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 πh (r2 + r22 + rr' ́). Here if r'= 0, we have v = r2h, the volume of a cone. If r'= r, we have v = } πh (r2 + r2 + r2) = πhr2, 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+2v+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 ƒ 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 + $2 + $3 + · + n − 2 f) 180.

...

+S is twice the number of edges, be

But s1 +82 +8 + cause each edge belongs to two faces.

.. the sum of the angles of all the faces is

(2 e 2) 180°, or (ef) 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 think

D

B

(3) the sum of the angles about the various points shown as inside of of which there are v- s1 points, about each of which the sum of the angles is 360°, making (v — s1) 360° in all.

or

or

Adding, we have

(81-2) 180°+(81−2) 180° + (v−s1) 360°= [(§1−2)+(v−s1)] 360°

=(v-2) 360°.

Equating the two sums already found, we have

(e — ƒ) 360° = (v − 2) 360°,

e-f=v-2,

e + 2 = v + f.

This proof is too abstract for most pupils in the high school, but it is more scientific than those found in any of the elementary textbooks, and teachers will find it of service in relieving their own minds of any question as to the legitimacy of the theorem.

Although this proposition is generally attributed to Euler, and was, indeed, rediscovered by him and published in 1752, it was known to the great French geometer Descartes, a fact that Leibnitz mentions.1

This theorem has a very practical application in the study of crystals, since it offers a convenient check on the count of faces, edges, and vertices. Some use of crystals, or even of polyhedrons cut from a piece of crayon, is desirable when studying Euler's proposition. The following illustrations of common forms of crystals may be used in this connection:

004

The first represents two truncated pyramids placed base to base. Here e = 20, ƒ=10, v = 12, so that e + 2 =ƒ + v. The second represents a crystal förmed 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.

=

1 For the historical bibliography consult G. Holzmüller, 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.

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