CHAPTER XIX

THE LEADING PROPOSITIONS OF BOOK VI

There have been numerous suggestions with respect to solid geometry, to the effect that it should be more closely connected with plane geometry. The attempt has been made, notably by Méray in France and de Paolis in Italy, to treat the corresponding propositions of plane and solid geometry together; as, for example, those relating to parallelograms and parallelepipeds, and those relating to plane and spherical triangles. Whatever the merits of this plan, it is not feasible in America at present, partly because of the nature of the college-entrance requirements. While it is true that to a boy or girl a solid is more concrete than a plane, it is not true that a geometric solid is more concrete than a geometric plane. Just as the world developed its solid geometry, as a science, long after it had developed its plane geometry, so the human mind grasps the ideas of plane figures earlier than those of the geometric solid.

There is, however, every reason for referring to the corresponding proposition of plane geometry when any given proposition of solid geometry is under consideration, and frequent references of this kind will be made in speaking of the propositions in this and the two succeeding chapters. Such reference has value in the apperception of the various laws of solid geometry, and it also adds an interest to the subject and creates some approach to power in the discovery of new facts in relation to figures of three dimensions.

The introduction to solid geometry should be made slowly. The pupil has been accustomed to seeing only plane figures, and therefore the drawing of a solid figure in the flat is confusing. The best way for the teacher to anticipate this difficulty is to have a few pieces of cardboard, a few knitting needles filed to sharp points, a pine board about a foot square, and some small corks. With the cardboard he can illustrate planes, whether alone, intersecting obliquely or at right angles, or parallel, and he can easily illustrate the figures given in the textbook in use. There are models of this kind for sale, but the simple ones made in a few seconds by the teacher or the pupil have much more meaning. The knitting needles may be stuck in the board to illustrate perpendicular or oblique lines, and if two or more are to meet in a point, they may be held together by sticking them in one of the small corks. Such homely apparatus, costing almost nothing, to be put together in class, seems much more real and is much more satisfactory than the German models.[87]

An extensive use of models is, however, unwise. The pupil must learn very early how to visualize a solid from the flat outline picture, just as a builder or a mechanic learns to read his working drawings. To have a model for each proposition, or even to have a photograph or a stereoscopic picture, is a very poor educational policy. A textbook may properly illustrate a few propositions by photographic aids, but after that the pupil should use the kind of figures that he must meet in his mathematical work. A child should not be kept in a perambulator all his life,--he must learn to walk if he is to be strong and grow to maturity; and it is so with a pupil in the use of models in solid geometry.[88]

The case is somewhat similar with respect to colored crayons. They have their value and their proper place, but they also have their strict limitations. It is difficult to keep their use within bounds; pupils come to use them to make pleasing pictures, and teachers unconsciously fall into the same habit. The value of colored crayons is two-fold: (1) they sometimes make two planes stand out more clearly, or they serve to differentiate some line that is under consideration from others that are not; (2) they enable a class to follow a demonstration more easily by hearing of "the red plane perpendicular to the blue one," instead of "the plane _MN_ perpendicular to the plane _PQ_." But it should always be borne in mind that in practical work we do not have colored ink or colored pencils commonly at hand, nor do we generally have colored crayons. Pupils should therefore become accustomed to the pencil and the white crayon as the regulation tools, and in general they should use them. The figures may not be as striking, but they are more quickly made and they are more practical.

The definition of "plane" has already been discussed in Chapter XII, and the other definitions of Book VI are not of enough interest to call for special remark. The axioms are the same as in plane geometry, but there is at least one postulate that needs to be added, although it would be possible to state various analogues of the postulates of plane geometry if we cared unnecessarily to enlarge the number.

The most important postulate of solid geometry is as follows: _One plane, and only one, can be passed through two intersecting straight lines._ This is easily illustrated, as in most textbooks, as also are three important corollaries derived from it:

1. _A straight line and a point not in the line determine a plane._ Of course this may be made the postulate, as may also the next one, the postulate being placed among the corollaries, but the arrangement here adopted is probably the most satisfactory for educational purposes.

2. _Three points not in a straight line determine a plane._ The common question as to why a three-legged stool stands firmly, while a four-legged table often does not, will add some interest at this point.

3. _Two parallel lines determine a plane._ This requires a slight but informal proof to show that it properly follows as a corollary from the postulate, but a single sentence suffices.

While studying this book questions of the following nature may arise with an advanced class, or may be suggested to those who have had higher algebra:

How many straight lines are in general (that is, at the most) determined by _n_ points in space? Two points determine 1 line, a third point adds (in general, in all these cases) 2 more, a fourth point adds 3 more, and an _n_th point _n_ - 1 more. Hence the maximum is 1 + 2 + 3 + ... + (_n_ - 1), or _n_(_n_-1)/2, which the pupil will understand if he has studied arithmetical progression. The maximum number of intersection points of _n_ straight lines in the same plane is also _n_(_n_ - 1)/2.

How many straight lines are in general determined by _n_ planes? The answer is the same, _n_(_n_ - 1)/2.

How many planes are in general determined by _n_ points in space? Here the answer is 1 + 3 + 6 + 10 + ... + (_n_ - 2)(_n_ - 1)/2, or _n_(_n_ - 1)(_n_ - 2)/(1 × 2 × 3). The same number of points is determined by _n_ planes.

THEOREM. _If two planes cut each other, their intersection is a straight line._

Among the simple illustrations are the back edges of the pages of a book, the corners of the room, and the simple test as to whether the edge of a card is straight by testing it on a plane. It is well to call attention to the fact that if two intersecting straight lines move parallel to their original position, and so that their intersection rests on a straight line not in the plane of those lines, the figure generated will be that of this proposition. In general, if we cut through any figure of solid geometry in some particular way, we are liable to get the figure of a proposition in plane geometry, as will frequently be seen.

THEOREM. _If a straight line is perpendicular to each of two other straight lines at their point of intersection, it is perpendicular to the plane of the two lines._

If students have trouble in visualizing the figure in three dimensions, some knitting needles through a piece of cardboard will make it clear. Teachers should call attention to the simple device for determining if a rod is perpendicular to a board (or a pipe to a floor, ceiling, or wall), by testing it twice, only, with a carpenter's square. Similarly, it may be asked of a class, How shall we test to see if the corner (line) of a room is perpendicular to the floor, or if the edge of a box is perpendicular to one of the sides?

In some elementary and in most higher geometries the perpendicular is called a _normal_ to the plane.

THEOREM. _All the perpendiculars that can be drawn to a straight line at a given point lie in a plane which is perpendicular to the line at the given point._

Thus the hands of a clock pass through a plane as the hands revolve, if they are, as is usual, perpendicular to the axis; and the same is true of the spokes of a wheel, and of a string with a stone attached, swung as rapidly as possible about a boy's arm as an axis. A clock pendulum too swings in a plane, as does the lever in a pair of scales.

THEOREM. _Through a given point within or without a plane there can be one perpendicular to a given plane, and only one._

This theorem is better stated to a class as two theorems.

Thus a plumb line hanging from a point in the ceiling, without swinging, determines one definite point in the floor; and, conversely, if it touches a given point in the floor, it must hang from one definite point in the ceiling. It should be noticed that if we cut through this figure, on the perpendicular line, we shall have the figure of the corresponding proposition in plane geometry, namely, that there can be, under similar circumstances, only one perpendicular to a line.

THEOREM. _Oblique lines drawn from a point to a plane, meeting the plane at equal distances from the foot of the perpendicular, are equal, etc._

There is no objection to speaking of a right circular cone in connection with this proposition, and saying that the slant height is thus proved to be constant. The usual corollary, that if the obliques are equal they meet the plane in a circle, offers a new plan of drawing a circle. A plumb line that is a little too long to reach the floor will, if swung so as just to touch the floor, describe a circle. A 10-foot pole standing in a 9-foot room will, if it moves so as to touch constantly a fixed point on either the floor or the ceiling, describe a circle on the ceiling or floor respectively.

One of the corollaries states that the locus of points in space equidistant from the extremities of a straight line is the plane perpendicular to this line at its middle point. This has been taken by some writers as the definition of a plane, but it is too abstract to be usable. It is advisable to cut through the figure along the given straight line, and see that we come back to the corresponding proposition in plane geometry.

A good many ships have been saved from being wrecked by the principle involved in this proposition.

If a dangerous shoal _A_ is near a headland _H_, the angle _HAX_ is measured and is put down upon the charts as the "vertical danger angle." Ships coming near the headland are careful to keep far enough away, say at _S_, so that the angle _HSX_ shall be less than this danger angle. They are then sure that they will avoid the dangerous shoal.

Related to this proposition is the problem of supporting a tall iron smokestack by wire stays. Evidently three stays are needed, and they are preferably placed at the vertices of an equilateral triangle, the smokestack being in the center. The practical problem may be given of locating the vertices of the triangle and of finding the length of each stay.

THEOREM. _Two straight lines perpendicular to the same plane are parallel._

Here again we may cut through the figure by the plane of the two parallels, and we get the figure of plane geometry relating to lines that are perpendicular to the same line. The proposition shows that the opposite corners of a room are parallel, and that therefore they lie in the same plane, or are _coplanar_, as is said in higher geometry.

It is interesting to a class to have attention called to the corollary that if two straight lines are parallel to a third straight line, they are parallel to each other; and to have the question asked why it is necessary to prove this when the same thing was proved in plane geometry. In case the reason is not clear, let some student try to apply the proof used in plane geometry.

THEOREM. _Two planes perpendicular to the same straight line are parallel._

Besides calling attention to the corresponding proposition of plane geometry, it is well now to speak of the fact that in propositions involving planes and lines we may often interchange these words. For example, using "line" for "straight line," for brevity, we have:

One _line_ does not determine One _plane_ does not determine a _plane_. a _line_.

Two intersecting _lines_ Two intersecting _planes_ determine determine a _plane_. a _line_.

Two _lines_ perpendicular to Two _planes_ perpendicular to a _plane_ are parallel. a _line_ are parallel.

If one of two parallel _lines_ If one of two parallel _planes_ is perpendicular to a _plane_, the is perpendicular to a _line_, the other is also perpendicular to other is also perpendicular to the _plane_. the _line_.

If two _lines_ are parallel, every If two _planes_ are parallel, _plane_ containing one of the every _line_ in one of the _planes_ _lines_ is parallel to the other is parallel to the other _plane_. _line_.

THEOREM. _The intersections of two parallel planes by a third plane are parallel lines._

Thus one of the edges of a box is parallel to the next succeeding edge if the opposite faces are parallel, and in sawing diagonally through an ordinary board (with rectangular cross section) the section is a parallelogram.

THEOREM. _A straight line perpendicular to one of two parallel planes is perpendicular to the other also._

Notice (1) the corresponding proposition in plane geometry; (2) the proposition that results from interchanging "plane" and (straight) "line."

THEOREM. _If two intersecting straight lines are each parallel to a plane, the plane of these lines is parallel to that plane._

Interchanging "plane" and (straight) "line," we have: If two intersecting _planes_ are each parallel to a _line_, the _line_ of (intersection of) these _planes_ is parallel to that _line_. Is this true?

THEOREM. _If two angles not in the same plane have their sides respectively parallel and lying on the same side of the straight line joining their vertices, they are equal and their planes are parallel._

Questions like the following may be asked in connection with the proposition: What is the corresponding proposition in plane geometry? Why do we need another proof here? Try the plane-geometry proof here.

THEOREM. _If two straight lines are cut by three parallel planes, their corresponding segments are proportional._

Here, again, it is desirable to ask for the corresponding proposition of plane geometry, and to ask why the proof of that proposition will not suffice for this one. The usual figure may be varied in an interesting manner by having the two lines meet on one of the planes, or outside the planes, or by having them parallel, in which cases the proof of the plane-geometry proposition holds here. This proposition is not of great importance from the practical standpoint, and it is omitted from some of the standard syllabi at present, although included in certain others. It is easy, however, to frame some interesting questions depending upon it for their answers, such as the following: In a gymnasium swimming tank the water is 4 feet deep and the ceiling is 8 feet above the surface of the water. A pole 15 feet long touches the ceiling and the bottom of the tank. Required to know what length of the pole is in the water.

At this point in Book VI it is customary to introduce the dihedral angle. The word "dihedral" is from the Greek, _di-_ meaning "two," and _hedra_ meaning "seat." We have the root _hedra_ also in "trihedral" (three-seated), "polyhedral" (many-seated), and "cathedral" (a church having a bishop's seat). The word is also, but less properly, spelled without the _h_, "diedral," a spelling not favored by modern usage. It is not necessary to dwell at length upon the dihedral angle, except to show the analogy between it and the plane angle. A few illustrations, as of an open book, the wall and floor of a room, and a swinging door, serve to make the concept clear, while a plane at right angles to the edge shows the measuring plane angle. So manifest is this relationship between the dihedral angle and its measuring plane angle that some teachers omit the proposition that two dihedral angles have the same ratio as their plane angles.

THEOREM. _If two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other._

This and the related propositions allow of numerous illustrations taken from the schoolroom, as of door edges being perpendicular to the floor. The pretended applications of these propositions are usually fictitious, and the propositions are of value chiefly for their own interest and because they are needed in subsequent proofs.

THEOREM. _The locus of a point equidistant from the faces of a dihedral angle is the plane bisecting the angle._

By changing "plane" to "line," and by making other obvious changes to correspond, this reduces to the analogous proposition of plane geometry. The figure formed by the plane perpendicular to the edge is also the figure of that analogous proposition. This at once suggests that there are two planes in the locus, provided the planes of the dihedral angle are taken as indefinite in extent, and that these planes are perpendicular to each other. It may interest some of the pupils to draw this general figure, analogous to the one in plane geometry.

THEOREM. _The projection of a straight line not perpendicular to a plane upon that plane is a straight line._

In higher mathematics it would simply be said that the projection is a straight line, the special case of the projection of a perpendicular being considered as a line-segment of zero length. There is no advantage, however, of bringing in zero and infinity in the course in elementary geometry. The legitimate reason for the modern use of these terms is seldom understood by beginners.

This subject of projection (Latin _pro-_, "forth," and _jacere_, "to throw") is extensively used in modern mathematics and also in the elementary work of the draftsman, and it will be referred to a little later. At this time, however, it is well to call attention to the fact that the projection of a straight line on a plane is a straight line or a point; the projection of a curve may be a curve or it may be straight; the projection of a point is a point; and the projection of a plane (which is easily understood without defining it) may be a surface or it may be a straight line. An artisan represents a solid by drawing its projection upon two planes at right angles to each other, and a map maker (cartographer) represents the surface of the earth by projecting it upon a plane. A photograph of the class is merely the projection of the class upon a photographic plate (plane), and when we draw a figure in solid geometry, we merely project the solid upon the plane of the paper.

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 illustrations 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 class use.

The idea of symmetric solids may be illustrated by a pair of gloves, all their parts being mutually equal but arranged in opposite order. Our hands, feet, and ears afford other illustrations 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 pass 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 classroom in geometry, to be used when it seems advantageous.