THE INTRODUCTION TO GEOMETRY
There are two difficult crises in the geometry course, both for the
pupil and for the teacher. These crises are met at the beginning of the
subject and at the beginning of solid geometry. Once a class has fairly
got into Book I, if the interest in the subject can be maintained, there
are only the incidental difficulties of logical advance throughout the
plane geometry. When the pupil who has been seeing figures in one plane
for a year attempts to visualize solids from a flat drawing, the second
difficult place is reached. Teachers going over solid geometry from year
to year often forget this difficulty, but most of them can easily place
themselves in the pupil's position by looking at the working drawings of
any artisan,--usually simple cases in the so-called descriptive
geometry. They will then realize how difficult it is to visualize a
solid from an unfamiliar kind of picture. The trouble is usually avoided
by the help of a couple of pieces of heavy cardboard or box board, and a
few knitting needles with which to represent lines in space. If these
are judiciously used in class for a few days, until the figures are
understood, the second crisis is easily passed. The continued use of
such material, however, or the daily use of either models or
photographs, weakens the pupil, even as a child is weakened by being
kept too long in a perambulator. Such devices have their place; they are
useful when needed, but they are pernicious when unnecessary. Just as
the mechanic must be able to make and to visualize his working drawings,
so the student of solid geometry must be able to get on with pencil and
paper, representing his solid figures in the flat.
But the introduction to plane geometry is not so easily disposed of. The
pupil at that time is entering a field that is entirely unfamiliar. He
is only fourteen or fifteen years of age, and his thoughts are
distinctly not on geometry. Of logic he knows little and cares less. He
is not interested in a subject of which he knows nothing, not even the
meaning of its name. He asks, naturally and properly, what it all
signifies, what possible use there is for studying geometry, and why he
should have to prove what seems to him evident without proof. To pass
him successfully through this stage has taxed the ingenuity of every
real teacher from the time of Euclid to the present; and just as Euclid
remarked to King Ptolemy, his patron, that there is no royal road to
geometry, so we may affirm that there is no royal road to the teaching
of geometry.
Nevertheless the experience of teachers counts for a great deal, and
this experience has shown that, aside from the matter of technic in
handling the class, certain suggestions are of value, and a few of these
will now be set forth.
First, as to why geometry is studied, it is manifestly impossible
successfully to explain to a boy of fourteen or fifteen the larger
reasons for studying anything whatever. When we confess ourselves
honestly we find that these reasons, whether in mathematics, the natural
sciences, handwork, letters, the vocations, or the fine arts, are none
too clear in our own minds, in spite of any pretentious language that we
may use. It is therefore most satisfactory to anticipate the question at
once, and to set the pupils, for a few days, at using the compasses and
ruler in the drawing of geometric designs and of the most common figures
that they will use. This serves several purposes: it excites their
interest, it guards against the slovenly figures that so often lead them
to erroneous conclusions, it has a genuine value for the future artisan,
and it shows that geometry is something besides mere theory. Whether the
textbook provides for it or not, the teacher will find a few days of
such work well spent, it being a simple matter to supplement the book in
this respect. There was a time when some form of mechanical drawing was
generally taught in the schools, but this has given place to more
genuine art work, leaving it to the teacher of geometry to impart such
knowledge of drawing as is a necessary preliminary to the regular study
of the subject.
Such work in drawing should go so far, and only so far, as to arouse an
interest in geometric form without becoming wearisome, and to
familiarize the pupil with the use of the instruments. He should be
counseled about making fine lines, about being careful in setting the
point of his compasses on the exact center that he wishes to use, and
about representing a point by a very fine dot, or, preferably at first,
by two crossed lines. Unless these details are carefully considered, the
pupil will soon find that the lines of his drawings do not fit together,
and that the result is not pleasing to the eye. The figures here given
are good ones upon which to begin, the dotted construction lines being
erased after the work is completed. They may be constructed with the
compasses and ruler alone, or the draftsman's T-square, triangle, and
protractor may be used, although these latter instruments are not
necessary. We should constantly remember that there is a danger in the
slavish use of instruments and of such helps as squared paper.
Just as Euclid rode roughshod over the growing intellects of
boys and girls, so may instruments ride roughshod over their
growing perceptions by interfering with natural and healthy
intuitions, and making them the subject of laborious
measurement.[40]
[Illustration]
The pupil who cannot see the equality of vertical angles intuitively
better than by the use of the protractor is abnormal. Nevertheless it is
the pupil's interest that is at stake, together with his ability to use
the instruments of daily life. If, therefore, he can readily be
supplied with draftsmen's materials, and is not compelled to use them in
a foolish manner, so much the better. They will not hurt his geometry if
the teacher does not interfere, and they will help his practical
drawing; but for obvious reasons we cannot demand that the pupil
purchase what is not really essential to his study of the subject. The
most valuable single instrument of the three just mentioned is the
protractor, and since a paper one costs only a few cents and is often
helpful in the drawing of figures, it should be recommended to pupils.
There is also another line of work that often arouses a good deal of
interest, namely, the simple field measures that can easily be made
about the school grounds. Guarding against the ever-present danger of
doing too much of such work, of doing work that has no interest for the
pupil, of requiring it done in a way that seems unreal to a class, and
of neglecting the essence of geometry by a line of work that involves no
new principles,--such outdoor exercises in measurement have a positive
value, and a plentiful supply of suggestions in this line is given in
the subsequent chapters. The object is chiefly to furnish a motive for
geometry, and for many pupils this is quite unnecessary. For some,
however, and particularly for the energetic, restless boy, such work has
been successfully offered by various teachers as an alternative to some
of the book work. Because of this value a considerable amount of such
work will be suggested for teachers who may care to use it, the textbook
being manifestly not the place for occasional topics of this nature.
For the purposes of an introduction only a tape line need be purchased.
Wooden pins and a plumb line can easily be made. Even before he comes
to the propositions in mensuration in geometry the pupil knows, from his
arithmetic, how to find ordinary areas and volumes, and he may therefore
be set at work to find the area of the school ground, or of a field, or
of a city block. The following are among the simple exercises for a
beginner:
[Illustration]
1. Drive stakes at two corners, _A_ and _B_, of the school grounds,
putting a cross on top of each; or make the crosses on the sidewalk, so
as to get two points between which to measure. Measure from _A_ to _B_
by holding the tape taut and level, dropping perpendiculars when
necessary by means of the plumb line, as shown in the figure. Check the
work by measuring from _B_ back to _A_ in the same way. Pupils will find
that their work should always be checked, and they will be surprised to
see how the results will vary in such a simple measurement as this,
unless very great care is taken. If they learn the lesson of accuracy
thus early, they will have gained much.
[Illustration]
2. Take two stakes, _X_, _Y_, in a field, preferably two or three
hundred feet apart, always marked on top with crosses so as to have
exact points from which to work. Let it then be required to stake out or
"range" the line from _X_ to _Y_ by placing stakes at specified
distances. One boy stands at _Y_ and another at _X_, each with a plumb
line. A third one takes a plumb line and stands at _P_, the observer at
_X_ motioning to him to move his plumb line to the right or the left
until it is exactly in line with _X_ and _Y_. A stake is then driven at
_P_, and the pupil at _X_ moves on to the stake _P_. Then _Q_ is
located in the same way, and then _R_, and so on. The work is checked by
ranging back from _Y_ to _X_. In some of the simple exercises suggested
later it is necessary to range a line so that this work is useful in
making measurements. The geometric principle involved is that two points
determine a straight line.
[Illustration]
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3. To test a perpendicular or to draw one line perpendicular to another
in a field, we may take a stout cord twelve feet long, having a knot at
the end of every foot. If this is laid along four feet, the ends of this
part being fixed, and it is stretched as here shown, so that the next
vertex is five feet from one of these ends and three feet from the other
end, a right angle will be formed. A right angle can also be run by
making a simple instrument, such as is described in Chapter XV. Still
another plan of drawing a line perpendicular to another line _AB_, from
a point _P_, consists in swinging a tape from _P_, cutting _AB_ at _X_
and _Y_, and then bisecting _XY_ by doubling the tape. This fixes the
foot of the perpendicular.
[Illustration]
4. It is now possible to find the area of a field of irregular shape by
dividing it into triangles and trapezoids, as shown in the figure.
Pupils know from their work in arithmetic how to find the area of a
triangle or a trapezoid, so that the area of the field is easily found.
The work may be checked by comparing the results of different groups of
pupils, or by drawing another diagonal and dividing the field into other
triangles and trapezoids.
These are about as many types of field work as there is any advantage in
undertaking for the purpose of securing the interest of pupils as a
preliminary to the work in geometry. Whether any of it is necessary, and
for what pupils it is necessary, and how much it should trespass upon
the time of scientific geometry are matters that can be decided only by
the teacher of a particular class.
[Illustration]
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A second difficulty of the pupil is seen in his attitude of mind towards
proofs in general. He does not see why vertical angles should be proved
equal when he knows that they are so by looking at the figure. This
difficulty should also be anticipated by giving him some opportunity to
know the weakness of his judgment, and for this purpose figures like the
following should be placed before him. He should be asked which of these
lines is longer, _AB_ or _XY_. Two equal lines should then be arranged
in the form of a letter T, as here shown, and he should be asked which
is the longer, _AB_ or _CD_. A figure that is very deceptive,
particularly if drawn larger and with heavy cross lines, is this one in
which _AB_ and _CD_ are really parallel, but do not seem to be so. Other
interesting deceptions have to do with producing lines, as in these
figures, where it is quite difficult in advance to tell whether _AB_ and
_CD_ are in the same line, and similarly for _WX_ and _YZ_. Equally
deceptive is this figure, in which it is difficult to tell which line
_AB_ will lie along when produced. In the next figure _AB_ appears to be
curved when in reality it is straight, and _CD_ appears straight when in
reality it is curved. The first of the following circles seems to be
slightly flattened at the points _P_, _Q_, _R_, _S_, and in the second
one the distance _BD_ seems greater than the distance _AC_. There are
many equally deceptive figures, and a few of them will convince the
beginner that the proofs are necessary features of geometry.
It is interesting, in connection with the tendency to feel that a
statement is apparent without proof, to recall an anecdote related by
the French mathematician, Biot, concerning the great scientist, Laplace:
Once Laplace, having been asked about a certain point in his
"Celestial Mechanics," spent nearly an hour in trying to recall
the chain of reasoning which he had carelessly concealed by the
words "It is easy to see."
A third difficulty lies in the necessity for putting a considerable
number of definitions at the beginning of geometry, in order to get a
working vocabulary. Although practically all writers scatter the
definitions as much as possible, there must necessarily be some
vocabulary at the beginning. In order to minimize the difficulty of
remembering so many new terms, it is helpful to mingle with them a
considerable number of exercises in which these terms are employed, so
that they may become fixed in mind through actual use. Thus it is of
value to have a class find the complements of 27 deg., 32 deg. 20', 41 deg. 32' 48",
26.75 deg., 33 1/3 deg., and 0 deg.. It is true that into the pure geometry of
Euclid the measuring of angles in degrees does not enter, but it has
place in the practical applications, and it serves at this juncture to
fix the meaning of a new term like "complement."
The teacher who thus anticipates the question as to the reason for
studying geometry, the mental opposition to proving statements, and the
forgetfulness of the meaning of common terms will find that much of the
initial difficulty is avoided. If, now, great care is given to the first
half dozen propositions, the pupil will be well on his way in geometry.
As to these propositions, two plans of selection are employed. The first
takes a few preliminary propositions, easily demonstrated, and seeks
thus to introduce the pupil to the nature of a proof. This has the
advantage of inspiring confidence and the disadvantage of appearing to
prove the obvious. The second plan discards all such apparently obvious
propositions as those about the equality of right angles, and the sum of
two adjacent angles formed by one line meeting another, and begins at
once on things that seem to the pupil as worth the proving. In this
latter plan the introduction is usually made with the proposition
concerning vertical angles, and the two simplest cases of congruent
triangles.
Whichever plan of selection is taken, it is important to introduce a
considerable number of one-step exercises immediately, that is,
exercises that require only one significant step in the proof. In this
way the pupil acquires confidence in his own powers, he finds that
geometry is not mere memorizing, and he sees that each proposition makes
him the master of a large field. To delay the exercises to the end of
each book, or even to delay them for several lessons, is to sow seeds
that will result in the attempt to master geometry by the sheer process
of memorizing.
As to the nature of these exercises, however, the mistake must not be
made of feeling that only those have any value that relate to football
or the laying out of a tennis court. Such exercises are valuable, but
such exercises alone are one-sided. Moreover, any one who examines the
hundreds of suggested exercises that are constantly appearing in various
journals, or who, in the preparation of teachers, looks through the
thousands of exercises that come to him in the papers of his students,
comes very soon to see how hollow is the pretense of most of them. As
has already been said, there are relatively few propositions in geometry
that have any practical applications, applications that are even honest
in their pretense. The principle that the writer has so often laid down
in other works, that whatever pretends to be practical should really be
so, applies with much force to these exercises. When we can find the
genuine application, if it is within reasonable grasp of the pupil, by
all means let us use it. But to put before a class of girls some
technicality of the steam engine that only a skilled mechanic would be
expected to know is not education,--it is mere sham. There is a noble
dignity to geometry, a dignity that a large majority of any class comes
to appreciate when guided by an earnest teacher; but the best way to
destroy this dignity, to take away the appreciation of pure mathematics,
and to furnish weaker candidates than now for advance in this field is
to deceive our pupils and ourselves into believing that the ultimate
purpose of mathematics is to measure things in a way in which no one
else measures them or has ever measured them.
In the proof of the early propositions of plane geometry, and again at
the beginning of solid geometry, there is a little advantage in using
colored crayon to bring out more distinctly the equal parts of two
figures, or the lines outside the plane, or to differentiate one plane
from another. This device, however, like that of models in solid
geometry, can easily be abused, and hence should be used sparingly, and
only until the purpose is accomplished. The student of mathematics must
learn to grasp the meaning of a figure drawn in black on white paper,
or, more rarely, in white on a blackboard, and the sooner he is able to
do this the better for him. The same thing may be said of the
constructing of models for any considerable number of figures in solid
geometry; enough work of this kind to enable a pupil clearly to
visualize the solids is valuable, but thereafter the value is usually
more than offset by the time consumed and the weakened power to grasp
the meaning of a geometric drawing.
There is often a tendency on the part of teachers in their first years
of work to overestimate the logical powers of their pupils and to
introduce forms of reasoning and technical terms that experience has
proved to be unsuited to one who is beginning geometry. Usually but
little harm is done, because the enthusiasm of any teacher who would use
this work would carry the pupils over the difficulties without much
waste of energy on their part. In the long run, however, the attempt is
usually abandoned as not worth the effort. Such a term as
"contrapositive," such distinctions as that between the logical and the
geometric converse, or between perfect and partial geometric conversion,
and such pronounced formalism as the "syllogistic method,"--all these
are happily unknown to most teachers and might profitably be unknown to
all pupils. The modern American textbook in geometry does not begin to
be as good a piece of logic as Euclid's "Elements," and yet it is to be
observed that none of these terms is found in this classic work, so that
they cannot be thought to be necessary to a logical treatment of the
subject. We need the word "converse," and some reference to the law of
converse is therefore permissible; the meaning of the _reductio ad
absurdum_, of a necessary and sufficient condition, and of the terms
"synthesis" and "analysis" may properly form part of the pupil's
equipment because of their universal use; but any extended incursion
into the domain of logic will be found unprofitable, and it is liable to
be positively harmful to a beginner in geometry.
A word should be said as to the lettering of the figures in the early
stages of geometry. In general, it is a great aid to the eye if this is
carried out with some system, and the following suggestions are given as
in accord with the best authors who have given any attention to the
subject:
1. In general, letter a figure counterclockwise, for the reason that we
read angles in this way in higher mathematics, and it is as easy to form
this habit now as to form one that may have to be changed. Where two
triangles are congruent, however, but have their sides arranged in
opposite order, it is better to letter them so that their corresponding
parts appear in the same order, although this makes one read clockwise.
[Illustration]
2. For the same reason, read angles counterclockwise. Thus [L]_A_ is
read "_BAC_," the reflex angle on the outside of the triangle being read
"_CAB_." Of course this is not vital, and many authors pay no attention
to it; but it is convenient, and if the teacher habitually does it, the
pupils will also tend to do it. It is helpful in trigonometry, and it
saves confusion in the case of a reflex angle in a polygon. Designate an
angle by a single letter if this can conveniently be done.
3. Designate the sides opposite angles _A_, _B_, _C_, in a triangle, by
_a_, _b_, _c_, and use these letters in writing proofs.
4. In the case of two congruent triangles use the letters _A_, _B_, _C_
and _A'_, _B'_, _C'_, or _X_, _Y_, _Z_, instead of letters chosen at
random, like _D_, _K_, _L_. It is easier to follow a proof where some
system is shown in lettering the figures. Some teachers insist that a
pupil at the blackboard should not use the letters given in the
textbook, hoping thereby to avoid memorizing. While the danger is
probably exaggerated, it is easy to change with some system, using _P_,
_Q_, _R_ and _P'_, _Q'_, _R'_, for example.
5. Use small letters for lines, as above stated, and also place them
within angles, it being easier to speak of and to see [L]_m_ than
[L]_DEF_. The Germans have a convenient system that some American
teachers follow to advantage, but that a textbook has no right to
require. They use, as in the following figure, _A_ for the point, _a_
for the opposite side, and the Greek letter [alpha] (alpha) for the
angle. The learning of the first three Greek letters, alpha ([alpha]),
beta ([beta]), and gamma ([gamma]), is not a hardship, and they are
worth using, although Greek is so little known in this country to-day
that the alphabet cannot be demanded of teachers who do not care to use
it.
[Illustration]
6. Also use small letters to represent numerical values. For example,
write _c_ = 2[pi]_r_ instead of _C_ = 2[pi]_R_. This is in accord with
the usage in algebra to which the pupil is accustomed.
7. Use initial letters whenever convenient, as in the case of _a_ for
area, _b_ for base, _c_ for circumference, _d_ for diameter, _h_ for
height (altitude), and so on.
Many of these suggestions seem of slight importance in themselves, and
some teachers will be disposed to object to any attempt at lettering a
figure with any regard to system. If, however, they will notice how a
class struggles to follow a demonstration given with reference to a
figure on the blackboard, they will see how helpful it is to have some
simple standards of lettering. It is hardly necessary to add that in
demonstrating from a figure on a blackboard it is usually better to say
"this line," or "the red line," than to say, without pointing to it,
"the line _AB_." It is by such simplicity of statement and by such
efforts to help the class to follow demonstrations that pupils are led
through many of the initial discouragements of the subject.
FOOTNOTES:
[40] Carson, loc. cit., p. 13.