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 cla.s.s in geometry.
This proposition was first proved, so Archimedes a.s.serts, by Eudoxus of Cnidus, famous as an astronomer, geometer, physician, and lawgiver, born in humble circ.u.mstances 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 alt.i.tude is the alt.i.tude 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 princ.i.p.al 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 a.s.sumed 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 a.s.serted 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 a.s.signed to fire, the octahedron to air, the icosahedron to water, and the cube to earth, it being a.s.serted that the smallest const.i.tuent part of each of these substances had the form here a.s.signed 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."" Aristaeus, 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 cla.s.s 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 circ.u.mscribed sphere. This fact seems to have been known to the Greeks, perhaps to Theaetetus (_ca._ 400 B.C.) and Aristaeus (_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 a.s.signing 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 circ.u.mscribed 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 ----------------------------------------------------------
======================================================================== NUMBER | | OF FACES| 12 | 20 --------+----------------------------------+---------------------------- _r_ |(_e_/4)[sqrt]3([sqrt]5 + 1) |_e_[sqrt]((5 + [sqrt]5)/8) | | _r"_ |(_e_/2)[sqrt]((25 + 11[sqrt]5)/10)|(_e_[sqrt]3)/12([sqrt]5 + 3) | | _a_ |3_e_^2[sqrt](5(5 + 2[sqrt]5)) | (5_e_^2)[sqrt]3 | | _v_ |((_e_^3)/4)(15 + 7[sqrt]5) |((5_e_^3)/12)([sqrt]5 + 3) ------------------------------------------------------------------------
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:
[Ill.u.s.tration]
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:
[Ill.u.s.tration]
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 ill.u.s.trate 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 s.p.a.ce_. 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 a.s.sumption that the cylinder is the limit of a certain inscribed or circ.u.mscribed 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 a.s.sumption outright. This is one of several cases in geometry where a proof is less convincing than the a.s.sumed 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 alt.i.tude. Applications of this formula are easily found.
THEOREM. _The volume of a circular cylinder is equal to the product of its base by its alt.i.tude._
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 ill.u.s.tration 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 alt.i.tude. 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 ill.u.s.trated to a cla.s.s by taking a piece of paper and rolling it up into a cornucopia, the s.p.a.ce inclosed being a _conic s.p.a.ce_, 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 s.p.a.ce 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 pa.s.sing 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 a.n.a.lytic geometry, a subject in which the figures resemble the graphic work in algebra, this having been taken from "a.n.a.lytics," 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 const.i.tute 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 Menaechmus in the fourth century B.C. This discovery is mentioned by Proclus, who says, "Further, as to these sections, the conics were conceived by Menaechmus."
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 cla.s.s 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 a.n.a.lytic geometry. In designing elliptic arches, however, three circular arcs are often joined, as here shown, the result being approximately an elliptic arc.
[Ill.u.s.tration]
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 alt.i.tude._
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 ill.u.s.trious 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 circ.u.mferences of its bases multiplied by the slant height._
An interesting case for a cla.s.s 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 ill.u.s.tration 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.