The Teaching of Geometry Part 30

There are other projections than those formed by lines that are perpendicular to the plane. The lines may be oblique to the plane, and this is the case with most projections. A photograph, for example, is not formed by lines perpendicular to a plane, for they all converge in the camera. If the lines of projection are all perpendicular to the plane, the projection is said to be orthographic, from the Greek _ortho-_ (straight) and _graphein_ (to draw). A good example of orthographic projection may be seen in the shadow cast by an object upon a piece of paper that is held perpendicular to the sun's rays. A good example of oblique projection is a shadow on the floor of the schoolroom.

THEOREM. _Between two straight lines not in the same plane there can be one common perpendicular, and only one._

The usual corollary states that this perpendicular is the shortest line joining them. It is interesting to compare this with the case of two lines in the same plane. If they are parallel, there may be any number of common perpendiculars. If they intersect, there is still a common perpendicular, but this can hardly be said to be between them, except for its zero segment.

There are many simple ill.u.s.trations of this case. For example, what is the shortest line between any given edge of the ceiling and the various edges of the floor of the schoolroom? If two galleries in a mine are to be connected by an air shaft, how shall it be planned so as to save labor? Make a drawing of the plan.

At this point the polyhedral angle is introduced. The word is from the Greek _polys_ (many) and _hedra_ (seat). Students have more difficulty in grasping the meaning of the size of a polyhedral angle than is the case with dihedral and plane angles. For this reason it is not good policy to dwell much upon this subject unless the question arises, since it is better understood when the relation of the polyhedral angle and the spherical polygon is met. Teachers will naturally see that just as we may measure the plane angle by taking the ratio of an arc to the whole circle, and of a dihedral angle by taking the ratio of that part of the cylindric surface that is cut out by the planes to the whole surface, so we may measure a polyhedral angle by taking the ratio of the spherical polygon to the whole spherical surface. It should also be observed that just as we may have cross polygons in a plane, so we may have spherical polygons that are similarly tangled, and that to these will correspond polyhedral angles that are also cross, their representation by drawings being too complicated for cla.s.s use.

The idea of symmetric solids may be ill.u.s.trated by a pair of gloves, all their parts being mutually equal but arranged in opposite order. Our hands, feet, and ears afford other ill.u.s.trations of symmetric solids.

THEOREM. _The sum of the face angles of any convex polyhedral angle is less than four right angles._

There are several interesting points of discussion in connection with this proposition. For example, suppose the vertex _V_ to approach the plane that cuts the edges in _A_, _B_, _C_, _D_, ..., the edges continuing to pa.s.s through these as fixed points. The sum of the angles about _V_ approaches what limit? On the other hand, suppose _V_ recedes indefinitely; then the sum approaches what limit? Then what are the two limits of this sum? Suppose the polyhedral angle were concave, why would the proof not hold?

FOOTNOTES:

[87] These may be purchased through the Leipziger Lehrmittelanstalt, Leipzig, Germany, which will send catalogues to intending buyers.

[88] An excellent set of stereoscopic views of the figures of solid geometry, prepared by E. M. Langley of Bedford, England, is published by Underwood & Underwood, New York. Such a set may properly have place in a school library or in a cla.s.sroom in geometry, to be used when it seems advantageous.

CHAPTER XX

THE LEADING PROPOSITIONS OF BOOK VII

Book VII relates to polyhedrons, cylinders, and cones. It opens with the necessary definitions relating to polyhedrons, the etymology of the terms often proving interesting and valuable when brought into the work incidentally by the teacher. "Polyhedron" is from the Greek _polys_ (many) and _hedra_ (seat). The Greek plural, _polyhedra_, is used in early English works, but "polyhedrons" is the form now more commonly seen in America. "Prism" is from the Greek _prisma_ (something sawed, like a piece of wood sawed from a beam). "Lateral" is from the Latin _latus_ (side). "Parallelepiped" is from the Greek _parallelos_ (parallel) and _epipedon_ (a plane surface), from _epi_ (on) and _pedon_ (ground). By a.n.a.logy to "parallelogram" the word is often spelled "parallelopiped," but the best mathematical works now adopt the etymological spelling above given. "Truncate" is from the Latin _truncare_ (to cut off).

A few of the leading propositions are now considered.

THEOREM. _The lateral area of a prism is equal to the product of a lateral edge by the perimeter of the right section._

It should be noted that although some syllabi do not give the proposition that parallel sections are congruent, this is necessary for this proposition, because it shows that the right sections are all congruent and hence that any one of them may be taken.

It is, of course, possible to construct a prism so oblique and so low that a right section, that is, a section cutting all the lateral edges at right angles, is impossible. In this case the lateral faces must be extended, thus forming what is called a _prismatic s.p.a.ce_. This term may or may not be introduced, depending upon the nature of the cla.s.s.

This proposition is one of the most important in Book VII, because it is the basis of the mensuration of the cylinder as well as the prism.

Practical applications are easily suggested in connection with beams, corridors, and prismatic columns, such as are often seen in school buildings. Most geometries supply sufficient material in this line, however.

THEOREM. _An oblique prism is equivalent to a right prism whose base is equal to a right section of the oblique prism, and whose alt.i.tude is equal to a lateral edge of the oblique prism._

This is a fundamental theorem leading up to the mensuration of the prism. Attention should be called to the a.n.a.logous proposition in plane geometry relating to the area of the parallelogram and rectangle, and to the fact that if we cut through the solid figure by a plane parallel to one of the lateral edges, the resulting figure will be that of the proposition mentioned. As in the preceding proposition, so in this case, there may be a question raised that will make it helpful to introduce the idea of prismatic s.p.a.ce.

THEOREM. _The opposite lateral faces of a parallelepiped are congruent and parallel._

It is desirable to refer to the corresponding case in plane geometry, and to note again that the figure is obtained by pa.s.sing a plane through the parallelepiped parallel to a lateral edge. The same may be said for the proposition about the diagonal plane of a parallelepiped.

These two propositions are fundamental in the mensuration of the prism.

THEOREM. _Two rectangular parallelepipeds are to each other as the products of their three dimensions._

This leads at once to the corollary that the volume of a rectangular parallelepiped equals the product of its three dimensions, the fundamental law in the mensuration of all solids. It is preceded by the proposition a.s.serting that rectangular parallelepipeds having congruent bases are proportional to their alt.i.tudes. This includes the incommensurable case, but this case may be omitted.

The number of simple applications of this proposition is practically unlimited. In all such cases it is advisable to take a considerable number of numerical exercises in order to fix in mind the real nature of the proposition. Any good geometry furnishes a certain number of these exercises.

The following is an interesting property of the rectangular parallelepiped, often called the rectangular solid:

If the edges are _a_, _b_, and _c_, and the diagonal is _d_, then (_a_/_d_)^2 + (_b_/_d_)^2 + (_c_/_d_)^2 = 1. This property is easily proved by the Pythagorean Theorem, for _d_^2 = _a_^2 + _b_^2 + _c_^2, whence (_a_^2 + _b_^2 + _c_^2) / _d_^2 = 1.

In case _c_ = 0, this reduces to the Pythagorean Theorem. The property is the fundamental one of solid a.n.a.lytic geometry.

THEOREM. _The volume of any parallelepiped is equal to the product of its base by its alt.i.tude._

This is one of the few propositions in Book VII where a model is of any advantage. It is easy to make one out of pasteboard, or to cut one from wood. If a wooden one is made, it is advisable to take an oblique parallelepiped and, by properly sawing it, to transform it into a rectangular one instead of using three different solids.

On account of its awkward form, this figure is sometimes called the Devil's Coffin, but it is a name that it would be well not to perpetuate.

THEOREM. _The volume of any prism is equal to the product of its base by its alt.i.tude._

This is also one of the basal propositions of solid geometry, and it has many applications in practical mensuration. A first-cla.s.s textbook will give a sufficient list of problems involving numerical measurement, to fix the law in mind. For outdoor work, involving measurements near the school or within the knowledge of the pupils, the following problem is a type:

[Ill.u.s.tration]

If this represents the cross section of a railway embankment that is _l_ feet long, _h_ feet high, _b_ feet wide at the bottom, and _b'_ feet wide at the top, find the number of cubic feet in the embankment. Find the volume if _l_ = 300, _h_ = 8, _b_ = 60, and _b'_ = 28.

The mensuration of the volume of the prism, including the rectangular parallelepiped and cube, was known to the ancients. Euclid was not concerned with practical measurement, so that none of this part of geometry appears in his "Elements." We find, however, in the papyrus of Ahmes, directions for the measuring of bins, and the Egyptian builders, long before his time, must have known the mensuration of the rectangular parallelepiped. Among the Hindus, long before the Christian era, rules were known for the construction of altars, and among the Greeks the problem of constructing a cube with twice the volume of a given cube (the "duplication of the cube") was attacked by many mathematicians. The solution of this problem is impossible by elementary geometry.

If _e_ equals the edge of the given cube, then _e_^3 is its volume and 2_e_^3 is the volume of the required cube. Therefore the edge of the required cube is _e_[3root]2. Now if _e_ is given, it is not possible with the straightedge and compa.s.ses to construct a line equal to _e_[3root]2, although it is easy to construct one equal to _e_[sqrt]2.

The study of the pyramid begins at this point. In practical measurement we usually meet the regular pyramid. It is, however, a simple matter to consider the oblique pyramid as well, and in measuring volumes we sometimes find these forms.

THEOREM. _The lateral area of a regular pyramid is equal to half the product of its slant height by the perimeter of its base._

This leads to the corollary concerning the lateral area of the frustum of a regular pyramid. It should be noticed that the regular pyramid may be considered as a frustum with the upper base zero, and the proposition as a special case under the corollary. It is also possible, if we choose, to let the upper base of the frustum pa.s.s through the vertex and cut the lateral edges above that point, although this is too complicated for most pupils. If this case is considered, it is well to bring in the general idea of _pyramidal s.p.a.ce_, the infinite s.p.a.ce bounded on several sides by the lateral faces, of the pyramid. This pyramidal s.p.a.ce is double, extending on two sides of the vertex.

THEOREM. _If a pyramid is cut by a plane parallel to the base:_

1. _The edges and alt.i.tude are divided proportionally._ 2. _The section is a polygon similar to the base._

To get the a.n.a.logous proposition of plane geometry, pa.s.s a plane through the vertex so as to cut the base. We shall then have the sides and alt.i.tude of the triangle divided proportionally, and of course the section will merely be a line-segment, and therefore it is similar to the base line.

The cutting plane may pa.s.s through the vertex, or it may cut the pyramidal s.p.a.ce above the vertex. In either case the proof is essentially the same.

THEOREM. _The volume of a triangular pyramid is equal to one third of the product of its base by its alt.i.tude, and this is also true of any pyramid._

This is stated as two theorems in all textbooks, and properly so. It is explained to children who are studying arithmetic by means of a hollow pyramid and a hollow prism of equal base and equal alt.i.tude. The pyramid is filled with sand or grain, and the contents is poured into the prism.