alternate, and inverse ratio, and also such as composition, separation,
and conversion of ratio. These entered into such propositions as, "If
four magnitudes are proportional, they will also be proportional
alternately." In later works they appear in the form of "proportion by
composition," "by division," and "by composition and division." None of
these is to-day of much importance, since modern symbolism has greatly
simplified the ancient expressions, and in particular the proposition
concerning "composition and division" is no longer a basal theorem in
geometry. Indeed, if our course of study were properly arranged, we
might well relegate the whole theory of proportion to algebra, allowing
this to precede the work in geometry.
We shall now consider a few of the principal propositions of Book III.
THEOREM. _If a line is drawn through two sides of a triangle parallel to
the third side, it divides those sides proportionally._
In addition to the usual proof it is instructive to consider in class
the cases in which the parallel is drawn through the two sides produced,
either below the base or above the vertex, and also in which the
parallel is drawn through the vertex.
THEOREM. _The bisector of an angle of a triangle divides the opposite
side into segments which are proportional to the adjacent sides._
The proposition relating to the bisector of an exterior angle may be
considered as a part of this one, but it is usually treated separately
in order that the proof shall appear less involved, although the two are
discussed together at this time. The proposition relating to the
exterior angle was recognized by Pappus of Alexandria.
If _ABC_ is the given triangle, and _CP__{1}, _CP__{2} are
respectively the internal and external bisectors, then _AB_ is
divided harmonically by _P__{1} and _P__{2}.
[therefore]_AP__{1} : _P__{1}_B_ = _AP__{2} : _P__{2}_B_.
[therefore]_AP__{2} : _P__{2}_B_ =
_AP__{2} - _P__{1}_P__{2} : _P__{1}_P__{2} - _P__{2}_B_,
and this is the criterion for the harmonic progression still
seen in many algebras. For, letting _AP__{2} = _a_,
_P__{1}_P__{2} = _b_, _P__{2}_B_ = _c_, we have
_a_/_c_ = (_a_ - _b_)/(_b_ - _c_),
which is also derived from taking the reciprocals of _a_, _b_,
_c_, and placing them in an arithmetical progression, thus:
1/_b_ - 1/_a_ = 1/_c_ - 1/_b_,
whence (_a_ - _b_)/_ab_ = (_b_ - _c_)/_bc_,
or (_a_ - _b_)/(_b_ - _c_) = _ab_/_bc_ = _a_/_c_.
This is the reason why the line _AB_ is said to be divided
harmonically. The line _P__{1}_P__{2} is also called the
_harmonic mean_ between _AP__{2} and _P__{2}_B_, and the points
_A_, _P__{1}, _B_, _P__{2} are said to form an _harmonic
range_.
[Illustration]
It may be noted that [L]_P__{2}_CP__{1}, being made up of halves
of two supplementary angles, is a right angle. Furthermore, if
the ratio _CA_ : _CB_ is given, and _AB_ is given, then _P__{1}
and _P__{2} are both fixed. Hence _C_ must lie on a semicircle
with _P__{1}_P__{2} as a diameter, and therefore the locus of a
point such that its distances from two given points are in a
given ratio is a circle. This fact, Pappus tells us, was known
to Apollonius.
At this point it is customary to define similar polygons as such as have
their corresponding angles equal and their corresponding sides
proportional. Aristotle gave substantially this definition, saying that
such figures have "their sides proportional and their angles equal."
Euclid improved upon this by saying that they must "have their angles
severally equal and the sides about the equal angles proportional." Our
present phraseology seems clearer. Instead of "corresponding angles" we
may say "homologous angles," but there seems to be no reason for using
the less familiar word.
[Illustration]
[Illustration]
[Illustration]
It is more general to proceed by first considering similar figures
instead of similar polygons, thus including the most obviously similar
of all figures,--two circles; but such a procedure is felt to be too
difficult by many teachers. By this plan we first define similar sets of
points, _A__{1}, _A__{2}, _A__{3}, ..., and _B__{1}, _B__{2}, _B__{3},
..., as such that _A__{1}_A__{2}, _B__{1}_B__{2}, _C__{1}_C__{2}, ...
are concurrent in _O_, and _A__{1}_O_ : _A__{2}_O_ = _B__{1}_O_ :
_B__{2}_O_ = _C__{1}_O_ : _C__{2}_O_ = ... Here the constant ratio
_A__{1}_O_ : _A__{2}_O_ is called the _ratio of similitude_, and _O_ is
called the _center of similitude_. Having defined similar sets of
points, we then define similar figures as those figures whose points
form similar sets. Then the two circles, the four triangles, and the
three quadrilaterals respectively are similar figures. If the ratio of
similitude is 1, the similar figures become symmetric figures, and they
are therefore congruent. All of the propositions relating to similar
figures can be proved from this definition, but it is customary to use
the Greek one instead.
[Illustration]
Among the interesting applications of similarity is the case of a
shadow, as here shown, where the light is the center of similitude. It
is also well known to most high school pupils that in a camera the lens
reverses the image. The mathematical arrangement is here shown, the lens
inclosing the center of similitude. The proposition may also be applied
to the enlargement of maps and working drawings.
The propositions concerning similar figures have no particularly
interesting history, nor do they present any difficulties that call for
discussion. In schools where there is a little time for trigonometry,
teachers sometimes find it helpful to begin such work at this time,
since all of the trigonometric functions depend upon the properties of
similar triangles, and a brief explanation of the simplest trigonometric
functions may add a little interest to the work. In the present state of
our curriculum we cannot do more than mention the matter as a topic of
general interest in this connection.
It is a mistaken idea that geometry is a prerequisite to trigonometry.
We can get along very well in teaching trigonometry if we have three
propositions: (1) the one about the sum of the angles of a triangle; (2)
the Pythagorean Theorem; (3) the one that asserts that two right
triangles are similar if an acute angle of the one equals an acute angle
of the other. For teachers who may care to make a little digression at
this time, the following brief statement of a few of the facts of
trigonometry may be of value:
[Illustration]
In the right triangle _OAB_ we shall let _AB_ = _y_,
_OA_ = _x_, _OB_ = _r_, thus adopting the letters of higher
mathematics. Then, so long as [L]_O_ remains the same, such
ratios as _y_/_x_, _y_/_r_, etc., will remain the same,
whatever is the size of the triangle. Some of these ratios have
special names. For example, we call
_y_/_r_ the _sine_ of _O_, and we write sin _O_ = _y_/_r_;
_x_/_r_ the _cosine_ of _O_, and we write cos _O_ = _x_/_r_;
_y_/_x_ the _tangent_ of _O_, and we write tan _O_ = _y_/_x_.
Now because
sin _O_ = _y_/_r_, therefore _r_ sin _O_ = _y_;
and because cos _O_ = _x_/_r_, therefore _r_ cos _O_ = _x_;
and because tan _O_ = _y_/_x_, therefore _x_ tan _O_ = _y_.
Hence, if we knew the values of sin _O_, cos _O_, and tan _O_
for the various angles, we could find _x_, _y_, or _r_ if we
knew any one of them.
Now the values of the sine, cosine, and tangent (_functions_ of
the angles, as they are called) have been computed for the
various angles, and some interest may be developed by obtaining
them by actual measurement, using the protractor and squared
paper. Some of those needed for such angles as a pupil in
geometry is likely to use are as follows:
============================================================
ANGLE | SINE |COSINE|TANGENT|| ANGLE | SINE |COSINE|TANGENT
------+------+------+-------++-------+------+------+--------
5 deg. | .087 | .996 | .087 || 50 deg. | .766 | .643 | 1.192
------+------+------+-------++-------+------+------+--------
10 deg. | .174 | .985 | .176 || 55 deg. | .819 | .574 | 1.428
------+------+------+-------++-------+------+------+--------
15 deg. | .259 | .966 | .268 || 60 deg. | .866 | .500 | 1.732
------+------+------+-------++-------+------+------+--------
20 deg. | .342 | .940 | .364 || 65 deg. | .906 | .423 | 2.145
------+------+------+-------++-------+------+------+--------
25 deg. | .423 | .906 | .466 || 70 deg. | .940 | .342 | 2.748
------+------+------+-------++-------+------+------+--------
30 deg. | .500 | .866 | .577 || 75 deg. | .966 | .259 | 3.732
------+------+------+-------++-------+------+------+--------
35 deg. | .574 | .819 | .700 || 80 deg. | .985 | .174 | 5.671
------+------+------+-------++-------+------+------+--------
40 deg. | .643 | .766 | .839 || 85 deg. | .996 | .087 |11.430
------+------+------+-------++-------+------+------+--------
45 deg. | .707 | .707 | 1.000 || 90 deg. | 1.00 | .000 |[infinity]
============================================================
It will of course be understood that the values are correct
only to the nearest thousandth. Thus the cosine of 5 deg. is
0.99619, and the sine of 85 deg. is 0.99619. The entire table can
be copied by a class in five minutes if a teacher wishes to
introduce this phase of the work, and the author has frequently
assigned the computing of a simpler table as a class exercise.
Referring to the figure, if we know that _r_ = 30 and
[L]_O_ = 40 deg., then since _y_ = _r_ sin _O_, we have
_y_ = 30 x 0.643 = 19.29. If we know that _x_ = 60 and
[L]_O_ = 35 deg., then since _y_ = _x_ tan _O_, we have
_y_ = 60 x 0.7 = 42. We may also find _r_, for cos _O_ = _x_/_r_,
whence _r_ = _x_/(cos _O_) = 60/0.819 = 73.26.
Therefore, if we could easily measure [L]_O_ and could measure the
distance _x_, we could find the height of a building _y_. In
trigonometry we use a transit for measuring angles, but it is easy to
measure them with sufficient accuracy for illustrative purposes by
placing an ordinary paper protractor upon something level, so that the
center comes at the edge, and then sighting along a ruler held against
it, so as to find the angle of elevation of a building. We may then
measure the distance to the building and apply the formula _y_ = _x_ tan
_O_.
[Illustration: A QUADRANT OF THE SIXTEENTH CENTURY
Finaeus's "De re et praxi geometrica," Paris, 1556]
It should always be understood that expensive apparatus is not necessary
for such illustrative work. The telescope used on the transit is only
three hundred years old, and the world got along very well with its
trigonometry before that was invented. So a little ingenuity will enable
any one to make from cheap protractors about as satisfactory instruments
as the world used before 1600. In order that this may be the more fully
appreciated, a few illustrations are here given, showing the old
instruments and methods used in practical surveying before the
eighteenth century.
[Illustration: A QUADRANT OF THE SEVENTEENTH CENTURY]
The illustration on page 236 shows a simple form of the quadrant, an
instrument easily made by a pupil who may be interested in outdoor
work. It was the common surveying instrument of the early days. A more
elaborate example is seen in the illustration, on page 237, of a
seventeenth-century brass specimen in the author's collection.[77]
[Illustration: A QUADRANT OF THE SEVENTEENTH CENTURY
Bartoli's "Del modo di misurare," Venice, 1689]
Another type, easily made by pupils, is shown in the above illustration
from Bartoli, 1689. Such instruments were usually made of wood, brass,
or ivory.[78]
Instruments for the running of lines perpendicular to other lines were
formerly common, and are easily made. They suffice, as the following
illustration shows, for surveying an ordinary field.
[Illustration: SURVEYING INSTRUMENT OF THE EIGHTEENTH CENTURY
N. Bion's "Traite de la construction ... des instrumens de
mathematique," The Hague, 1723]
[Illustration: THE QUADRANT USED FOR ALTITUDES
Finaeus's "De re et praxi geometrica," Paris, 1556]
The quadrant was practically used for all sorts of outdoor measuring.
For example, the illustration from Finaeus, on this page, shows how it
was used for altitudes, and the one reproduced on page 240 shows how it
was used for measuring depths.
A similar instrument from the work of Bettinus is given on page 241, the
distance of a ship being found by constructing an isosceles triangle. A
more elaborate form, with a pendulum attachment, is seen in the
illustration from De Judaeis, which also appears on page 241.
[Illustration: THE QUADRANT USED FOR DEPTHS
Finaeus's "Protomathesis," Paris, 1532]
[Illustration: A QUADRANT OF THE SIXTEENTH CENTURY
De Judaeis's "De quadrante geometrico," Nuernberg, 1594]
[Illustration: THE QUADRANT USED FOR DISTANCES
Bettinus's "Apiaria universae philosophiae mathematicae," Bologna,
1645]
The quadrant finally developed into the octant, as shown in the
following illustration from Hoffmann, and this in turn developed into
the sextant, which is now used by all navigators.
[Illustration: THE OCTANT
Hoffmann's "De Octantis," Jena, 1612]
In connection with this general subject the use of the speculum (mirror)
in measuring heights should be mentioned. The illustration given on page
243 shows how in early days a simple device was used for this purpose.
Two similar triangles are formed in this way, and we have only to
measure the height of the eye above the ground, and the distances of the
mirror from the tower and the observer, to have three terms of a
proportion.
All of these instruments are easily made. The mirror is always at hand,
and a paper protractor on a piece of board, with a plumb line attached,
serves as a quadrant. For a few cents, and by the expenditure of an hour
or so, a school can have almost as good instruments as the ordinary
surveyor had before the nineteenth century.
[Illustration: THE SPECULUM
Finaeus's "De re et praxi geometrica," Paris, 1556]
A well-known method of measuring the distance across a stream is
illustrated in the figure below, where the distance from _A_ to some
point _P_ is required.
[Illustration]
Run a line from _A_ to _C_ by standing at _C_ in line with _A_
and _P_. Then run two perpendiculars from _A_ and _C_ by any of
the methods already given,--sighting on a protractor or along
the edge of a book if no better means are at hand. Then sight
from some point _D_, on _CD_, to _P_, putting a stake at _B_.
Then run the perpendicular _BE_. Since _DE_ : _EB_ = _BA_ :
_AP_, and since we can measure _DE_, _EB_, and _BA_ with the
tape, we can compute the distance _AP_.
There are many variations of this scheme of measuring distances by
means of similar triangles, and pupils may be encouraged to try some of
them. Other figures are suggested on page 244, and the triangles need
not be confined to those having a right angle.
A very simple illustration of the use of similar triangles is found in
one of the stories told of Thales. It is related that he found the
height of the pyramids by measuring their shadow at the instant when his
own shadow just equaled his height. He thus had the case of two similar
isosceles triangles. This is an interesting exercise which may be tried
about the time that pupils are leaving school in the afternoon.
[Illustration]
Another application of the same principle is seen in a method often
taken for measuring the height of a tree.
[Illustration]
The observer has a large right triangle made of wood. Such a
triangle is shown in the picture, in which _AB_ = _BC_. He
holds _AB_ level and walks toward the tree until he just sees
the top along _AC_. Then because
_AB_ = _BC_,
and _AB_ : _BC_ = _AD_ : _DE_,
the height above _D_ will equal the distance _AD_.
Questions like the following may be given to the class:
1. What is the height of the tree in the picture if the
triangle is 5 ft. 4 in. from the ground, and _AD_ is 23 ft. 8
in.?
2. Suppose a triangle is used which has _AB_ = twice _BC_. What
is the height if _AD_ = 75 ft.?
There are many variations of this principle. One consists in measuring
the shadows of a tree and a staff at the same time. The height of the
staff being known, the height of the tree is found by proportion.
Another consists in sighting from the ground, across a mark on an
upright staff, to the top of the tree. The height of the mark being
known, and the distances from the eye to the staff and to the tree being
measured, the height of the tree is found.
[Illustration]
An instrument sold by dealers for the measuring of heights is known as
the hypsometer. It is made of brass, and is of the form here shown. The
base is graduated in equal divisions, say 50, and the upright bar is
similarly divided. At the ends of the hinged radius are two sights. If
the observer stands 50 feet from a tree and sights at the top, so that
the hinged radius cuts the upright bar at 27, then he knows at once that
the tree is 27 feet high. It is easy for a class to make a fairly good
instrument of this kind out of stiff pasteboard.
An interesting application of the theorem relating to similar triangles
is this: Extend your arm and point to a distant object, closing your
left eye and sighting across your finger tip with your right eye. Now
keep the finger in the same position and sight with your left eye. The
finger will then seem to be pointing to an object some distance to the
right of the one at which you were pointing. If you can estimate the
distance between these two objects, which can often be done with a fair
degree of accuracy when there are houses intervening, then you will be
able to tell approximately your distance from the objects, for it will
be ten times the estimated distance between them. The finding of the
reason for this by measuring the distance between the pupils of the two
eyes, and the distance from the eye to the finger tip, and then drawing
the figure, is an interesting exercise.
Perhaps some pupil who has read Thoreau's descriptions of outdoor life
may be interested in what he says of his crude mathematics. He writes,
"I borrowed the plane and square, level and dividers, of a carpenter,
and with a shingle contrived a rude sort of a quadrant, with pins for
sights and pivots." With this he measured the heights of a cliff on the
Massachusetts coast, and with similar home-made or school-made
instruments a pupil in geometry can measure most of the heights and
distances in which he is interested.
THEOREM. _If in a right triangle a perpendicular is drawn from the
vertex of the right angle to the hypotenuse:_
1. _The triangles thus formed are similar to the given triangle, and are
similar to each other._
2. _The perpendicular is the mean proportional between the segments of
the hypotenuse._
3. _Each of the other sides is the mean proportional between the
hypotenuse and the segment of the hypotenuse adjacent to that side._
To this important proposition there is one corollary of particular
interest, namely, _The perpendicular from any point on a circle to a
diameter is the mean proportional between the segments of the diameter_.
By means of this corollary we can easily construct a line whose
numerical value is the square root of any number we please.
Thus we may make _AD_ = 2 in., _DB_ = 3 in., and erect _DC_
[perp] to _AB_. Then the length of _DC_ will be [sqrt]6 in.,
and we may find [sqrt]6 approximately by measuring _DC_.
[Illustration]
Furthermore, if we introduce negative magnitudes into geometry,
and let _DB_ = +3 and _DA_ = -2, then _DC_ will equal [sqrt](-6).
In other words, we have a justification for representing
imaginary quantities by lines perpendicular to the line on
which we represent real quantities, as is done in the graphic
treatment of imaginaries in algebra.
It is an interesting exercise to have a class find, to one decimal
place, by measuring as above, the value of [sqrt]2, [sqrt]3, [sqrt]5,
and [sqrt]9, the last being integral. If, as is not usually the case,
the class has studied the complex number, the absolute value of
[sqrt](-6), [sqrt](-7), ..., may be found in the same way.
A practical illustration of the value of the above theorem is seen in a
method for finding distances that is frequently described in early
printed books. It seems to have come from the Roman surveyors.
[Illustration]
If a carpenter's square is put on top of an upright stick, as
here shown, and an observer sights along the arms to a distant
point _B_ and a point _A_ near the stick, then the two
triangles are similar. Hence _AD_ : _DC_ = _DC_ : _DB_. Hence,
if _AD_ and _DC_ are measured, _DB_ can be found. The
experiment is an interesting and instructive one for a class,
especially as the square can easily be made out of heavy
pasteboard.
THEOREM. _If two chords intersect within a circle, the product of the
segments of the one is equal to the product of the segments of the
other._
THEOREM. _If from a point without a circle a secant and a tangent are
drawn, the tangent is the mean proportional between the secant and its
external segment._
COROLLARY. _If from a point without a circle a secant is drawn, the
product of the secant and its external segment is constant in whatever
direction the secant is drawn._
These two propositions and the corollary are all parts of one general
proposition: _If through a point a line is drawn cutting a circle, the
product of the segments of the line is constant_.
[Illustration]
If _P_ is within the circle, then _xx'_ = _yy'_; if _P_ is on
the circle, then _x_ and _y_ become 0, and 0 . _x'_ = 0 . _y'_
= 0; if _P_ is at _P__{3}, then _x_ and _y_, having passed
through 0, may be considered negative if we wish, although the
two negative signs would cancel out in the equation; if _P_ is
at _P__{4}, then _y_ = _y'_ and we have _xx'_ = _y_^2,
or _x_ : _y_ = _y_ : _x'_, as stated in the proposition.
We thus have an excellent example of the Principle of Continuity, and
classes are always interested to consider the result of letting _P_
assume various positions. Among the possible cases is the one of two
tangents from an external point, and the one where _P_ is at the center
of the circle.
Students should frequently be questioned as to the meaning of "product
of lines." The Greeks always used "rectangle of lines," but it is
entirely legitimate to speak of "product of lines," provided we define
the expression consistently. Most writers do this, saying that by the
product of lines is meant the product of their numerical values, a
subject already discussed at the beginning of this chapter.
THEOREM. _The square on the bisector of an angle of a triangle is equal
to the product of the sides of this angle diminished by the product of
the segments made by the bisector upon the third side of the triangle._
This proposition enables us to compute the length of a bisector of a
triangle if the lengths of the sides are known.
[Illustration]
For, in this figure, let _a_ = 3, _b_ = 5, and _c_ = 6.
Then [because] _x_ : _y_ = _b_ : _a_, and _y_ = 6 - _x_,
we have _x_/(6 - _x_) = 5/3.
[therefore] 3_x_ = 30 - 5_x_.
[therefore] _x_ = 3 3/4, _y_ = 2 1/4.
By the theorem, _z_^2 = _ab_ - _xy_
= 15 - (8 7/16) = 6 9/16.
[therefore] _z_ = [sqrt](6 9/16) = 1/4 [sqrt]105 = 2.5+.
THEOREM. _In any triangle the product of two sides is equal to the
product of the diameter of the circumscribed circle by the altitude upon
the third side._
This enables us, after the Pythagorean Theorem has been studied, to
compute the length of the diameter of the circumscribed circle in terms
of the three sides.
[Illustration]
For if we designate the sides by _a_, _b_, and _c_, as usual,
and let _CD_ = _d_ and _PB_ = _x_, then
(_CP_)^2 = _a_^2 - _x_^2
= _b_^2 - (_c_ - _x_)^2.
[therefore] _a_^2 - _x_^2 = _b_^2 - _c_^2 + 2_cx_ - _x_^2.
[therefore] _x_ = (_a_^2 - _b_^2 + _c_^2) / 2_c_.
[therefore] (_CP_)^2 = _a_^2 - ((_a_^2 - _b_^2 + _c_^2) / 2_c_)^2.
But _CP_ . _d_ = _ab_.
[therefore] _d_ = 2_abc_ / [sqrt](4_a_^2_c_^2 - (_a_^2 - _b_^2 + _c_^2)^2).
This is not available at this time, however, because the Pythagorean
Theorem has not been proved.
These two propositions are merely special cases of the following general
theorem, which may be given as an interesting exercise:
_If ABC is an inscribed triangle, and through C there are drawn two
straight lines CD, meeting AB in D, and CP, meeting the circle in P,
with angles ACD and PCB equal, then AC x BC will equal CD x CP._
[Illustration: FIG. 1]
[Illustration: FIG. 2]
[Illustration: FIG. 3]
[Illustration: FIG. 4]
Fig. 1 is the general case where _D_ falls between _A_ and _B_.
If _CP_ is a diameter, it reduces to the second figure given on
page 249. If _CP_ bisects [L]_ACB_, we have Fig. 3, from which
may be proved the proposition given at the foot of page 248. If
_D_ lies on _BA_ produced, we have Fig. 2. If _D_ lies on _AB_
produced, we have Fig. 4.
This general proposition is proved by showing that
[triangles]_ADC_ and _PBC_ are similar, exactly as in the
second proposition given on page 249.
These theorems are usually followed by problems of construction, of
which only one has great interest, namely, _To divide a given line in
extreme and mean ratio._
The purpose of this problem is to prepare for the construction of the
regular decagon and pentagon. The division of a line in extreme and mean
ratio is called "the golden section," and is probably "the section"
mentioned by Proclus when he says that Eudoxus "greatly added to the
number of the theorems which Plato originated regarding the section."
The expression "golden section" is not old, however, and its origin is
uncertain.
If a line _AB_ is divided in golden section at _P_, we have
_AB_ x _PB_ = (_AP_)^2.
Therefore, if _AB_ = _a_, and _AP_ = _x_, we have
_a_(_a_ - _x_) = _x_^2,
or _x_^2 + _ax_ - _a_^2 = 0;
whence _x_ = - _a_/2 +- _a_/2[sqrt]5
= _a_(1.118 - 0.5)
= 0.618_a_,
the other root representing the external point.
That is, _x_ = about 0.6_a_, and _a_ - _x_ = about 0.4_a_, and
_a_ is therefore divided in about the ratio of 2 : 3.
There has been a great deal written upon the aesthetic features of the
golden section. It is claimed that a line is most harmoniously divided
when it is either bisected or divided in extreme and mean ratio. A
painting has the strong feature in the center, or more often at a point
about 0.4 of the distance from one side, that is, at the golden section
of the width of the picture. It is said that in nature this same harmony
is found, as in the division of the veins of such leaves as the ivy and
fern.
FOOTNOTES:
[76] For a very full discussion of these four definitions see Heath's
"Euclid," Vol. II, p. 116, and authorities there cited.
[77] These two and several which follow are from Stark, loc. cit.
[78] The author has a beautiful ivory specimen of the Sixteenth century.
CHAPTER XVII
THE LEADING PROPOSITIONS OF BOOK IV
Book IV treats of the area of polygons, and offers a large number of
practical applications. Since the number of applications to the
measuring of areas of various kinds of polygons is unlimited, while in
the first three books these applications are not so obvious, less effort
is made in this chapter to suggest practical problems to the teachers.
The survey of the school grounds or of vacant lots in the vicinity
offers all the outdoor work that is needed to make Book IV seem very
important.
THEOREM. _Two rectangles having equal altitudes are to each other as
their bases._
Euclid's statement (Book VI, Proposition 1) was as follows: _Triangles
and parallelograms which are under the same height are to one another as
their bases_. Our plan of treating the two figures separately is
manifestly better from the educational standpoint.
In the modern treatment by limits the proof is divided into two parts:
first, for commensurable bases; and second, for incommensurable ones. Of
these the second may well be omitted, or merely be read over by the
teacher and class and the reasons explained. In general, it is doubtful
if the majority of an American class in geometry get much out of the
incommensurable case. Of course, with a bright class a teacher may well
afford to take it as it is given in the textbook, but the important
thing is that the commensurable case should be proved and the
incommensurable one recognized.
Euclid's treatment of proportion was so rigorous that no special
treatment of the incommensurable was necessary. The French geometer,
Legendre, gave a rigorous proof by _reductio ad absurdum_. In America
the pupils are hardly ready for these proofs, and so our treatment by
limits is less rigorous than these earlier ones.
THEOREM. _The area of a rectangle is equal to the product of its base by
its altitude._
The easiest way to introduce this is to mark a rectangle, with
commensurable sides, on squared paper, and count up the squares; or,
what is more convenient, to draw the rectangle and mark the area off in
squares.
It is interesting and valuable to a class to have its attention called
to the fact that the perimeter of a rectangle is no criterion as to the
area. Thus, if a rectangle has an area of 1 square foot and is only
1/440 of an inch high, the perimeter is over 2 miles. The story of how
Indians were induced to sell their land by measuring the perimeter is a
very old one. Proclus speaks of travelers who described the size of
cities by the perimeters, and of men who cheated others by pretending to
give them as much land as they themselves had, when really they made
only the perimeters equal. Thucydides estimated the size of Sicily by
the time it took to sail round it. Pupils will be interested to know in
this connection that of polygons having the same perimeter and the same
number of sides, the one having equal sides and equal angles is the
greatest, and that of plane figures having the same perimeter, the
circle is the greatest. These facts were known to the Greek writers,
Zenodorus (_ca._ 150 B.C.) and Proclus (410-485 A.D.).
The surfaces of rectangular solids may now be found, there being an
advantage in thus incidentally connecting plane and solid geometry
wherever it is natural to do so.
THEOREM. _The area of a parallelogram is equal to the product of its
base by its altitude._
The best way to introduce this theorem is to cut a parallelogram from
paper, and then, with the class, separate it into two parts by a cut
perpendicular to the base. The two parts may then be fitted together to
make a rectangle. In particular, if we cut off a triangle from one end
and fit it on the other, we have the basis for the proof of the
textbooks. The use of squared paper for such a proposition is not wise,
since it makes the measurement appear to be merely an approximation. The
cutting of the paper is in every way more satisfactory.
THEOREM. _The area of a triangle is equal to half the product of its
base by its altitude._
Of course, the Greeks would never have used the wording of either of
these two propositions. Euclid, for example, gives this one as follows:
_If a parallelogram have the same base with a triangle and be in the
same parallels, the parallelogram is double of the triangle._ As to the
parallelogram, he simply says it is equal to a parallelogram of equal
base and "in the same parallels," which makes it equal to a rectangle of
the same base and the same altitude.
The number of applications of these two theorems is so great that the
teacher will not be at a loss to find genuine ones that appeal to the
class. Teachers may now introduce pyramids, requiring the areas of the
triangular faces to be found.
The Ahmes papyrus (_ca._ 1700 B.C.) gives the area of an isosceles
triangle as 1/2 _bs_, where _s_ is one of the equal sides, thus taking
_s_ for the altitude. This shows the primitive state of geometry at that
time.
THEOREM. _The area of a trapezoid is equal to half the sum of its bases
multiplied by the altitude._
[Illustration]
An interesting variation of the ordinary proof is made by
placing a trapezoid _T'_, congruent to _T_, in the position
here shown. The parallelogram formed equals _a_(_b_ + _b'_),
and therefore
_T_ = _a_ . (_b_ + _b'_)/2.
The proposition should be discussed for the case _b_ = _b'_,
when it reduces to the one about the area of a parallelogram.
If _b'_= 0, the trapezoid reduces to a triangle, and
_T_ = _a_ . _b_/2.
This proposition is the basis of the theory of land surveying, a piece
of land being, for purposes of measurement, divided into trapezoids and
triangles, the latter being, as we have seen, a kind of special
trapezoid.
The proposition is not in Euclid, but is given by Proclus in the fifth
century.
The term "isosceles trapezoid" is used to mean a trapezoid with two
opposite sides equal, but not parallel. The area of such a figure was
incorrectly given by the Ahmes papyrus as 1/2(_b_ + _b'_)_s_, where _s_
is one of the equal sides. This amounts to taking _s_ = _a_.
The proposition is particularly important in the surveying of an
irregular field such as is found in hilly districts. It is customary to
consider the field as a polygon, and to draw a meridian line, letting
fall perpendiculars upon it from the vertices, thus forming triangles
and trapezoids that can easily be measured. An older plan, but one
better suited to the use of pupils who may be working only with the
tape, is given on page 99.
THEOREM. _The areas of two triangles which have an angle of the one
equal to an angle of the other are to each other as the products of the
sides including the equal angles._
This proposition may be omitted as far as its use in plane geometry is
concerned, for we can prove the next proposition here given without
using it. In solid geometry it is used only in a proposition relating to
the volumes of two triangular pyramids having a common trihedral angle,
and this is usually omitted. But the theorem is so simple that it takes
but little time, and it adds greatly to the student's appreciation of
similar triangles. It not only simplifies the next one here given, but
teachers can at once deduce the latter from it as a special case by
asking to what it reduces if a second angle of one triangle is also
equal to a second angle of the other triangle.
It is helpful to give numerical values to the sides of a few triangles
having such equal angles, and to find the numerical ratio of the areas.
THEOREM. _The areas of two similar triangles are to each other as the
squares on any two corresponding sides._
[Illustration]
This may be proved independently of the preceding proposition
by drawing the altitudes _p_ and _p'_. Then
[triangle]_ABC_/[triangle]_A'B'C'_ = _cp_/_c'p'_.
But _c_/_c'_ = _p_/_p'_,
by similar triangles.
[therefore] [triangle]_ABC_/[triangle]_A'B'C'_ = _c_^2/_c'_^2,
and so for other sides.
This proof is unnecessarily long, however, because of the
introduction of the altitudes.
In this and several other propositions in Book IV occurs the expression
"the square _on_ a line." We have, in our departure from Euclid, treated
a line either as a geometric figure or as a number (the length of the
line), as was the more convenient. Of course if we are speaking of a
line, the preferable expression is "square _on_ the line," whereas if we
speak of a number, we say "square _of_ the number." In the case of a
rectangle of two lines we have come to speak of the "product of the
lines," meaning the product of their numerical values. We are therefore
not as accurate in our phraseology as Euclid, and we do not pretend to
be, for reasons already given. But when it comes to "square _on_ a line"
or "square _of_ a line," the former is the one demanding no explanation
or apology, and it is even better understood than the latter.
THEOREM. _The areas of two similar polygons are to each other as the
squares on any two corresponding sides._
This is a proposition of great importance, and in due time the pupil
sees that it applies to circles, with the necessary change of the word
"sides" to "lines." It is well to ask a few questions like the
following: If one square is twice as high as another, how do the areas
compare? If the side of one equilateral triangle is three times as long
as that of another, how do the perimeters compare? how do the areas
compare? If the area of one square is twenty-five times the area of
another square, the side of the first is how many times as long as the
side of the second? If a photograph is enlarged so that a tree is four
times as high as it was before, what is the ratio of corresponding
dimensions? The area of the enlarged photograph is how many times as
great as the area of the original?
THEOREM. _The square on the hypotenuse of a right triangle is equivalent
to the sum of the squares on the other two sides._
Of all the propositions of geometry this is the most famous and perhaps
the most valuable. Trigonometry is based chiefly upon two facts of plane
geometry: (1) in similar triangles the corresponding sides are
proportional, and (2) this proposition. In mensuration, in general, this
proposition enters more often than any others, except those on the
measuring of the rectangle and triangle. It is proposed, therefore, to
devote considerable space to speaking of the history of the theorem, and
to certain proofs that may profitably be suggested from time to time to
different classes for the purpose of adding interest to the work.
Proclus, the old Greek commentator on Euclid, has this to say of the
history: "If we listen to those who wish to recount ancient history, we
may find some of them referring this theorem to Pythagoras and saying
that he sacrificed an ox in honor of his discovery. But for my part,
while I admire those who first observed the truth of this theorem, I
marvel more at the writer of the 'Elements' (Euclid), not only because
he made it fast by a most lucid demonstration, but because he compelled
assent to the still more general theorem by the irrefragable arguments
of science in Book VI. For in that book he proves, generally, that in
right triangles the figure on the side subtending the right angle is
equal to the similar and similarly placed figures described on the sides
about the right angle." Now it appears from this that Proclus, in the
fifth century A.D., thought that Pythagoras discovered the proposition
in the sixth century B.C., that the usual proof, as given in most of
our American textbooks, was due to Euclid, and that the generalized
form was also due to the latter. For it should be made known to students
that the proposition is true not only for squares, but for any similar
figures, such as equilateral triangles, parallelograms, semicircles, and
irregular figures, provided they are similarly placed on the three sides
of the right triangle.
Besides Proclus, Plutarch testifies to the fact that Pythagoras was the
discoverer, saying that "Pythagoras sacrificed an ox on the strength of
his proposition as Apollodotus says," but saying that there were two
possible propositions to which this refers. This Apollodotus was
probably Apollodorus, surnamed Logisticus (the Calculator), whose date
is quite uncertain, and who speaks in some verses of a "famous
proposition" discovered by Pythagoras, and all tradition makes this the
one. Cicero, who comments upon these verses, does not question the
discovery, but doubts the story of the sacrifice of the ox. Of other
early writers, Diogenes Laertius, whose date is entirely uncertain
(perhaps the second century A.D.), and Athenaeus (third century A.D.) may
be mentioned as attributing the theorem to Pythagoras, while Heron
(first century A.D.) says that he gave a rule for forming right
triangles with rational integers for the sides, like 3, 4, 5, where
3^2 + 4^2 = 5^2. It should be said, however, that the Pythagorean origin
has been doubted, notably in an article by H. Vogt, published in the
_Bibliotheca Mathematica_ in 1908 (Vol. IX (3), p. 15), entitled "Die
Geometrie des Pythagoras," and by G. Junge, in his work entitled "Wann
haben die Griechen das Irrationale entdeckt?" (Halle, 1907). These
writers claim that all the authorities attributing the proposition to
Pythagoras are centuries later than his time, and are open to grave
suspicion. Nevertheless it is hardly possible that such a general
tradition, and one so universally accepted, should have arisen without
good foundation. The evidence has been carefully studied by Heath in his
"Euclid," who concludes with these words: "On the whole, therefore, I
see no sufficient reason to question the tradition that, so far as Greek
geometry is concerned ..., Pythagoras was the first to introduce the
theorem ... and to give a general proof of it." That the fact was known
earlier, probably without the general proof, is recognized by all modern
writers.
[Illustration]
Pythagoras had studied in Egypt and possibly in the East before he
established his school at Crotona, in southern Italy. In Egypt, at any
rate, he could easily have found that a triangle with the sides 3, 4, 5,
is a right triangle, and Vitruvius (first century B.C.) tells us that he
taught this fact. The Egyptian _harpedonaptae_ (rope stretchers)
stretched ropes about pegs so as to make such a triangle for the purpose
of laying out a right angle in their surveying, just as our surveyors do
to-day. The great pyramids have an angle of slope such as is given by
this triangle. Indeed, a papyrus of the twelfth dynasty, lately
discovered at Kahun, in Egypt, refers to four of these triangles, such
as 1^2 + (3/4)^2 = (1 1/4)^2. This property seems to have been a matter
of common knowledge long before Pythagoras, even as far east as China.
He was, therefore, naturally led to attempt to prove the general
property which had already been recognized for special cases, and in
particular for the isosceles right triangle.
How Pythagoras proved the proposition is not known. It has been thought
that he used a proof by proportion, because Proclus says that Euclid
gave a new style of proof, and Euclid does not use proportion for this
purpose, while the subject, in incomplete form, was highly esteemed by
the Pythagoreans. Heath suggests that this is among the possibilities:
[Illustration]
[triangles]_ABC_ and _APC_ are similar.
[therefore] _AB_ x _AP_ = (_AC_)^2.
Similarly, _AB_ x _PB_ = (_BC_)^2.
[therefore] _AB_(_AP_ + _PB_) = (_AC_)^2 + (_BC_)^2,
or (_AB_)^2 = (_AC_)^2 + (_BC_)^2.
Others have thought that Pythagoras derived his proof from dissecting a
square and showing that the square on the hypotenuse must equal the sum
of the squares on the other two sides, in some such manner as this:
[Illustration: FIG. 1]
[Illustration: FIG. 2]
Here Fig. 1 is evidently _h_^2 + 4 [triangles].
Fig. 2 is evidently _a_^2 + _b_^2 + 4 [triangles].
[therefore] _h_^2 + 4 [triangles] = _a_^2 + _b_^2 + 4
[triangles], the [triangles] all being congruent.
[therefore] _h_^2 = _a_^2 + _b_^2.
The great Hindu mathematician, Bhaskara (born 1114 A.D.), proceeds in a
somewhat similar manner. He draws this figure, but gives no proof. It is
evident that he had in mind this relation:
[Illustration]
_h_^2 = 4 . _ab_/2 + (_b_ - _a_)^2 = _a_^2 + _b_^2.
A somewhat similar proof can be based upon the following figure:
[Illustration]
If the four triangles, 1 + 2 + 3 + 4, are taken away, there
remains the square on the hypotenuse. But if we take away the
two shaded rectangles, which equal the four triangles, there
remain the squares on the two sides. Therefore the square on
the hypotenuse must equal the sum of these two squares.
[Illustration]
It has long been thought that the truth of the proposition was first
observed by seeing the tiles on the floors of ancient temples. If they
were arranged as here shown, the proposition would be evident for the
special case of an isosceles right triangle.
The Hindus knew the proposition long before Bhaskara, however, and
possibly before Pythagoras. It is referred to in the old religious poems
of the Brahmans, the "Sulvasutras," but the date of these poems is so
uncertain that it is impossible to state that they preceded the sixth
century B.C.,[79] in which Pythagoras lived. The "Sulvasutra" of
Apastamba has a collection of rules, without proofs, for constructing
various figures. Among these is one for constructing right angles by
stretching cords of the following lengths: 3, 4, 5; 12, 16, 20; 15, 20,
25 (the two latter being multiples of the first); 5, 12, 13; 15, 36, 39;
8, 15, 17; 12, 35, 37. Whatever the date of these "Sulvasutras," there
is no evidence that the Indians had a definite proof of the theorem,
even though they, like the early Egyptians, recognized the general fact.
It is always interesting to a class to see more than one proof of a
famous theorem, and many teachers find it profitable to ask their pupils
to work out proofs that are (to them) original, often suggesting the
figure. Two of the best known historic proofs are here given.
The first makes the Pythagorean Theorem a special case of a proposition
due to Pappus (fourth century A.D.), relating to any kind of a triangle.
[Illustration]
Somewhat simplified, this proposition asserts that if _ABC_ is
_any_ kind of triangle, and _MC_, _NC_ are parallelograms on
_AC_, _BC_, the opposite sides being produced to meet at _P_;
and if _PC_ is produced making _QR_ = _PC_; and if the
parallelogram _AT_ is constructed, then _AT_ = _MC_ + _NC_.
For _MC_ = _AP_ = _AR_, having equal bases and equal altitudes.
Similarly, _NC_ = _QT_.
Adding, _MC_ + _NC_ = _AT_.
If, now, _ABC_ is a right triangle, and if _MC_ and _NC_ are
squares, it is easy to show that _AT_ is a square, and the
proposition reduces to the Pythagorean Theorem.
The Arab writer, Al-Nair[=i]z[=i] (died about 922 A.D.), attributes to
Th[=a]bit ben Qurra (826-901 A.D.) a proof substantially as follows:
[Illustration]
The four triangles _T_ can be proved congruent. Then if we take
from the whole figure _T_ and _T'_, we have left the squares on
the two sides of the right angle. If we take away the other two
triangles instead, we have left the square on the hypotenuse.
Therefore the former is equivalent to the latter.
A proof attributed to the great artist, Leonardo da Vinci (1452-1519),
is as follows:
[Illustration]
The construction of the following figure is evident. It is
easily shown that the four quadrilaterals _ABMX_, _XNCA_,
_SBCP_, and _SRQP_ are congruent.
[therefore] _ABMXNCA_ equals _SBCPQRS_ but is not congruent to
it, the congruent quadrilaterals being differently arranged.
Subtract the congruent triangles _MXN_, _ABC_, _RAQ_, and the
proposition is proved.[80]
The following is an interesting proof of the proposition:
Let _ABC_ be the original triangle, with _AB_ < _BC_. Turn the
triangle about _B_, through 90 deg., until it comes into the position
_A'BC'_. Then because it has been turned through 90 deg., _C'A'P_ will
be perpendicular to _AC_. Then
1/2(_AB_)^2 = [triangle]_ABA'_,
and 1/2(_BC'_)^2 = [triangle]_BC'C_,
because _BC_ = _BC'_.
[therefore] 1/2((_AB_)^2 + (_BC_)^2) =
[triangle]_ABA'_ + [triangle]_BC'C_.
[therefore] 1/2((_AB_)^2 + (_BC_)^2)
= [triangle]_AC'A'_ + [triangle]_A'C'C_
[Illustration]
(For [triangle]_ABA'_ + [triangle]_BC'A'_ + [triangle]_A'C'C_
is the second member of both equations.)
= 1/2_A'C'_ . _AP_ + 1/2_A'C'_ . _PC_
= 1/2_A'C'_ . _AC_
= 1/2(_AC_)^2.
[therefore] (_AB_)^2 + (_BC_)^2 = (_AC_)^2.
The Pythagorean Theorem, as it is generally called, has had other names.
It is not uncommonly called the _pons asinorum_ (see page 174) in
France. The Arab writers called it the Figure of the Bride, although the
reason for this name is unknown; possibly two being joined in one has
something to do with it. It has also been called the Bride's Chair, and
the shape of the Euclid figure is not unlike the chair that a slave
carries on his back, in which the Eastern bride is sometimes transported
to the wedding ceremony. Schopenhauer, the German philosopher, referring
to the figure, speaks of it as "a proof walking on stilts," and as "a
mouse-trap proof."
An interesting theory suggested by the proposition is that of computing
the sides of right triangles so that they shall be represented by
rational numbers. Pythagoras seems to have been the first to take up
this theory, although such numbers were applied to the right triangle
before his time, and Proclus tells us that Plato also contributed to it.
The rule of Pythagoras, put in modern symbols, was as follows:
_n_^2 + ((_n_^2 - 1)/2)^2 = ((_n_^2 + 1)/2)^2,
the sides being _n_, (_n_^2 - 1)/2, and (_n_^2 + 1)/2. If for
_n_ we put 3, we have 3, 4, 5. If we take the various odd
numbers, we have
_n_ = 1, 3, 5, 7, 9, ...,
(_n_^2 - 1)/2 = 0, 4, 12, 24, 40, ...,
(_n_^2 + 1)/2 = 1, 5, 13, 25, 41, ....
Of course _n_ may be even, giving fractional values. Thus, for _n_ = 2
we have for the three sides, 2, 1 1/2, 2 1/2. Other formulas are also
known. Plato's, for example, is as follows:
(2_n_)^2 + (_n_^2 - 1)^2 = (_n_^2 + 1)^2.
If 2_n_ = 2, 4, 6, 8, 10, ...,
then _n_^2 - 1 = 0, 3, 8, 15, 24, ...,
and _n_^2 + 1 = 2, 5, 10, 17, 26, ....
This formula evidently comes from that of Pythagoras by doubling the
sides of the squares.[81]
THEOREM. _In any triangle the square of the side opposite an acute angle
is equal to the sum of the squares of the other two sides diminished by
twice the product of one of those sides by the projection of the other
upon that side._
THEOREM. _A similar statement for the obtuse triangle._
These two propositions are usually proved by the help of the Pythagorean
Theorem. Some writers, however, actually construct the squares and give
a proof similar to the one in that proposition. This plan goes back at
least to Gregoire de St. Vincent (1647).
[Illustration]
It should be observed that
_a_^2 = _b_^2 + _c_^2 - 2_b'c_.
If [L]_A_ = 90 deg., then _b'_ = 0, and this becomes
_a_^2 = _b_^2 + _c_^2.
If [L]_A_ is obtuse, then _b'_ passes through 0 and becomes
negative, and _a_^2 = _b_^2 + _c_^2 + 2_b'c_.
Thus we have three propositions in one.
[Illustration]
At the close of Book IV many geometries give as an exercise, and some
give as a regular proposition, the celebrated problem that bears the
name of Heron of Alexandria, namely, to compute the area of a triangle
in terms of its sides. The result is the important formula
Area = [sqrt](_s_(_s_ - _a_)(_s_ - _b_)(_s_ - _c_)),
where _a_, _b_, and _c_ are the sides, and _s_ is the semiperimeter
1/2(_a_ + _b_ + _c_). As a practical application the class may be able
to find a triangular piece of land, as here shown, and to measure the
sides. If the piece is clear, the result may be checked by measuring the
altitude and applying the formula _a_ = 1/2_bh_.
It may be stated to the class that Heron's formula is only a special
case of the more general one developed about 640 A.D., by a famous
Hindu mathematician, Brahmagupta. This formula gives the area of an
inscribed quadrilateral as
[sqrt]((_s_ - _a_)(_s_ - _b_)(_s_ - _c_)(_s_ - _d_)), where
_a_, _b_, _c_, and _d_ are the sides and _s_ is the semiperimeter. If
_d_ = 0, the quadrilateral becomes a triangle and we have Heron's
formula.[82]
At the close of Book IV, also, the geometric equivalents of the
algebraic formulas for (_a_ + _b_)^2, (_a_ - _b_)^2, and
(_a_ + _b_)(_a_ - _b_) are given. The class may like to know that
Euclid had no algebra and was compelled to prove such relations as these
by geometry, while we do it now much more easily by algebraic
multiplication.
FOOTNOTES:
[79] See, for example, G. B. Kaye, "The Source of Hindu Mathematics," in
the _Journal of the Royal Asiatic Society_, July, 1910.
[80] An interesting Japanese proof of this general character may be seen
in Y. Mikami, "Mathematical Papers from the Far East," p. 127, Leipzig,
1910.
[81] Special recognition of indebtedness to H. A. Naber's "Das Theorem
des Pythagoras" (Haarlem, 1908), Heath's "Euclid," Gow's "History of
Greek Mathematics," and Cantor's "Geschichte" is due in connection with
the Pythagorean Theorem.
[82] The rule was so ill understood that Bhaskara (twelfth century) said
that Brahmagupta was a "blundering devil" for giving it ("Lilavati," Sec.
172).
CHAPTER XVIII
THE LEADING PROPOSITIONS OF BOOK V
[Illustration]
Book V treats of regular polygons and circles, and includes the
computation of the approximate value of [pi]. It opens with a definition
of a regular polygon as one that is both equilateral and equiangular.
While in elementary geometry the only regular polygons studied are
convex, it is interesting to a class to see that there are also regular
cross polygons. Indeed, the regular cross pentagon was the badge of the
Pythagoreans, as Lucian (_ca._ 100 B.C.) and an unknown commentator on
Aristophanes (_ca._ 400 B.C.) tell us. At the vertices of this polygon
the Pythagoreans placed the Greek letters signifying "health."
Euclid was not interested in the measure of the circle, and there is
nothing in his "Elements" on the value of [pi]. Indeed, he expressly
avoided numerical measures of all kinds in his geometry, wishing the
science to be kept distinct from that form of arithmetic known to the
Greeks as logistic, or calculation. His Book IV is devoted to the
construction of certain regular polygons, and his propositions on this
subject are now embodied in Book V as it is usually taught in America.
If we consider Book V as a whole, we are struck by three features. Of
these the first is the pure geometry involved, and this is the essential
feature to be emphasized. The second is the mensuration of the circle, a
relatively unimportant piece of theory in view of the fact that the
pupil is not ready for incommensurables, and a feature that imparts no
information that the pupil did not find in arithmetic. The third is the
somewhat interesting but mathematically unimportant application of the
regular polygons to geometric design.
As to the mensuration of the circle it is well for us to take a broad
view before coming down to details. There are only four leading
propositions necessary for the mensuration of the circle and the
determination of the value of [pi]. These are as follows: (1) The
inscribing of a regular hexagon, or any other regular polygon of which
the side is easily computed in terms of the radius. We may start with a
square, for example, but this is not so good as the hexagon because its
side is incommensurable with the radius, and its perimeter is not as
near the circumference. (2) The perimeters of similar regular polygons
are proportional to their radii, and their areas to the squares of the
radii. It is now necessary to state, in the form of a postulate if
desired, that the circle is the limit of regular inscribed and
circumscribed polygons as the number of sides increases indefinitely,
and hence that (2) holds for circles. (3) The proposition relating to
the area of a regular polygon, and the resulting proposition relating to
the circle. (4) Given the side of a regular inscribed polygon, to find
the side of a regular inscribed polygon of double the number of sides.
It will thus be seen that if we were merely desirous of approximating
the value of [pi], and of finding the two formulas _c_ = 2[pi]_r_ and
_a_ = [pi]_r_^2, we should need only four propositions in this book upon
which to base our work. It is also apparent that even if the
incommensurable cases are generally omitted, the notion of _limit_ is
needed at this time, and that it must briefly be reviewed before
proceeding further.
There is, however, a much more worthy interest than the mere mensuration
of the circle, namely, the construction of such polygons as can readily
be formed by the use of compasses and straightedge alone. The pleasure
of constructing such figures and of proving that the construction is
correct is of itself sufficient justification for the work. As to the
use of such figures in geometric design, some discussion will be offered
at the close of this chapter.
The first few propositions include those that lead up to the mensuration
of the circle. After they are proved it is assumed that the circle is
the limit of the regular inscribed and circumscribed polygons as the
number of sides increases indefinitely. This may often be proved with
some approach to rigor by a few members of an elementary class, but it
is the experience of teachers that the proof is too difficult for most
beginners, and so the assumption is usually made in the form of an
unproved theorem.
The following are some of the leading propositions of this book:
THEOREM. _Two circumferences have the same ratio as their radii._
This leads to defining the ratio of the circumference to the diameter as
[pi]. Although this is a Greek letter, it was not used by the Greeks to
represent this ratio. Indeed, it was not until 1706 that an English
writer, William Jones, in his "Synopsis Palmariorum Matheseos," used it
in this way, it being the initial letter of the Greek word for
"periphery." After establishing the properties that _c_ = 2[pi]_r_, and
_a_ = [pi]_r_^2, the textbooks follow the Greek custom and proceed to
show how to inscribe and circumscribe various regular polygons, the
purpose being to use these in computing the approximate numerical value
of [pi]. Of these regular polygons two are of special interest, and
these will now be considered.
PROBLEM. _To inscribe a regular hexagon in a circle._
That the side of a regular inscribed hexagon equals the radius must have
been recognized very early. The common divisions of the circle in
ancient art are into four, six, and eight equal parts. No draftsman
could have worked with a pair of compasses without quickly learning how
to effect these divisions, and that compasses were early used is
attested by the specimens of these instruments often seen in museums.
There is a tradition that the ancient Babylonians considered the circle
of the year as made up of 360 days, whence they took the circle as
composed of 360 steps or grades (degrees). This tradition is without
historic foundation, however, there being no authority in the
inscriptions for this assumption of the 360-division by the Babylonians,
who seem rather to have preferred 8, 12, 120, 240, and 480 as their
division numbers. The story of 360 deg. in the Babylonian circle seems to
start with Achilles Tatius, an Alexandrian grammarian of the second or
third century A.D. It is possible, however, that the Babylonians got
their favorite number 60 (as in 60 seconds make a minute, 60 minutes
make an hour or degree) from the hexagon in a circle (1/6 of 360 deg. = 60 deg.),
although the probabilities seem to be that there is no such
connection.[83]
The applications of this problem to mensuration are numerous. The fact
that we may use for tiles on a floor three regular polygons--the
triangle, square, and hexagon--is noteworthy, a fact that Proclus tells
us was recognized by Pythagoras. The measurement of the regular
hexagon, given one side, may be used in computing sections of hexagonal
columns, in finding areas of flower beds, and in other similar cases.
This review of the names of the polygons offers an opportunity to
impress their etymology again on the mind. In this case, for example, we
have "hexagon" from the Greek words for "six" and "angle."
PROBLEM. _To inscribe a regular decagon in a given circle._
Euclid states the problem thus: _To construct an isosceles triangle
having each of the angles at the base double of the remaining one._ This
makes each base angle 72 deg. and the vertical angle 36 deg., the latter being
the central angle of a regular decagon,--essentially our present method.
This proposition seems undoubtedly due to the Pythagoreans, as tradition
has always asserted. Proclus tells us that Pythagoras discovered "the
construction of the cosmic figures," or the five regular polyhedrons,
and one of these (the dodecahedron) involves the construction of the
regular pentagon.
Iamblichus (_ca._ 325 A.D.) tells us that Hippasus, a Pythagorean, was
said to have been drowned for daring to claim credit for the
construction of the regular dodecahedron, when by the rules of the
brotherhood all credit should have been assigned to Pythagoras.
If a regular polygon of _s_ sides can be inscribed, we may bisect the
central angles, and therefore inscribe one of 2_s_ sides, and then of
4_s_ sides, and then of 8_s_ sides, and in general of 2^{_n_}_s_ sides. This
includes the case of _s_ = 2 and _n_ = 0, for we can inscribe a regular
polygon of two sides, the angles being, by the usual formula,
2(2 - 2)/2 = 0, although, of course, we never think of two equal and
coincident lines as forming what we might call a _digon_.
We therefore have the following regular polygons:
From the equilateral triangle, regular polygons of 2^_n_ . 3 sides;
From the square, regular polygons of 2^_n_ sides;
From the regular pentagon, regular polygons of 2^_n_ . 5 sides;
From the regular pentedecagon, regular polygons of 2^_n_ . 15 sides.
This gives us, for successive values of _n_, the following regular
polygons of less than 100 sides:
From 2^_n_ . 3, 3, 6, 12, 24, 48, 96;
From 2^_n_, 2, 4, 8, 16, 32, 64;
From 2^_n_ . 5, 5, 10, 20, 40, 80;
From 2^_n_ . 15, 15, 30, 60.
[Illustration: ROMAN MOSAIC FOUND AT POMPEII]
Gauss (1777-1855), a celebrated German mathematician, proved (in 1796)
that it is possible also to inscribe a regular polygon of 17 sides, and
hence polygons of 2^_n_ . 17 sides, or 17, 34, 68, ..., sides, and also
3 . 17 = 51 and 5 . 17 = 85 sides, by the use of the compasses and
straightedge, but the proof is not adapted to elementary geometry. In
connection with the study of the regular polygons some interest attaches
to the reference to various forms of decorative design. The mosaic
floor, parquetry, Gothic windows, and patterns of various kinds often
involve the regular figures. If the teacher uses such material, care
should be taken to exemplify good art. For example, the equilateral
triangle and its relation to the regular hexagon is shown in the picture
of an ancient Roman mosaic floor on page 274.[84] In the next
illustration some characteristic Moorish mosaic work appears, in which
it will be seen that the basal figure is the square, although at first
sight this would not seem to be the case.[85] This is followed by a
beautiful Byzantine mosaic, the original of which was in five colors of
marble. Here it will be seen that the equilateral triangle and the
regular hexagon are the basal figures, and a few of the properties of
these polygons might be derived from the study of such a design. In the
Arabic pattern on page 276 the dodecagon appears as the basis, and the
remarkable powers of the Arab designer are shown in the use of symmetry
without employing regular figures.
[Illustration: MOSAIC FROM DAMASCUS]
[Illustration: MOSAIC FROM AN ANCIENT BYZANTINE CHURCH]
PROBLEM. _Given the side and the radius of a regular inscribed polygon,
to find the side of the regular inscribed polygon of double the number
of sides._
[Illustration: ARABIC PATTERN]
The object of this proposition is, of course, to prepare the way for
finding the perimeter of a polygon of 2_n_ sides, knowing that of _n_
sides. The Greek plan was generally to use both an inscribed and a
circumscribed polygon, thus approaching the circle as a limit both from
without and within. This is more conclusive from the ultrascientific
point of view, but it is, if anything, less conclusive to a beginner,
because he does not so readily follow the proof. The plan of using the
two polygons was carried out by Archimedes of Syracuse (287-212 B.C.) in
his famous method of approximating the value of [pi], although before
him Antiphon (fifth century B.C.) had inscribed a square (or equilateral
triangle) as a basis for the work, and Bryson (his contemporary) had
attacked the problem by circumscribing as well as inscribing a regular
polygon.
PROBLEM. _To find the numerical value of the ratio of the circumference
of a circle to its diameter._
As already stated, the usual plan of the textbooks is in part the method
followed by Archimedes. It is possible to start with any regular polygon
of which the side can conveniently be found in terms of the radius. In
particular we might begin with an inscribed square instead of a regular
hexagon. In this case we should have
_Length of Side_ _Perimeter_
_s__{4} = 1.414... = 1.41 5.66
_s__{8} = [sqrt](2 - [sqrt](4 - 1.414^2)) = 0.72 5.76
and so on.
It is a little easier to start with the hexagon, however, for we are
already nearer the circle, and the side and perimeter are both
commensurable with the radius. It is not, of course, intended that
pupils should make the long numerical calculations. They may be required
to compute _s__{12} and possibly _s__{24}, but aside from this they are
expected merely to know the process.
If it were possible to find the value of [pi] exactly, we could find the
circumference exactly in terms of the radius, since c = 2[pi]_r_. If we
could find the circumference exactly, we could find the area exactly,
since _a_ = [pi]_r_^2. If we could find the area exactly in this form,
[pi] times a square, we should have a rectangle, and it is easy to
construct a square equivalent to any rectangle. Therefore, if we could
find the value of [pi] exactly, we could construct a square with area
equivalent to the area of the circle; in other words, we could "square
the circle." We could also, as already stated, construct a straight line
equivalent to the circumference; in other words, we could "rectify the
circumference." These two problems have attracted the attention of the
world for over two thousand years, but on account of their interrelation
they are usually spoken of as a single problem, "to square the circle."
Since we can construct [sqrt]_a_ by means of the straightedge and
compasses, it would be possible for us to square the circle if we could
express [pi] by a finite number of square roots. Conversely, every
geometric construction reduces to the intersection of two straight
lines, of a straight line and a circle, or of two circles, and is
therefore equivalent to a rational operation or to the extracting of a
square root. Hence a geometric construction cannot be effected by the
straightedge and compasses unless it is equivalent to a series of
rational operations or to the extracting of a finite number of square
roots. It was proved by a German professor, Lindemann, in 1882, that
[pi] cannot be expressed as an algebraic number, that is, as the root of
an equation with rational coefficients, and hence it cannot be found by
the above operations, and, furthermore, that the solution of this famous
problem is impossible by elementary geometry.[86]
It should also be pointed out to the student that for many practical
purposes one of the limits of [pi] stated by Archimedes, namely, 3 1/7,
is sufficient. For more accurate work 3.1416 is usually a satisfactory
approximation. Indeed, the late Professor Newcomb stated that "ten
decimal places are sufficient to give the circumference of the earth to
the fraction of an inch, and thirty decimal places would give the
circumference of the whole visible universe to a quantity imperceptible
with the most powerful microscope."
Probably the earliest approximation of the value of [pi] was 3. This
appears very commonly in antiquity, as in I Kings vii, 23, and 2
Chronicles iv, 2. In the Ahmes papyrus (_ca._ 1700 B.C.) there is a rule
for finding the area of the circle, expressed in modern symbols as
(8/9)^2_d_^2, which makes [pi] = 256/81 or 3.1604....
Archimedes, using a plan somewhat similar to ours, found that [pi] lay
between 3 1/7 and 3 10/71. Ptolemy, the great Greek astronomer,
expressed the value as 3 17/120, or 3.14166.... The fact that Ptolemy
divided his diameter into 120 units and his circumference into 360 units
probably shows, however, the influence of the ancient value 3.
In India an approximate value appears in a certain poem written before
the Christian era, but the date is uncertain. About 500 A.D. Aryabhatta
(or possibly a later writer of the same name) gave the value
62832/20000, or 3.1416. Brahmagupta, another Hindu (born 598 A.D.), gave
[sqrt](10), and this also appears in the writings of the Chinese
mathematician Chang Heng (78-139 A.D.). A little later in China, Wang
Fan (229-267) gave 142 / 45, or 3.1555...; and one of his
contemporaries, Lui Hui, gave 157 / 50, or 3.14. In the fifth century
Ch'ung-chih gave as the limits of [pi], 3.1415927 and 3.1415926, from
which he inferred that 22/7 and 355/113 were good approximations,
although he does not state how he came to this conclusion.
In the Middle Ages the greatest mathematician of Italy, Leonardo
Fibonacci, or Leonardo of Pisa (about 1200 A.D.), found as limits
3.1427... and 3.1410.... About 1600 the Chinese value 355/113 was
rediscovered by Adriaen Anthonisz (1527-1607), being published by his
son, who is known as Metius (1571-1635), in the year 1625. About the
same period the French mathematician Vieta (1540-1603) found the value
of [pi] to 9 decimal places, and Adriaen van Rooman (1561-1615) carried
it to 17 decimal places, and Ludolph van Ceulen (1540-1610) to 35
decimal places. It was carried to 140 decimal places by Georg Vega (died
in 1793), to 200 by Zacharias Dase (died in 1844), to 500 by Richter
(died in 1854), and more recently by Shanks to 707 decimal places.
There have been many interesting formulas for [pi], among them being the
following:
[pi]/2 = 2/1 . 2/3 . 4/3 . 4/5 . 6/5 . 6/7 . 8/7 . 8/9 . ....
(Wallis, 1616-1703)
4/[pi] = 1 + 1/2
+ 9/2
+ 25/2
+ 49/2
+ .... (Brouncker, 1620-1684)
[pi]/4 = 1 - 1/3 + 1/5 - 1/7 + .... (Gregory, 1638-1675)
[pi]/6 =
[sqrt](1/3) . (1 - 1/(3 . 3) + 1/(3^2 . 5) - 1/(3^3 . 7) + ...).
[pi]/2 = (log _i_) / _i_. (Bernoulli)
[pi]/(2[sqrt](3)) = 1 - 1/5 + 1/7 - 1/11 + 1/13 - 1/17 + 1/19...,
thus connecting the primes.
[pi]^2/16 = 1 - 1/2^2 - 1/3^2 + 1/4^2 - 1/5^2 + 1/6^2 - 1/7^2 -
1/8^2 + 1/9^2 + ....
[pi]/2 = _x_/2 + sin _x_ + (sin^2 _x_) / 2 + (sin^3 _x_) / 3 + ....
(0 < _x_ < 2[pi])
[pi]/4 = 3/4 + 1/(2 . 3 . 4) - 1/(4 . 5 . 6) + 1/(6 . 7 . 8) - ....
2[pi]^2/3 = 7 - (1/(1 . 3) + 1/(3 . 6) + 1/(6 . 10) + ...).
[pi] =
2^_n_[sqrt](2 - [sqrt](2 + [sqrt](2 + [sqrt](2 + [sqrt](2...))))).
Students of elementary geometry are not prepared to appreciate it, but
teachers will be interested in the remarkable formula discovered by
Euler (1707-1783), the great Swiss mathematician, namely,
1 + _e_^{_i_[pi]} = 0. In this relation are included the five most
interesting quantities in mathematics,--zero, the unit, the base of the
so-called Napierian logarithms, _i_ = [sqrt](-1), and [pi]. It was by
means of this relation that the transcendence of _e_ was proved by the
French mathematician Hermite, and the transcendence of [pi] by the
German Lindemann.
[Illustration]
There should be introduced at this time, if it has not already been
done, the proposition of the lunes of Hippocrates (_ca._ 470 B.C.), who
proved a theorem that asserts, in somewhat more general form, that if
three semicircles be described on the sides of a right triangle as
diameters, as shown, the lunes _L_ + _L'_ are together equivalent to the
triangle _T_.
[Illustration]
In the use of the circle in design one of the simplest forms suggested
by Book V is the trefoil (three-leaf), as here shown, with the necessary
construction lines. This is a very common ornament in architecture, both
with rounded ends and with the ends slightly pointed.
The trefoil is closely connected with hexagonal designs, since the
regular hexagon is formed from the inscribed equilateral triangle by
doubling the number of sides. The following are designs that are easily
made:
[Illustration]
It is not very profitable, because it is manifestly unreal, to measure
the parts of such figures, but it offers plenty of practice in numerical
work.
[Illustration: CHOIR OF LINCOLN CATHEDRAL]
[Illustration: PORCH OF LINCOLN CATHEDRAL]
In the illustrations of the Gothic windows given in Chapter XV only the
square and circle were generally involved. Teachers who feel it
necessary or advisable to go outside the regular work of geometry for
the purpose of increasing the pupil's interest or of training his hand
in the drawing of figures will find plenty of designs given in any
pictures of Gothic cathedrals. For example, this picture of the noble
window in the choir of Lincoln Cathedral shows the use of the square,
hexagon, and pentagon. In the porch of the same cathedral, shown in the
next illustration, the architect has made use of the triangle, square,
and pentagon in planning his ornamental stonework. It is possible to add
to the work in pure geometry some work in the mensuration of the
curvilinear figures shown in these designs. This form of mensuration is
not of much value, however, since it places before the pupil a problem
that he sees at once is fictitious, and that has no human interest.
[Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]
[Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND SQUARES]
[Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND THE EQUILATERAL
TRIANGLE]
[Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND THE REGULAR HEXAGON]
The designs given on page 283 involve chiefly the square as a basis, but
it will be seen from one of the figures that the equilateral triangle
and the hexagon also enter. The possibilities of endless variation of a
single design are shown in the illustration on page 284, the basis in
this case being the square. The variations in the use of the triangle
and hexagon have been the object of study of many designers of Gothic
windows, and some examples of these forms are shown on page 285. In
more simple form this ringing of the changes on elementary figures is
shown on page 286. Some teachers have used color work with such designs
for the purpose of increasing the interest of their pupils, but the
danger of thus using the time with no serious end in view will be
apparent.
[Illustration]
In the matter of the mensuration of the circle the annexed design has
some interest. The figure is not uncommon in decoration, and it is
interesting to show, as a matter of pure geometry, that the area of the
circle is divided into three equal portions by means of the four
interior semicircles.
[Illustration]
An important application of the formula _a_ = [pi]_r_^2 is seen in the
area of the annulus, or ring, the formula being _a_ =
[pi]_r_^2 - [pi]_r'_^2 = [pi](_r_^2 - _r'_^2) =
[pi](_r_ + _r'_)(_r_ - _r'_).
It is used in finding the area of the cross section of
pipes, and this is needed when we wish to compute the volume of the iron
used.
Another excellent application is that of finding the area of the surface
of a cylinder, there being no reason why such simple cases from solid
geometry should not furnish working material for plane geometry,
particularly as they have already been met by the pupils in arithmetic.
A little problem that always has some interest for pupils is one that
Napoleon is said to have suggested to his staff on his voyage to Egypt:
To divide a circle into four equal parts by the use of circles alone.
[Illustration]
Here the circles _B_ are tangent to the circle _A_ at the
points of division. Furthermore, considering areas, and taking
_r_ as the radius of _A_, we have _A_ = [pi]_r_^2, and
_B_ = [pi](_r_/2)^2. Hence _B_ = 1/4_A_, or the sum of the areas of
the four circles _B_ equals the area of _A_. Hence the four
_D'_s must equal the four _C'_s, and _D_ = _C_. The rest of the
argument is evident. The problem has some interest to pupils
aside from the original question suggested by Napoleon.
At the close of plane geometry teachers may find it helpful to have the
class make a list of the propositions that are actually used in proving
other propositions, and to have it appear what ones are proved by them.
This forms a kind of genealogical tree that serves to fix the parent
propositions in mind. Such a work may also be carried on at the close of
each book, if desired. It should be understood, however, that certain
propositions are used in the exercises, even though they are not
referred to in subsequent propositions, so that their omission must not
be construed to mean that they are not important.
An exercise of distinctly less value is the classification of the
definitions. For example, the classification of polygons or of
quadrilaterals, once so popular in textbook making, has generally been
abandoned as tending to create or perpetuate unnecessary terms. Such
work is therefore not recommended.
FOOTNOTES:
[83] Bosanquet and Sayre, "The Babylonian Astronomy," _Monthly Notices
of the Royal Asiatic Society_, Vol. XL, p. 108.
[84] This and the three illustrations following are from Kolb, loc. cit.
[85] This was in five colors of marble.
[86] The proof is too involved to be given here. The writer has set it
forth in a chapter on the transcendency of [pi] in a work soon to be
published by Professor Young of The University of Chicago.
CHAPTER XIX
THE LEADING PROPOSITIONS OF BOOK VI
There have been numerous suggestions with respect to solid geometry, to
the effect that it should be more closely connected with plane geometry.
The attempt has been made, notably by Meray in France and de Paolis in
Italy, to treat the corresponding propositions of plane and solid
geometry together; as, for example, those relating to parallelograms and
parallelepipeds, and those relating to plane and spherical triangles.
Whatever the merits of this plan, it is not feasible in America at
present, partly because of the nature of the college-entrance
requirements. While it is true that to a boy or girl a solid is more
concrete than a plane, it is not true that a geometric solid is more
concrete than a geometric plane. Just as the world developed its solid
geometry, as a science, long after it had developed its plane geometry,
so the human mind grasps the ideas of plane figures earlier than those
of the geometric solid.
There is, however, every reason for referring to the corresponding
proposition of plane geometry when any given proposition of solid
geometry is under consideration, and frequent references of this kind
will be made in speaking of the propositions in this and the two
succeeding chapters. Such reference has value in the apperception of the
various laws of solid geometry, and it also adds an interest to the
subject and creates some approach to power in the discovery of new
facts in relation to figures of three dimensions.
The introduction to solid geometry should be made slowly. The pupil has
been accustomed to seeing only plane figures, and therefore the drawing
of a solid figure in the flat is confusing. The best way for the teacher
to anticipate this difficulty is to have a few pieces of cardboard, a
few knitting needles filed to sharp points, a pine board about a foot
square, and some small corks. With the cardboard he can illustrate
planes, whether alone, intersecting obliquely or at right angles, or
parallel, and he can easily illustrate the figures given in the textbook
in use. There are models of this kind for sale, but the simple ones made
in a few seconds by the teacher or the pupil have much more meaning. The
knitting needles may be stuck in the board to illustrate perpendicular
or oblique lines, and if two or more are to meet in a point, they may be
held together by sticking them in one of the small corks. Such homely
apparatus, costing almost nothing, to be put together in class, seems
much more real and is much more satisfactory than the German models.[87]
An extensive use of models is, however, unwise. The pupil must learn
very early how to visualize a solid from the flat outline picture, just
as a builder or a mechanic learns to read his working drawings. To have
a model for each proposition, or even to have a photograph or a
stereoscopic picture, is a very poor educational policy. A textbook may
properly illustrate a few propositions by photographic aids, but after
that the pupil should use the kind of figures that he must meet in his
mathematical work. A child should not be kept in a perambulator all his
life,--he must learn to walk if he is to be strong and grow to maturity;
and it is so with a pupil in the use of models in solid geometry.[88]
The case is somewhat similar with respect to colored crayons. They have
their value and their proper place, but they also have their strict
limitations. It is difficult to keep their use within bounds; pupils
come to use them to make pleasing pictures, and teachers unconsciously
fall into the same habit. The value of colored crayons is two-fold: (1)
they sometimes make two planes stand out more clearly, or they serve to
differentiate some line that is under consideration from others that are
not; (2) they enable a class to follow a demonstration more easily by
hearing of "the red plane perpendicular to the blue one," instead of
"the plane _MN_ perpendicular to the plane _PQ_." But it should always
be borne in mind that in practical work we do not have colored ink or
colored pencils commonly at hand, nor do we generally have colored
crayons. Pupils should therefore become accustomed to the pencil and the
white crayon as the regulation tools, and in general they should use
them. The figures may not be as striking, but they are more quickly made
and they are more practical.
The definition of "plane" has already been discussed in Chapter XII, and
the other definitions of Book VI are not of enough interest to call for
special remark. The axioms are the same as in plane geometry, but there
is at least one postulate that needs to be added, although it would be
possible to state various analogues of the postulates of plane geometry
if we cared unnecessarily to enlarge the number.
The most important postulate of solid geometry is as follows: _One
plane, and only one, can be passed through two intersecting straight
lines._ This is easily illustrated, as in most textbooks, as also are
three important corollaries derived from it:
1. _A straight line and a point not in the line determine a plane._ Of
course this may be made the postulate, as may also the next one, the
postulate being placed among the corollaries, but the arrangement here
adopted is probably the most satisfactory for educational purposes.
2. _Three points not in a straight line determine a plane._ The common
question as to why a three-legged stool stands firmly, while a
four-legged table often does not, will add some interest at this point.
3. _Two parallel lines determine a plane._ This requires a slight but
informal proof to show that it properly follows as a corollary from the
postulate, but a single sentence suffices.
While studying this book questions of the following nature may arise
with an advanced class, or may be suggested to those who have had higher
algebra:
How many straight lines are in general (that is, at the most) determined
by _n_ points in space? Two points determine 1 line, a third point adds
(in general, in all these cases) 2 more, a fourth point adds 3 more, and
an _n_th point _n_ - 1 more. Hence the maximum is
1 + 2 + 3 + ... + (_n_ - 1), or _n_(_n_-1)/2, which the pupil will
understand if he has studied arithmetical progression. The maximum
number of intersection points of _n_ straight lines in the same plane is
also _n_(_n_ - 1)/2.
How many straight lines are in general determined by _n_ planes? The
answer is the same, _n_(_n_ - 1)/2.
How many planes are in general determined by _n_ points in space? Here
the answer is 1 + 3 + 6 + 10 + ... + (_n_ - 2)(_n_ - 1)/2, or
_n_(_n_ - 1)(_n_ - 2)/(1 x 2 x 3). The same number of points is
determined by _n_ planes.
THEOREM. _If two planes cut each other, their intersection is a straight
line._
Among the simple illustrations are the back edges of the pages of a
book, the corners of the room, and the simple test as to whether the
edge of a card is straight by testing it on a plane. It is well to call
attention to the fact that if two intersecting straight lines move
parallel to their original position, and so that their intersection
rests on a straight line not in the plane of those lines, the figure
generated will be that of this proposition. In general, if we cut
through any figure of solid geometry in some particular way, we are
liable to get the figure of a proposition in plane geometry, as will
frequently be seen.
THEOREM. _If a straight line is perpendicular to each of two other
straight lines at their point of intersection, it is perpendicular to
the plane of the two lines._
If students have trouble in visualizing the figure in three dimensions,
some knitting needles through a piece of cardboard will make it clear.
Teachers should call attention to the simple device for determining if a
rod is perpendicular to a board (or a pipe to a floor, ceiling, or
wall), by testing it twice, only, with a carpenter's square. Similarly,
it may be asked of a class, How shall we test to see if the corner
(line) of a room is perpendicular to the floor, or if the edge of a box
is perpendicular to one of the sides?
In some elementary and in most higher geometries the perpendicular is
called a _normal_ to the plane.
THEOREM. _All the perpendiculars that can be drawn to a straight line at
a given point lie in a plane which is perpendicular to the line at the
given point._
Thus the hands of a clock pass through a plane as the hands revolve, if
they are, as is usual, perpendicular to the axis; and the same is true
of the spokes of a wheel, and of a string with a stone attached, swung
as rapidly as possible about a boy's arm as an axis. A clock pendulum
too swings in a plane, as does the lever in a pair of scales.
THEOREM. _Through a given point within or without a plane there can be
one perpendicular to a given plane, and only one._
This theorem is better stated to a class as two theorems.
Thus a plumb line hanging from a point in the ceiling, without swinging,
determines one definite point in the floor; and, conversely, if it
touches a given point in the floor, it must hang from one definite point
in the ceiling. It should be noticed that if we cut through this figure,
on the perpendicular line, we shall have the figure of the corresponding
proposition in plane geometry, namely, that there can be, under similar
circumstances, only one perpendicular to a line.
THEOREM. _Oblique lines drawn from a point to a plane, meeting the plane
at equal distances from the foot of the perpendicular, are equal, etc._
There is no objection to speaking of a right circular cone in connection
with this proposition, and saying that the slant height is thus proved
to be constant. The usual corollary, that if the obliques are equal they
meet the plane in a circle, offers a new plan of drawing a circle. A
plumb line that is a little too long to reach the floor will, if swung
so as just to touch the floor, describe a circle. A 10-foot pole
standing in a 9-foot room will, if it moves so as to touch constantly a
fixed point on either the floor or the ceiling, describe a circle on the
ceiling or floor respectively.
One of the corollaries states that the locus of points in space
equidistant from the extremities of a straight line is the plane
perpendicular to this line at its middle point. This has been taken by
some writers as the definition of a plane, but it is too abstract to be
usable. It is advisable to cut through the figure along the given
straight line, and see that we come back to the corresponding
proposition in plane geometry.
A good many ships have been saved from being wrecked by the principle
involved in this proposition.
[Illustration]
If a dangerous shoal _A_ is near a headland _H_, the angle
_HAX_ is measured and is put down upon the charts as the
"vertical danger angle." Ships coming near the headland are
careful to keep far enough away, say at _S_, so that the angle
_HSX_ shall be less than this danger angle. They are then sure
that they will avoid the dangerous shoal.
Related to this proposition is the problem of supporting a tall iron
smokestack by wire stays. Evidently three stays are needed, and they
are preferably placed at the vertices of an equilateral triangle, the
smokestack being in the center. The practical problem may be given of
locating the vertices of the triangle and of finding the length of each
stay.
THEOREM. _Two straight lines perpendicular to the same plane are
parallel._
Here again we may cut through the figure by the plane of the two
parallels, and we get the figure of plane geometry relating to lines
that are perpendicular to the same line. The proposition shows that the
opposite corners of a room are parallel, and that therefore they lie in
the same plane, or are _coplanar_, as is said in higher geometry.
It is interesting to a class to have attention called to the corollary
that if two straight lines are parallel to a third straight line, they
are parallel to each other; and to have the question asked why it is
necessary to prove this when the same thing was proved in plane
geometry. In case the reason is not clear, let some student try to apply
the proof used in plane geometry.
THEOREM. _Two planes perpendicular to the same straight line are
parallel._
Besides calling attention to the corresponding proposition of plane
geometry, it is well now to speak of the fact that in propositions
involving planes and lines we may often interchange these words. For
example, using "line" for "straight line," for brevity, we have:
One _line_ does not determine One _plane_ does not determine
a _plane_. a _line_.
Two intersecting _lines_ Two intersecting _planes_ determine
determine a _plane_. a _line_.
Two _lines_ perpendicular to Two _planes_ perpendicular to
a _plane_ are parallel. a _line_ are parallel.
If one of two parallel _lines_ If one of two parallel _planes_
is perpendicular to a _plane_, the is perpendicular to a _line_, the
other is also perpendicular to other is also perpendicular to
the _plane_. the _line_.
If two _lines_ are parallel, every If two _planes_ are parallel,
_plane_ containing one of the every _line_ in one of the _planes_
_lines_ is parallel to the other is parallel to the other _plane_.
_line_.
THEOREM. _The intersections of two parallel planes by a third plane are
parallel lines._
Thus one of the edges of a box is parallel to the next succeeding edge
if the opposite faces are parallel, and in sawing diagonally through an
ordinary board (with rectangular cross section) the section is a
parallelogram.
THEOREM. _A straight line perpendicular to one of two parallel planes is
perpendicular to the other also._
Notice (1) the corresponding proposition in plane geometry; (2) the
proposition that results from interchanging "plane" and (straight)
"line."
THEOREM. _If two intersecting straight lines are each parallel to a
plane, the plane of these lines is parallel to that plane._
Interchanging "plane" and (straight) "line," we have: If two
intersecting _planes_ are each parallel to a _line_, the _line_ of
(intersection of) these _planes_ is parallel to that _line_. Is this
true?
THEOREM. _If two angles not in the same plane have their sides
respectively parallel and lying on the same side of the straight line
joining their vertices, they are equal and their planes are parallel._
Questions like the following may be asked in connection with the
proposition: What is the corresponding proposition in plane geometry?
Why do we need another proof here? Try the plane-geometry proof here.
THEOREM. _If two straight lines are cut by three parallel planes, their
corresponding segments are proportional._
Here, again, it is desirable to ask for the corresponding proposition of
plane geometry, and to ask why the proof of that proposition will not
suffice for this one. The usual figure may be varied in an interesting
manner by having the two lines meet on one of the planes, or outside the
planes, or by having them parallel, in which cases the proof of the
plane-geometry proposition holds here. This proposition is not of great
importance from the practical standpoint, and it is omitted from some of
the standard syllabi at present, although included in certain others. It
is easy, however, to frame some interesting questions depending upon it
for their answers, such as the following: In a gymnasium swimming tank
the water is 4 feet deep and the ceiling is 8 feet above the surface of
the water. A pole 15 feet long touches the ceiling and the bottom of the
tank. Required to know what length of the pole is in the water.
At this point in Book VI it is customary to introduce the dihedral
angle. The word "dihedral" is from the Greek, _di-_ meaning "two," and
_hedra_ meaning "seat." We have the root _hedra_ also in "trihedral"
(three-seated), "polyhedral" (many-seated), and "cathedral" (a church
having a bishop's seat). The word is also, but less properly, spelled
without the _h_, "diedral," a spelling not favored by modern usage. It
is not necessary to dwell at length upon the dihedral angle, except to
show the analogy between it and the plane angle. A few illustrations, as
of an open book, the wall and floor of a room, and a swinging door,
serve to make the concept clear, while a plane at right angles to the
edge shows the measuring plane angle. So manifest is this relationship
between the dihedral angle and its measuring plane angle that some
teachers omit the proposition that two dihedral angles have the same
ratio as their plane angles.
THEOREM. _If two planes are perpendicular to each other, a straight line
drawn in one of them perpendicular to their intersection is
perpendicular to the other._
This and the related propositions allow of numerous illustrations taken
from the schoolroom, as of door edges being perpendicular to the floor.
The pretended applications of these propositions are usually fictitious,
and the propositions are of value chiefly for their own interest and
because they are needed in subsequent proofs.
THEOREM. _The locus of a point equidistant from the faces of a dihedral
angle is the plane bisecting the angle._