became the great architect of St. Paul's Cathedral in London, has a Latin inscription which means, "Reader, if you would see his monument, look about you." So it is with practical examples in Book VII. Appended to this Book, or more often to the course in solid geometry, is frequently found a proposition known as Euler's Theorem. This is often considered too difficult for the average pupil and is therefore omitted. On account of its importance, however, in the theory of polyhedrons, some reference to it at this time may be helpful to the teacher. The theorem asserts that in any convex polyhedron the number of edges increased by two is equal to the number of vertices increased by the number of faces. In other words, that _e_ + 2 = _v_ + _f_. On account of its importance a proof will be given that differs from the one ordinarily found in textbooks.