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.
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 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:
[Illustration]
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.'" 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 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 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 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
----------------------------------------------------------
========================================================================
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:
[Illustration]
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:
[Illustration]
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
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 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.
[Illustration]
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 deg.,
therefore the sum of the angles of all the faces is
(_s__{1} + _s__{2} + _s__{3} + ... + _s__{_n_} - 2_f_)180 deg..
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 deg., or (_e_ - _f_)360 deg..
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 deg.,
plus
(2) the sum of the angles on the other side of _s__{1}, or
(_s__{1} - 2)180 deg., 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 deg., making
(_v_ - _s__{1})360 deg. in all.
[Illustration]
Adding, we have
(_s__{1} - 2)180 deg. + (_s__{1} - 2)180 deg. + (_v_ - _s__{1})360 deg.
= [(_s__{1} - 2) + (_v_ - _s__{1})]360 deg.
= (_v_ - 2)360 deg..
Equating the two sums already found, we have
(_e_ - _f_)360 deg. = (_v_ - 2)360 deg.,
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:
[Illustration]
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. Holzmueller, _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 space 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
passing 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.
Analogous 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 illustrating 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 analogue in
plane geometry, where there is one. If here we pass 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
passing 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 circumscribed 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
analogous proposition of plane geometry, the circle circumscribed 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 analogue 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 analogous 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_ x _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 analogies 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]
[Illustration]
THEOREM. _The sum of the sides of a spherical polygon is less than
360 deg.._
In all such cases the relation to the polyhedral angle should be made
clear. This is done in the proofs usually given in the textbooks. It is
easily seen that this is true only with the limitation set forth in most
textbooks, that the spherical polygons considered are convex. Thus we
might have a spherical triangle that is concave, with its base 359 deg., and
its other two sides each 90 deg., the sum of the sides being 539 deg..
THEOREM. _The sum of the angles of a spherical triangle is greater than
180 deg. and less than 540 deg.._
It is for the purpose of proving this important fact that polar
triangles are introduced. This proposition shows the relation of the
spherical to the plane triangle. If our planes were in reality slightly
curved, being small portions of enormous spherical surfaces, then the
sum of the angles of a triangle would not be exactly 180 deg., but would
exceed 180 deg. by some amount depending on the curvature of the surface.
Just as a being may be imagined as having only two dimensions, and
living always on a plane surface (in a space of two dimensions), and
having no conception of a space of three dimensions, so we may think of
ourselves as living in a space of three dimensions but surrounded by a
space of four dimensions. The flat being could not point to a third
dimension because he could not get out of his plane, and we cannot point
to the fourth dimension because we cannot get out of our space. Now what
the flat being thinks is his plane may be the surface of an enormous
sphere in our three dimensions; in other words, the space he lives in
may curve through some higher space without his being conscious of it.
So our space may also curve through some higher space without our being
conscious of it. If our planes have really some curvature, then the sum
of the angles of our triangles has a slight excess over 180 deg.. All this
is mere speculation, but it may interest some student to know that the
idea of fourth and higher dimensions enters largely into mathematical
investigation to-day.
THEOREM. _Two symmetric spherical triangles are equivalent._
While it is not a subject that has any place in a school, save perhaps
for incidental conversation with some group of enthusiastic students, it
may interest the teacher to consider this proposition in connection with
the fourth dimension just mentioned. Consider these triangles, where
[L]_A_ = [L]_A'_, _AB_ = _A'B'_, _AC_ = _A'C'_. We prove them congruent
by superposition, turning one over and placing it upon the other. But
suppose we were beings in Flatland, beings with only two dimensions and
without the power to point in any direction except in the plane we lived
in. We should then be unable to turn [triangle]_A'B'C'_ over so that it
could coincide with [triangle]_ABC_, and we should have to prove these
triangles equivalent in some other way, probably by dividing them into
isosceles triangles that could be superposed.
[Illustration]
[Illustration]
Now it is the same thing with symmetric spherical triangles; we cannot
superpose them. But might it not be possible to do so if we could turn
them through the fourth dimension exactly as we turn the Flatlander's
triangle through our third dimension? It is interesting to think about
this possibility even though we carry it no further, and in these side
lights on mathematics lies much of the fascination of the subject.
THEOREM. _The shortest line that can be drawn on the surface of a sphere
between two points is the minor arc of a great circle joining the two
points._
It is always interesting to a class to apply this practically. By taking
a terrestrial globe and drawing a great circle between the southern
point of Ireland and New York City, we represent the shortest route for
ships crossing to England. Now if we notice where this great-circle arc
cuts the various meridians and mark this on an ordinary Mercator's
projection map, such as is found in any schoolroom, we shall find that
the path of the ship does not make a straight line. Passengers at sea
often do not understand why the ship's course on the map is not a
straight line; but the chief reason is that the ship is taking a
great-circle arc, and this is not, in general, a straight line on a
Mercator projection. The small circles of latitude are straight lines,
and so are the meridians and the equator, but other great circles are
represented by curved lines.
THEOREM. _The area of the surface of a sphere is equal to the product of
its diameter by the circumference of a great circle._
This leads to the remarkable formula, _a_ = 4[pi]_r_^2. That the area of
the sphere, a curved surface, should exactly equal the sum of the areas
of four great circles, plane surfaces, is the remarkable feature. This
was one of the greatest discoveries of Archimedes (_ca._ 287-212 B.C.),
who gives it as the thirty-fifth proposition of his treatise on the
"Sphere and the Cylinder," and who mentions it specially in a letter to
his friend Dositheus, a mathematician of some prominence. Archimedes
also states that the surface of a sphere is two thirds that of the
circumscribed cylinder, or the same as the curved surface of this
cylinder. This is evident, since the cylindric surface of the cylinder
is 2[pi]_r_ x 2_r_, or 4[pi]_r_^2, and the two bases have an area
[pi]_r_^2 + [pi]_r_^2, making the total area 6[pi]_r_^2.
THEOREM. _The area of a spherical triangle is equal to the area of a
lune whose angle is half the triangle's spherical excess._
This theorem, so important in finding areas on the earth's surface,
should be followed by a considerable amount of computation of triangular
areas, else it will be rather meaningless. Students tend to memorize a
proof of this character, and in order to have the proposition mean what
it should to them, they should at once apply it. The same is true of the
following proposition on the area of a spherical polygon. It is probable
that neither of these propositions is very old; at any rate, they do not
seem to have been known to the writers on elementary mathematics among
the Greeks.
THEOREM. _The volume of a sphere is equal to the product of the area of
its surface by one third of its radius._
This gives the formula _v_ = (4/3)[pi]_r_^3. This is one of the greatest
discoveries of Archimedes. He also found as a result that the volume of
a sphere is two thirds the volume of the circumscribed cylinder. This is
easily seen, since the volume of the cylinder is [pi]_r_^2 x 2_r_, or
2[pi]_r_^3, and (4/3)[pi]_r_^3 is 2/3 of 2[pi]_r_^3. It was because of
these discoveries on the sphere and cylinder that Archimedes wished
these figures engraved upon his tomb, as has already been stated. The
Roman general Marcellus conquered Syracuse in 212 B.C., and at the sack
of the city Archimedes was killed by an ignorant soldier. Marcellus
carried out the wishes of Archimedes with respect to the figures on his
tomb.
The volume of a sphere can also be very elegantly found by means of a
proposition known as Cavalieri's Theorem. This asserts that if two
solids lie between parallel planes, and are such that the two sections
made by any plane parallel to the given planes are equal in area, the
solids are themselves equal in volume. Thus, if these solids have the
same altitude, _a_, and if _S_ and _S'_ are equal sections made by a
plane parallel to _MN_, then the solids have the same volume. The proof
is simple, since prisms of the same altitude, say _a_/_n_, and on the
bases _S_ and _S'_ are equivalent, and the sums of _n_ such prisms are
the given solids; and as _n_ increases, the sums of the prisms approach
the solids as their limits; hence the volumes are equal.
[Illustration]
This proposition, which will now be applied to finding the volume of the
sphere, was discovered by Bonaventura Cavalieri (1591 or 1598-1647). He
was a Jesuit professor in the University of Bologna, and his best known
work is his "Geometria Indivisilibus," which he wrote in 1626, at least
in part, and published in 1635 (second edition, 1647). By means of the
proposition it is also possible to prove several other theorems, as that
the volumes of triangular pyramids of equivalent bases and equal
altitudes are equal.
[Illustration]
To find the volume of a sphere, take the quadrant _OPQ_, in the
square _OPRQ_. Then if this figure is revolved about _OP_,
_OPQ_ will generate a hemisphere, _OPR_ will generate a cone of
volume (1/3)[pi]_r_^3, and _OPRQ_ will generate a cylinder of
volume [pi]_r_^3. Hence the figure generated by _ORQ_ will have
a volume [pi]_r_^3 - (1/3)[pi]_r_^3, or (2/3)[pi]_r_^3, which
we will call _x_.
Now _OA_ = _AB_, and _OC_ = _AD_; also (_OC_)^2 - (_OA_)^2 = (_AC_)^2,
so that (_AD_)^2 - (_AB_)^2 = (_AC_)^2,
and [pi](_AD_)^2 - [pi](_AB_)^2 = [pi](_AC_)^2.
But [pi](_AD_)^2 - [pi](_AB_)^2 is the area of the ring
generated by _BD_, a section of _x_, and [pi](_AC_)^2 is the
corresponding section of the hemisphere. Hence, by Cavalieri's
Theorem,
(2/3)[pi]_r_^3 = the volume of the hemisphere.
[therefore] (4/3)[pi]_r_^3 = the volume of the sphere.
In connection with the sphere some easy work in quadratics may be
introduced even if the class has had only a year in algebra.
For example, suppose a cube is inscribed in a hemisphere of
radius _r_ and we wish to find its edge, and thereby its
surface and its volume.
If _x_ = the edge of the cube, the diagonal of the base must be
_x_[sqrt]2, and the projection of _r_ (drawn from the center of
the base to one of the vertices) on the base is half of this
diagonal, or (_x_[sqrt]2)/2.
Hence, by the Pythagorean Theorem,
_r_^2 = _x_^2 + ((_x_[sqrt]2)/2)^2 = (3/2)_x_^2
[therefore] _x_ = _r_[sqrt](2/3),
and the total surface is 6_x_^2 = 4_r_^2,
and the volume is _x_^3 = (2/3)_r_^3[sqrt](2/3).
FOOTNOTES:
[92] The illustration is from Dupin, loc. cit.
L'ENVOI
In the Valley of Youth, through which all wayfarers must pass on their
journey from the Land of Mystery to the Land of the Infinite, there is a
village where the pilgrim rests and indulges in various excursions for
which the valley is celebrated. There also gather many guides in this
spot, some of whom show the stranger all the various points of common
interest, and others of whom take visitors to special points from which
the views are of peculiar significance. As time has gone on new paths
have opened, and new resting places have been made from which these
views are best obtained. Some of the mountain peaks have been neglected
in the past, but of late they too have been scaled, and paths have been
hewn out that approach the summits, and many pilgrims ascend them and
find that the result is abundantly worth the effort and the time.
The effect of these several improvements has been a natural and usually
friendly rivalry in the body of guides that show the way. The mountains
have not changed, and the views are what they have always been. But
there are not wanting those who say, "My mountain may not be as lofty as
yours, but it is easier to ascend"; or "There are quarries on my peak,
and points of view from which a building may be seen in process of
erection, or a mill in operation, or a canal, while your mountain shows
only a stretch of hills and valleys, and thus you will see that mine is
the more profitable to visit." Then there are guides who are themselves
often weak of limb, and who are attached to numerous sand dunes, and
these say to the weaker pilgrims, "Why tire yourselves climbing a rocky
mountain when here are peaks whose summits you can reach with ease and
from which the view is just as good as that from the most famous
precipice?" The result is not wholly disadvantageous, for many who pass
through the valley are able to approach the summits of the sand dunes
only, and would make progress with greatest difficulty should they
attempt to scale a real mountain, although even for them it would be
better to climb a little way where it is really worth the effort instead
of spending all their efforts on the dunes.
Then, too, there have of late come guides who have shown much ingenuity
by digging tunnels into some of the greatest mountains. These they have
paved with smooth concrete, and have arranged for rubber-tired cars that
run without jar to the heart of some mountain. Arrived there the pilgrim
has a glance, as the car swiftly turns in a blaze of electric light, at
a roughly painted panorama of the view from the summit, and he is
assured by the guide that he has accomplished all that he would have
done, had he laboriously climbed the peak itself.
In the midst of all the advocacy of sand-dune climbing, and of
rubber-tired cars to see a painted view, the great body of guides still
climb their mountains with their little groups of followers, and the
vigor of the ascent and the magnificence of the view still attract all
who are strong and earnest, during their sojourn in the Valley of Youth.
Among the mountains that have for ages attracted the pilgrims is Mons
Latinus, usually called in the valley by the more pleasing name Latina.
Mathematica, and Rhetorica, and Grammatica are also among the best
known. A group known as Montes Naturales comprises Physica, Biologica,
and Chemica, and one great peak with minor peaks about it is called by
the people Philosophia. There are those who claim that these great
masses of rock are too old to be climbed, as if that affected the view;
while others claim that the ascent is too difficult and that all who do
not favor the sand dunes are reactionary. But this affects only a few
who belong to the real mountains, and the others labor diligently to
improve the paths and to lessen unnecessary toil, but they seek not to
tear off the summits nor do they attend to the amusing attempts of those
who sit by the hillocks and throw pebbles at the rocky sides of the
mountains upon which they work.
* * * * *
Geometry is a mountain. Vigor is needed for its ascent. The views all
along the paths are magnificent. The effort of climbing is stimulating.
A guide who points out the beauties, the grandeur, and the special
places of interest commands the admiration of his group of pilgrims.
One who fails to do this, who does not know the paths, who puts
unnecessary burdens upon the pilgrim, or who blindfolds him in his
progress, is unworthy of his position. The pretended guide who says that
the painted panorama, seen from the rubber-tired car, is as good as the
view from the summit is simply a fakir and is generally recognized as
such. The mountain will stand; it will not be used as a mere commercial
quarry for building stone; it will not be affected by pellets thrown
from the little hillocks about; but its paths will be freed from
unnecessary flints, they will be straightened where this can
advantageously be done, and new paths on entirely novel plans will be
made as time goes on, but these paths will be hewed out of rock, not
made out of the dreams of a day. Every worthy guide will assist in all
these efforts at betterment, and will urge the pilgrim at least to
ascend a little way because of the fact that the same view cannot be
obtained from other peaks; but he will not take seriously the efforts of
the fakir, nor will he listen with more than passing interest to him who
proclaims the sand heap to be a Matterhorn.
INDEX
Ahmes, 27, 254, 278, 306
Alexandroff, 164
Algebra, 37, 84
Al-Khowarazmi, 37
Allman, G. J., 29
Almagest, 35
Al-Nair[=i]z[=i], 171, 193, 214, 264
Al-Qif[t.][=i], 49
Analysis, 41, 161
Angle, 142, 155;
trisection of, 31, 215
Anthonisz, Adriaen, 279
Antiphon, 31, 32, 276
Apollodotus (Apollodorus), 259
Apollonius, 34, 214, 231
Applied problems, 75, 103, 178, 186, 192, 195, 203, 204, 209, 215, 217,
242, 267, 295, 317
Appreciation of geometry, 19
Arab geometry, 37, 51
Archimedes, 34, 42, 48, 139, 141, 215, 276, 278, 314, 327, 328
Aristaeus, 310
Aristotle, 33, 42, 134, 135, 137, 145, 154, 177, 209
Aryabhatta, 36, 279
Associations, syllabi of, 58, 60, 64
Assumptions, 116
Astrolabe, 172
Athelhard of Bath, 37, 51
Athenaeus, 259
Axioms, 31, 41, 116
Babylon, 26, 272
Bartoli, 10, 44, 238
Belli, 10, 44, 172
Beltinus, 239, 241
Beltrami, 127
Bennett, J., 224
Bernoulli, 280
Bertrand, 62
Betz, 131
Bezout, 62
Bhaskara, 232, 268
Billings, R. W., 222
Billingsley, 52
Bion, 192, 239
Boethius, 43, 50
Bolyai, 128
Bonola, 128
Books of geometry, 165, 167, 201, 227, 252, 269, 289, 303, 321
Bordas-Demoulin, 24
Borel, 11, 67, 196
Bosanquet, 272
Bossut, 23
Bourdon, 62
Bourlet, 67, 165, 196
Brahmagupta, 36, 268, 279
Bretschneider, C. A., 30
Brouncker, 280
Bruce, W. N., 199
Bryson, 31, 32, 276
Cajori, 46
Calandri, 30
Campanus, 37, 51, 135
Cantor, M., 29, 46
Capella, 50, 135
Capra, 44
Carson, G. W. L., 18, 96, 114
Casey, J., 38
Cassiodorius, 50
Cataneo, 10, 44
Cavalieri, 136, 329
Chinese values of [pi], 279
Church schools, 43
Cicero, 34, 50, 259, 314
Circle, 145, 201, 270, 287;
squaring the, 31, 32, 277
Circumference, 145
Cissoid, 34
Class in geometry, 108
Clavius, 121
Colleges, geometry in the, 46
Collet, 24
Commensurable magnitudes, 206, 207
Conchoid, 34
Condorcet, 23
Cone, 315
Congruent, 151
Conic sections, 33, 315
Continuity, 212
Converse proposition, 175, 190, 191
Crelle, 142
Cube, duplicating the, 32, 307
Cylinder, 313
D'Alembert, 24, 67
Dase, 279
Decagon, 273
Definitions, 41, 132
De Judaeis, 239, 241
De Morgan, A., 58
De Paolis, 67
Descartes, 38, 84, 320
Diameter, 146
Dihedral, 298
Diocles, 34
Diogenes Laertius, 259
Diorismus, 41
Direction, 150
Distance, 154
Doyle, Conan, 8
Drawing, 95, 221, 281
Duality, 173
Duhamel, 164
Dupin, 11, 217
Duplication problem, 32, 307
Duerer, 10
Educational problems, 1
Egypt, 26, 40
Eisenlohr, 27
Engel and Staeckel, 128
England, 14, 46, 58, 60
Epicureans, 188
Equal, 151, 153
Equilateral, 147
Equivalent, 151
Eratosthenes, 48
Euclid, 33, 42, 43, 44, 119, 125, 135, 156, 165, 167 ff., 201 ff.,
et passim;
editions of, 47, 52;
efforts at improving, 57;
life of, 47;
nature of his "Elements," 52, 55;
opinions of, 8
Eudemus, 33, 168, 171, 185, 216, 309
Eudoxus, 32, 41, 48, 227, 308, 314, 317
Euler, 38, 280, 318
Eutocius, 184
Exercises, nature of, 74, 103;
how to attack, 160
Exhaustions, method of, 31
Extreme and mean ratio, 250
Figures in geometry, 104, 107, 113
Finaeus, 44, 239, 240, 243
Fourier, 142
Fourth dimension, 326
Frankland, 56, 117, 127, 135, 159
Fusion, of algebra and geometry, 84;
of geometry and trigonometry, 91
Gargioli, 44
Gauss, 140, 274
Geminus, 126, 128, 149
Geometry, books of, 165, 167, 201, 227, 252, 269, 289, 303, 321;
compared with other subjects, 14;
introduction to, 93;
modern, 38;
of motion, 68, 196;
reasons for teaching, 7, 15, 20;
related to algebra, 84;
textbooks in, 70
Gerbert, 43
Gherard of Cremona, 37, 51
Gnomon, 212
Golden section, 250
Gothic windows, 75, 221 ff., 274, 282
Gow, J., 29, 56
Greece, 28, 40
Gregoire de St. Vincent, 267
Gregory, 280
Grevy, 67
Gymnasia, geometry in the, 45
Hadamard, 164
Hamilton, W., 14
Harmonic division, 231
Harpedonaptae, 28
Harriot, 37
Harvard syllabus, 63
Heath, T. L., 49, 56, 119, 126, 127, 135, 149, 159, 170, 175, 228, 261
Hebrews, 26
Henrici, O., 11, 14, 25, 164, 196
Henrici and Treutlein, 68, 164, 196
Hermite, 281
Herodotus, 28
Heron, 35, 137, 139, 141, 209, 259, 267
Hexagon, regular, 272
High schools, geometry in the, 45
Hilbert, 119, 131
Hipparchus, 35
Hippasus, 273, 309
Hippias, 31, 215
Hippocrates, 31, 41, 281
History of geometry, 26
Hobson, 166
Hoffmann, 242
Holzmueller, 320
Hughes, Justice, 9
Hypatia, 36
Hypsicles, 34
Hypsometer, 245
Iamblichus, 273, 309
Illusions, optical, 100
Ingrami, 128
Instruments, 96, 178, 236
Introduction to geometry, 93
Ionic school, 28
Jackson, C. S., 12
Jones, W., 271
Junge, 259
Karagiannides, 128
Karpinski, 37
Kaye, G. B., 232
Kepler, 24, 149
Kingsley, C., 36
Klein, F., 68, 89
Kolb, 222
Lacroix, 24, 46, 62, 66
Langley, E. M., 291
Laplace, 101
Legendre, 10, 45, 62, 127, 128, 152
Leibnitz, 140, 150
Leon, 41
Leonardo da Vinci, 264
Leonardo of Pisa, 37, 43, 279
Lettering figures, 105
Limits, 207
Lindemann, 278, 281
Line defined, 137
Lobachevsky, 128
Loci, 163, 198
Locke, W. J., 13
Lodge, A., 14
Logic, 17, 104
Loomis, 164
Ludolph van Ceulen, 279
Lycees, geometry in, 45
M'Clelland, 38
McCormack, T. J., 11
Measured by, 208
Memorizing, 12, 79
Menaechmus, 33, 316
Menelaus, 35
Meray, 67, 68, 196, 289
Methods, 41, 42, 115
Metius, 279
Mikami, 264
Minchin, 14
Models, 93, 290
Modern geometry, 38
Mohammed ibn Musa, 37
Moore, E. H., 131
Mosaics, 274
Mueller, H., 68
Muensterberg, 22
Napoleon, 24, 287
Newton, 24
Nicomedes, 34, 215
Octant, 242
Oenopides, 31, 212, 216
Optical illusions, 100
Oughtred, 37
Paciuolo, 86
Pamphilius, 185
Pappus, 36, 230, 263
Parallelepiped, 303
Parallels, 149, 181
Parquetry, 222, 274
Pascal, 24, 159
Peletier, 169
Perigon, 151
Perry, J., 13, 14
Petersen, 164
Philippus of Mende, 32, 185
Philo, 178
Philolaus, 309
[pi], 26, 27, 34, 36, 271, 278, 280
Plane, 140
Plato, 25, 31, 41, 48, 129, 136, 137, 309, 310
Playfair, 128
Pleasure of geometry, 16
Plimpton, G. A., 51, 52
Plutarch, 259
Poinsot, 311
Point, 135
Polygons, 156, 252, 269, 274
Polyhedrons, 301, 303, 310
Pomodoro, 179
_Pons asinorum_, 174, 265
Posidonius, 128, 149
Postulates, 31, 41, 116, 125, 292
Practical geometry, 3, 7, 9, 44
Printing, effect of, 44
Prism, 303
Problems, applied, 75, 103, 178, 186, 192, 195, 203, 204, 209,
215, 217, 242, 267, 295, 317
Proclus, 36, 47, 48, 52, 71, 127, 128, 136, 137, 139, 140, 149,
155, 186, 188, 197, 212, 214, 253, 258, 310, 311
Projections, 300
Proofs in full, 79
Proportion, 32, 227
Psychology, 12, 20
Ptolemy, C., 35, 278;
king, 48, 49
Pyramid, 307
Pythagoras, 29, 40, 258, 272, 273, 310
Pythagorean Theorem, 28, 36, 258
Pythagorean numbers, 32, 36, 263, 266
Pythagoreans, 136, 137, 185, 227, 269, 309
Quadrant, 236
Quadratrix, 31, 215
Quadrilaterals, 148, 157
Quadrivium, 42
Questions at issue, 3
Rabelais, 24
Radius, 153
Ratio, 205, 227
Real problem defined, 75, 103
Reasons for studying geometry, 7, 100
Rebiere, 25
Reciprocal propositions, 173
Recitation in geometry, 113
Recorde, 37
Rectilinear figures, 146
Reductio ad absurdum, 41, 177
Regular polygons, 269
"Rhind Papyrus," 27
Rhombus and rhomboid, 148
Riccardi, 47
Richter, 279
Roman surveyors, 247
Saccheri, 127
Sacrobosco, 43
San Giovanni, 192
Sauvage, 164
Sayre, 272
Scalene, 147
Schlegel, 68
Schopenhauer, 121, 265
Schotten, 46, 135, 149
Sector, 154, 156
Segment, 154
Semicircle, 146
Shanks, 279
Similar figures, 232
Simon, 38, 56, 135
Simson, 142
Sisam, 11
Smith, D. E., 25, 37, 51, 52, 119, 131, 135, 159
Solid geometry, 289
Speusippus, 32
Sphere, 321
Square on a line, 257
Squaring the circle, 31, 32, 277
Staeckel, 128
Stamper, 46, 83
Stark, W. E., 172, 238
Stereoscopic slides, 291
Stobaeus, 8
Straight angle, 151
Straight line, 138
Suggested proofs, 81
Sulvasutras, 232
Superposition, 169
Surface, 140
Swain, G. F., 13
Syllabi, 58, 60, 63, 64, 66, 67, 80, 82
Sylvester II, 43
Synthetic method, 161
Tangent, 154
Tartaglia, 153
Tatius, Achilles, 272
Teaching geometry, reasons for, 7, 15, 20;
development of, 40
Textbooks, 32, 33, 41, 70, 80, 82
Thales, 28, 168, 171, 185, 210
Theaetetus, 48, 310
Theon of Alexandria, 36
Thibaut, 185
Thoreau, 246
Trapezium, 148
Treutlein, 68, 164
Triangle, 147
Trigonometry, 234
Trisection problem, 31, 215
Trivium, 42
Universities, geometry in the, 43
Uselessness of mathematics, 13
Veblen, 131, 159
Vega, 279
Veronese, 68
Vieta, 279
Vogt, 259
Wallis, 127, 280
Young, J. W. A., 25, 131, 159, 277
Zamberti, 52
Zenodorus, 34, 253
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Transcribers notes
On page 30: Pythagoras fled to Megapontum has been left as printed,
though the author probably meant Metapontum.
On page 269: 100 B.C. has been left as it was printed, though it is
probably a typo for 100 A.D.
End of Project Gutenberg's The Teaching of Geometry, by David Eugene Smith