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.
Let s1, s2, ···, sn be the number of sides of the various faces, and f the number of faces. Now since the sum of the angles of a polygon of s sides is (s - 2)180°, therefore the sum of the angles of all the faces is (s1 + s2 + s3 + ··· + sn - 2f)180°.
But s1 + s2 + s3 + ··· + sn is twice the number of edges, because each edge belongs to two faces.
∴ the sum of the angles of all the faces is
(2e - 2f)180°, or (e - f)360°.
Since the polyhedron is convex, it is possible to find some outside point of view, P, from which some face, as ABCDE, covers up the whole figure, as in this illustration. If we think of all the vertices projected on ABCDE, by lines through P, the sum of the angles of all the faces will be the same as the sum of the angles of all their projections on ABCDE. Calling ABCDE s1, and thinking of the projections as traced by dotted lines on the opposite side of s1, this sum is evidently equal to
(1) the sum of the angles in s1, or (s1 - 2) 180°, plus
(2) the sum of the angles on the other side of s1, or (s1 - 2)180°, plus
(3) the sum of the angles about the various points shown as inside of s1, of which there are v - s1 points, about each of which the sum of the angles is 360°, making (v - s1)360° in all.