If one needs examples in mensuration beyond those given in a first-cla.s.s 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 a.s.serts 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 _s__{1}, _s__{2}, , _s__{_n_} 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 (_s__{1} + _s__{2} + _s__{3} + + _s__{_n_} - 2_f_)180.
But _s__{1} + _s__{2} + _s__{3} + + _s__{_n_} is twice the number of edges, because each edge belongs to two faces.
[therefore] the sum of the angles of all the faces is
(2_e_ - 2_f_)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 ill.u.s.tration. 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_ _s__{1}, and thinking of the projections as traced by dotted lines on the opposite side of _s__{1}, this sum is evidently equal to
(1) the sum of the angles in _s__{1}, or (_s__{1} - 2) 180, plus
(2) the sum of the angles on the other side of _s__{1}, or (_s__{1} - 2)180, plus
(3) the sum of the angles about the various points shown as inside of _s__{1}, of which there are _v_ - _s__{1} points, about each of which the sum of the angles is 360, making (_v_ - _s__{1})360 in all.
[Ill.u.s.tration]
Adding, we have
(_s__{1} - 2)180 + (_s__{1} - 2)180 + (_v_ - _s__{1})360
= [(_s__{1} - 2) + (_v_ - _s__{1})]360
= (_v_ - 2)360.
Equating the two sums already found, we have
(_e_ - _f_)360 = (_v_ - 2)360,
or _e_ - _f_ = _v_ - 2,
or _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.[91]
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 ill.u.s.trations of common forms of crystals may be used in this connection:
[Ill.u.s.tration]
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 ill.u.s.tration is from Dupin, loc. cit.
[91] For the historical bibliography consult G. Holzmuller, _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 s.p.a.ce 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 pa.s.sing 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.
a.n.a.logous 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 ill.u.s.trating 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 a.n.a.logue in plane geometry, where there is one. If here we pa.s.s 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 pa.s.sing 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 circ.u.mscribed 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 a.n.a.logous proposition of plane geometry, the circle circ.u.mscribed 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 a.n.a.logue 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 a.n.a.logous 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_ _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 a.n.a.logies 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.
An application of the proposition is seen in the case of an eclipse, where the sphere _O"_ represents the moon, _O_ the earth, and _S_ the sun. It is also seen in the case of the full moon, when _S_ is on the other side of the earth. In this case the part _MIN_ is fully illuminated by the moon, but the zone _ABNM_ is only partly illuminated, as the figure shows.[92]
[Ill.u.s.tration]
THEOREM. _The sum of the sides of a spherical polygon is less than 360._