Book VII relates to polyhedrons, cylinders, and cones. It opens with the

Chapter 314,912 wordsPublic domain

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 analogy 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 space_. This term may or may not be introduced, depending upon the nature of the class.

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 altitude 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 analogous 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 space.

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 passing 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 asserting that rectangular parallelepipeds having congruent bases are proportional to their altitudes. 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 analytic geometry.

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

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 altitude._

This is also one of the basal propositions of solid geometry, and it has many applications in practical mensuration. A first-class 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:

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 compasses 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 pass 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 space_, the infinite space bounded on several sides by the lateral faces, of the pyramid. This pyramidal space 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 altitude are divided proportionally._ 2. _The section is a polygon similar to the base._

To get the analogous proposition of plane geometry, pass a plane through the vertex so as to cut the base. We shall then have the sides and altitude 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 pass through the vertex, or it may cut the pyramidal space 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 altitude, 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 altitude. The pyramid is filled with sand or grain, and the contents is poured into the prism. This is repeated, and again repeated, showing that the volume of the prism is three times the volume of the pyramid. It sometimes varies the work to show this to a class in geometry.

This proposition was first proved, so Archimedes asserts, by Eudoxus of Cnidus, famous as an astronomer, geometer, physician, and lawgiver, born in humble circumstances about 407 B.C. He studied at Athens and in Egypt, and founded a famous school of geometry at Cyzicus. His discovery also extended to the volume of the cone, and it was his work that gave the beginning to the science of stereometry, the mensuration part of solid geometry.

THEOREM. _The volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum._

Attention should be called to the fact that this formula _v_ = 1/3 _a_(_b_ + _b'_ + [sqrt](_bb'_)) applies to the pyramid by letting _b'_ = 0, to the prism by letting _b_ = _b'_, and also to the parallelepiped and cube, these being special forms of the prism. This formula is, therefore, a very general one, relating to all the polyhedrons that are commonly met in mensuration.

THEOREM. _There cannot be more than five regular convex polyhedrons._

Eudemus of Rhodes, one of the principal pupils of Aristotle, in his history of geometry of which Proclus preserves some fragments, tells us that Pythagoras discovered the construction of the "mundane figures," meaning the five regular polyhedrons. Iamblichus speaks of the discovery of the dodecahedron in these words:

As to Hippasus, who was a Pythagorean, they say that he perished in the sea on account of his impiety, inasmuch as he boasted that he first divulged the knowledge of the sphere with the twelve pentagons. Hippasus assumed the glory of the discovery to himself, whereas everything belongs to Him, for thus they designate Pythagoras, and do not call Him by name.

Iamblichus here refers to the dodecahedron inscribed in the sphere. The Pythagoreans looked upon these five solids as fundamental forms in the structure of the universe. In particular Plato tells us that they asserted that the four elements of the real world were the tetrahedron, octahedron, icosahedron, and cube, and Plutarch ascribes this doctrine to Pythagoras himself. Philolaus, who lived in the fifth century B.C., held that the elementary nature of bodies depended on their form. The tetrahedron was assigned to fire, the octahedron to air, the icosahedron to water, and the cube to earth, it being asserted that the smallest constituent part of each of these substances had the form here assigned to it. Although Eudemus attributes all five to Pythagoras, it is certain that the tetrahedron, cube, and octahedron were known to the Egyptians, since they appear in their architectural decorations. These solids were studied so extensively in the school of Plato that Proclus also speaks of them as the Platonic bodies, saying that Euclid "proposed to himself the construction of the so-called Platonic bodies as the final aim of his arrangement of the 'Elements.'" Aristæus, probably a little older than Euclid, wrote a book upon these solids.

As an interesting amplification of this proposition, the centers of the faces (squares) of a cube may be connected, an inscribed octahedron being thereby formed. Furthermore, if the vertices of the cube are _A_, _B_, _C_, _D_, _A'_, _B'_, _C'_, _D'_, then by drawing _AC_, _CD'_, _D'A_, _D'B'_, _B'A_, and _B'C_, a regular tetrahedron will be formed. Since the construction of the cube is a simple matter, this shows how three of the five regular solids may be constructed. The actual construction of the solids is not suited to elementary geometry.[89]

It is not difficult for a class to find the relative areas of the cube and the inscribed tetrahedron and octahedron. If _s_ is the side of the cube, these areas are 6_s_^2, (1/2)_s_^2[sqrt]3, and _s_^2[sqrt]3; that is, the area of the octahedron is twice that of the tetrahedron inscribed in the cube.

Somewhat related to the preceding paragraph is the fact that the edges of the five regular solids are incommensurable with the radius of the circumscribed sphere. This fact seems to have been known to the Greeks, perhaps to Theætetus (_ca._ 400 B.C.) and Aristæus (_ca._ 300 B.C.), both of whom wrote on incommensurables.

Just as we may produce the sides of a regular polygon and form a regular cross polygon or stellar polygon, so we may have stellar polyhedrons. Kepler, the great astronomer, constructed some of these solids in 1619, and Poinsot, a French mathematician, carried the constructions so far in 1801 that several of these stellar polyhedrons are known as Poinsot solids. There is a very extensive literature upon this subject.

The following table may be of some service in assigning problems in mensuration in connection with the regular polyhedrons, although some of the formulas are too difficult for beginners to prove. In the table _e_ = edge of the polyhedron, _r_ = radius of circumscribed sphere, _r'_ = radius of inscribed sphere, _a_ = total area, _v_ = volume.

========================================================== NUMBER | | | OF FACES| 4 | 6 | 8 --------+-----------------+--------------+---------------- _r_ | _e_[sqrt](3/8) |(_e_/2)[sqrt]3| _e_[sqrt](1/2) | | | _r'_ | _e_[sqrt](1/24) | _e_/2 | _e_[sqrt](1/6) | | | _a_ | _e_^2[sqrt]3 | 6_e_^2 | 2_e_^2[sqrt]3 | | | _v_ |(_e_^3/12)[sqrt]2| _e_^3 |(_e_^3/3)[sqrt]2 ----------------------------------------------------------

Some interest is added to the study of polyhedrons by calling attention to their occurrence in nature, in the form of crystals. The computation of the surfaces and volumes of these forms offers an opportunity for applying the rules of mensuration, and the construction of the solids by paper folding or by the cutting of crayon or some other substance often arouses a considerable interest. The following are forms of crystals that are occasionally found:

They show how the cube is modified by having its corners cut off. A cube may be inscribed in an octahedron, its vertices being at the centers of the faces of the octahedron. If we think of the cube as expanding, the faces of the octahedron will cut off the corners of the cube as seen in the first figure, leaving the cube as shown in the second figure. If the corners are cut off still more, we have the third figure.

Similarly, an octahedron may be inscribed in a cube, and by letting it expand a little, the faces of the cube will cut off the corners of the octahedron. This is seen in the following figures:

This is a form that is found in crystals, and the computation of the surface and volume is an interesting exercise. The quartz crystal, an hexagonal pyramid on an hexagonal prism, is found in many parts of the country, or is to be seen in the school museum, and this also forms an interesting object of study in this connection.

The properties of the cylinder are next studied. The word is from the Greek _kylindros_, from _kyliein_ (to roll). In ancient mathematics circular cylinders were the only ones studied, but since some of the properties are as easily proved for the case of a noncircular directrix, it is not now customary to limit them in this way. It is convenient to begin by a study of the cylindric surface, and a piece of paper may be curved or rolled up to illustrate this concept. If the paper is brought around so that the edges meet, whatever curve may form a cross section the surface is said to inclose a _cylindric space_. This concept is sometimes convenient, but it need be introduced only as necessity for using it arises. The other definitions concerning the cylinder are so simple as to require no comment.

The mensuration of the volume of a cylinder depends upon the assumption that the cylinder is the limit of a certain inscribed or circumscribed prism as the number of sides of the base is indefinitely increased. It is possible to give a fairly satisfactory and simple proof of this fact, but for pupils of the age of beginners in geometry in America it is better to make the assumption outright. This is one of several cases in geometry where a proof is less convincing than the assumed statement.

THEOREM. _The lateral area of a circular cylinder is equal to the product of the perimeter of a right section of the cylinder by an element._

For practical purposes the cylinder of revolution (right circular cylinder) is the one most frequently used, and the important formula is therefore _l_ = 2[pi]_rh_ where _l_ = the lateral area, _r_ = the radius, and _h_ = the altitude. Applications of this formula are easily found.

THEOREM. _The volume of a circular cylinder is equal to the product of its base by its altitude._

Here again the important case is that of the cylinder of revolution, where _v_ = [pi]_r_^2_h_.

The number of applications of this proposition is, of course, very great. In architecture and in mechanics the cylinder is constantly seen, and the mensuration of the surface and the volume is important. A single illustration of this type of problem will suffice.

A machinist is making a crank pin (a kind of bolt) for an engine, according to this drawing. He considers it as weighing the same as three steel cylinders having the diameters and lengths in inches as here shown, where 7 3/4" means 7 3/4 inches. He has this formula for the weight (_w_) of a steel cylinder where _d_ is the diameter and _l_ is the length: _w_ = 0.07[pi]_d_^2_l_. Taking [pi] = 3 1/7, find the weight of the pin.

The most elaborate study of the cylinder, cone, and sphere (the "three round bodies") in the Greek literature is that of Archimedes of Syracuse (on the island of Sicily), who lived in the third century B.C. Archimedes tells us, however, that Eudoxus (born _ca._ 407 B.C.) discovered that any cone is one third of a cylinder of the same base and the same altitude. Tradition says that Archimedes requested that a sphere and a cylinder be carved upon his tomb, and that this was done. Cicero relates that he discovered the tomb by means of these symbols. The tomb now shown to visitors in ancient Syracuse as that of Archimedes cannot be his, for it bears no such figures, and is not "outside the gate of Agrigentum," as Cicero describes.

The cone is now introduced. A conic surface is easily illustrated to a class by taking a piece of paper and rolling it up into a cornucopia, the space inclosed being a _conic space_, a term that is sometimes convenient. The generation of a conic surface may be shown by taking a blackboard pointer and swinging it around by its tip so that the other end moves in a curve. If we consider a straight line as the limit of a curve, then the pointer may swing in a plane, and so a plane is the limit of a conic surface. If we swing the pointer about a point in the middle, we shall generate the two nappes of the cone, the conic space now being double.

In practice the right circular cone, or cone of revolution, is the important type, and special attention should be given to this form.

THEOREM. _Every section of a cone made by a plane passing through its vertex is a triangle._

At this time, or in speaking of the preliminary definitions, reference should be made to the conic sections. Of these there are three great types: (1) the ellipse, where the cutting plane intersects all the elements on one side of the vertex; a circle is a special form of the ellipse; (2) the parabola, where the plane is parallel to an element; (3) the hyperbola, where the plane cuts some of the elements on one side of the vertex, and the rest on the other side; that is, where it cuts both nappes. It is to be observed that the ellipse may vary greatly in shape, from a circle to a very long ellipse, as the cutting plane changes from being perpendicular to the axis to being nearly parallel to an element. The instant it becomes parallel to an element the ellipse changes suddenly to a parabola. If the plane tips the slightest amount more, the section becomes an hyperbola.

While these conic sections are not studied in elementary geometry, the terms should be known for general information, particularly the ellipse and parabola. The study of the conic sections forms a large part of the work of analytic geometry, a subject in which the figures resemble the graphic work in algebra, this having been taken from "analytics," as the higher subject is commonly called. The planets move about the sun in elliptic orbits, and Halley's comet that returned to view in 1909-1910 has for its path an enormous ellipse. Most comets seem to move in parabolas, and a body thrown into the air would take a parabolic path if it were not for the resistance of the atmosphere. Two of the sides of the triangle in this proposition constitute a special form of the hyperbola.

The study of conic sections was brought to a high state by the Greeks. They were not known to the Pythagoreans, but were discovered by Menæchmus in the fourth century B.C. This discovery is mentioned by Proclus, who says, "Further, as to these sections, the conics were conceived by Menæchmus."

Since if the cutting plane is perpendicular to the axis the section is a circle, and if oblique it is an ellipse, a parabola, or an hyperbola, it follows that if light proceeds from a point, the shadow of a circle is a circle, an ellipse, a parabola, or an hyperbola, depending on the position of the plane on which the shadow falls. It is interesting and instructive to a class to see these shadows, but of course not much time can be allowed for such work. At this point the chief thing is to have the names "ellipse" and "parabola," so often met in reading, understood.

It is also of interest to pupils to see at this time the method of drawing an ellipse by means of a pencil stretching a string band that moves about two pins fastened in the paper. This is a practical method, and is familiar to all teachers who have studied analytic geometry. In designing elliptic arches, however, three circular arcs are often joined, as here shown, the result being approximately an elliptic arc.

Here _O_ is the center of arc _BC_, _M_ of arc _AB_, and _N_ of arc _CD_. Since _XY_ is perpendicular to _BM_ and _BO_, it is tangent to arcs _AB_ and _BC_, so there is no abrupt turning at _B_, and similarly for _C_.[90]

THEOREM. _The volume of a circular cone is equal to one third the product of its base by its altitude._

It is easy to prove this for noncircular cones as well, but since they are not met commonly in practice, they may be omitted in elementary geometry. The important formula at this time is _v_ = 1/3[pi]_r_^2_h_. As already stated, this proposition was discovered by Eudoxus of Cnidus (born _ca._ 407 B.C., died _ca._ 354 B.C.), a man who, as already stated, was born poor, but who became one of the most illustrious and most highly esteemed of all the Greeks of his time.

THEOREM. _The lateral area of a frustum of a cone of revolution is equal to half the sum of the circumferences of its bases multiplied by the slant height._

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_ = (1/3)[pi]_h_(_r_^2 + _r'_^2 + _rr'_). Here if _r'_ = 0, we have _v_ = (1/3)[pi]_r_^2_h_, the volume of a cone. If _r'_ = _r_, we have _v_ = (1/3)[pi]_h_(_r_^2 + _r_^2 + _r_^2) = [pi]_hr_^2, 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_ + 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 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_ _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.

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 illustrations of common forms of crystals may be used in this connection:

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 illustration is from Dupin, loc. cit.

[91] For the historical bibliography consult G. Holzmüller, _Elemente der Stereometrie_, Vol. I, p. 181, Leipzig, 1900.