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Title: The Teaching of Geometry
Author: David Eugene Smith
Release Date: October 10, 2011 [EBook #37681]
Language: English
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THE TEACHING OF
GEOMETRY
BY
DAVID EUGENE SMITH
GINN AND COMPANY
BOSTON · NEW YORK · CHICAGO · LONDON
COPYRIGHT, 1911, BY DAVID EUGENE SMITH
ALL RIGHTS RESERVED
911.6
The Athenæum Press
GINN AND COMPANY · PROPRIETORS
BOSTON · U.S.A.
PREFACE
A book upon the teaching of geometry may be planned in divers ways. It
may be written to exploit a new theory of geometry, or a new method of
presenting the science as we already have it. On the other hand, it may
be ultraconservative, making a plea for the ancient teaching and the
ancient geometry. It may be prepared for the purpose of setting forth
the work as it now is, or with the tempting but dangerous idea of
prophecy. It may appeal to the iconoclast by its spirit of destruction,
or to the disciples of _laissez faire_ by its spirit of conserving what
the past has bequeathed. It may be written for the few who always lead,
or think they lead, or for the many who are ranked by the few as
followers. And in view of these varied pathways into the joint domain of
geometry and education, a writer may well afford to pause before he sets
his pen to paper, and to decide with care the route that he will take.
At present in America we have a fairly well-defined body of matter in
geometry, and this occupies a fairly well-defined place in the
curriculum. There are not wanting many earnest teachers who would change
both the matter and the place in a very radical fashion. There are not
wanting others, also many in number, who are content with things as they
find them. But by far the largest part of the teaching body is of a mind
to welcome the natural and gradual evolution of geometry toward better
things, contributing to this evolution as much as it can, glad to know
the best that others have to offer, receptive of ideas that make for
better teaching, but out of sympathy with either the extreme of
revolution or the extreme of stagnation.
It is for this larger class, the great body of progressive teachers,
that this book is written. It stands for vitalizing geometry in every
legitimate way; for improving the subject matter in such manner as not
to destroy the pupil's interest; for so teaching geometry as to make it
appeal to pupils as strongly as any other subject in the curriculum; but
for the recognition of geometry for geometry's sake and not for the sake
of a fancied utility that hardly exists. Expressing full appreciation of
the desirability of establishing a motive for all studies, so as to have
the work proceed with interest and vigor, it does not hesitate to
express doubt as to certain motives that have been exploited, nor to
stand for such a genuine, thought-compelling development of the science
as is in harmony with the mental powers of the pupils in the American
high school.
For this class of teachers the author hopes that the book will prove of
service, and that through its perusal they will come to admire the
subject more and more, and to teach it with greater interest. It offers
no panacea, it champions no single method, but it seeks to set forth
plainly the reasons for teaching a geometry of the kind that we have
inherited, and for hoping for a gradual but definite improvement in the
science and in the methods of its presentation.
DAVID EUGENE SMITH
CONTENTS
CHAPTER PAGE
I. CERTAIN QUESTIONS NOW AT ISSUE 1
II. WHY GEOMETRY IS STUDIED 7
III. A BRIEF HISTORY OF GEOMETRY 26
IV. DEVELOPMENT OF THE TEACHING OF GEOMETRY 40
V. EUCLID 47
VI. EFFORTS AT IMPROVING EUCLID 57
VII. THE TEXTBOOK IN GEOMETRY 70
VIII. THE RELATION OF ALGEBRA TO GEOMETRY 84
IX. THE INTRODUCTION TO GEOMETRY 93
X. THE CONDUCT OF A CLASS IN GEOMETRY 108
XI. THE AXIOMS AND POSTULATES 116
XII. THE DEFINITIONS OF GEOMETRY 132
XIII. HOW TO ATTACK THE EXERCISES 160
XIV. BOOK I AND ITS PROPOSITIONS 165
XV. THE LEADING PROPOSITIONS OF BOOK II 201
XVI. THE LEADING PROPOSITIONS OF BOOK III 227
XVII. THE LEADING PROPOSITIONS OF BOOK IV 252
XVIII. THE LEADING PROPOSITIONS OF BOOK V 269
XIX. THE LEADING PROPOSITIONS OF BOOK VI 289
XX. THE LEADING PROPOSITIONS OF BOOK VII 303
XXI. THE LEADING PROPOSITIONS OF BOOK VIII 321
INDEX 335
THE TEACHING OF GEOMETRY
CHAPTER I
CERTAIN QUESTIONS NOW AT ISSUE
It is commonly said at the present time that the opening of the
twentieth century is a period of unusual advancement in all that has to
do with the school. It would be pleasant to feel that we are living in
such an age, but it is doubtful if the future historian of education
will find this to be the case, or that biographers will rank the leaders
of our generation relatively as high as many who have passed away, or
that any great movements of the present will be found that measure up to
certain ones that the world now recognizes as epoch-making. Every
generation since the invention of printing has been a period of
agitation in educational matters, but out of all the noise and
self-assertion, out of all the pretense of the chronic revolutionist,
out of all the sham that leads to dogmatism, so little is remembered
that we are apt to feel that the past had no problems and was content
simply to accept its inheritance. In one sense it is not a misfortune
thus to be blinded by the dust of present agitation and to be deafened
by the noisy clamor of the agitator, since it stirs us to action at
finding ourselves in the midst of the skirmish; but in another sense it
is detrimental to our progress, since we thereby tend to lose the idea
of perspective, and the coin comes to appear to our vision as large as
the moon.
In considering a question like the teaching of geometry, we at once find
ourselves in the midst of a skirmish of this nature. If we join
thoughtlessly in the noise, we may easily persuade ourselves that we are
waging a mighty battle, fighting for some stupendous principle, doing
deeds of great valor and of personal sacrifice. If, on the other hand,
we stand aloof and think of the present movement as merely a chronic
effervescence, fostered by the professional educator at the expense of
the practical teacher, we are equally shortsighted. Sir Conan Doyle
expressed this sentiment most delightfully in these words:
The dead are such good company that one may come to think too
little of the living. It is a real and pressing danger with
many of us that we should never find our own thoughts and our
own souls, but be ever obsessed by the dead.
In every generation it behooves the open-minded, earnest, progressive
teacher to seek for the best in the way of improvement, to endeavor to
sift the few grains of gold out of the common dust, to weigh the values
of proposed reforms, and to put forth his efforts to know and to use the
best that the science of education has to offer. This has been the
attitude of mind of the real leaders in the school life of the past, and
it will be that of the leaders of the future.
With these remarks to guide us, it is now proposed to take up the issues
of the present day in the teaching of geometry, in order that we may
consider them calmly and dispassionately, and may see where the
opportunities for improvement lie.
At the present time, in the educational circles of the United States,
questions of the following type are causing the chief discussion among
teachers of geometry:
1. Shall geometry continue to be taught as an application of logic, or
shall it be treated solely with reference to its applications?
2. If the latter is the purpose in view, shall the propositions of
geometry be limited to those that offer an opportunity for real
application, thus contracting the whole subject to very narrow
dimensions?
3. Shall a subject called geometry be extended over several years, as is
the case in Europe,[1] or shall the name be applied only to serious
demonstrative geometry[2] as given in the second year of the four-year
high school course in the United States at present?
4. Shall geometry be taught by itself, or shall it be either mixed with
algebra (say a day of one subject followed by a day of the other) or
fused with it in the form of a combined mathematics?
5. Shall a textbook be used in which the basal propositions are proved
in full, the exercises furnishing the opportunity for original work and
being looked upon as the most important feature, or shall one be
employed in which the pupil is expected to invent the proofs for the
basal propositions as well as for the exercises?
6. Shall the terminology and the spirit of a modified Euclid and
Legendre prevail in the future as they have in the past, or shall there
be a revolution in the use of terms and in the general statements of the
propositions?
7. Shall geometry be made a strong elective subject, to be taken only by
those whose minds are capable of serious work? Shall it be a required
subject, diluted to the comprehension of the weakest minds? Or is it
now, by proper teaching, as suitable for all pupils as is any other
required subject in the school curriculum? And in any case, will the
various distinct types of high schools now arising call for distinct
types of geometry?
This brief list might easily be amplified, but it is sufficiently
extended to set forth the trend of thought at the present time, and to
show that the questions before the teachers of geometry are neither
particularly novel nor particularly serious. These questions and others
of similar nature are really side issues of two larger questions of far
greater significance: (1) Are the reasons for teaching demonstrative
geometry such that it should be a required subject, or at least a
subject that is strongly recommended to all, whatever the type of high
school? (2) If so, how can it be made interesting?
The present work is written with these two larger questions in mind,
although it considers from time to time the minor ones already
mentioned, together with others of a similar nature. It recognizes that
the recent growth in popular education has brought into the high school
a less carefully selected type of mind than was formerly the case, and
that for this type a different kind of mathematical training will
naturally be developed. It proceeds upon the theory, however, that for
the normal mind,--for the boy or girl who is preparing to win out in the
long run,--geometry will continue to be taught as demonstrative
geometry, as a vigorous thought-compelling subject, and along the
general lines that the experience of the world has shown to be the best.
Soft mathematics is not interesting to this normal mind, and a sham
treatment will never appeal to the pupil; and this book is written for
teachers who believe in this principle, who believe in geometry for the
sake of geometry, and who earnestly seek to make the subject so
interesting that pupils will wish to study it whether it is required or
elective. The work stands for the great basal propositions that have
come down to us, as logically arranged and as scientifically proved as
the powers of the pupils in the American high school will permit; and it
seeks to tell the story of these propositions and to show their possible
and their probable applications in such a way as to furnish teachers
with a fund of interesting material with which to supplement the book
work of their classes.
After all, the problem of teaching any subject comes down to this: Get a
subject worth teaching and then make every minute of it interesting.
Pupils do not object to work if they like a subject, but they do object
to aimless and uninteresting tasks. Geometry is particularly fortunate
in that the feeling of accomplishment comes with every proposition
proved; and, given a class of fair intelligence, a teacher must be
lacking in knowledge and enthusiasm who cannot foster an interest that
will make geometry stand forth as the subject that brings the most
pleasure, and that seems the most profitable of all that are studied in
the first years of the high school.
Continually to advance, continually to attempt to make mathematics
fascinating, always to conserve the best of the old and to sift out and
use the best of the new, to believe that "mankind is better served by
nature's quiet and progressive changes than by earthquakes,"[3] to
believe that geometry as geometry is so valuable and so interesting that
the normal mind may rightly demand it,--this is to ally ourselves with
progress. Continually to destroy, continually to follow strange gods,
always to decry the best of the old, and to have no well-considered aim
in the teaching of a subject,--this is to join the forces of reaction,
to waste our time, to be recreant to our trust, to blind ourselves to
the failures of the past, and to confess our weakness as teachers. It is
with the desire to aid in the progressive movement, to assist those who
believe that real geometry should be recommended to all, and to show
that geometry is both attractive and valuable that this book is written.
FOOTNOTES:
[1] And really, though not nominally, in the United States, where the
first concepts are found in the kindergarten, and where an excellent
course in mensuration is given in any of our better class of
arithmetics. That we are wise in not attempting serious demonstrative
geometry much earlier seems to be generally conceded.
[2] The third stage of geometry as defined in the recent circular (No.
711) of the British Board of Education, London, 1909.
[3] The closing words of a sensible review of the British Board of
Education circular (No. 711), on "The Teaching of Geometry" (London,
1909), by H. S. Hall in the _School World_, 1909, p. 222.
CHAPTER II
WHY GEOMETRY IS STUDIED
With geometry, as with other subjects, it is easier to set forth what
are not the reasons for studying it than to proceed positively and
enumerate the advantages. Although such a negative course is not
satisfying to the mind as a finality, it possesses definite advantages
in the beginning of such a discussion as this. Whenever false prophets
arise, and with an attitude of pained superiority proclaim unworthy aims
in human life, it is well to show the fallacy of their position before
proceeding to a constructive philosophy. Taking for a moment this
negative course, let us inquire as to what are not the reasons for
studying geometry, or, to be more emphatic, as to what are not the
worthy reasons.
In view of a periodic activity in favor of the utilities of geometry, it
is well to understand, in the first place, that geometry is not studied,
and never has been studied, because of its positive utility in
commercial life or even in the workshop. In America we commonly allow at
least a year to plane geometry and a half year to solid geometry; but
all of the facts that a skilled mechanic or an engineer would ever need
could be taught in a few lessons. All the rest is either obvious or is
commercially and technically useless. We prove, for example, that the
angles opposite the equal sides of a triangle are equal, a fact that is
probably quite as obvious as the postulate that but one line can be
drawn through a given point parallel to a given line. We then prove,
sometimes by the unsatisfactory process of _reductio ad absurdum_, the
converse of this proposition,--a fact that is as obvious as most other
facts that come to our consciousness, at least after the preceding
proposition has been proved. And these two theorems are perfectly fair
types of upwards of one hundred sixty or seventy propositions comprising
Euclid's books on plane geometry. They are generally not useful in daily
life, and they were never intended to be so. There is an oft-repeated
but not well-authenticated story of Euclid that illustrates the feeling
of the founders of geometry as well as of its most worthy teachers. A
Greek writer, Stobæus, relates the story in these words:
Some one who had begun to read geometry with Euclid, when he
had learned the first theorem, asked, "But what shall I get by
learning these things?" Euclid called his slave and said, "Give
him three obols, since he must make gain out of what he
learns."
Whether true or not, the story expresses the sentiment that runs through
Euclid's work, and not improbably we have here a bit of real
biography,--practically all of the personal Euclid that has come down to
us from the world's first great textbook maker. It is well that we read
the story occasionally, and also such words as the following, recently
uttered[4] by Sir Conan Doyle,--words bearing the same lesson, although
upon a different theme:
In the present utilitarian age one frequently hears the
question asked, "What is the use of it all?" as if every noble
deed was not its own justification. As if every action which
makes for self-denial, for hardihood, and for endurance was
not in itself a most precious lesson to mankind. That people
can be found to ask such a question shows how far materialism
has gone, and how needful it is that we insist upon the value
of all that is nobler and higher in life.
An American statesman and jurist, speaking upon a similar occasion[5],
gave utterance to the same sentiments in these words:
When the time comes that knowledge will not be sought for its
own sake, and men will not press forward simply in a desire of
achievement, without hope of gain, to extend the limits of
human knowledge and information, then, indeed, will the race
enter upon its decadence.
There have not been wanting, however, in every age, those whose zeal is
in inverse proportion to their experience, who were possessed with the
idea that it is the duty of the schools to make geometry practical. We
have them to-day, and the world had them yesterday, and the future shall
see them as active as ever.
These people do good to the world, and their labors should always be
welcome, for out of the myriad of suggestions that they make a few have
value, and these are helpful both to the mathematician and the artisan.
Not infrequently they have contributed material that serves to make
geometry somewhat more interesting, but it must be confessed that most
of their work is merely the threshing of old straw, like the work of
those who follow the will-o'-the-wisp of the circle squarers. The
medieval astrologers wished to make geometry more practical, and so they
carried to a considerable length the study of the star polygon, a figure
that they could use in their profession. The cathedral builders, as
their art progressed, found that architectural drawings were more exact
if made with a single opening of the compasses, and it is probable that
their influence led to the development of this phase of geometry in the
Middle Ages as a practical application of the science. Later, and about
the beginning of the sixteenth century, the revival of art, and
particularly the great development of painting, led to the practical
application of geometry to the study of perspective and of those
curves[6] that occur most frequently in the graphic arts. The sixteenth
and seventeenth centuries witnessed the publication of a large number of
treatises on practical geometry, usually relating to the measuring of
distances and partly answering the purposes of our present trigonometry.
Such were the well-known treatises of Belli (1569), Cataneo (1567), and
Bartoli (1589).[7]
The period of two centuries from about 1600 to about 1800 was quite as
much given to experiments in the creation of a practical geometry as is
the present time, and it was no doubt as much by way of protest against
this false idea of the subject as a desire to improve upon Euclid that
led the great French mathematician, Legendre, to publish his geometry in
1794,--a work that soon replaced Euclid in the schools of America.
It thus appears that the effort to make geometry practical is by no
means new. Euclid knew of it, the Middle Ages contributed to it, that
period vaguely styled the Renaissance joined in the movement, and the
first three centuries of printing contributed a large literature to the
subject. Out of all this effort some genuine good remains, but
relatively not very much.[8] And so it will be with the present
movement; it will serve its greatest purpose in making teachers think
and read, and in adding to their interest and enthusiasm and to the
interest of their pupils; but it will not greatly change geometry,
because no serious person ever believed that geometry was taught chiefly
for practical purposes, or was made more interesting or valuable through
such a pretense. Changes in sequence, in definitions, and in proofs will
come little by little; but that there will be any such radical change in
these matters in the immediate future, as some writers have anticipated,
is not probable.[9]
A recent writer of much acumen[10] has summed up this thought in these
words:
Not one tenth of the graduates of our high schools ever enter
professions in which their algebra and geometry are applied to
concrete realities; not one day in three hundred sixty-five is
a high school graduate called upon to "apply," as it is called,
an algebraic or a geometrical proposition.... Why, then, do we
teach these subjects, if this alone is the sense of the word
"practical"!... To me the solution of this paradox consists in
boldly confronting the dilemma, and in saying that our
conception of the practical utility of those studies must be
readjusted, and that we have frankly to face the truth that the
"practical" ends we seek are in a sense _ideal_ practical ends,
yet such as have, after all, an eminently utilitarian value in
the intellectual sphere.
He quotes from C. S. Jackson, a progressive contemporary teacher of
mechanics in England, who speaks of pupils confusing millimeters and
centimeters in some simple computation, and who adds:
There is the enemy! The real enemy we have to fight against,
whatever we teach, is carelessness, inaccuracy, forgetfulness,
and slovenliness. That battle has been fought and won with
diverse weapons. It has, for instance, been fought with Latin
grammar before now, and won. I say that because we must be very
careful to guard against the notion that there is any one
panacea for this sort of thing. It borders on quackery to say
that elementary physics will cure everything.
And of course the same thing may be said for mathematics. Nevertheless it
is doubtful if we have any other subject that does so much to bring to
the front this danger of carelessness, of slovenly reasoning, of
inaccuracy, and of forgetfulness as this science of geometry, which has
been so polished and perfected as the centuries have gone on.
There have been those who did not proclaim the utilitarian value of
geometry, but who fell into as serious an error, namely, the advocating
of geometry as a means of training the memory. In times not so very far
past, and to some extent to-day, the memorizing of proofs has been
justified on this ground. This error has, however, been fully exposed by
our modern psychologists. They have shown that the person who memorizes
the propositions of Euclid by number is no more capable of memorizing
other facts than he was before, and that the learning of proofs verbatim
is of no assistance whatever in retaining matter that is helpful in
other lines of work. Geometry, therefore, as a training of the memory is
of no more value than any other subject in the curriculum.
If geometry is not studied chiefly because it is practical, or because
it trains the memory, what reasons can be adduced for its presence in
the courses of study of every civilized country? Is it not, after all, a
mere fetish, and are not those virulent writers correct who see nothing
good in the subject save only its utilities?[11] Of this type one of the
most entertaining is William J. Locke,[12] whose words upon the subject
are well worth reading:
... I earned my living at school slavery, teaching to children
the most useless, the most disastrous, the most soul-cramping
branch of knowledge wherewith pedagogues in their insensate
folly have crippled the minds and blasted the lives of
thousands of their fellow creatures--elementary mathematics.
There is no more reason for any human being on God's earth to
be acquainted with the binomial theorem or the solution of
triangles, unless he is a professional scientist,--when he can
begin to specialize in mathematics at the same age as the
lawyer begins to specialize in law or the surgeon in
anatomy,--than for him to be expert in Choctaw, the Cabala, or
the Book of Mormon. I look back with feelings of shame and
degradation to the days when, for a crust of bread, I
prostituted my intelligence to wasting the precious hours of
impressionable childhood, which could have been filled with so
many beautiful and meaningful things, over this utterly futile
and inhuman subject. It trains the mind,--it teaches boys to
think, they say. It doesn't. In reality it is a cut-and-dried
subject, easy to fit into a school curriculum. Its
sacrosanctity saves educationalists an enormous amount of
trouble, and its chief use is to enable mindless young men from
the universities to make a dishonest living by teaching it to
others, who in their turn may teach it to a future generation.
To be fair we must face just such attacks, and we must recognize that
they set forth the feelings of many honest people. One is tempted to
inquire if Mr. Locke could have written in such an incisive style if he
had not, as was the case, graduated with honors in mathematics at one of
the great universities. But he might reply that if his mind had not been
warped by mathematics, he would have written more temperately, so the
honors in the argument would be even. Much more to the point is the fact
that Mr. Locke taught mathematics in the schools of England, and that
these schools do not seem to the rest of the world to furnish a good
type of the teaching of elementary mathematics. No country goes to
England for its model in this particular branch of education, although
the work is rapidly changing there, and Mr. Locke pictures a local
condition in teaching rather than a general condition in mathematics.
Few visitors to the schools of England would care to teach mathematics
as they see it taught there, in spite of their recognition of the
thoroughness of the work and the earnestness of many of the teachers. It
is also of interest to note that the greatest protests against formal
mathematics have come from England, as witness the utterances of such
men as Sir William Hamilton and Professors Perry, Minchin, Henrici, and
Alfred Lodge. It may therefore be questioned whether these scholars are
not unconsciously protesting against the English methods and curriculum
rather than against the subject itself. When Professor Minchin says that
he had been through the six books of Euclid without really understanding
an angle, it is Euclid's text and his own teacher that are at fault, and
not geometry.
Before considering directly the question as to why geometry should be
taught, let us turn for a moment to the other subjects in the secondary
curriculum. Why, for example, do we study literature? "It does not
lower the price of bread," as Malherbe remarked in speaking of the
commentary of Bachet on the great work of Diophantus. Is it for the
purpose of making authors? Not one person out of ten thousand who study
literature ever writes for publication. And why do we allow pupils to
waste their time in physical education? It uses valuable hours, it
wastes money, and it is dangerous to life and limb. Would it not be
better to set pupils at sawing wood? And why do we study music? To give
pleasure by our performances? How many who attempt to play the piano or
to sing give much pleasure to any but themselves, and possibly their
parents? The study of grammar does not make an accurate writer, nor the
study of rhetoric an orator, nor the study of meter a poet, nor the
study of pedagogy a teacher. The study of geography in the school does
not make travel particularly easier, nor does the study of biology tend
to populate the earth. So we might pass in review the various subjects
that we study and ought to study, and in no case would we find utility
the moving cause, and in every case would we find it difficult to state
the one great reason for the pursuit of the subject in question,--and so
it is with geometry.
What positive reasons can now be adduced for the study of a subject that
occupies upwards of a year in the school course, and that is, perhaps
unwisely, required of all pupils? Probably the primary reason, if we do
not attempt to deceive ourselves, is pleasure. We study music because
music gives us pleasure, not necessarily our own music, but good music,
whether ours, or, as is more probable, that of others. We study
literature because we derive pleasure from books; the better the book
the more subtle and lasting the pleasure. We study art because we
receive pleasure from the great works of the masters, and probably we
appreciate them the more because we have dabbled a little in pigments or
in clay. We do not expect to be composers, or poets, or sculptors, but
we wish to appreciate music and letters and the fine arts, and to derive
pleasure from them and to be uplifted by them. At any rate, these are
the nobler reasons for their study.
So it is with geometry. We study it because we derive pleasure from
contact with a great and an ancient body of learning that has occupied
the attention of master minds during the thousands of years in which it
has been perfected, and we are uplifted by it. To deny that our pupils
derive this pleasure from the study is to confess ourselves poor
teachers, for most pupils do have positive enjoyment in the pursuit of
geometry, in spite of the tradition that leads them to proclaim a
general dislike for all study. This enjoyment is partly that of the
game,--the playing of a game that can always be won, but that cannot be
won too easily. It is partly that of the æsthetic, the pleasure of
symmetry of form, the delight of fitting things together. But probably
it lies chiefly in the mental uplift that geometry brings, the contact
with absolute truth, and the approach that one makes to the Infinite. We
are not quite sure of any one thing in biology; our knowledge of geology
is relatively very slight, and the economic laws of society are
uncertain to every one except some individual who attempts to set them
forth; but before the world was fashioned the square on the hypotenuse
was equal to the sum of the squares on the other two sides of a right
triangle, and it will be so after this world is dead; and the inhabitant
of Mars, if he exists, probably knows its truth as we know it. The
uplift of this contact with absolute truth, with truth eternal, gives
pleasure to humanity to a greater or less degree, depending upon the
mental equipment of the particular individual; but it probably gives an
appreciable amount of pleasure to every student of geometry who has a
teacher worthy of the name. First, then, and foremost as a reason for
studying geometry has always stood, and will always stand, the pleasure
and the mental uplift that comes from contact with such a great body of
human learning, and particularly with the exact truth that it contains.
The teacher who is imbued with this feeling is on the road to success,
whatever method of presentation he may use; the one who is not imbued
with it is on the road to failure, however logical his presentation or
however large his supply of practical applications.
Subordinate to these reasons for studying geometry are many others,
exactly as with all other subjects of the curriculum. Geometry, for
example, offers the best developed application of logic that we have, or
are likely to have, in the school course. This does not mean that it
always exemplifies perfect logic, for it does not; but to the pupil who
is not ready for logic, per se, it offers an example of close reasoning
such as his other subjects do not offer. We may say, and possibly with
truth, that one who studies geometry will not reason more clearly on a
financial proposition than one who does not; but in spite of the results
of the very meager experiments of the psychologists, it is probable that
the man who has had some drill in syllogisms, and who has learned to
select the essentials and to neglect the nonessentials in reaching his
conclusions, has acquired habits in reasoning that will help him in
every line of work. As part of this equipment there is also a terseness
of statement and a clearness in arrangement of points in an argument
that has been the subject of comment by many writers.
Upon this same topic an English writer, in one of the sanest of recent
monographs upon the subject,[13] has expressed his views in the
following words:
The statement that a given individual has received a sound
geometrical training implies that he has segregated from the
whole of his sense impressions a certain set of these
impressions, that he has then eliminated from their
consideration all irrelevant impressions (in other words,
acquired a subjective command of these impressions), that he
has developed on the basis of these impressions an ordered and
continuous system of logical deduction, and finally that he is
capable of expressing the nature of these impressions and his
deductions therefrom in terms simple and free from ambiguity.
Now the slightest consideration will convince any one not
already conversant with the idea, that the same sequence of
mental processes underlies the whole career of any individual
in any walk of life if only he is not concerned entirely with
manual labor; consequently a full training in the performance
of such sequences must be regarded as forming an essential part
of any education worthy of the name. Moreover, the full
appreciation of such processes has a higher value than is
contained in the mental training involved, great though this
be, for it induces an appreciation of intellectual unity and
beauty which plays for the mind that part which the
appreciation of schemes of shape and color plays for the
artistic faculties; or, again, that part which the appreciation
of a body of religious doctrine plays for the ethical
aspirations. Now geometry is not the sole possible basis for
inculcating this appreciation. Logic is an alternative for
adults, provided that the individual is possessed of sufficient
wide, though rough, experience on which to base his reasoning.
Geometry is, however, highly desirable in that the objective
bases are so simple and precise that they can be grasped at an
early age, that the amount of training for the imagination is
very large, that the deductive processes are not beyond the
scope of ordinary boys, and finally that it affords a better
basis for exercise in the art of simple and exact expression
than any other possible subject of a school course.
Are these results really secured by teachers, however, or are they
merely imagined by the pedagogue as a justification for his existence?
Do teachers have any such appreciation of geometry as has been
suggested, and even if they have it, do they impart it to their pupils?
In reply it may be said, probably with perfect safety, that teachers of
geometry appreciate their subject and lead their pupils to appreciate it
to quite as great a degree as obtains in any other branch of education.
What teacher appreciates fully the beauties of "In Memoriam," or of
"Hamlet," or of "Paradise Lost," and what one inspires his pupils with
all the nobility of these world classics? What teacher sees in biology
all the grandeur of the evolution of the race, or imparts to his pupils
the noble lessons of life that the study of this subject should suggest?
What teacher of Latin brings his pupils to read the ancient letters with
full appreciation of the dignity of style and the nobility of thought
that they contain? And what teacher of French succeeds in bringing a
pupil to carry on a conversation, to read a French magazine, to see the
history imbedded in the words that are used, to realize the charm and
power of the language, or to appreciate to the full a single classic? In
other words, none of us fully appreciates his subject, and none of us
can hope to bring his pupils to the ideal attitude toward any part of
it. But it is probable that the teacher of geometry succeeds relatively
better than the teacher of other subjects, because the science has
reached a relatively higher state of perfection. The body of truth in
geometry has been more clearly marked out, it has been more
successfully fitted together, its lesson is more patent, and the
experience of centuries has brought it into a shape that is more usable
in the school. While, therefore, we have all kinds of teaching in all
kinds of subjects, the very nature of the case leads to the belief that
the class in geometry receives quite as much from the teacher and the
subject as the class in any other branch in the school curriculum.
But is this not mere conjecture? What are the results of scientific
investigation of the teaching of geometry? Unfortunately there is little
hope from the results of such an inquiry, either here or in other
fields. We cannot first weigh a pupil in an intellectual or moral
balance, then feed him geometry, and then weigh him again, and then set
back his clock of time and begin all over again with the same
individual. There is no "before taking" and "after taking" of a subject
that extends over a year or two of a pupil's life. We can weigh
utilities roughly, we can estimate the pleasure of a subject relatively,
but we cannot say that geometry is worth so many dollars, and history so
many, and so on through the curriculum. The best we can do is to ask
ourselves what the various subjects, with teachers of fairly equal
merit, have done for us, and to inquire what has been the experience of
other persons. Such an investigation results in showing that, with few
exceptions, people who have studied geometry received as much of
pleasure, of inspiration, of satisfaction, of what they call training
from geometry as from any other subject of study,--given teachers of
equal merit,--and that they would not willingly give up the something
which geometry brought to them. If this were not the feeling, and if
humanity believed that geometry is what Mr. Locke's words would seem to
indicate, it would long ago have banished it from the schools, since
upon this ground rather than upon the ground of utility the subject has
always stood.
These seem to be the great reasons for the study of geometry, and to
search for others would tend to weaken the argument. At first sight they
may not seem to justify the expenditure of time that geometry demands,
and they may seem unduly to neglect the argument that geometry is a
stepping-stone to higher mathematics. Each of these points, however, has
been neglected purposely. A pupil has a number of school years at his
disposal; to what shall they be devoted? To literature? What claim has
letters that is such as to justify the exclusion of geometry? To music,
or natural science, or language? These are all valuable, and all should
be studied by one seeking a liberal education; but for the same reason
geometry should have its place. What subject, in fine, can supply
exactly what geometry does? And if none, then how can the pupil's time
be better expended than in the study of this science?[14] As to the
second point, that a claim should be set forth that geometry is a _sine
qua non_ to higher mathematics, this belief is considerably exaggerated
because there are relatively few who proceed from geometry to a higher
branch of mathematics. This argument would justify its status as an
elective rather than as a required subject.
Let us then stand upon the ground already marked out, holding that the
pleasure, the culture, the mental poise, the habits of exact reasoning
that geometry brings, and the general experience of mankind upon the
subject are sufficient to justify us in demanding for it a reasonable
amount of time in the framing of a curriculum. Let us be fair in our
appreciation of all other branches, but let us urge that every student
may have an opportunity to know of real geometry, say for a single year,
thereafter pursuing it or not, according as we succeed in making its
value apparent, or fail in our attempt to present worthily an ancient
and noble science to the mind confided to our instruction.
The shortsightedness of a narrow education, of an education that teaches
only machines to a prospective mechanic, and agriculture to a
prospective farmer, and cooking and dressmaking to the girl, and that
would exclude all mathematics that is not utilitarian in the narrow
sense, cannot endure.
The community has found out that such schemes may be well
fitted to give the children a good time in school, but lead
them to a bad time afterward. Life is hard work, and if they
have never learned in school to give their concentrated
attention to that which does not appeal to them and which does
not interest them immediately, they have missed the most
valuable lesson of their school years. The little practical
information they could have learned at any time; the energy of
attention and concentration can no longer be learned if the
early years are wasted. However narrow and commercial the
standpoint which is chosen may be, it can always be found that
it is the general education which pays best, and the more the
period of cultural work can be expanded the more efficient will
be the services of the school for the practical services of the
nation.[15]
Of course no one should construe these remarks as opposing in the
slightest degree the laudable efforts that are constantly being put
forth to make geometry more interesting and to vitalize it by
establishing as strong motives as possible for its study. Let the home,
the workshop, physics, art, play,--all contribute their quota of motive
to geometry as to all mathematics and all other branches. But let us
never forget that geometry has a _raison d'être_ beyond all this, and
that these applications are sought primarily for the sake of geometry,
and that geometry is not taught primarily for the sake of these
applications.
When we consider how often geometry is attacked by those who profess to
be its friends, and how teachers who have been trained in mathematics
occasionally seem to make of the subject little besides a mongrel course
in drawing and measuring, all the time insisting that they are
progressive while the champions of real geometry are reactionary, it is
well to read some of the opinions of the masters. The following
quotations may be given occasionally in geometry classes as showing the
esteem in which the subject has been held in various ages, and at any
rate they should serve to inspire the teacher to greater love for his
subject.
The enemies of geometry, those who know it only imperfectly,
look upon the theoretical problems, which constitute the most
difficult part of the subject, as mental games which consume
time and energy that might better be employed in other ways.
Such a belief is false, and it would block the progress of
science if it were credible. But aside from the fact that the
speculative problems, which at first sight seem barren, can
often be applied to useful purposes, they always stand as among
the best means to develop and to express all the forces of the
human intelligence.--ABBÉ BOSSUT.
The sailor whom an exact observation of longitude saves from
shipwreck owes his life to a theory developed two thousand
years ago by men who had in mind merely the speculations of
abstract geometry.--CONDORCET.
If mathematical heights are hard to climb, the fundamental
principles lie at every threshold, and this fact allows them to
be comprehended by that common sense which Descartes declared
was "apportioned equally among all men."--COLLET.
It may seem strange that geometry is unable to define the terms
which it uses most frequently, since it defines neither
movement, nor number, nor space,---the three things with which
it is chiefly concerned. But we shall not be surprised if we
stop to consider that this admirable science concerns only the
most simple things, and the very quality that renders these
things worthy of study renders them incapable of being defined.
Thus the very lack of definition is rather an evidence of
perfection than a defect, since it comes not from the obscurity
of the terms, but from the fact that they are so very well
known.--PASCAL.
God eternally geometrizes.--PLATO.
God is a circle of which the center is everywhere and the
circumference nowhere.--RABELAIS.
Without mathematics no one can fathom the depths of philosophy.
Without philosophy no one can fathom the depths of mathematics.
Without the two no one can fathom the depths of
anything.--BORDAS-DEMOULIN.
We may look upon geometry as a practical logic, for the truths
which it studies, being the most simple and most clearly
understood of all truths, are on this account the most
susceptible of ready application in reasoning.--D'ALEMBERT.
The advance and the perfecting of mathematics are closely
joined to the prosperity of the nation.--NAPOLEON.
Hold nothing as certain save what can be demonstrated.--NEWTON.
To measure is to know.--KEPLER.
The method of making no mistake is sought by every one. The
logicians profess to show the way, but the geometers alone ever
reach it, and aside from their science there is no genuine
demonstration.--PASCAL.
The taste for exactness, the impossibility of contenting one's
self with vague notions or of leaning upon mere hypotheses, the
necessity for perceiving clearly the connection between certain
propositions and the object in view,--these are the most
precious fruits of the study of mathematics.--LACROIX.
=Bibliography.= Smith, The Teaching of Elementary Mathematics,
p. 234, New York, 1900; Henrici, Presidential Address before
the British Association, _Nature_, Vol. XXVIII, p. 497; Hill,
Educational Value of Mathematics, _Educational Review_, Vol.
IX, p. 349; Young, The Teaching of Mathematics, p. 9, New York,
1907. The closing quotations are from Rebière, Mathématiques et
Mathématiciens, Paris, 1893.
FOOTNOTES:
[4] In an address in London, June 15, 1909, at a dinner to Sir Ernest
Shackelton.
[5] Governor Hughes, now Justice Hughes, of New York, at the Peary
testimonial on February 8, 1910, at New York City.
[6] The first work upon this subject, and indeed the first printed
treatise on curves in general, was written by the famous artist of
Nürnberg, Albrecht Dürer.
[7] Several of these writers are mentioned in Chapter IV.
[8] If any reader chances upon George Birkbeck's English translation of
Charles Dupin's "Mathematics Practically Applied," Halifax, 1854, he
will find that Dupin gave more good applications of geometry than all of
our American advocates of practical geometry combined.
[9] See, for example, Henrici's "Congruent Figures," London, 1879, and
the review of Borel's "Elements of Mathematics," by Professor Sisam in
the _Bulletin of the American Mathematical Society_, July, 1910, a
matter discussed later in this work.
[10] T. J. McCormack, "Why do we study Mathematics: a Philosophical and
Historical Retrospect," p. 9, Cedar Rapids, Iowa, 1910.
[11] Of the fair and candid arguments against the culture value of
mathematics, one of the best of the recent ones is that by G. F. Swain,
in the _Atti del IV Congresso Internazionale dei Matematici_, Rome,
1909, Vol. III, p. 361. The literature of this school is quite
extensive, but Perry's "England's Neglect of Science," London, 1900, and
"Discussion on the Teaching of Mathematics," London, 1901, are typical.
[12] In his novel, "The Morals of Marcus Ordeyne."
[13] G. W. L. Carson, "The Functions of Geometry as a Subject of
Education," p. 3, Tonbridge, 1910.
[14] It may well be, however, that the growing curriculum may justify
some reduction in the time formerly assigned to geometry, and any
reasonable proposition of this nature should be fairly met by teachers
of mathematics.
[15] Professor Münsterberg, in the _Metropolitan Magazine_ for July,
1910.
CHAPTER III
A BRIEF HISTORY OF GEOMETRY
The geometry of very ancient peoples was largely the mensuration of
simple areas and solids, such as is taught to children in elementary
arithmetic to-day. They early learned how to find the area of a
rectangle, and in the oldest mathematical records that have come down to
us there is some discussion of the area of triangles and the volume of
solids.
The earliest documents that we have relating to geometry come to us from
Babylon and Egypt. Those from Babylon are written on small clay tablets,
some of them about the size of the hand, these tablets afterwards having
been baked in the sun. They show that the Babylonians of that period
knew something of land measures, and perhaps had advanced far enough to
compute the area of a trapezoid. For the mensuration of the circle they
later used, as did the early Hebrews, the value [pi] = 3. A tablet in
the British Museum shows that they also used such geometric forms as
triangles and circular segments in astrology or as talismans.
The Egyptians must have had a fair knowledge of practical geometry long
before the date of any mathematical treatise that has come down to us,
for the building of the pyramids, between 3000 and 2400 B.C., required
the application of several geometric principles. Some knowledge of
surveying must also have been necessary to carry out the extensive
plans for irrigation that were executed under Amenemhat III, about 2200
B.C.
The first definite knowledge that we have of Egyptian mathematics comes
to us from a manuscript copied on papyrus, a kind of paper used about
the Mediterranean in early times. This copy was made by one Aah-mesu
(The Moon-born), commonly called Ahmes, who probably flourished about
1700 B.C. The original from which he copied, written about 2300 B.C.,
has been lost, but the papyrus of Ahmes, written nearly four thousand
years ago, is still preserved, and is now in the British Museum. In this
manuscript, which is devoted chiefly to fractions and to a crude
algebra, is found some work on mensuration. Among the curious rules are
the incorrect ones that the area of an isosceles triangle equals half
the product of the base and one of the equal sides; and that the area of
a trapezoid having bases _b_, _b'_, and the nonparallel sides each equal
to _a_, is 1/2_a_(_b_ + _b'_). One noteworthy advance appears, however.
Ahmes gives a rule for finding the area of a circle, substantially as
follows: Multiply the square on the radius by (16/9)^2, which is
equivalent to taking for [pi] the value 3.1605. This papyrus also
contains some treatment of the mensuration of solids, particularly with
reference to the capacity of granaries. There is also some slight
mention of similar figures, and an extensive treatment of unit
fractions,--fractions that were quite universal among the ancients. In
the line of algebra it contains a brief treatment of the equation of the
first degree with one unknown, and of progressions.[16]
Herodotus tells us that Sesostris, king of Egypt,[17] divided the land
among his people and marked out the boundaries after the overflow of the
Nile, so that surveying must have been well known in his day. Indeed,
the _harpedonaptæ_, or rope stretchers, acquired their name because they
stretched cords, in which were knots, so as to make the right triangle
3, 4, 5, when they wished to erect a perpendicular. This is a plan
occasionally used by surveyors to-day, and it shows that the practical
application of the Pythagorean Theorem was known long before Pythagoras
gave what seems to have been the first general proof of the proposition.
From Egypt, and possibly from Babylon, geometry passed to the shores of
Asia Minor and Greece. The scientific study of the subject begins with
Thales, one of the Seven Wise Men of the Grecian civilization. Born at
Miletus, not far from Smyrna and Ephesus, about 640 B.C., he died at
Athens in 548 B.C. He spent his early manhood as a merchant,
accumulating the wealth that enabled him to spend his later years in
study. He visited Egypt, and is said to have learned such elements of
geometry as were known there. He founded a school of mathematics and
philosophy at Miletus, known from the country as the Ionic School. How
elementary the knowledge of geometry then was may be understood from the
fact that tradition attributes only about four propositions to
Thales,--(1) that vertical angles are equal, (2) that equal angles lie
opposite the equal sides of an isosceles triangle, (3) that a triangle
is determined by two angles and the included side, (4) that a diameter
bisects the circle, and possibly the propositions about the angle-sum
of a triangle for special cases, and the angle inscribed in a
semicircle.[18]
The greatest pupil of Thales, and one of the most remarkable men of
antiquity, was Pythagoras. Born probably on the island of Samos, just
off the coast of Asia Minor, about the year 580 B.C., Pythagoras set
forth as a young man to travel. He went to Miletus and studied under
Thales, probably spent several years in Egypt, very likely went to
Babylon, and possibly went even to India, since tradition asserts this
and the nature of his work in mathematics suggests it. In later life he
went to a Greek colony in southern Italy, and at Crotona, in the
southeastern part of the peninsula, he founded a school and established
a secret society to propagate his doctrines. In geometry he is said to
have been the first to demonstrate the proposition that the square on
the hypotenuse is equal to the sum of the squares upon the other two
sides of a right triangle. The proposition was known in India and Egypt
before his time, at any rate for special cases, but he seems to have
been the first to prove it. To him or to his school seems also to have
been due the construction of the regular pentagon and of the five
regular polyhedrons. The construction of the regular pentagon requires
the dividing of a line into extreme and mean ratio, and this problem is
commonly assigned to the Pythagoreans, although it played an important
part in Plato's school. Pythagoras is also said to have known that six
equilateral triangles, three regular hexagons, or four squares, can be
placed about a point so as just to fill the 360°, but that no other
regular polygons can be so placed. To his school is also due the proof
for the general case that the sum of the angles of a triangle equals two
right angles, the first knowledge of the size of each angle of a regular
polygon, and the construction of at least one star-polygon, the
star-pentagon, which became the badge of his fraternity. The brotherhood
founded by Pythagoras proved so offensive to the government that it was
dispersed before the death of the master. Pythagoras fled to Megapontum,
a seaport lying to the north of Crotona, and there he died about 501
B.C.[19]
[Illustration: FANCIFUL PORTRAIT OF PYTHAGORAS Calandri's Arithmetic,
1491]
For two centuries after Pythagoras geometry passed through a period of
discovery of propositions. The state of the science may be seen from
the fact that Oenopides of Chios, who flourished about 465 B.C., and who
had studied in Egypt, was celebrated because he showed how to let fall a
perpendicular to a line, and how to make an angle equal to a given
angle. A few years later, about 440 B.C., Hippocrates of Chios wrote the
first Greek textbook on mathematics. He knew that the areas of circles
are proportional to the squares on their radii, but was ignorant of the
fact that equal central angles or equal inscribed angles intercept equal
arcs.
Antiphon and Bryson, two Greek scholars, flourished about 430 B.C. The
former attempted to find the area of a circle by doubling the number of
sides of a regular inscribed polygon, and the latter by doing the same
for both inscribed and circumscribed polygons. They thus approximately
exhausted the area between the polygon and the circle, and hence this
method is known as the method of exhaustions.
About 420 B.C. Hippias of Elis invented a certain curve called the
quadratrix, by means of which he could square the circle and trisect any
angle. This curve cannot be constructed by the unmarked straightedge and
the compasses, and when we say that it is impossible to square the
circle or to trisect any angle, we mean that it is impossible by the
help of these two instruments alone.
During this period the great philosophic school of Plato (429-348 B.C.)
flourished at Athens, and to this school is due the first systematic
attempt to create exact definitions, axioms, and postulates, and to
distinguish between elementary and higher geometry. It was at this time
that elementary geometry became limited to the use of the compasses and
the unmarked straightedge, which took from this domain the possibility
of constructing a square equivalent to a given circle ("squaring the
circle"), of trisecting any given angle, and of constructing a cube that
should have twice the volume of a given cube ("duplicating the cube"),
these being the three famous problems of antiquity. Plato and his school
interested themselves with the so-called Pythagorean numbers, that is,
with numbers that would represent the three sides of a right triangle
and hence fulfill the condition that _a_^2 + _b_^2 = _c_^2. Pythagoras
had already given a rule that would be expressed in modern form, as
1/4(_m_^2 + 1)^2 = _m_^2 + 1/4(_m_^2 - 1)^2. The school of Plato found
that [(1/2_m_)^2 + 1]^2 = _m_^2 + [(1/2_m_)^2 - 1]^2. By giving various
values to _m_, different Pythagorean numbers may be found. Plato's
nephew, Speusippus (about 350 B.C.), wrote upon this subject. Such
numbers were known, however, both in India and in Egypt, long before
this time.
One of Plato's pupils was Philippus of Mende, in Egypt, who flourished
about 380 B.C. It is said that he discovered the proposition relating to
the exterior angle of a triangle. His interest, however, was chiefly in
astronomy.
Another of Plato's pupils was Eudoxus of Cnidus (408-355 B.C.). He
elaborated the theory of proportion, placing it upon a thoroughly
scientific foundation. It is probable that Book V of Euclid, which is
devoted to proportion, is essentially the work of Eudoxus. By means of
the method of exhaustions of Antiphon and Bryson he proved that the
pyramid is one third of a prism, and the cone is one third of a
cylinder, each of the same base and the same altitude. He wrote the
first textbook known on solid geometry.
The subject of conic sections starts with another pupil of Plato's,
Menæchmus, who lived about 350 B.C. He cut the three forms of conics
(the ellipse, parabola, and hyperbola) out of three different forms of
cone,--the acute-angled, right-angled, and obtuse-angled,--not noticing
that he could have obtained all three from any form of right circular
cone. It is interesting to see the far-reaching influence of Plato.
While primarily interested in philosophy, he laid the first scientific
foundations for a system of mathematics, and his pupils were the leaders
in this science in the generation following his greatest activity.
The great successor of Plato at Athens was Aristotle, the teacher of
Alexander the Great. He also was more interested in philosophy than in
mathematics, but in natural rather than mental philosophy. With him
comes the first application of mathematics to physics in the hands of a
great man, and with noteworthy results. He seems to have been the first
to represent an unknown quantity by letters. He set forth the theory of
the parallelogram of forces, using only rectangular components, however.
To one of his pupils, Eudemus of Rhodes, we are indebted for a history
of ancient geometry, some fragments of which have come down to us.
The first great textbook on geometry, and the greatest one that has ever
appeared, was written by Euclid, who taught mathematics in the great
university at Alexandria, Egypt, about 300 B.C. Alexandria was then
practically a Greek city, having been named in honor of Alexander the
Great, and being ruled by the Greeks.
In his work Euclid placed all of the leading propositions of plane
geometry then known, and arranged them in a logical order. Most
geometries of any importance written since his time have been based upon
Euclid, improving the sequence, symbols, and wording as occasion
demanded. He also wrote upon other branches of mathematics besides
elementary geometry, including a work on optics. He was not a great
creator of mathematics, but was rather a compiler of the work of others,
an office quite as difficult to fill and quite as honorable.
Euclid did not give much solid geometry because not much was known then.
It was to Archimedes (287-212 B.C.), a famous mathematician of Syracuse,
on the island of Sicily, that some of the most important propositions of
solid geometry are due, particularly those relating to the sphere and
cylinder. He also showed how to find the approximate value of [pi] by a
method similar to the one we teach to-day, proving that the real value
lay between 3 1/7 and 3 10/71. The story goes that the sphere and
cylinder were engraved upon his tomb, and Cicero, visiting Syracuse many
years after his death, found the tomb by looking for these symbols.
Archimedes was the greatest mathematical physicist of ancient times.
The Greeks contributed little more to elementary geometry, although
Apollonius of Perga, who taught at Alexandria between 250 and 200 B.C.,
wrote extensively on conic sections, and Hypsicles of Alexandria, about
190 B.C., wrote on regular polyhedrons. Hypsicles was the first Greek
writer who is known to have used sexagesimal fractions,--the degrees,
minutes, and seconds of our angle measure. Zenodorus (180 B.C.) wrote on
isoperimetric figures, and his contemporary, Nicomedes of Gerasa,
invented a curve known as the conchoid, by means of which he could
trisect any angle. Another contemporary, Diocles, invented the cissoid,
or ivy-shaped curve, by means of which he solved the famous problem of
duplicating the cube, that is, constructing a cube that should have
twice the volume of a given cube.
The greatest of the Greek astronomers, Hipparchus (180-125 B.C.), lived
about this period, and with him begins spherical trigonometry as a
definite science. A kind of plane trigonometry had been known to the
ancient Egyptians. The Greeks usually employed the chord of an angle
instead of the half chord (sine), the latter having been preferred by
the later Arab writers.
The most celebrated of the later Greek physicists was Heron of
Alexandria, formerly supposed to have lived about 100 B.C., but now
assigned to the first century A.D. His contribution to geometry was the
formula for the area of a triangle in terms of its sides a, b, and c,
with s standing for the semiperimeter 1/2(_a_ + _b_ + _c_). The formula
is [sqrt](_s_(_s_-_a_)(_s_-_b_)(_s_-_c_)).
Probably nearly contemporary with Heron was Menelaus of Alexandria, who
wrote a spherical trigonometry. He gave an interesting proposition
relating to plane and spherical triangles, their sides being cut by a
transversal. For the plane triangle _ABC_, the sides _a_, _b_, and _c_
being cut respectively in _X_, _Y_, and _Z_, the theorem asserts
substantially that
(_AZ_/_BZ_) · (_BX_/_CX_) · (_CY_/_AY_) = 1.
The most popular writer on astronomy among the Greeks was Ptolemy
(Claudius Ptolemaeus, 87-165 A.D.), who lived at Alexandria. He wrote a
work entitled "Megale Syntaxis" (The Great Collection), which his
followers designated as _Megistos_ (greatest), on which account the Arab
translators gave it the name "Almagest" (_al_ meaning "the"). He
advanced the science of trigonometry, but did not contribute to
geometry.
At the close of the third century Pappus of Alexandria (295 A.D.) wrote
on geometry, and one of his theorems, a generalized form of the
Pythagorean proposition, is mentioned in Chapter XVI of this work. Only
two other Greek writers on geometry need be mentioned. Theon of
Alexandria (370 A.D.), the father of the Hypatia who is the heroine of
Charles Kingsley's well-known novel, wrote a commentary on Euclid to
which we are indebted for some historical information. Proclus (410-485
A.D.) also wrote a commentary on Euclid, and much of our information
concerning the first Book of Euclid is due to him.
The East did little for geometry, although contributing considerably to
algebra. The first great Hindu writer was Aryabhatta, who was born in
476 A.D. He gave the very close approximation for [pi], expressed in
modern notation as 3.1416. He also gave rules for finding the volume of
the pyramid and sphere, but they were incorrect, showing that the Greek
mathematics had not yet reached the Ganges. Another Hindu writer,
Brahmagupta (born in 598 A.D.), wrote an encyclopedia of mathematics. He
gave a rule for finding Pythagorean numbers, expressed in modern symbols
as follows:
1/4((_p_^2/_q_) + _q_)^2 = 1/4((_p_^2/_q_) - _q_)^2 + _p_^2.
He also generalized Heron's formula by asserting that the area of an
inscribed quadrilateral of sides _a_, _b_, _c_, _d_, and semiperimeter
_s_, is [sqrt]((_s_ - _a_)(_s_ - _b_)(_s_ - _c_)(_s_ - _d_)).
The Arabs, about the time of the "Arabian Nights Tales" (800 A.D.), did
much for mathematics, translating the Greek authors into their language
and also bringing learning from India. Indeed, it is to them that modern
Europe owed its first knowledge of Euclid. They contributed nothing of
importance to elementary geometry, however.
The greatest of the Arab writers was Mohammed ibn Musa al-Khowarazmi
(820 A.D.). He lived at Bagdad and Damascus. Although chiefly interested
in astronomy, he wrote the first book bearing the name "algebra"
("Al-jabr wa'l-muq[=a]balah," Restoration and Equation), composed an
arithmetic using the Hindu numerals,[20] and paid much attention to
geometry and trigonometry.
Euclid was translated from the Arabic into Latin in the twelfth century,
Greek manuscripts not being then at hand, or being neglected because of
ignorance of the language. The leading translators were Athelhard of
Bath (1120), an English monk; Gherard of Cremona (1160), an Italian
monk; and Johannes Campanus (1250), chaplain to Pope Urban IV.
The greatest European mathematician of the Middle Ages was Leonardo of
Pisa[21] (_ca._ 1170-1250). He was very influential in making the
Hindu-Arabic numerals known in Europe, wrote extensively on algebra, and
was the author of one book on geometry. He contributed nothing to the
elementary theory, however. The first edition of Euclid was printed in
Latin in 1482, the first one in English appearing in 1570.
Our symbols are modern, + and - first appearing in a German work in
1489; = in Recorde's "Whetstone of Witte" in 1557; > and < in the works
of Harriot (1560-1621); and × in a publication by Oughtred (1574-1660).
The most noteworthy advance in geometry in modern times was made by the
great French philosopher Descartes, who published a small work entitled
"La Géométrie" in 1637. From this springs the modern analytic geometry,
a subject that has revolutionized the methods of all mathematics. Most
of the subsequent discoveries in mathematics have been in higher
branches. To the great Swiss mathematician Euler (1707-1783) is due,
however, one proposition that has found its way into elementary
geometry, the one showing the relation between the number of edges,
vertices, and faces of a polyhedron.
There has of late arisen a modern elementary geometry devoted chiefly to
special points and lines relating to the triangle and the circle, and
many interesting propositions have been discovered. The subject is so
extensive that it cannot find any place in our crowded curriculum, and
must necessarily be left to the specialist.[22] Some idea of the nature
of the work may be obtained from a mention of a few propositions:
The medians of a triangle are concurrent in the centroid, or center of
gravity of the triangle.
The bisectors of the various interior and exterior angles of a triangle
are concurrent by threes in the incenter or in one of the three
excenters of the triangle.
The common chord of two intersecting circles is a special case of their
radical axis, and tangents to the circles from any point on the radical
axis are equal.
If _O_ is the orthocenter of the triangle _ABC_, and _X_, _Y_, _Z_ are
the feet of the perpendiculars from _A_, _B_, _C_ respectively, and _P_,
_Q_, _R_ are the mid-points of _a_, _b_, _c_ respectively, and _L_, _M_,
_N_ are the mid-points of _OA_, _OB_, _OC_ respectively; then the points
_L_, _M_, _N_; _P_, _Q_, _R_; _X_, _Y_, _Z_ all lie on a circle, the
"nine points circle."
In the teaching of geometry it adds a human interest to the subject to
mention occasionally some of the historical facts connected with it. For
this reason this brief sketch will be supplemented by many notes upon
the various important propositions as they occur in the several books
described in the later chapters of this work.
FOOTNOTES:
[16] It was published in German translation by A. Eisenlohr, "Ein
mathematisches Handbuch der alten Aegypter," Leipzig, 1877, and in
facsimile by the British Museum, under the title, "The Rhind Papyrus,"
in 1898.
[17] Generally known as Rameses II. He reigned in Egypt about 1350 B.C.
[18] Two excellent works on Thales and his successors, and indeed the
best in English, are the following: G. J. Allman, "Greek Geometry from
Thales to Euclid," Dublin, 1889; J. Gow, "A History of Greek
Mathematics," Cambridge, 1884. On all mathematical subjects the best
general history is that of M. Cantor, "Geschichte der Mathematik," 4
vols, Leipzig, 1880-1908.
[19] Another good work on Greek geometry, with considerable material on
Pythagoras, is by C. A. Bretschneider, "Die Geometrie und die Geometer
vor Eukleides," Leipzig, 1870.
[20] Smith and Karpinski, "The Hindu-Arabic Numerals," Boston, 1911.
[21] For a sketch of his life see Smith and Karpinski, loc. cit.
[22] Those who care for a brief description of this phase of the subject
may consult J. Casey, "A Sequel to Euclid," Dublin, fifth edition, 1888;
W. J. M'Clelland, "A Treatise on the Geometry of the Circle," New York,
1891; M. Simon, "Über die Entwicklung der Elementar-Geometrie im XIX.
Jahrhundert," Leipzig, 1906.
CHAPTER IV
DEVELOPMENT OF THE TEACHING OF GEOMETRY
We know little of the teaching of geometry in very ancient times, but we
can infer its nature from the teaching that is still seen in the native
schools of the East. Here a man, learned in any science, will have a
group of voluntary students sitting about him, and to them he will
expound the truth. Such schools may still be seen in India, Persia, and
China, the master sitting on a mat placed on the ground or on the floor
of a veranda, and the pupils reading aloud or listening to his words of
exposition.
In Egypt geometry seems to have been in early times mere mensuration,
confined largely to the priestly caste. It was taught to novices who
gave promise of success in this subject, and not to others, the idea of
general culture, of training in logic, of the cultivation of exact
expression, and of coming in contact with truth being wholly wanting.
In Greece it was taught in the schools of philosophy, often as a general
preparation for philosophic study. Thus Thales introduced it into his
Ionic school, Pythagoras made it very prominent in his great school at
Crotona in southern Italy (Magna Græcia), and Plato placed above the
door of his _Academia_ the words, "Let no one ignorant of geometry enter
here,"--a kind of entrance examination for his school of philosophy. In
these gatherings of students it is probable that geometry was taught in
much the way already mentioned for the schools of the East, a small
group of students being instructed by a master. Printing was unknown,
papyrus was dear, parchment was only in process of invention. Paper such
as we know had not yet appeared, so that instruction was largely oral,
and geometric figures were drawn by a pointed stick on a board covered
with fine sand, or on a tablet of wax.
But with these crude materials there went an abundance of time, so that
a number of great results were accomplished in spite of the difficulties
attending the study of the subject. It is said that Hippocrates of Chios
(_ca._ 440 B.C.) wrote the first elementary textbook on mathematics and
invented the method of geometric reduction, the replacing of a
proposition to be proved by another which, when proved, allows the first
one to be demonstrated. A little later Eudoxus of Cnidus (_ca._ 375
B.C.), a pupil of Plato's, used the _reductio ad absurdum_, and Plato is
said to have invented the method of proof by analysis, an elaboration of
the plan used by Hippocrates. Thus these early philosophers taught their
pupils not facts alone, but methods of proof, giving them power as well
as knowledge. Furthermore, they taught them how to discuss their
problems, investigating the conditions under which they are capable of
solution. This feature of the work they called the _diorismus_, and it
seems to have started with Leon, a follower of Plato.
Between the time of Plato (_ca._ 400 B.C.) and Euclid (_ca._ 300 B.C.)
several attempts were made to arrange the accumulated material of
elementary geometry in a textbook. Plato had laid the foundations for
the science, in the form of axioms, postulates, and definitions, and he
had limited the instruments to the straightedge and the compasses.
Aristotle (_ca._ 350 B.C.) had paid special attention to the history of
the subject, thus finding out what had already been accomplished, and
had also made much of the applications of geometry. The world was
therefore ready for a good teacher who should gather the material and
arrange it scientifically. After several attempts to find the man for
such a task, he was discovered in Euclid, and to his work the next
chapter is devoted.
After Euclid, Archimedes (_ca._ 250 B.C.) made his great contributions.
He was not a teacher like his illustrious predecessor, but he was a
great discoverer. He has left us, however, a statement of his methods of
investigation which is helpful to those who teach. These methods were
largely experimental, even extending to the weighing of geometric forms
to discover certain relations, the proof being given later. Here was
born, perhaps, what has been called the laboratory method of the
present.
Of the other Greek teachers we have but little information as to methods
of imparting instruction. It is not until the Middle Ages that there is
much known in this line. Whatever of geometry was taught seems to have
been imparted by word of mouth in the way of expounding Euclid, and this
was done in the ancient fashion.
The early Church leaders usually paid no attention to geometry, but as
time progressed the _quadrivium_, or four sciences of arithmetic, music,
geometry, and astronomy, came to rank with the _trivium_ (grammar,
rhetoric, dialectics), the two making up the "seven liberal arts." All
that there was of geometry in the first thousand years of Christianity,
however, at least in the great majority of Church schools, was summed
up in a few definitions and rules of mensuration. Gerbert, who became
Pope Sylvester II in 999 A.D., gave a new impetus to geometry by
discovering a manuscript of the old Roman surveyors and a copy of the
geometry of Boethius, who paraphrased Euclid about 500 A.D. He thereupon
wrote a brief geometry, and his elevation to the papal chair tended to
bring the study of mathematics again into prominence.
Geometry now began to have some place in the Church schools, naturally
the only schools of high rank in the Middle Ages. The study of the
subject, however, seems to have been merely a matter of memorizing.
Geometry received another impetus in the book written by Leonardo of
Pisa in 1220, the "Practica Geometriae." Euclid was also translated into
Latin about this time (strangely enough, as already stated, from the
Arabic instead of the Greek), and thus the treasury of elementary
geometry was opened to scholars in Europe. From now on, until the
invention of printing (_ca._ 1450), numerous writers on geometry appear,
but, so far as we know, the method of instruction remained much as it
had always been. The universities began to appear about the thirteenth
century, and Sacrobosco, a well-known medieval mathematician, taught
mathematics about 1250 in the University of Paris. In 1336 this
university decreed that mathematics should be required for a degree. In
the thirteenth century Oxford required six books of Euclid for one who
was to teach, but this amount of work seems to have been merely nominal,
for in 1450 only two books were actually read. The universities of
Prague (founded in 1350) and Vienna (statutes of 1389) required most of
plane geometry for the teacher's license, although Vienna demanded but
one book for the bachelor's degree. So, in general, the universities of
the thirteenth, fourteenth, and fifteenth centuries required less for
the degree of master of arts than we now require from a pupil in our
American high schools. On the other hand, the university students were
younger than now, and were really doing only high school work.
The invention of printing made possible the study of geometry in a new
fashion. It now became possible for any one to study from a book,
whereas before this time instruction was chiefly by word of mouth,
consisting of an explanation of Euclid. The first Euclid was printed in
1482, at Venice, and new editions and variations of this text came out
frequently in the next century. Practical geometries became very
popular, and the reaction against the idea of mental discipline
threatened to abolish the old style of text. It was argued that geometry
was uninteresting, that it was not sufficient in itself, that boys
needed to see the practical uses of the subject, that only those
propositions that were capable of application should be retained, that
there must be a fusion between the demands of culture and the demands of
business, and that every man who stood for mathematical ideals
represented an obsolete type. Such writers as Finæus (1556), Bartoli
(1589), Belli (1569), and Cataneo (1567), in the sixteenth century, and
Capra (1678), Gargiolli (1655), and many others in the seventeenth
century, either directly or inferentially, took this attitude towards
the subject,--exactly the attitude that is being taken at the present
time by a number of teachers in the United States. As is always the
case, to such an extreme did this movement lead that there was a
reaction that brought the Euclid type of book again to the front, and
it has maintained its prominence even to the present.
The study of geometry in the high schools is relatively recent. The
Gymnasium (classical school preparatory to the university) at Nürnberg,
founded in 1526, and the Cathedral school at Württemberg (as shown by
the curriculum of 1556) seem to have had no geometry before 1600,
although the Gymnasium at Strassburg included some of this branch of
mathematics in 1578, and an elective course in geometry was offered at
Zwickau, in Saxony, in 1521. In the seventeenth century geometry is
found in a considerable number of secondary schools, as at Coburg
(1605), Kurfalz (1615, elective), Erfurt (1643), Gotha (1605), Giessen
(1605), and numerous other places in Germany, although it appeared but
rarely in the secondary schools of France before the eighteenth century.
In Germany the Realschulen--schools with more science and less classics
than are found in the Gymnasium--came into being in the eighteenth
century, and considerable effort was made to construct a course in
geometry that should be more practical than that of the modified Euclid.
At the opening of the nineteenth century the Prussian schools were
reorganized, and from that time on geometry has had a firm position in
the secondary schools of all Germany. In the eighteenth century some
excellent textbooks on geometry appeared in France, among the best being
that of Legendre (1794), which influenced in such a marked degree the
geometries of America. Soon after the opening of the nineteenth century
the _lycées_ of France became strong institutions, and geometry, chiefly
based on Legendre, was well taught in the mathematical divisions. A
worthy rival of Legendre's geometry was the work of Lacroix, who called
attention continually to the analogy between the theorems of plane and
solid geometry, and even went so far as to suggest treating the related
propositions together in certain cases.
In England the preparatory schools, such as Rugby, Harrow, and Eton, did
not commonly teach geometry until quite recently, leaving this work for
the universities. In Christ's Hospital, London, however, geometry was
taught as early as 1681, from a work written by several teachers of
prominence. The highest class at Harrow studied "Euclid and vulgar
fractions" one period a week in 1829, but geometry was not seriously
studied before 1837. In the Edinburgh Academy as early as 1885, and in
Rugby by 1839, plane geometry was completed.
Not until 1844 did Harvard require any plane geometry for entrance. In
1855 Yale required only two books of Euclid. It was therefore from 1850
to 1875 that plane geometry took a definite place in the American high
school. Solid geometry has not been generally required for entrance to
any eastern college, although in the West this is not the case. The East
teaches plane geometry more thoroughly, but allows a pupil to enter
college or to go into business with no solid geometry. Given a year to
the subject, it is possible to do little more than cover plane geometry;
with a year and a half the solid geometry ought easily to be covered
also.
=Bibliography.= Stamper, A History of the Teaching of
Elementary Geometry, New York, 1909, with a very full
bibliography of the subject; Cajori, The Teaching of
Mathematics in the United States, Washington, 1890; Cantor,
Geschichte der Mathematik, Vol. IV, p. 321, Leipzig, 1908;
Schotten, Inhalt und Methode des planimetrischen Unterrichts,
Leipzig, 1890.
CHAPTER V
EUCLID
It is fitting that a chapter in a book upon the teaching of this subject
should be devoted to the life and labors of the greatest of all textbook
writers, Euclid,--a man whose name has been, for more than two thousand
years, a synonym for elementary plane geometry wherever the subject has
been studied. And yet when an effort is made to pick up the scattered
fragments of his biography, we are surprised to find how little is known
of one whose fame is so universal. Although more editions of his work
have been printed than of any other book save the Bible,[23] we do not
know when he was born, or in what city, or even in what country, nor do
we know his race, his parentage, or the time of his death. We should not
feel that we knew much of the life of a man who lived when the Magna
Charta was wrested from King John, if our first and only source of
information was a paragraph in the works of some historian of to-day;
and yet this is about the situation in respect to Euclid. Proclus of
Alexandria, philosopher, teacher, and mathematician, lived from 410 to
485 A.D., and wrote a commentary on the works of Euclid. In his
writings, which seem to set forth in amplified form his lectures to the
students in the Neoplatonist School of Alexandria, Proclus makes this
statement, and of Euclid's life we have little else:
Not much younger than these[24] is Euclid, who put together the
"Elements," collecting many of the theorems of Eudoxus,
perfecting many of those of Theætetus, and also demonstrating
with perfect certainty what his predecessors had but
insufficiently proved. He flourished in the time of the first
Ptolemy, for Archimedes, who closely followed this ruler,[25]
speaks of Euclid. Furthermore it is related that Ptolemy one
time demanded of him if there was in geometry no shorter way
than that of the "Elements," to whom he replied that there was
no royal road to geometry.[26] He was therefore younger than
the pupils of Plato, but older than Eratosthenes and
Archimedes; for the latter were contemporary with one another,
as Eratosthenes somewhere says.[27]
Thus we have in a few lines, from one who lived perhaps seven or eight
hundred years after Euclid, nearly all that is known of the most famous
teacher of geometry that ever lived. Nevertheless, even this little
tells us about when he flourished, for Hermotimus and Philippus were
pupils of Plato, who died in 347 B.C., whereas Archimedes was born about
287 B.C. and was writing about 250 B.C. Furthermore, since Ptolemy I
reigned from 306 to 283 B.C., Euclid must have been teaching about 300
B.C., and this is the date that is generally assigned to him.
Euclid probably studied at Athens, for until he himself assisted in
transferring the center of mathematical culture to Alexandria, it had
long been in the Grecian capital, indeed since the time of Pythagoras.
Moreover, numerous attempts had been made at Athens to do exactly what
Euclid succeeded in doing,--to construct a logical sequence of
propositions; in other words, to write a textbook on plane geometry. It
was at Athens, therefore, that he could best have received the
inspiration to compose his "Elements."[28] After finishing his education
at Athens it is quite probable that he, like other savants of the
period, was called to Alexandria by Ptolemy Soter, the king, to assist
in establishing the great school which made that city the center of the
world's learning for several centuries. In this school he taught, and
here he wrote the "Elements" and numerous other works, perhaps ten in
all.
Although the Greek writers who may have known something of the life of
Euclid have little to say of him, the Arab writers, who could have known
nothing save from Greek sources, have allowed their imaginations the
usual latitude in speaking of him and of his labors. Thus
Al-Qif[t.][=i], who wrote in the thirteenth century, has this to say in
his biographical treatise "Ta'r[=i]kh al-[H.]ukam[=a]":
Euclid, son of Naucrates, grandson of Zenarchus, called the
author of geometry, a Greek by nationality, domiciled at
Damascus, born at Tyre, most learned in the science of
geometry, published a most excellent and most useful work
entitled "The Foundation or Elements of Geometry," a subject in
which no more general treatise existed before among the Greeks;
nay, there was no one even of later date who did not walk in
his footsteps and frankly profess his doctrine.
This is rather a specimen of the Arab tendency to manufacture history
than a serious contribution to the biography of Euclid, of whose
personal history we have only the information given by Proclus.
[Illustration: EUCLID
From an old print]
Euclid's works at once took high rank, and they are mentioned by various
classical authors. Cicero knew of them, and Capella (_ca._ 470 A.D.),
Cassiodorius (_ca._ 515 A.D.), and Boethius (_ca._ 480-524 A.D.) were
all more or less familiar with the "Elements." With the advance of the
Dark Ages, however, learning was held in less and less esteem, so that
Euclid was finally forgotten, and manuscripts of his works were either
destroyed or buried in some remote cloister. The Arabs, however, whose
civilization assumed prominence from about 750 A.D. to about 1500,
translated the most important treatises of the Greeks, and Euclid's
"Elements" among the rest. One of these Arabic editions an English monk
of the twelfth century, one Athelhard (Æthelhard) of Bath, found and
translated into Latin (_ca._ 1120 A.D.). A little later Gherard of
Cremona (1114-1187) made a new translation from the Arabic, differing in
essential features from that of Athelhard, and about 1260 Johannes
Campanus made still a third translation, also from Arabic into
Latin.[29] There is reason to believe that Athelhard, Campanus, and
Gherard may all have had access to an earlier Latin translation, since
all are quite alike in some particulars while diverging noticeably in
others. Indeed, there is an old English verse that relates:
The clerk Euclide on this wyse hit fonde
Thys craft of gemetry yn Egypte londe ...
Thys craft com into England, as y yow say,
Yn tyme of good Kyng Adelstone's day.
If this be true, Euclid was known in England as early as 924-940 A.D.
Without going into particulars further, it suffices to say that the
modern knowledge of Euclid came first through the Arabic into the Latin,
and the first printed edition of the "Elements" (Venice, 1482) was the
Campanus translation. Greek manuscripts now began to appear, and at the
present time several are known. There is a manuscript of the ninth
century in the Bodleian library at Oxford, one of the tenth century in
the Vatican, another of the tenth century in Florence, one of the
eleventh century at Bologna, and two of the twelfth century at Paris.
There are also fragments containing bits of Euclid in Greek, and going
back as far as the second and third century A.D. The first modern
translation from the Greek into the Latin was made by Zamberti (or
Zamberto),[30] and was printed at Venice in 1513. The first translation
into English was made by Sir Henry Billingsley and was printed in 1570,
sixteen years before he became Lord Mayor of London.
Proclus, in his commentary upon Euclid's work, remarks:
In the whole of geometry there are certain leading theorems,
bearing to those which follow the relation of a principle,
all-pervading, and furnishing proofs of many properties. Such
theorems are called by the name of _elements_, and their
function may be compared to that of the letters of the alphabet
in relation to language, letters being indeed called by the
same name in Greek [[Greek: stoicheia], stoicheia].[31]
This characterizes the work of Euclid, a collection of the basic
propositions of geometry, and chiefly of plane geometry, arranged in
logical sequence, the proof of each depending upon some preceding
proposition, definition, or assumption (axiom or postulate). The number
of the propositions of plane geometry included in the "Elements" is not
entirely certain, owing to some disagreement in the manuscripts, but it
was between one hundred sixty and one hundred seventy-five. It is
possible to reduce this number by about thirty or forty, because Euclid
included a certain amount of geometric algebra; but beyond this we
cannot safely go in the way of elimination, since from the very nature
of the "Elements" these propositions are basic. The efforts at revising
Euclid have been generally confined, therefore, to rearranging his
material, to rendering more modern his phraseology, and to making a book
that is more usable with beginners if not more logical in its
presentation of the subject. While there has been an improvement upon
Euclid in the art of bookmaking, and in minor matters of phraseology and
sequence, the educational gain has not been commensurate with the effort
put forth. With a little modification of Euclid's semi-algebraic Book II
and of his treatment of proportion, with some scattering of the
definitions and the inclusion of well-graded exercises at proper places,
and with attention to the modern science of bookmaking, the "Elements"
would answer quite as well for a textbook to-day as most of our modern
substitutes, and much better than some of them. It would, moreover, have
the advantage of being a classic,--somewhat the same advantage that
comes from reading Homer in the original instead of from Pope's metrical
translation. This is not a plea for a return to the Euclid text, but for
a recognition of the excellence of Euclid's work.
The distinctive feature of Euclid's "Elements," compared with the modern
American textbook, is perhaps this: Euclid begins a book with what seems
to him the easiest proposition, be it theorem or problem; upon this he
builds another; upon these a third, and so on, concerning himself but
little with the classification of propositions. Furthermore, he arranges
his propositions so as to construct his figures before using them. We,
on the other hand, make some little attempt to classify our propositions
within each book, and we make no attempt to construct our figures before
using them, or at least to prove that the constructions are correct.
Indeed, we go so far as to study the properties of figures that we
cannot construct, as when we ask for the size of the angle of a regular
heptagon. Thus Euclid begins Book I by a problem, to construct an
equilateral triangle on a given line. His object is to follow this by
problems on drawing a straight line equal to a given straight line, and
cutting off from the greater of two straight lines a line equal to the
less. He now introduces a theorem, which might equally well have been
his first proposition, namely, the case of the congruence of two
triangles, having given two sides and the included angle. By means of
his third and fourth propositions he is now able to prove the _pons
asinorum_, that the angles at the base of an isosceles triangle are
equal. We, on the other hand, seek to group our propositions where this
can conveniently be done, putting the congruent propositions together,
those about inequalities by themselves, and the propositions about
parallels in one set. The results of the two arrangements are not
radically different, and the effect of either upon the pupil's mind does
not seem particularly better than that of the other. Teachers who have
used both plans quite commonly feel that, apart from Books II and V,
Euclid is nearly as easily understood as our modern texts, if presented
in as satisfactory dress.
The topics treated and the number of propositions in the plane geometry
of the "Elements" are as follows:
Book I. Rectilinear figures 48
Book II. Geometric algebra 14
Book III. Circles 37
Book IV. Problems about circles 16
Book V. Proportion 25
Book VI. Applications of proportion 33
----
173
Of these we now omit Euclid's Book II, because we have an algebraic
symbolism that was unknown in his time, although he would not have used
it in geometry even had it been known. Thus his first proposition in
Book II is as follows:
If there be two straight lines, and one of them be cut into any
number of segments whatever, the rectangle contained by the two
straight lines is equal to the rectangles contained by the
uncut straight line and each of the segments.
This amounts to saying that if _x_ = _p_ + _q_ + _r_ + ···, then
_ax_ = _ap_ + _aq_ + _ar_ + ···. We also materially simplify Euclid's
Book V. He, for example, proves that "If four magnitudes be
proportional, they will also be proportional alternately." This he
proves generally for any kind of magnitude, while we merely prove it for
numbers having a common measure. We say that we may substitute for the
older form of proportion, namely,
_a_ : _b_ = _c_ : _d_,
the fractional form _a_/_b_ = _c_/_d_.
From this we have _ad_ = _bc_.
Whence _a_/_c_ = _b_/_d_.
In this work we assume that we may multiply equals by _b_ and _d_. But
suppose _b_ and _d_ are cubes, of which, indeed, we do not even know the
approximate numerical measure; what shall we do? To Euclid the
multiplication by a cube or a polygon or a sphere would have been
entirely meaningless, as it always is from the standpoint of pure
geometry. Hence it is that our treatment of proportion has no serious
standing in geometry as compared with Euclid's, and our only
justification for it lies in the fact that it is easier. Euclid's
treatment is much more rigorous than ours, but it is adapted to the
comprehension of only advanced students, while ours is merely a
confession, and it should be a frank confession, of the weakness of our
pupils, and possibly, at times, of ourselves.
If we should take Euclid's Books II and V for granted, or as
sufficiently evident from our study of algebra, we should have remaining
only one hundred thirty-four propositions, most of which may be
designated as basal propositions of plane geometry. Revise Euclid as we
will, we shall not be able to eliminate any large number of his
fundamental truths, while we might do much worse than to adopt these one
hundred thirty-four propositions _in toto_ as the bases, and indeed as
the definition, of elementary plane geometry.
=Bibliography.= Heath, The Thirteen Books of Euclid's Elements,
3 vols., Cambridge, 1908; Frankland, The First Book of Euclid,
Cambridge, 1906; Smith, Dictionary of Greek and Roman
Biography, article Eukleides; Simon, Euclid und die sechs
planimetrischen Bücher, Leipzig, 1901; Gow, History of Greek
Mathematics, Cambridge, 1884, and any of the standard histories
of mathematics. Both Heath and Simon give extensive
bibliographies. The latest standard Greek and Latin texts are
Heiberg's, published by Teubner of Leipzig.
FOOTNOTES:
[23] Riccardi, Saggio di una bibliografia Euclidea, Part I, p. 3,
Bologna, 1887. Riccardi lists well towards two thousand editions.
[24] Hermotimus of Colophon and Philippus of Mende.
[25] Literally, "Who closely followed the first," i.e. the first
Ptolemy.
[26] Menæchmus is said to have replied to a similar question of
Alexander the Great: "O King, through the country there are royal roads
and roads for common citizens, but in geometry there is one road for
all."
[27] This is also shown in a letter from Archimedes to Eratosthenes,
recently discovered by Heiberg.
[28] On this phase of the subject, and indeed upon Euclid and his
propositions and works in general, consult T. L. Heath, "The Thirteen
Books of Euclid's Elements," 3 vols., Cambridge, 1908, a masterly
treatise of which frequent use has been made in preparing this work.
[29] A contemporary copy of this translation is now in the library of
George A. Plimpton, Esq., of New York. See the author's "Rara
Arithmetica," p. 433, Boston, 1909.
[30] A beautiful vellum manuscript of this translation is in the library
of George A. Plimpton, Esq., of New York. See the author's "Rara
Arithmetica," p. 481, Boston, 1909.
[31] Heath, loc. cit., Vol. I, p. 114.
CHAPTER VI
EFFORTS AT IMPROVING EUCLID
From time to time an effort is made by some teacher, or association of
teachers, animated by a serious desire to improve the instruction in
geometry, to prepare a new syllabus that shall mark out some "royal
road," and it therefore becomes those who are interested in teaching to
consider with care the results of similar efforts in recent years. There
are many questions which such an attempt suggests: What is the real
purpose of the movement? What will the teaching world say of the result?
Shall a reckless, ill-considered radicalism dominate the effort,
bringing in a distasteful terminology and symbolism merely for its
novelty, insisting upon an ultralogical treatment that is beyond the
powers of the learner, rearranging the subject matter to fit some narrow
notion of the projectors, seeking to emasculate mathematics by looking
only to the applications, riding some little hobby in the way of some
particular class of exercises, and cutting the number of propositions to
a minimum that will satisfy the mere demands of the artisan? Such are
some of the questions that naturally arise in the mind of every one who
wishes well for the ancient science of geometry.
It is not proposed in this chapter to attempt to answer these questions,
but rather to assist in understanding the problem by considering the
results of similar attempts. If it shall be found that syllabi have
been prepared under circumstances quite as favorable as those that
obtain at present, and if these syllabi have had little or no real
influence, then it becomes our duty to see if new plans may be worked
out so as to be more successful than their predecessors. If the older
attempts have led to some good, it is well to know what is the nature of
this good, to the end that new efforts may also result in something of
benefit to the schools.
It is proposed in this chapter to call attention to four important
syllabi, setting forth briefly their distinguishing features and drawing
some conclusions that may be helpful in other efforts of this nature.
In England two noteworthy attempts have been made within a century,
looking to a more satisfactory sequence and selection of propositions
than is found in Euclid. Each began with a list of propositions arranged
in proper sequence, and each was thereafter elaborated into a textbook.
Neither accomplished fully the purpose intended, but each was
instrumental in provoking healthy discussion and in improving the texts
from which geometry is studied.
The first of these attempts was made by Professor Augustus de Morgan,
under the auspices of the Society for the Diffusion of Useful Knowledge,
and it resulted in a textbook, including "plane, solid, and spherical"
geometry, in six books. According to De Morgan's plan, plane geometry
consisted of three books, the number of propositions being as follows:
Book I. Rectilinear figures 60
Book II. Ratio, proportion, applications 69
Book III. The circle 65
----
Total for plane geometry 194
Of the 194 propositions De Morgan selected 114 with their corollaries as
necessary for a beginner who is teaching himself.
In solid geometry the plan was as follows:
Book IV. Lines in different planes, solids
contained by planes 52
Book V. Cylinder, cone, sphere 25
Book VI. Figures on a sphere 42
----
Total for solid geometry 119
Of these 119 propositions De Morgan selected 76 with their corollaries
as necessary for a beginner, thus making 190 necessary propositions out
of 305 desirable ones, besides the corollaries in plane and solid
geometry. In other words, of the desirable propositions he considered
that about two thirds are absolutely necessary.
It is interesting to note, however, that he summed up the results of his
labors by saying:
It will be found that the course just laid down, excepting the
sixth book of it only, is not of much greater extent, nor very
different in point of matter from that of Euclid, whose
"Elements" have at all times been justly esteemed a model not
only of easy and progressive instruction in geometry, but of
accuracy and perspicuity in reasoning.
De Morgan's effort, essentially that of a syllabus-maker rather than a
textbook writer, although it was published under the patronage of a
prominent society with which were associated the names of men like Henry
Hallam, Rowland Hill, Lord John Russell, and George Peacock, had no
apparent influence on geometry either in England or abroad. Nevertheless
the syllabus was in many respects excellent; it rearranged the matter,
it classified the propositions, it improved some of the terminology, and
it reduced the number of essential propositions; it had the assistance
of De Morgan's enthusiasm and of the society with which he was so
prominently connected, and it was circulated with considerable
generosity throughout the English-speaking world; but in spite of all
this it is to-day practically unknown.
A second noteworthy attempt in England was made about a quarter of a
century ago by a society that was organized practically for this very
purpose, the Association for the Improvement of Geometrical Teaching.
This society was composed of many of the most progressive teachers in
England, and it included in its membership men of high standing in
mathematics in the universities. As a result of their labors a syllabus
was prepared, which was elaborated into a textbook, and in 1889 a
revised syllabus was issued.
As to the arrangement of matter, the syllabus departs from Euclid
chiefly by separating the problems from the theorems, as is the case in
our American textbooks, and in improving the phraseology. The course is
preceded by some simple exercises in the use of the compasses and ruler,
a valuable plan that is followed by many of the best teachers
everywhere. Considerable attention is paid to logical processes before
beginning the work, such terms as "contrapositive" and "obverse," and
such rules as the "rule of conversion" and the "rule of identity" being
introduced before any propositions are considered.
The arrangement of the work and the number of propositions in plane
geometry are as follows:
Book I. The straight line 51
Book II. Equality of areas 19
Book III. The circle 42
Book IV. Ratio and proportion 32
Book V. Proportion 24
----
Total for plane geometry 168
Here, then, is the result of several years of labor by a somewhat
radical organization, fostered by excellent mathematicians, and carried
on in a country where elementary geometry is held in highest esteem, and
where Euclid was thought unsuited to the needs of the beginner. The
number of propositions remains substantially the same as in Euclid, and
the introduction of some unusable logic tends to counterbalance the
improvement in sequence of the propositions. The report provoked
thought; it shook the Euclid stronghold; it was probably instrumental in
bringing about the present upheaval in geometry in England, but as a
working syllabus it has not appealed to the world as the great
improvement upon Euclid's "Elements" that was hoped by many of its early
advocates.
The same association published later, and republished in 1905, a "Report
on the Teaching of Geometry," in which it returned to Euclid, modifying
the "Elements" by omitting certain propositions, changing the order and
proof of others, and introducing a few new theorems. It seems to reduce
the propositions to be proved in plane geometry to about one hundred
fifteen, and it recommends the omission of the incommensurable case.
This number is, however, somewhat misleading, for Euclid frequently puts
in one proposition what we in America, for educational reasons, find it
better to treat in two, or even three, propositions. This report,
therefore, reaches about the same conclusion as to the geometric facts
to be mastered as is reached by our later textbook writers in America.
It is not extreme, and it stands for good mathematics.
In the United States the influence of our early wars with England, and
the sympathy of France at that time, turned the attention of our
scholars of a century ago from Cambridge to Paris as a mathematical
center. The influx of French mathematics brought with it such works as
Legendre's geometry (1794) and Bourdon's algebra, and made known the
texts of Lacroix, Bertrand, and Bezout. Legendre's geometry was the
result of the efforts of a great mathematician at syllabus-making, a
natural thing in a country that had early broken away from Euclid.
Legendre changed the Greek sequence, sought to select only propositions
that are necessary to a good understanding of the subject, and added a
good course in solid geometry. His arrangement, with the number of
propositions as given in the Davies translation, is as follows:
Book I. Rectilinear figures 31
Book II. Ratio and proportion 14
Book III. The circle 48
Book IV. Proportions of figures and areas 51
Book V. Polygons and circles 17
----
Total for plane geometry 161
Legendre made, therefore, practically no reduction in the number of
Euclid's propositions, and his improvement on Euclid consisted chiefly
in his separation of problems and theorems, and in a less rigorous
treatment of proportion which boys and girls could comprehend.
D'Alembert had demanded that the sequence of propositions should be
determined by the order in which they had been discovered, but Legendre
wisely ignored such an extreme and gave the world a very usable book.
The principal effect of Legendre's geometry in America was to make every
textbook writer his own syllabus-maker, and to put solid geometry on a
more satisfactory footing. The minute we depart from a standard text
like Euclid's, and have no recognized examining body, every one is free
to set up his own standard, always within the somewhat uncertain
boundary prescribed by public opinion and by the colleges. The efforts
of the past few years at syllabus-making have been merely attempts to
define this boundary more clearly.
Of these attempts two are especially worthy of consideration as having
been very carefully planned and having brought forth such definite
results as to appeal to a large number of teachers. Other syllabi have
been made and are familiar to many teachers, but in point of clearness
of purpose, conciseness of expression, and form of publication they have
not been such as to compare with the two in question.
The first of these is the Harvard syllabus, which is placed in the hands
of students for reference when trying the entrance examinations of that
university, a plan not followed elsewhere. It sets forth the basal
propositions that should form the essential part of the student's
preparation, and that are necessary and sufficient for proving any
"original proposition" (to take the common expression) that may be set
on the examination. The propositions are arranged by books as follows:
Book I. Angles, triangles, parallels 25
Book II. The circle, angle measure 18
Book III. Similar polygons 10
Book IV. Area of polygons 8
Book V. Polygons and circle measure 11
Constructions 21
Ratio and proportion 6
----
Total for plane geometry 99
The total for solid geometry is 79 propositions, or 178 for both plane
and solid geometry. This is perhaps the most successful attempt that
has been made at reaching a minimum number of propositions. It might
well be further reduced, since it includes the proposition about two
adjacent angles formed by one line meeting another, and the one about
the circle as the limit of the inscribed and circumscribed regular
polygons. The first of these leads a beginner to doubt the value of
geometry, and the second is beyond the powers of the majority of
students. As compared with the syllabus reported by a Wisconsin
committee in 1904, for example, here are 99 propositions against 132. On
the other hand, a committee appointed by the Central Association of
Science and Mathematics Teachers reported in 1909 a syllabus with what
seems at first sight to be a list of only 59 propositions in plane
geometry. This number is fictitious, however, for the reason that
numerous converses are indicated with the propositions, and are not
included in the count, and directions are given to include "related
theorems" and "problems dealing with the length and area of a circle,"
so that in some cases one proposition is evidently intended to cover
several others. This syllabus is therefore lacking in definiteness, so
that the Harvard list stands out as perhaps the best of its type.
The second noteworthy recent attempt in America is that made by a
committee of the Association of Mathematical Teachers in New England.
This committee was organized in 1904. It held sixteen meetings and
carried on a great deal of correspondence. As a result, it prepared a
syllabus arranged by topics, the propositions of solid geometry being
grouped immediately after the corresponding ones of plane geometry. For
example, the nine propositions on congruence in a plane are followed by
nine on congruence in space. As a result, the following summarizes the
work in plane geometry:
Congruence in a plane 9
Equivalence 3
Parallels and perpendiculars 9
Symmetry 20
Angles 15
Tangents 4
Similar figures 18
Inequalities 8
Lengths and areas 17
Loci 2
Concurrent lines 5
----
Total for plane geometry 110
Not so conventional in arrangement as the Harvard syllabus, and with a
few propositions that are evidently not basal to the same extent as the
rest, the list is nevertheless a very satisfactory one, and the
parallelism shown between plane and solid geometry is suggestive to both
student and teacher.
On the whole, however, the Harvard selection of basal propositions is
perhaps as satisfactory as any that has been made, even though it
appears to lack a "factor of safety," and it is probable that any
further reduction would be unwise.
What, now, has been the effect of all these efforts? What teacher or
school would be content to follow any one of these syllabi exactly? What
textbook writer would feel it safe to limit his regular propositions to
those in any one syllabus? These questions suggest their own answers,
and the effect of all this effort seems at first thought to have been so
slight as to be entirely out of proportion to the end in view. This
depends, however, on what this end is conceived to be. If the purpose
has been to cut out a very large number of the propositions that are
found in Euclid's plane geometry, the effort has not been successful. We
may reduce this number to about one hundred thirty, but in general,
whatever a syllabus may give as a minimum, teachers will favor a larger
number than is suggested by the Harvard list, for the purpose of
exercise in the reading of mathematics if for no other reason. The
French geometer, Lacroix, who wrote more than a century ago, proposed to
limit the propositions to those needed to prove other important ones,
and those needed in practical mathematics. If to this we should add
those that are used in treating a considerable range of exercises, we
should have a list of about one hundred thirty.
But this is not the real purpose of these syllabi, or at most it seems
like a relatively unimportant one. The purpose that has been attained is
to stop the indefinite increase in the number of propositions that would
follow from the recent developments in the geometry of the triangle and
circle, and of similar modern topics, if some such counter-movement as
this did not take place. If the result is, as it probably will be, to
let the basal propositions of Euclid remain about as they always have
been, as the standards for beginners, the syllabi will have accomplished
a worthy achievement. If, in addition, they furnish an irreducible
minimum of propositions to which a student may have access if he desires
it, on an examination, as was intended in the case of the Harvard and
the New England Association syllabi, the achievement may possibly be
still more worthy.
In preparing a syllabus, therefore, no one should hope to bring the
teaching world at once to agree to any great reduction in the number of
basal propositions, nor to agree to any radical change of terminology,
symbolism, or sequence. Rather should it be the purpose to show that we
have enough topics in geometry at present, and that the number of
propositions is really greater than is absolutely necessary, so that
teachers shall not be led to introduce any considerable number of
propositions out of the large amount of new material that has recently
been accumulating. Such a syllabus will always accomplish a good
purpose, for at least it will provoke thought and arouse interest, but
any other kind is bound to be ephemeral.[32]
Besides the evolutionary attempts at rearranging and reducing in number
the propositions of Euclid, there have been very many revolutionary
efforts to change his treatment of geometry entirely. The great French
mathematician, D'Alembert, for example, in the eighteenth century,
wished to divide geometry into three branches: (1) that dealing with
straight lines and circles, apparently not limited to a plane; (2) that
dealing with surfaces; and (3) that dealing with solids. So Méray in
France and De Paolis[33] in Italy have attempted to fuse plane and solid
geometry, but have not produced a system that has been particularly
successful. More recently Bourlet, Grévy, Borel, and others in France
have produced several works on the elements of mathematics that may lead
to something of value. They place intuition to the front, favor as much
applied mathematics as is reasonable, to all of which American teachers
would generally agree, but they claim that the basis of elementary
geometry in the future must be the "investigation of the group of
motions." It is, of course, possible that certain of the notions of the
higher mathematical thought of the nineteenth century may be so
simplified as to be within the comprehension of the tyro in geometry,
and we should be ready to receive all efforts of this kind with open
mind. These writers have not however produced the ideal work, and it may
seriously be questioned whether a work based upon their ideas will prove
to be educationally any more sound and usable than the labors of such
excellent writers as Henrici and Treutlein, and H. Müller, and Schlegel
a few years ago in Germany, and of Veronese in Italy. All such efforts,
however, should be welcomed and tried out, although so far as at present
appears there is nothing in sight to replace a well-arranged, vitalized,
simplified textbook based upon the labors of Euclid and Legendre.
The most broad-minded of the great mathematicians who have recently
given attention to secondary problems is Professor Klein of Göttingen.
He has had the good sense to look at something besides the mere question
of good mathematics.[34] Thus he insists upon the psychologic point of
view, to the end that the geometry shall be adapted to the mental
development of the pupil,--a thing that is apparently ignored by Méray
(at least for the average pupil), and, it is to be feared, by the other
recent French writers. He then demands a careful selection of the
subject matter, which in our American schools would mean the elimination
of propositions that are not basal, that is, that are not used for most
of the exercises that one naturally meets in elementary geometry and in
applied work. He further insists upon a reasonable correlation with
practical work to which every teacher will agree so long as the work is
really or even potentially practical. And finally he asks that we look
with favor upon the union of plane and solid geometry, and of algebra
and geometry. He does not make any plea for extreme fusion, but
presumably he asks that to which every one of open mind would agree,
namely, that whenever the opportunity offers in teaching plane geometry,
to open the vision to a generalization in space, or to the measurement
of well-known solids, or to the use of the algebra that the pupil has
learned, the opportunity should be seized.
FOOTNOTES:
[32] The author is a member of a committee that has for more than a year
been considering a syllabus in geometry. This committee will probably
report sometime during the year 1911. At the present writing it seems
disposed to recommend about the usual list of basal propositions.
[33] "Elementi di Geometria," Milan, 1884.
[34] See his "Elementarmathematik vom höheren Standpunkt aus," Part II,
Leipzig, 1909.
CHAPTER VII
THE TEXTBOOK IN GEOMETRY
In considering the nature of the textbook in geometry we need to bear in
mind the fact that the subject is being taught to-day in America to a
class of pupils that is not composed like the classes found in other
countries or in earlier generations. In general, in other countries,
geometry is not taught to mixed classes of boys and girls. Furthermore,
it is generally taught to a more select group of pupils than in a
country where the high school and college are so popular with people in
all the walks of life. In America it is not alone the boy who is
interested in education in general, or in mathematics in particular, who
studies geometry, and who joins with others of like tastes in this
pursuit, but it is often the boy and the girl who are not compelled to
go out and work, and who fill the years of youth with a not
over-strenuous school life. It is therefore clear that we cannot hold
the interest of such pupils by the study of Euclid alone. Geometry must,
for them, be less formal than it was half a century ago. We cannot
expect to make our classes enthusiastic merely over a logical sequence
of proved propositions. It becomes necessary to make the work more
concrete, and to give a much larger number of simple exercises in order
to create the interest that comes from independent work, from a feeling
of conquest, and from a desire to do something original. If we would
"cast a glamor over the multiplication table," as an admirer of Macaulay
has said that the latter could do, we must have the facilities for so
doing.
It therefore becomes necessary in weighing the merits of a textbook to
consider: (1) if the number of proved propositions is reduced to a safe
minimum; (2) if there is reasonable opportunity to apply the theory, the
actual applications coming best, however, from the teacher as an outside
interest; (3) if there is an abundance of material in the way of simple
exercises, since such material is not so readily given by the teacher as
the seemingly local applications of the propositions to outdoor
measurements; (4) if the book gives a reasonable amount of introductory
work in the use of simple and inexpensive instruments, not at that time
emphasizing the formal side of the subject; (5) if there is afforded
some opportunity to see the recreative side of the subject, and to know
a little of the story of geometry as it has developed from ancient to
modern times.
But this does not mean that there is to be a geometric cataclysm. It
means that we must have the same safe, conservative evolution in
geometry that we have in other subjects. Geometry is not going to
degenerate into mere measuring, nor is the ancient sequence going to
become a mere hodge-podge without system and with no incentive to
strenuous effort. It is now about fifteen hundred years since Proclus
laid down what he considered the essential features of a good textbook,
and in all of our efforts at reform we cannot improve very much upon his
statement. "It is essential," he says, "that such a treatise should be
rid of everything superfluous, for the superfluous is an obstacle to the
acquisition of knowledge; it should select everything that embraces the
subject and brings it to a focus, for this is of the highest service to
science; it must have great regard both to clearness and to conciseness,
for their opposites trouble our understanding; it must aim to generalize
its theorems, for the division of knowledge into small elements renders
it difficult of comprehension."
It being prefaced that we must make the book more concrete in its
applications, either directly or by suggesting seemingly practical
outdoor work; that we must increase the number of simple exercises
calling for original work; that we must reasonably reduce the number of
proved propositions; and that we must not allow the good of the ancient
geometry to depart, let us consider in detail some of the features of a
good, practical, common-sense textbook.
The early textbooks in geometry contained only the propositions, with
the proofs in full, preceded by lists of definitions and assumptions
(axioms and postulates). There were no exercises, and the proofs were
given in essay form. Then came treatises with exercises, these exercises
being grouped at the end of the work or at the close of the respective
books. The next step was to the unit page, arranged in steps to aid the
eye, one proposition to a page whenever this was possible. Some effort
was made in this direction in France about two hundred years ago, but
with no success. The arrangement has so much to commend it, however, the
proof being so much more easily followed by the eye than was the case in
the old-style works, that it has of late been revived. In this respect
the Wentworth geometry was a pioneer in America, and so successful was
the effort that this type of page has been adopted, as far as the
various writers were able to adopt it, in all successful geometries that
have appeared of late years in this country. As a result, the American
textbooks on this subject are more helpful and pleasing to the eye than
those found elsewhere.
The latest improvements in textbook-making have removed most of the
blemishes of arrangement that remained, scattering the exercises through
the book, grading them with greater care, and making them more modern in
character. But the best of the latest works do more than this. They
reduce the number of proved theorems and increase the number of
exercises, and they simplify the proofs whenever possible and eliminate
the most difficult of the exercises of twenty-five years ago. It would
be possible to carry this change too far by putting in only half as
many, or a quarter as many, regular propositions, but it should not be
the object to see how the work can be cut down, but to see how it can be
improved.
What should be the basis of selection of propositions and exercises?
Evidently the selection must include the great basal propositions that
are needed in mensuration and in later mathematics, together with others
that are necessary to prove them. Euclid's one hundred seventy-three
propositions of plane geometry were really upwards of one hundred
eighty, because he several times combined two or more in one. These we
may reduce to about one hundred thirty with perfect safety, or less than
one a day for a school year, but to reduce still further is undesirable
as well as unnecessary. It would not be difficult to dispense with a few
more; indeed, we might dispense with thirty more if we should set about
it, although we must never forget that a goodly number in addition to
those needed for the logical sequence are necessary for the wide range
of exercises that are offered. But let it be clear that if we teach 100
instead of 130, our results are liable to be about 100/130 as
satisfactory. We may theorize on pedagogy as we please, but geometry
will pay us about in proportion to what we give.
And as to the exercises, what is the basis of selection? In general, let
it be said that any exercise that pretends to be real should be so, and
that words taken from science or measurements do not necessarily make
the problem genuine. To take a proposition and apply it in a manner that
the world never sanctions is to indulge in deceit. On the other hand,
wholly to neglect the common applications of geometry to handwork of
various kinds is to miss one of our great opportunities to make the
subject vital to the pupil, to arouse new interest, and to give a
meaning to it that is otherwise wanting. It should always be remembered
that mental discipline, whatever the phrase may mean, can as readily be
obtained from a genuine application of a theorem as from a mere
geometric puzzle. On the other hand, it is evident that not more than 25
per cent of propositions have any genuine applications outside of
geometry, and that if we are to attempt any applications at all, these
must be sought mainly in the field of pure geometry. In the exercises,
therefore, we seek to-day a sane and a balanced book, giving equal
weight to theory and to practice, to the demands of the artisan and to
those of the mathematician, to the applications of concrete science and
to those of pure geometry, thus making a fusion of pure and applied
mathematics, with the latter as prominent as the supply of genuine
problems permits. The old is not all bad and the new is not all good,
and a textbook is a success in so far as it selects boldly the good that
is in the old and rejects with equal boldness the bad that is in the
new.
Lest the nature of the exercises of geometry may be misunderstood, it is
well that we consider for a moment what constitutes a genuine
application of the subject. It is the ephemeral fashion just at present
in America to call these genuine applications by the name of "real
problems." The name is an unfortunate importation, but that is not a
matter of serious moment. The important thing is that we should know
what makes a problem "real" to the pupil of geometry, especially as the
whole thing is coming rapidly into disrepute through the mistaken zeal
of some of its supporters.
A real problem is a problem that the average citizen may sometime be
called upon to solve; that, if so called upon, he will solve in the
manner indicated; and that is expressed in terms that are familiar to
the pupil.
This definition, which seems fairly to state the conditions under which
a problem can be called "real" in the schoolroom, involves three points:
(1) people must be liable to meet such a problem; (2) in that case they
will solve it in the way suggested by the book; (3) it must be clothed
in language familiar to the pupil. For example, let the problem be to
find the dimensions of a rectangular field, the data being the area of
the field and the area of a road four rods wide that is cut from three
sides of the field. As a real problem this is ridiculous, since no one
would ever meet such a case outside the puzzle department of a
schoolroom. Again, if by any stretch of a vigorous imagination any human
being should care to find the area of a piece of glass, bounded by the
arcs of circles, in a Gothic window in York Minster, it is fairly
certain that he would not go about it in the way suggested in some of
the earnest attempts that have been made by several successful teachers
to add interest to geometry. And for the third point, a problem is not
real to a pupil simply because it relates to moments of inertia or the
tensile strength of a steel bar. Indeed, it is unreal precisely because
it does talk of these things at a time when they are unfamiliar, and
properly so, to the pupil.
It must not be thought that puzzle problems, and unreal problems
generally, have no value. All that is insisted upon is that such
problems as the above are not "real," and that about 90 per cent of
problems that go by this name are equally lacking in the elements that
make for reality in this sense of the word. For the other 10 per cent of
such problems we should be thankful, and we should endeavor to add to
the number. As for the great mass, however, they are no better than
those that have stood the test of generations, and by their pretense
they are distinctly worse.
It is proper, however, to consider whether a teacher is not justified in
relating his work to those geometric forms that are found in art, let us
say in floor patterns, in domes of buildings, in oilcloth designs, and
the like, for the purpose of arousing interest, if for no other reason.
The answer is apparent to any teacher: It is certainly justifiable to
arouse the pupil's interest in his subject, and to call his attention to
the fact that geometric design plays an important part in art; but we
must see to it that our efforts accomplish this purpose. To make a
course in geometry one on oilcloth design would be absurd, and nothing
more unprofitable or depressing could be imagined in connection with
this subject. Of course no one would advocate such an extreme, but it
sometimes seems as if we are getting painfully near it in certain
schools.
A pupil has a passing interest in geometric design. He should learn to
use the instruments of geometry, and he learns this most easily by
drawing a few such patterns. But to keep him week after week on
questions relating to such designs of however great variety, and
especially to keep him upon designs relating to only one or two types,
is neither sound educational policy nor even common sense. That this
enthusiastic teacher or that one succeeds by such a plan is of no
significance; it is the enthusiasm that succeeds, not the plan.
The experience of the world is that pupils of geometry like to use the
subject practically, but that they are more interested in the pure
theory than in any fictitious applications, and this is why pure
geometry has endured, while the great mass of applied geometry that was
brought forward some three hundred years ago has long since been
forgotten. The question of the real applications of the subject is
considered in subsequent chapters.
In Chapter VI we considered the question of the number of regular
propositions to be expected in the text, and we have just considered the
nature of the exercises which should follow those propositions. It is
well to turn our attention next to the nature of the proofs of the basal
theorems. Shall they appear in full? Shall they be merely suggested
demonstrations? Shall they be only a series of questions that lead to
the proof? Shall the proofs be omitted entirely? Or shall there be some
combination of these plans?
The natural temptation in the nervous atmosphere of America is to listen
to the voice of the mob and to proceed at once to lynch Euclid and every
one who stands for that for which the "Elements" has stood these two
thousand years. This is what some who wish to be considered as educators
tend to do; in the language of the mob, to "smash things"; to call
reactionary that which does not conform to their ephemeral views. It is
so easy to be an iconoclast, to think that _cui bono_ is a conclusive
argument, to say so glibly that Raphael was not a great painter,--to do
anything but construct. A few years ago every one must take up with the
heuristic method developed in Germany half a century back and containing
much that was commendable. A little later one who did not believe that
the Culture Epoch Theory was vital in education was looked upon with
pity by a considerable number of serious educators. A little later the
man who did not think that the principle of Concentration in education
was a _regula aurea_ was thought to be hopeless. A little later it may
have been that Correlation was the saving factor, to be looked upon in
geometry teaching as a guiding beacon, even as the fusion of all
mathematics is the temporary view of a few enthusiasts to-day.[35]
And just now it is vocational training that is the catch phrase, and to
many this phrase seems to sound the funeral knell of the standard
textbook in geometry. But does it do so? Does this present cry of the
pedagogical circle really mean that we are no longer to have geometry
for geometry's sake? Does it mean that a panacea has been found for the
ills of memorizing without understanding a proof in the class of a
teacher who is so inefficient as to allow this kind of work to go on?
Does it mean that a teacher who does not see the human side of
geometry, who does not know the real uses of geometry, and who has no
faculty of making pupils enthusiastic over geometry,--that this teacher
is to succeed with some scrappy, weak, pretending apology for a real
work on the subject?
No one believes in stupid teaching, in memorizing a textbook, in having
a book that does all the work for a pupil, or in any of the other ills
of inefficient instruction. On the other hand, no fair-minded person can
condemn a type of book that has stood for generations until something
besides the mere transient experiments of the moment has been suggested
to replace it. Let us, for example, consider the question of having the
basal propositions proved in full, a feature that is so easy to condemn
as leading to memorizing.
The argument in favor of a book with every basal proposition proved in
full, or with most of them so proved, the rest having only suggestions
for the proof, is that the pupil has before him standard forms
exhibiting the best, most succinct, most clearly stated demonstrations
that geometry contains. The demonstrations stand for the same thing that
the type problems stand for in algebra, and are generally given in full
in the same way. The argument against the plan is that it takes away the
pupil's originality by doing all the work for him, allowing him to
merely memorize the work. Now if all there is to geometry were in the
basal propositions, this argument might hold, just as it would hold in
algebra in case there were only those exercises that are solved in full.
But just as this is not the case in algebra, the solved exercises
standing as types or as bases for the pupil's real work, so the
demonstrated proposition forms a relatively small part of geometry,
standing as a type, a basis for the more important part of the work.
Moreover, a pupil who uses a syllabus is exposed to a danger that should
be considered, namely, that of dishonesty. Any textbook in geometry will
furnish the proofs of most of the propositions in a syllabus, whatever
changes there may be in the sequence, and it is not a healthy condition
of mind that is induced by getting the proofs surreptitiously. Unless a
teacher has more time for the course than is usually allowed, he cannot
develop the new work as much as is necessary with only a syllabus, and
the result is that a pupil gets more of his work from other books and
has less time for exercises. The question therefore comes to this: Is it
better to use a book containing standard forms of proof for the basal
propositions, and have time for solving a large number of original
exercises and for seeking the applications of geometry? Or is it better
to use a book that requires more time on the basal propositions, with
the danger of dishonesty, and allows less time for solving originals? To
these questions the great majority of teachers answer in favor of the
textbook with most of the basal propositions fully demonstrated. In
general, therefore, it is a good rule to use the proofs of the basal
propositions as models, and to get the original work from the exercises.
Unless we preserve these model proofs, or unless we supply them with a
syllabus, the habit of correct, succinct self-expression, which is one
of the chief assets of geometry, will tend to become atrophied. So
important is this habit that "no system of education in which its
performance is neglected can hope or profess to evolve men and women who
are competent in the full sense of the word. So long as teachers of
geometry neglect the possibilities of the subject in this respect, so
long will the time devoted to it be in large part wasted, and so long
will their pupils continue to imbibe the vicious idea that it is much
more important to be able to do a thing than to say how it can be
done."[36]
It is here that the chief danger of syllabus-teaching lies, and it is
because of this patent fact that a syllabus without a carefully selected
set of model proofs, or without the unnecessary expenditure of time by
the class, is a dangerous kind of textbook.
What shall then be said of those books that merely suggest the proofs,
or that give a series of questions that lead to the demonstrations?
There is a certain plausibility about such a plan at first sight. But it
is easily seen to have only a fictitious claim to educational value. In
the first place, it is merely an attempt on the part of the book to take
the place of the teacher and to "develop" every lesson by the heuristic
method. The questions are so framed as to admit, in most cases, of only
a single answer, so that this answer might just as well be given instead
of the question. The pupil has therefore a proof requiring no more
effort than is the case in the standard form of textbook, but not given
in the clear language of a careful writer. Furthermore, the pupil is
losing here, as when he uses only a syllabus, one of the very things
that he should be acquiring, namely, the habit of reading mathematics.
If he met only syllabi without proofs, or "suggestive" geometries, or
books that endeavored to question every proof out of him, he would be in
a sorry plight when he tried to read higher mathematics, or even other
elementary treatises. It is for reasons such as these that the heuristic
textbook has never succeeded for any great length of time or in any wide
territory.
And finally, upon this point, shall the demonstrations be omitted
entirely, leaving only the list of propositions,--in other words, a pure
syllabus? This has been sufficiently answered above. But there is a
modification of the pure syllabus that has much to commend itself to
teachers of exceptional strength and with more confidence in themselves
than is usually found. This is an arrangement that begins like the
ordinary textbook and, after the pupil has acquired the form of proof,
gradually merges into a syllabus, so that there is no temptation to go
surreptitiously to other books for help. Such a book, if worked out with
skill, would appeal to an enthusiastic teacher, and would accomplish the
results claimed for the cruder forms of manual already described. It
would not be in general as safe a book as the standard form, but with
the right teacher it would bring good results.
In conclusion, there are two types of textbook that have any hope of
success. The first is the one with all or a large part of the basal
propositions demonstrated in full, and with these propositions not
unduly reduced in number. Such a book should give a large number of
simple exercises scattered through the work, with a relatively small
number of difficult ones. It should be modern in its spirit, with
figures systematically lettered, with each page a unit as far as
possible, and with every proof a model of clearness of statement and
neatness of form. Above all, it should not yield to the demand of a few
who are always looking merely for something to change, nor should it in
a reactionary spirit return to the old essay form of proof, which
hinders the pupil at this stage.
The second type is the semisyllabus, otherwise with all the spirit of
the first type. In both there should be an honest fusion of pure and
applied geometry, with no exercises that pretend to be practical
without being so, with no forced applications that lead the pupil to
measure things in a way that would appeal to no practical man, with no
merely narrow range of applications, and with no array of difficult
terms from physics and engineering that submerge all thought of
mathematics in the slough of despond of an unknown technical vocabulary.
Outdoor exercises, even if somewhat primitive, may be introduced, but it
should be perfectly understood that such exercises are given for the
purpose of increasing the interest in geometry, and they should be
abandoned if they fail of this purpose.
=Bibliography.= For a list of standard textbooks issued prior
to the present generation, consult the bibliography in Stamper,
History of the Teaching of Geometry, New York, 1908.
FOOTNOTES:
[35] For some classes of schools and under certain circumstances courses
in combined mathematics are very desirable. All that is here insisted
upon is that any general fusion all along the line would result in weak,
insipid, and uninteresting mathematics. A beginning, inspirational
course in combined mathematics has a good reason for being in many high
schools in spite of its manifest disadvantages, and such a course may be
developed to cover all of the required mathematics given in certain
schools.
[36] Carson, loc. cit., p. 15.
CHAPTER VIII
THE RELATION OF ALGEBRA TO GEOMETRY
From the standpoint of theory there is or need be no relation whatever
between algebra and geometry. Algebra was originally the science of the
equation, as its name[37] indicates. This means that it was the science
of finding the value of an unknown quantity in a statement of equality.
Later it came to mean much more than this, and Newton spoke of it as
universal arithmetic, and wrote an algebra with this title. At present
the term is applied to the elements of a science in which numbers are
represented by letters and in which certain functions are studied,
functions which it is not necessary to specify at this time. The work
relates chiefly to functions involving the idea of number. In geometry,
on the other hand, the work relates chiefly to form. Indeed, in pure
geometry number plays practically no part, while in pure algebra form
plays practically no part.
In 1687 the great French philosopher, Descartes, wishing to picture
certain algebraic functions, wrote a work of about a hundred pages,
entitled "La Géométrie," and in this he showed a correspondence between
the numbers of algebra (which may be expressed by letters) and the
concepts of geometry. This was the first great step in the analytic
geometry that finally gave us the graph in algebra. Since then there
have been brought out from time to time other analogies between algebra
and geometry, always to the advantage of each science. This has led to a
desire on the part of some teachers to unite algebra and geometry into
one science, having simply a class in mathematics without these special
names.
It is well to consider the advantages and the disadvantages of such a
plan, and to decide as to the rational attitude to be taken by teachers
concerning the question at issue. On the side of advantages it is
claimed that there is economy of time and of energy. If a pupil is
studying formulas, let the formulas of geometry be studied; if he is
taking up ratio and proportion; let him do so for algebra and geometry
at the same time; if he is solving quadratics, let him apply them at
once to certain propositions concerning secants; and if he is proving
that (_a_ + _b_)^2 equals _a_^2 + 2_ab_ + _b_^2, let him do so by
algebra and by geometry simultaneously. It is claimed that not only is
there economy in this arrangement, but that the pupil sees mathematics
as a whole, and thus acquires more of a mastery than comes by our
present "tandem arrangement."
On the side of disadvantages it may be asked if the same arguments would
not lead us to teach Latin and Greek together, or Latin and French, or
all three simultaneously? If pupils should decline nouns in all three
languages at the same time, learn to count in all at the same time, and
begin to translate in all simultaneously, would there not be an economy
of time and effort, and would there not be developed a much broader view
of language? Now the fusionist of algebra and geometry does not like
this argument, and he says that the cases are not parallel, and he tries
to tell why they are not. He demands that his opponent abandon argument
by analogy and advance some positive reason why algebra and geometry
should not be fused. Then his opponent says that it is not for him to
advance any reason for what already exists, the teaching of the two
separately; that he has only to refute the fusionist's arguments, and
that he has done so. He asserts that algebra and geometry are as
distinct as chemistry and biology; that they have a few common points,
but not enough to require teaching them together. He claims that to
begin Latin and Greek at the same time has always proved to be
confusing, and that the same is true of algebra and geometry. He grants
that unified knowledge is desirable, but he argues that when the fine
arts of music and color work fuse, and when the natural sciences of
chemistry and physics are taught in the same class, and when we follow
the declension of a German noun by that of a French noun and a Latin
noun, and when we teach drawing and penmanship together, then it is well
to talk of mixing algebra and geometry.
It is well, before deciding such a question for ourselves (for evidently
we cannot decide it for the world), to consider what has been the result
of experience. Algebra and geometry were always taught together in early
times, as were trigonometry and astronomy. The Ahmes papyrus contains
both primitive algebra and primitive geometry. Euclid's "Elements"
contains not only pure geometry, but also a geometric algebra and the
theory of numbers. The early works of the Hindus often fused geometry
and arithmetic, or geometry and algebra. Even the first great printed
compendium of mathematics, the "S[=u]ma" of Paciuolo (1494) contained
all of the branches of mathematics. Much of this later attempt was not,
however, an example of perfect fusion, but rather of assigning one set
of chapters to algebra, another to geometry, and another to arithmetic.
So fusion, more or less perfect, has been tried over long periods, and
abandoned as each subject grew more complete in itself, with its own
language and its peculiar symbols.
But it is asserted that fusion is being carried on successfully to-day
by more than one enthusiastic teacher, and that this proves the
contention that the plan is a good one. Books are cited to show that the
arrangement is feasible, and classes are indicated where the work is
progressing along this line.
What, then, is the conclusion? That is a question for the teacher to
settle, but it is one upon which a writer on the teaching of mathematics
should not fear to express his candid opinion.
It is a fact that the Greek and Latin fusion is a fair analogy. There
are reasons for it, but there are many more against it, the chief one
being the confusion of beginning two languages at once, and the learning
simultaneously of two vocabularies that must be kept separate. It is
also a fact that algebra and geometry are fully as distinct as physics
and chemistry, or chemistry and biology. Life may be electricity, and a
brief cessation of oxidization in the lungs brings death, but these
facts are no reasons for fusing the sciences of physics, biology, and
chemistry. Algebra is primarily a theory of certain elementary
functions, a generalized arithmetic, while geometry is primarily a
theory of form with a highly refined logic to be used in its mastery.
They have a few things in common, as many other subjects have, but they
have very many more features that are peculiar to the one or the other.
The experience of the world has led it away from a simultaneous
treatment, and the contrary experience of a few enthusiastic teachers of
to-day proves only their own powers to succeed with any method. It is
easy to teach logarithms in the seventh school year, but it is not good
policy to do so under present conditions. So the experience of the world
is against the plan of strict fusion, and no arguments have as yet been
advanced that are likely to change the world's view. No one has written
a book combining algebra and geometry in this fashion that has helped
the cause of fusion a particle; on the contrary, every such work that
has appeared has damaged that cause by showing how unscientific a result
has come from the labor of an enthusiastic supporter of the movement.
But there is one feature that has not been considered above, and that is
a serious handicap to any effort at combining the two sciences in the
high school, and this is the question of relative difficulty. It is
sometimes said, in a doctrinaire fashion, that geometry is easier than
algebra, since form is easier to grasp than function, and that therefore
geometry should precede algebra. But every teacher of mathematics knows
better than this. He knows that the simplest form is easier to grasp
than the simplest function, but nevertheless that plane geometry, as we
understand the term to-day, is much more difficult than elementary
algebra for a pupil of fourteen. The child studies form in the
kindergarten before he studies number, and this is sound educational
policy. He studies form, in mensuration, throughout his course in
arithmetic, and this, too, is good educational policy. This kind of
geometry very properly precedes algebra. But the demonstrations of
geometry, the study by pupils of fourteen years of a geometry that was
written for college students and always studied by them until about
fifty years ago,--that is by no means as easy as the study of a simple
algebraic symbolism and its application to easy equations. If geometry
is to be taught for the same reasons as at present, it cannot
advantageously be taught earlier than now without much simplification,
and it cannot successfully be fused with algebra save by some teacher
who is willing to sacrifice an undue amount of energy to no really
worthy purpose. When great mathematicians like Professor Klein speak of
the fusion of all mathematics, they speak from the standpoint of
advanced students, not for the teacher of elementary geometry.
It is therefore probable that simple mensuration will continue, as a
part of arithmetic, to precede algebra, as at present; and that algebra
into or through quadratics will precede geometry,[38] drawing upon the
mensuration of arithmetic as may be needed; and that geometry will
follow this part of algebra, using its principles as far as possible to
assist in the demonstrations and to express and manipulate its formulas.
Plane geometry, or else a year of plane and solid geometry, will
probably, in this country, be followed by algebra, completing quadratics
and studying progressions; and by solid geometry, or a supplementary
course in plane and solid geometry, this work being elective in many, if
not all, schools.[39] It is also probable that a general review of
mathematics, where the fusion idea may be carried out, will prove to be
a feature of the last year of the high school, and one that will grow
in popularity as time goes on. Such a plan will keep algebra and
geometry separate, but it will allow each to use all of the other that
has preceded it, and will encourage every effort in this direction. It
will accomplish all that a more complete fusion really hopes to
accomplish, and it will give encouragement to all who seek to modernize
the spirit of each of these great branches of mathematics.
There is, however, a chance for fusion in two classes of school, neither
of which is as yet well developed in this country. The first is the
technical high school that is at present coming into some prominence. It
is not probable even here that the best results can be secured by
eliminating all mathematics save only what is applicable in the shop,
but if this view should prevail for a time, there would be so little
left of either algebra or geometry that each could readily be joined to
the other. The actual amount of algebra needed by a foreman in a machine
shop can be taught in about four lessons, and the geometry or
mensuration that he needs can be taught in eight lessons at the most.
The necessary trigonometry may take eight more, so that it is entirely
feasible to unite these three subjects. The boy who takes such a course
would know as much about mathematics as a child who had read ten pages
in a primer would know about literature, but he would have enough for
his immediate needs, even though he had no appreciation of mathematics
as a science. If any one asks if this is not all that the school should
give him, it might be well to ask if the school should give only the
ability to read, without the knowledge of any good literature; if it
should give only the ability to sing, without the knowledge of good
music; if it should give only the ability to speak, without any training
in the use of good language; and if it should give a knowledge of home
geography, without any intimation that the world is round,--an atom in
the unfathomable universe about us.
The second opportunity for fusion is possibly (for it is by no means
certain) to be found in a type of school in which the only required
courses are the initial ones. These schools have some strong advocates,
it being claimed that every pupil should be introduced to the large
branches of knowledge and then allowed to elect the ones in which he
finds himself the most interested. Whether or not this is sound
educational policy need not be discussed at this time; but if such a
plan were developed, it might be well to offer a somewhat superficial
(in the sense of abridged) course that should embody a little of
algebra, a little of geometry, and a little of trigonometry. This would
unconsciously become a bait for students, and the result would probably
be some good teaching in the class in question. It is to be hoped that
we may have some strong, well-considered textbooks upon this phase of
the work.
As to the fusion of trigonometry and plane geometry little need be said,
because the subject is in the doctrinaire stage. Trigonometry naturally
follows the chapter on similar triangles, but to put it there means, in
our crowded curriculum, to eliminate something from geometry. Which,
then, is better,--to give up the latter portion of geometry, or part of
it at least, or to give up trigonometry? Some advocates have entered a
plea for two or three lessons in trigonometry at this point, and this is
a feature that any teacher may introduce as a bit of interest, as is
suggested in Chapter XVI, just as he may give a popular talk to his
class upon the fourth dimension or the non-Euclidean geometry. The
lasting impression upon the pupil will be exactly the same as that of
four lessons in Sanskrit while he is studying Latin. He might remember
each with pleasure, Latin being related, as it is, to Sanskrit, and
trigonometry being an outcome of the theory of similar triangles. But
that either of these departures from the regular sequence is of any
serious mathematical or linguistic significance no one would feel like
asserting. Each is allowable on the score of interest, but neither will
add to the pupil's power in any essential feature.
Each of these subjects is better taught by itself, each using the other
as far as possible and being followed by a review that shall make use of
all. It is not improbable that we may in due time have high schools that
give less extended courses in algebra and geometry, adding brief
practical courses in trigonometry and the elements of the calculus; but
even in such schools it is likely to be found that geometry is best
taught by itself, making use of all the mathematics that has preceded
it.
It will of course be understood that the fusion of algebra and geometry
as here understood has nothing to do with the question of teaching the
two subjects simultaneously, say two days in the week for one and three
days for the other. This plan has many advocates, although on the whole
it has not been well received in this country. But what is meant here is
the actual fusing of algebra and geometry day after day,--a plan that
has as yet met with only a sporadic success, but which may be developed
for beginning classes in due time.
FOOTNOTES:
[37] _Al-jabr wa'l-muq[=a]balah_: "restoration and equation" is a fairly
good translation of the Arabic.
[38] Or be carried along at the same time as a distinct topic.
[39] With a single year for required geometry it would be better from
every point of view to cut the plane geometry enough to admit a fair
course in solid geometry.
CHAPTER IX
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]
[Illustration]
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]
[Illustration]
[Illustration]
[Illustration]
[Illustration]
[Illustration]
[Illustration]
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°, 32° 20', 41° 32' 48",
26.75°, 33 1/3°, and 0°. 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.
CHAPTER X
THE CONDUCT OF A CLASS IN GEOMETRY
No definite rules can be given for the detailed conduct of a class in
any subject. If it were possible to formulate such rules, all the
personal magnetism of the teacher, all the enthusiasm, all the
originality, all the spirit of the class, would depart, and we should
have a dull, dry mechanism. There is no one best method of teaching
geometry or anything else. The experience of the schools has shown that
a few great principles stand out as generally accepted, but as to the
carrying out of these principles there can be no definite rules.
Let us first consider the general question of the employment of time in
a recitation in geometry. We might all agree on certain general
principles, and yet no two teachers ever would or ever should divide the
period even approximately in the same way. First, a class should have an
opportunity to ask questions. A teacher here shows his power at its
best, listening sympathetically to any good question, quickly seeing the
essential point, and either answering it or restating it in such a way
that the pupil can answer it for himself. Certain questions should be
answered by the teacher; he is there for that purpose. Others can at
once be put in such a light that the pupil can himself answer them.
Others may better be answered by the class. Occasionally, but more
rarely, a pupil may be told to "look that up for to-morrow," a plan
that is commonly considered by students as a confession of weakness on
the part of the teacher, as it probably is. Of course a class will waste
time in questioning a weak teacher, but a strong one need have no fear
on this account. Five minutes given at the opening of a recitation to
brisk, pointed questions by the class, with the same credit given to a
good question as to a good answer, will do a great deal to create a
spirit of comradeship, of frankness, and of honesty, and will reveal to
a sympathetic teacher the difficulties of a class much better than the
same amount of time devoted to blackboard work. But there must be no
dawdling, and the class must feel that it has only a limited time, say
five minutes at the most, to get the help it needs.
Next in order of the division of the time may be the teacher's report on
any papers that the class has handed in. It is impossible to tell how
much of this paper work should be demanded. The local school conditions,
the mental condition of the class, and the time at the disposal of the
teacher are all factors in the case. In general, it may be said that
enough of this kind of work is necessary to see that pupils are neat and
accurate in setting down their demonstrations. On the other hand, paper
work gives an opportunity for dishonesty, and it consumes a great deal
of the teacher's time that might better be given to reading good books
on the subject that he is teaching. If, however, any papers have been
submitted, about five minutes may well be given to a rapid review of the
failures and the successes. In general, it is good educational policy to
speak of the errors and failures impersonally, but occasionally to
mention by name any one who has done a piece of work that is worthy of
special comment. Pupils may better be praised in public and blamed in
private. There is such a thing, however, as praising too much, when
nothing worthy of note has been done, just as there is danger of blaming
too much, resulting in mere "nagging."
The third division of the recitation period may profitably go to
assigning the advance lesson. The class questions and the teacher's
report on written work have shown the mental status of the pupils, so
that the teacher now knows what he may expect for the next lesson. If he
assigns his lesson at the beginning of the period, he does not have this
information. If he waits to the end, he may be too hurried to give any
"development" that the new lesson may require. There can be no rule as
to how to assign a new lesson; it all depends upon what the lesson is,
upon the mental state of the class, and not a little upon the
idiosyncrasy of the teacher. The German educator, Herbart, laid down
certain formal steps in developing a new lesson, and his successors have
elaborated these somewhat as follows:
1. _Aim._ Always take a class into your confidence. Tell the members at
the outset the goal. No one likes to be led blindfolded.
2. _Preparation._ A few brief questions to bring the class to think of
what is to be considered.
3. _Presentation of the new._ Preferably this is done by questions, the
answers leading the members of the class to discover the new truth for
themselves.
4. _Apperception._ Calling attention to the fact that this new fact was
known before, in part, and that it relates to a number of things already
in the mind. The more the new can be tied up to the old the more
tenaciously it will be held.
5. _Generalization and application._
It is evident at once that a great deal of time may be wasted in always
following such a plan, perhaps in ever following it consciously. But, on
the other hand, probably every good teacher, whether he has heard of
Herbart or not, naturally covers these points in substantially this
order. For an inexperienced teacher it is helpful to be familiar with
them, that he may call to mind the steps, arranged in a psychological
sequence, that he would do well to follow. It must always be remembered
that there is quite as much danger in "developing" too much as in taking
the opposite extreme. A mechanical teacher may develop a new lesson
where there is need for only a question or two or a mere suggestion. It
should also be recognized that students need to learn to read
mathematics for themselves, and that always to take away every
difficulty by explanations given in advance is weakening to any one.
Therefore, in assigning the new lesson we may say "Take the next two
pages," and thus discourage most of the class. On the other hand, we may
spend an unnecessary amount of time and overdevelop the work of those
same pages, and have the whole lesson lose all its zest. It is here that
the genius of the teacher comes forth to find the happy mean.
The fourth division of the hour should be reached, in general, in about
ten minutes. This includes the recitation proper. But as to the nature
of this work no definite instructions should be attempted. To a good
teacher they would be unnecessary, to a poor one they would be harmful.
Part of the class may go to the board, and as they are working, the rest
may be reciting. Those at the board should be limited as to time, for
otherwise a premium is placed on mere dawdling. They should be so
arranged as to prevent copying, but the teacher's eye is the best
preventive of this annoying feature. Those at their seats may be called
upon one at a time to demonstrate at the blackboard, the rest being
called upon for quick responses, as occasion demands. The European plan
of having small blackboards is in many respects better than ours, since
pupils cannot so easily waste time. They have to work rapidly and talk
rapidly, or else take their seats.
What should be put on the board, whether the figure alone, or the figure
and the proof, depends upon the proposition. In general, there should be
a certain number of figures put on the board for the sake of rapid work
and as a basis for the proofs of the day. There should also be a certain
amount of written work for the sake of commending or of criticizing
adversely the proof used. There are some figures that are so complicated
as to warrant being put upon sheets of paper and hung before the class.
Thus there is no rule upon the subject, and the teacher must use his
judgment according to the circumstances and the propositions.
If the early "originals" are one-step exercises, and a pupil is required
to recite rapidly, a habit of quick expression is easily acquired that
leads to close attention on the part of all the class. Students as a
rule recite slower than they need to, from mere habit. Phlegmatic as we
think the German is, and nervous as is the American temperament, a
student in geometry in a German school will usually recite more quickly
and with more vigor than one with us. Our extensive blackboards have
something to do with this, allowing so many pupils to be working at the
board that a teacher cannot attend to them all. The result is a habit of
wasting the minutes that can only be overcome by the teacher setting a
definite but reasonable time limit, and holding the pupil responsible if
the work is not done in the time specified. If this matter is taken in
hand the first day, and special effort made in the early weeks of the
year, much of the difficulty can be overcome.
As to the nature of the recitation to be expected from the pupil, no
definite rule can be laid down, since it varies so much with the work of
the day. In general, however, a pupil should state the theorem quickly,
state exactly what is given and what is to be proved, with respect to
the figure, and then give the proof. At first it is desirable that he
should give the authorities in full, and later give only the essential
part in a few words. It is better to avoid the expression "by previous
proposition," for it soon comes to be abused, and of course the learning
of section numbers in a book is a barbarism. It is only by continually
stating the propositions used that a pupil comes to have well fixed in
his memory the basal theorems of geometry, and without these he cannot
make progress in his subsequent mathematics. In general, it is better to
allow a pupil to finish his proof before asking him any questions, the
constant interruptions indulged in by some teachers being the cause of
no little confusion and hesitancy on the part of pupils. Sometimes it is
well to have a figure drawn differently from the one in the book, or
lettered differently, so as to make sure that the pupil has not
memorized the proof, but in general such devices are unnecessary, for a
teacher can easily discover whether the proof is thoroughly understood,
either by the manner of the pupil or by some slight questioning. A good
textbook has the figures systematically lettered in some helpful way
that is easily followed by the class that is listening to the
recitation, and it is not advisable to abandon this for a random set of
letters arranged in no proper order.
It is good educational policy for the teacher to commend at least as
often as he finds fault when criticizing a recitation at the blackboard
and when discussing the pupils' papers. Optimism, encouragement,
sympathy, the genuine desire to help, the putting of one's self in the
pupil's place, the doing to the pupil as the teacher would that he
should do in return,--these are educational policies that make for
better geometry as they make for better life.
The prime failure in teaching geometry lies unquestionably in the lack
of interest on the part of the pupil, and this has been brought about by
the ancient plan of simply reading and memorizing proofs. It is to get
away from this that teachers resort to some such development of the
lesson in advance, as has been suggested above. It is usually a good
plan to give the easier propositions as exercises before they are
reached in the text, where this can be done. An English writer has
recently contributed this further idea:
It might be more stimulating to encourage investigation than to
demand proofs of stated facts; that is to say, "Here is a
figure drawn in this way, find out anything you can about it."
Some such exercises having been performed jointly by teachers
and pupils, the lust of investigation and healthy competition
which is present in every normal boy or girl might be awakened
so far as to make such little researches really attractive;
moreover, the training thus given is of far more value than
that obtained by proving facts which are stated in advance, for
it is seldom, if ever, that the problems of adult life present
themselves in this manner. The spirit of the question, "What is
true?" is positive and constructive, but that involved in "Is
this true?" is negative and destructive.[41]
When the question is asked, "How shall I teach?" or "What is the
Method?" there is no answer such as the questioner expects. A Japanese
writer, Motowori, a great authority upon the Shinto faith of his people,
once wrote these words: "To have learned that there is no way to be
learned and practiced is really to have learned the way of the gods."
FOOTNOTES:
[41] Carson, loc. cit., p. 12.
CHAPTER XI
THE AXIOMS AND POSTULATES
The interest as well as the value of geometry lies chiefly in the fact
that from a small number of assumptions it is possible to deduce an
unlimited number of conclusions. With the truth of these assumptions we
are not so much concerned as with the reasoning by which we draw the
conclusions, although it is manifestly desirable that the assumptions
should not be false, and that they should be as few as possible.
It would be natural, and in some respects desirable, to call these
foundations of geometry by the name "assumptions," since they are simply
statements that are assumed to be true. The real foundation principles
cannot be proved; they are the means by which we prove other statements.
But as with most names of men or things, they have received certain
titles that are time-honored, and that it is not worth the while to
attempt to change. In English we call them axioms and postulates, and
there is no more reason for attempting to change these terms than there
is for attempting to change the names of geometry[42] and of
algebra.[43]
Since these terms are likely to continue, it is necessary to
distinguish between them more carefully than is often done, and to
consider what assumptions we are justified in including under each. In
the first place, these names do not go back to Euclid, as is ordinarily
supposed, although the ideas and the statements are his. "Postulate" is
a Latin form of the Greek [Greek: aitêma] (_aitema_), and appears only
in late translations. Euclid stated in substance, "Let the following be
assumed." "Axiom" ([Greek: axiôma], _axioma_) dates perhaps only from
Proclus (fifth century A.D.), Euclid using the words "common notions"
([Greek: koinai ennoiai], _koinai ennoiai_) for "axioms," as Aristotle
before him had used "common things," "common principles," and "common
opinions."
The distinction between axiom and postulate was not clearly made by
ancient writers. Probably what was in Euclid's mind was the Aristotelian
distinction that an axiom was a principle common to all sciences,
self-evident but incapable of proof, while the postulates were the
assumptions necessary for building up the particular science under
consideration, in this case geometry.[44]
We thus come to the modern distinction between axiom and postulate, and
say that a general statement admitted to be true without proof is an
axiom, while a postulate in geometry is a geometric statement admitted
to be true, without proof. For example, when we say "If equals are added
to equals, the sums are equal," we state an assumption that is taken
also as true in arithmetic, in algebra, and in elementary mathematics in
general. This is therefore an axiom. At one time such a statement was
defined as "a self-evident truth," but this has in recent years been
abandoned, since what is evident to one person is not necessarily
evident to another, and since all such statements are mere matters of
assumption in any case. On the other hand, when we say, "A circle may be
described with any given point as a center and any given line as a
radius," we state a special assumption of geometry, and this assumption
is therefore a geometric postulate. Some few writers have sought to
distinguish between axiom and postulate by saying that the former was an
assumed theorem and the latter an assumed problem, but there is no
standard authority for such a distinction, and indeed the difference
between a theorem and a problem is very slight. If we say, "A circle may
be passed through three points not in the same straight line," we state
a theorem; but if we say, "Required to pass a circle through three
points," we state a problem. The mental process of handling the two
propositions is, however, practically the same in spite of the minor
detail of wording. So with the statement, "A straight line may be
produced to any required length." This is stated in the form of a
theorem, but it might equally well be stated thus: "To produce a
straight line to any required length." It is unreasonable to call this
an axiom in one case and a postulate in the other. However stated, it is
a geometric postulate and should be so classed.
What, now, are the axioms and postulates that we are justified in
assuming, and what determines their number and character? It seems
reasonable to agree that they should be as few as possible, and that for
educational purposes they should be so clear as to be intelligible to
beginners. But here we encounter two conflicting ideas. To get the
"irreducible minimum" of assumptions is to get a set of statements quite
unintelligible to students beginning geometry or any other branch of
elementary mathematics. Such an effort is laudable when the results are
intended for advanced students in the university, but it is merely
suggestive to teachers rather than usable with pupils when it touches
upon the primary steps of any science. In recent years several such
attempts have been made. In particular, Professor Hilbert has given a
system[45] of congruence postulates, but they are rather for the
scientist than for the student of elementary geometry.
In view of these efforts it is well to go back to Euclid and see what
this great teacher of university men[46] had to suggest. The following
are the five "common notions" that Euclid deemed sufficient for the
purposes of elementary geometry.
1. _Things equal to the same thing are also equal to each other._ This
axiom has persisted in all elementary textbooks. Of course it is a
simple matter to attempt criticism,--to say that -2 is the square root
of 4, and +2 is also the square root of 4, whence -2 = +2; but it is
evident that the argument is not sound, and that it does not invalidate
the axiom. Proclus tells us that Apollonius attempted to prove the axiom
by saying, "Let _a_ equal _b_, and _b_ equal _c_. I say that _a_ equals
_c_. For, since _a_ equals _b_, _a_ occupies the same space as _b_.
Therefore _a_ occupies the same space as _c_. Therefore _a_ equals
_c_." The proof is of no value, however, save as a curiosity.
2. _And if to equals equals are added, the wholes are equal._
3. _If equals are subtracted from equals, the remainders are equal._
Axioms 2 and 3 are older than Euclid's time, and are the only ones given
by him relating to the solution of the equation. Certain other axioms
were added by later writers, as, "Things which are double of the same
thing are equal to one another," and "Things which are halves of the
same thing are equal to one another." These two illustrate the ancient
use of _duplatio_ (doubling) and _mediatio_ (halving), the primitive
forms of multiplication and division. Euclid would not admit the
multiplication axiom, since to him this meant merely repeated addition.
The partition (halving) axiom he did not need, and if needed, he would
have inferred its truth. There are also the axioms, "If equals are added
to unequals, the wholes are unequal," and "If equals are subtracted from
unequals, the remainders are unequal," neither of which Euclid would
have used because he did not define "unequals." The modern arrangement
of axioms, covering addition, subtraction, multiplication, division,
powers, and roots, sometimes of unequals as well as equals, comes from
the development of algebra. They are not all needed for geometry, but in
so far as they show the relation of arithmetic, algebra, and geometry,
they serve a useful purpose. There are also other axioms concerning
unequals that are of advantage to beginners, even though unnecessary
from the standpoint of strict logic.
4. _Things that coincide with one another are equal to one another._
This is no longer included in the list of axioms. It is rather a
definition of "equal," or of "congruent," to take the modern term. If
not a definition, it is certainly a postulate rather than an axiom,
being purely geometric in character. It is probable that Euclid included
it to show that superposition is to be considered a legitimate form of
proof, but why it was not placed among the postulates is not easily
seen. At any rate it is unfortunately worded, and modern writers
generally insert the postulate of motion instead,--that a figure may be
moved about in space without altering its size or shape. The German
philosopher, Schopenhauer (1844), criticized Euclid's axiom as follows:
"Coincidence is either mere tautology or something entirely empirical,
which belongs not to pure intuition but to external sensuous experience.
It presupposes, in fact, the mobility of figures."
5. _The whole is greater than the part._ To this Clavius (1574) added,
"The whole is equal to the sum of its parts," which may be taken to be a
definition of "whole," but which is helpful to beginners, even if not
logically necessary. Some writers doubt the genuineness of this axiom.
Having considered the axioms of Euclid, we shall now consider the axioms
that are needed in the study of elementary geometry. The following are
suggested, not from the standpoint of pure logic, but from that of the
needs of the teacher and pupil.
1. _If equals are added to equals, the sums are equal._ Instead of this
axiom, the one numbered 8 below is often given first. For convenience in
memorizing, however, it is better to give the axioms in the following
order: (1) addition, (2) subtraction, (3) multiplication, (4) division,
(5) powers and roots,--all of equal quantities.
2. _If equals are subtracted from equals, the remainders are equal._
3. _If equals are multiplied by equals, the products are equal._
4. _If equals are divided by equals, the quotients are equal._
5. _Like powers or like positive roots of equals are equal._ Formerly
students of geometry knew nothing of algebra, and in particular nothing
of negative quantities. Now, however, in American schools a pupil
usually studies algebra a year before he studies demonstrative geometry.
It is therefore better, in speaking of roots, to limit them to positive
numbers, since the two square roots of 4 (+2 and -2), for example, are
not equal. If the pupil had studied complex numbers before he began
geometry, it would have been advisable to limit the roots still further
to real roots, since the four fourth roots of 1 (+1, -1, +[sqrt](-1),
-[sqrt](-1)), for example, are not equal save in absolute value. It is
well, however, to eliminate these fine distinctions as far as possible,
since their presence only clouds the vision of the beginner.
It should also be noted that these five axioms might be combined in one,
namely, _If equals are operated on by equals in the same way, the
results are equal_. In Axiom 1 this operation is addition, in Axiom 2 it
is subtraction, and so on. Indeed, in order to reduce the number of
axioms two are already combined in Axiom 5. But there is a good reason
for not combining the first four with the fifth, and there is also a
good reason for combining two in Axiom 5. The reason is that these are
the axioms continually used in equations, and to combine them all in one
would be to encourage laxness of thought on the part of the pupil. He
would always say "by Axiom 1" whatever he did to an equation, and the
teacher would not be certain whether the pupil was thinking definitely
of dividing equals by equals, or had a hazy idea that he was
manipulating an equation in some other way that led to an answer. On the
other hand, Axiom 5 is not used as often as the preceding four, and the
interchange of integral and fractional exponents is relatively common,
so that the joining of these two axioms in one for the purpose of
reducing the total number is justifiable.
6. _If unequals are operated on by positive equals in the same way, the
results are unequal in the same order._ This includes in a single
statement the six operations mentioned in the preceding axioms; that is,
if _a_ > _b_ and if _x_ = _y_, then _a_ + _x_ > _b_ + _y_,
_a_ - _x_ > _b_ - _y_, _ax_ > _by_, etc. The reason for thus combining
six axioms in one in the case of inequalities is apparent. They are
rarely used in geometry, and if a teacher is in doubt as to the pupil's
knowledge, he can easily inquire in the few cases that arise, whereas it
would consume a great deal of time to do this for the many equations
that are met. The axiom is stated in such a way as to exclude
multiplying or dividing by negative numbers, this case not being needed.
7. _If unequals are added to unequals in the same order, the sums are
unequal in the same order; if unequals are subtracted from equals, the
remainders are unequal in the reverse order._ These are the only cases
in which unequals are necessarily combined with unequals, or operate
upon equals in geometry, and the axiom is easily explained to the class
by the use of numbers.
8. _Quantities that are equal to the same quantity or to equal
quantities are equal to each other._ In this axiom the word "quantity"
is used, in the common manner of the present time, to include number and
all geometric magnitudes (length, area, volume).
9. _A quantity may be substituted for its equal in an equation or in an
inequality._ This axiom is tacitly assumed by all writers, and is very
useful in the proofs of geometry. It is really the basis of several
other axioms, and if we were seeking the "irreducible minimum," it would
replace them. Since, however, we are seeking only a reasonably abridged
list of convenient assumptions that beginners will understand and use,
this axiom has much to commend it. If we consider the equations
(1) _a_ = _x_ and (2) _b_ = _x_, we see that for _x_ in equation (1) we
may substitute _b_ from equation (2) and have _a_ = _b_; in other words,
that Axiom 8 is included in Axiom 9. Furthermore, if (1) _a_ = _b_ and
(2) _x_ = _y_, then since _a_ + _x_ is the same as _a_ + _x_, we may, by
substituting, say that _a_ + _x_ = _a_ + _x_ = _b_ + _x_ = _b_ + _y_. In
other words, Axiom 1 is included in Axiom 9. Thus an axiom that includes
others has a legitimate place, because a beginner would be too much
confused by seeing its entire scope, and because he will make frequent
use of it in his mathematical work.
10. _If the first of three quantities is greater than the second, and
the second is greater than the third, then the first is greater than the
third._ This axiom is needed several times in geometry. The case in
which _a_ > _b_ and _b_ = _c_, therefore _a_ > _c_, is provided for in
Axiom 9.
11. _The whole is greater than any of its parts and is equal to the sum
of all its parts._ The latter part of this axiom is really only the
definition of "whole," and it would be legitimate to state a definition
accordingly and refer to it where the word is employed. Where, however,
we wish to speak of a polygon, for example, and wish to say that the
area is equal to the combined areas of the triangles composing it, it is
more satisfactory to have this axiom to which to refer. It will be
noticed that two related axioms are here combined in one, for a reason
similar to the one stated under Axiom 5.
In the case of the postulates we are met by a problem similar to the one
confronting us in connection with the axioms,--the problem of the
"irreducible minimum" as related to the question of teaching. Manifestly
Euclid used postulates that he did not state, and proved some statements
that he might have postulated.[47]
The postulates given by Euclid under the name [Greek:
aitêmata](_aitemata_) were requests made by the teacher to his pupil
that certain things be conceded. They were five in number, as follows:
1. _Let the following be conceded: to draw a straight line from any
point to any point._
Strictly speaking, Euclid might have been required to postulate that
points and straight lines exist, but he evidently considered this
statement sufficient. Aristotle had, however, already called attention
to the fact that a mere definition was sufficient only to show what a
concept is, and that this must be followed by a proof that the thing
exists. We might, for example, define _x_ as a line that bisects an
angle without meeting the vertex, but this would not show that an _x_
exists, and indeed it does not exist. Euclid evidently intended the
postulate to assert that this line joining two points is unique, which
is only another way of saying that two points determine a straight line,
and really includes the idea that two straight lines cannot inclose
space. For purposes of instruction, the postulate would be clearer if it
read, _One straight line, and only one, can be drawn through two given
points_.
2. _To produce a finite straight line continuously in a straight line._
In this postulate Euclid practically assumes that a straight line can be
produced only in a straight line; in other words, that two different
straight lines cannot have a common segment. Several attempts have been
made to prove this fact, but without any marked success.
3. _To describe a circle with any center and radius._
4. _That all right angles are equal to one another._
While this postulate asserts the essential truth that a right
angle is a _determinate magnitude_ so that it really serves as
an invariable standard by which other (acute and obtuse) angles
may be measured, much more than this is implied, as will easily
be seen from the following consideration. If the statement is
to be _proved_, it can only be proved by the method of applying
one pair of right angles to another and so arguing their
equality. But this method would not be valid unless on the
assumption of the _invariability of figures_, which would have
to be asserted as an antecedent postulate. Euclid preferred to
assert as a postulate, directly, the fact that all right angles
are equal; and hence his postulate must be taken as equivalent
to the principle of _invariability of figures_, or its
equivalent, the _homogeneity_ of space.[48]
It is better educational policy, however, to assert this fact more
definitely, and to state the additional assumption that figures may be
moved about in space without deformation. The fourth of Euclid's
postulates is often given as an axiom, following the idea of the Greek
philosopher Geminus (who flourished in the first century B.C.), but this
is because Euclid's distinction between axiom and postulate is not
always understood. Proclus (410-485 A.D.) endeavored to prove the
postulate, and a later and more scientific effort was made by the
Italian geometrician Saccheri (1667-1733). It is very commonly taken as
a postulate that all straight angles are equal, this being more evident
to the senses, and the equality of right angles is deduced as a
corollary. This method of procedure has the sanction of many of our best
modern scholars.
5. _That, if a straight line falling on two straight lines make the
interior angle on the same side less than two right angles, the two
straight lines, if produced indefinitely, meet on that side on which are
the angles less than the two right angles._
This famous postulate, long since abandoned in teaching the beginner in
geometry, is a remarkable evidence of the clear vision of Euclid. For
two thousand years mathematicians sought to prove it, only to
demonstrate the wisdom of its author in placing it among the
assumptions.[49] Every proof adduced contains some assumption that
practically conceals the postulate itself. Thus the great English
mathematician John Wallis (1616-1703) gave a proof based upon the
assumption that "given a figure, another figure is possible which is
similar to the given one, and of any size whatever." Legendre
(1752-1833) did substantially the same at one time, and offered several
other proofs, each depending upon some equally unprovable assumption.
The definite proof that the postulate cannot be demonstrated is due to
the Italian Beltrami (1868).
Of the alternative forms of the postulate, that of Proclus is generally
considered the best suited to beginners. As stated by Playfair (1795),
this is, "Through a given point only one parallel can be drawn to a
given straight line"; and as stated by Proclus, "If a straight line
intersect one of two parallels, it will intersect the other also."
Playfair's form is now the common "postulate of parallels," and is the
one that seems destined to endure.
Posidonius and Geminus, both Stoics of the first century B.C., gave as
their alternative, "There exist straight lines everywhere equidistant
from one another." One of Legendre's alternatives is, "There exists a
triangle in which the sum of the three angles is equal to two right
angles." One of the latest attempts to suggest a substitute is that of
the Italian Ingrami (1904), "Two parallel straight lines intercept, on
every transversal which passes through the mid-point of a segment
included between them, another segment the mid-point of which is the
mid-point of the first."
Of course it is entirely possible to assume that through a point more
than one line can be drawn parallel to a given straight line, in which
case another type of geometry can be built up, equally rigorous with
Euclid's. This was done at the close of the first quarter of the
nineteenth century by Lobachevsky (1793-1856) and Bolyai (1802-1860),
resulting in the first of several "non-Euclidean" geometries.[50]
Taking the problem to be that of stating a reasonably small number of
geometric assumptions that may form a basis to supplement the general
axioms, that shall cover the most important matters to which the student
must refer, and that shall be so simple as easily to be understood by a
beginner, the following are recommended:
1. _One straight line, and only one, can be drawn through two given
points._ This should also be stated for convenience in the form, _Two
points determine a straight line_. From it may also be drawn this
corollary, _Two straight lines can intersect in only one point_, since
two points would determine a straight line. Such obvious restatements of
or corollaries to a postulate are to be commended, since a beginner is
often discouraged by having to prove what is so obvious that a
demonstration fails to commend itself to his mind.
2. _A straight line may be produced to any required length._ This, like
Postulate 1, requires the use of a straightedge for drawing the physical
figure. The required length is attained by using the compasses to
measure the distance. The straightedge and the compasses are the only
two drawing instruments recognized in elementary geometry.[51] While
this involves more than Euclid's postulate, it is a better working
assumption for beginners.
3. _A straight line is the shortest path between two points._ This is
easily proved by the method of Euclid[52] for the case where the paths
are broken lines, but it is needed as a postulate for the case of curve
paths. It is a better statement than the common one that a straight line
is the shortest _distance_ between two points; for distance is measured
on a line, but it is not itself a line. Furthermore, there are
scientific objections to using the word "distance" any more than is
necessary.
4. _A circle may be described with any given point as a center and any
given line as a radius._ This involves the use of the second of the two
geometric instruments, the compasses.
5. _Any figure may be moved from one place to another without altering
the size or shape._ This is the postulate of the homogeneity of space,
and asserts that space is such that we may move a figure as we please
without deformation of any kind. It is the basis of all cases of
superposition.
6. _All straight angles are equal._ It is possible to prove this, and
therefore, from the standpoint of strict logic, it is unnecessary as a
postulate. On the other hand, it is poor educational policy for a
beginner to attempt to prove a thing that is so obvious. The attempt
leads to a loss of interest in the subject, the proposition being (to
state a paradox) hard because it is so easy. It is, of course, possible
to postulate that straight angles are equal, and to draw the conclusion
that their halves (right angles) are equal; or to proceed in the
opposite direction, and postulate that all right angles are equal, and
draw the conclusion that their doubles (straight angles) are equal. Of
the two the former has the advantage, since it is probably more obvious
that all straight angles are equal. It is well to state the following
definite corollaries to this postulate: (1) _All right angles are
equal_; (2) _From a point in a line only one perpendicular can be drawn
to the line_, since two perpendiculars would make the whole (right
angle) equal to its part; (3) _Equal angles have equal complements,
equal supplements, and equal conjugates_; (4) _The greater of two
angles has the less complement, the less supplement, and the less
conjugate._ All of these four might appear as propositions, but, as
already stated, they are so obvious as to be more harmful than useful to
beginners when given in such form.
The postulate of parallels may properly appear in connection with that
topic in Book I, and it is accordingly treated in Chapter XIV.
There is also another assumption that some writers are now trying to
formulate in a simple fashion. We take, for example, a line segment
_AB_, and describe circles with _A_ and _B_ respectively as centers, and
with a radius _AB_. We say that the circles will intersect as at _C_ and
_D_. But how do we know that they intersect? We assume it, just as we
assume that an indefinite straight line drawn from a point inclosed by a
circle will, if produced far enough, cut the circle twice. Of course a
pupil would not think of this if his attention was not called to it, and
the harm outweighs the good in doing this with one who is beginning the
study of geometry.
With axioms and with postulates, therefore, the conclusion is the same:
from the standpoint of scientific geometry there is an irreducible
minimum of assumptions, but from the standpoint of practical teaching
this list should give place to a working set of axioms and postulates
that meet the needs of the beginner.
=Bibliography.= Smith, Teaching of Elementary Mathematics, New
York, 1900; Young, The Teaching of Mathematics, New York, 1901;
Moore, On the Foundations of Mathematics, _Bulletin of the
American Mathematical Society_, 1903, p. 402; Betz, Intuition
and Logic in Geometry, _The Mathematics Teacher_, Vol. II, p.
3; Hilbert, The Foundations of Geometry, Chicago, 1902; Veblen,
A System of Axioms for Geometry, _Transactions of the American
Mathematical Society_, 1904, p. 343.
FOOTNOTES:
[42] From the Greek [Greek: gê], _ge_ (earth), + [Greek: metrein],
_metrein_ (to measure), although the science has not had to do directly
with the measure of the earth for over two thousand years.
[43] From the Arabic _al_ (the) + _jabr_ (restoration), referring to
taking a quantity from one side of an equation and then restoring the
balance by taking it from the other side (see page 37).
[44] One of the clearest discussions of the subject is in W. B.
Frankland, "The First Book of Euclid's 'Elements,'" p. 26, Cambridge,
1905.
[45] "Grundlagen der Geometrie," Leipzig, 1899. See Heath's "Euclid,"
Vol. I, p. 229, for an English version; also D. E. Smith, "Teaching of
Elementary Mathematics," p. 266, New York, 1900.
[46] We need frequently to recall the fact that Euclid's "Elements" was
intended for advanced students who went to Alexandria as young men now
go to college, and that the book was used only in university instruction
in the Middle Ages and indeed until recent times.
[47] For example, he moves figures without deformation, but states no
postulate on the subject; and he proves that one side of a triangle is
less than the sum of the other two sides, when he might have postulated
that a straight line is the shortest path between two points. Indeed,
his followers were laughed at for proving a fact so obvious as this one
concerning the triangle.
[48] T. L. Heath, "Euclid," Vol. I, p. 200.
[49] For a résumé of the best known attempts to prove this postulate,
see Heath, "Euclid," Vol. I, p. 202; W. B. Frankland, "Theories of
Parallelism," Cambridge, 1910.
[50] For the early history of this movement see Engel and Stäckel, "Die
Theorie der Parallellinien von Euklid bis auf Gauss," Leipzig, 1895;
Bonola, Sulla teoria delle parallele e sulle geometrie non-euclidee, in
his "Questioni riguardanti la geometria elementare," 1900;
Karagiannides, "Die nichteuklidische Geometrie vom Alterthum bis zur
Gegenwart," Berlin, 1893.
[51] This limitation upon elementary geometry was placed by Plato (died
347 B.C.), as already stated.
[52] Book I, Proposition 20.
CHAPTER XII
THE DEFINITIONS OF GEOMETRY
When we consider the nature of geometry it is evident that more
attention must be paid to accuracy of definitions than is the case in
most of the other sciences. The essence of all geometry worthy of
serious study is not the knowledge of some fact, but the proof of that
fact; and this proof is always based upon preceding proofs, assumptions
(axioms or postulates), or definitions. If we are to prove that one line
is perpendicular to another, it is essential that we have an exact
definition of "perpendicular," else we shall not know when we have
reached the conclusion of the proof.
The essential features of a definition are that the term defined shall
be described in terms that are simpler than, or at least better known
than, the thing itself; that this shall be done in such a way as to
limit the term to the thing defined; and that the description shall not
be redundant. It would not be a good definition to say that a right
angle is one fourth of a perigon and one half of a straight angle,
because the concept "perigon" is not so simple, and the term "perigon"
is not so well known, as the term and the concept "right angle," and
because the definition is redundant, containing more than is necessary.
It is evident that satisfactory definitions are not always possible; for
since the number of terms is limited, there must be at least one that is
at least as simple as any other, and this cannot be described in terms
simpler than itself. Such, for example, is the term "angle." We can
easily explain the meaning of this word, and we can make the concept
clear, but this must be done by a certain amount of circumlocution and
explanation, not by a concise and perfect definition. Unless a beginner
in geometry knows what an angle is before he reads the definition in a
textbook, he will not know from the definition. This fact of the
impossibility of defining some of the fundamental concepts will be
evident when we come to consider certain attempts that have been made in
this direction.
It should also be understood in this connection that a definition makes
no assertion as to the existence of the thing defined. If we say that a
tangent to a circle is an unlimited straight line that touches the
circle in one point, and only one, we do not assert that it is possible
to have such a line; that is a matter for proof. Not in all cases,
however, can this proof be given, as in the existence of the simplest
concepts. We cannot, for example, prove that a point or a straight line
exists after we have defined these concepts. We therefore tacitly or
explicitly assume (postulate) the existence of these fundamentals of
geometry. On the other hand, we can prove that a tangent exists, and
this may properly be considered a legitimate proposition or corollary of
elementary geometry. In relation to geometric proof it is necessary to
bear in mind, therefore, that we are permitted to define any term we
please; for example, "a seven-edged polyhedron" or Leibnitz's "ten-faced
regular polyhedron," neither of which exists; but, strictly speaking, we
have no right to make use of a definition in a proof until we have shown
or postulated that the thing defined has an existence. This is one of
the strong features of Euclid's textbook. Not being able to prove that a
point, a straight line, and a circle exists, he practically postulates
these facts; but he uses no other definition in a proof without showing
that the thing defined exists, and this is his reason for mingling his
problems with his theorems. At the present time we confessedly sacrifice
his logic in this respect for the reason that we teach geometry to
pupils who are too young to appreciate that logic.
It was pointed out by Aristotle, long before Euclid, that it is not a
satisfactory procedure to define a thing by means of terms that are
strictly not prior to it, as when we attempt to define something by
means of its opposite. Thus to define a curve as "a line, no part of
which is straight," would be a bad definition unless "straight" had
already been explicitly defined; and to define "bad" as "not good" is
unsatisfactory for the reason that "bad" and "good" are concepts that
are evolved simultaneously. But all this is only a detail under the
general principle that a definition must employ terms that are better
understood than the one defined.
It should be understood that some definitions are much more important
than others, considered from the point of view of the logic of geometry.
Those that enter into geometric proofs are basal; those that form part
of the conversational language of geometry are not. Euclid gave
twenty-three definitions in Book I, and did not make use of even all of
these terms. Other terms, those not employed in his proofs, he assumed
to be known, just as he assumed a knowledge of any other words in his
language. Such procedure would not be satisfactory under modern
conditions, but it is of great importance that the teacher should
recognize that certain definitions are basal, while others are merely
informational.
It is now proposed to consider the basal definitions of geometry, first,
that the teacher may know what ones are to be emphasized and learned;
and second, that he may know that the idea that the standard definitions
can easily be improved is incorrect. It is hoped that the result will be
the bringing into prominence of the basal concepts, and the discouraging
of attempts to change in unimportant respects the definitions in the
textbook used by the pupil.
In order to have a systematic basis for work, the definitions of two
books of Euclid will first be considered.[53]
1. POINT. _A point is that which has no part._ This was incorrectly
translated by Capella in the fifth century, "Punctum est cuius pars
nihil est" (a point is that of which a part is nothing), which is as
much as to say that the point itself is nothing. It generally appears,
however, as in the Campanus edition,[54] "Punctus est cuius pars non
est," which is substantially Euclid's wording. Aristotle tells of the
definitions of point, line, and surface that prevailed in his time,
saying that they all defined the prior by means of the posterior.[55]
Thus a point was defined as "an extremity of a line," a line as "the
extremity of a surface," and a surface as "the extremity of a
solid,"--definitions still in use and not without their value. For it
must not be assumed that scientific priority is necessarily priority in
fact; a child knows of "solid" before he knows of "point," so that it
may be a very good way to explain, if not to define, by beginning with
solid, passing thence to surface, thence to line, and thence to point.
The first definition of point of which Proclus could learn is attributed
by him to the Pythagoreans, namely, "a monad having position," the early
form of our present popular definition of a point as "position without
magnitude." Plato defined it as "the beginning of a line," thus
presupposing the definition of "line"; and, strangely enough, he
anticipated by two thousand years Cavalieri, the Italian geometer, by
speaking of points as "indivisible lines." To Aristotle, who protested
against Plato's definitions, is due the definition of a point as
"something indivisible but having position."
Euclid's definition is essentially that of Aristotle, and is followed by
most modern textbook writers, except as to its omission of the reference
to position. It has been criticized as being negative, "which has _no_
part"; but it is generally admitted that a negative definition is
admissible in the case of the most elementary concepts. For example,
"blind" must be defined in terms of a negation.
At present not much attention is given to the definition of "point,"
since the term is not used as the basis of a proof, but every effort is
made to have the concept clear. It is the custom to start from a small
solid, conceive it to decrease in size, and think of the point as the
limit to which it is approaching, using these terms in their usual sense
without further explanation.
2. LINE. _A line is breadthless length._ This is usually modified in
modern textbooks by saying that "a line is that which has length without
breadth or thickness," a statement that is better understood by
beginners. Euclid's definition is thought to be due to Plato, and is
only one of many definitions that have been suggested. The Pythagoreans
having spoken of the point as a monad naturally were led to speak of the
line as dyadic, or related to two. Proclus speaks of another definition,
"magnitude in one dimension," and he gives an excellent illustration of
line as "the edge of a shadow," thus making it real but not material.
Aristotle speaks of a line as a magnitude "divisible in one way only,"
as contrasted with a surface which is divisible in two ways, and with a
solid which is divisible in three ways. Proclus also gives another
definition as the "flux of a point," which is sometimes rendered as the
path of a moving point. Aristotle had suggested the idea when he wrote,
"They say that a line by its motion produces a surface, and a point by
its motion a line."
Euclid did not deem it necessary to attempt a classification of lines,
contenting himself with defining only a straight line and a circle, and
these are really the only lines needed in elementary geometry. His
commentators, however, made the attempt. For example. Heron (first
century A.D.) probably followed his definition of line by this
classification:
{ Straight
Lines { { Circular circumferences
{ Not straight { Spiral shaped
{ Curved (generally)
Proclus relates that both Plato and Aristotle divided lines into
"straight," "circular," and "a mixture of the two," a statement which
is not quite exact, but which shows the origin of a classification not
infrequently found in recent textbooks. Geminus (_ca._ 50 B.C.) is said
by Proclus to have given two classifications, of which one will suffice
for our purposes:
{ Composite (broken line forming an angle)
{
Lines { { Forming a figure, or determinate. (Circle,
{ { ellipse, cissoid.)
{ Incomposite { Not forming a figure, or indeterminate and
{ extending without a limit. (Straight
{ line, parabola, hyperbola, conchoid.)
Of course his view of the cissoid, the curve represented by the equation
_y_^2(_a_ + _x_) = (_a_ - _x_)^3, is not the modern view.
3. _The extremities of a line are points._ This is not a definition in
the sense of its two predecessors. A modern writer would put it as a
note under the definition of line. Euclid did not wish to define a point
as the extremity of a line, for Aristotle had asserted that this was not
scientific; so he defined point and line, and then added this statement
to show the relation of one to the other. Aristotle had improved upon
this by stating that the "division" of a line, as well as an extremity,
is a point, as is also the intersection of two lines. These statements,
if they had been made by Euclid, would have avoided the objection made
by Proclus, that some lines have no extremities, as, for example, a
circle, and also a straight line extending infinitely in both
directions.
4. STRAIGHT LINE. _A straight line is that which lies evenly with
respect to the points on itself._ This is the least satisfactory of all
of the definitions of Euclid, and emphasizes the fact that the straight
line is the most difficult to define of the elementary concepts of
geometry. What is meant by "lies evenly"? Who would know what a
straight line is, from this definition, if he did not know in advance?
The ancients suggested many definitions of straight line, and it is well
to consider a few in order to appreciate the difficulties involved.
Plato spoke of it as "that of which the middle covers the ends," meaning
that if looked at endways, the middle would make it impossible to see
the remote end. This is often modified to read that "a straight line
when looked at endways appears as a point,"--an idea that involves the
postulate that our line of sight is straight. Archimedes made the
statement that "of all the lines which have the same extremities, the
straight line is the least," and this has been modified by later writers
into the statement that "a straight line is the shortest distance
between two points." This is open to two objections as a definition: (1)
a line is not distance, but distance is the _length_ of a line,--it is
measured on a line; (2) it is merely stating a property of a straight
line to say that "a straight line is the shortest path between two
points,"--a proper postulate but not a good definition. Equally
objectionable is one of the definitions suggested by both Heron and
Proclus, that "a straight line is a line that is stretched to its
uttermost"; for even then it is reasonable to think of it as a catenary,
although Proclus doubtless had in mind the Archimedes statement. He also
stated that "a straight line is a line such that if any part of it is in
a plane, the whole of it is in the plane,"--a definition that runs in a
circle, since plane is defined by means of straight line. Proclus also
defines it as "a uniform line, capable of sliding along itself," but
this is also true of a circle.
Of the various definitions two of the best go back to Heron, about the
beginning of our era. Proclus gives one of them in this form, "That line
which, when its ends remain fixed, itself remains fixed." Heron proposed
to add, "when it is, as it were, turned round in the same plane." This
has been modified into "that which does not change its position when it
is turned about its extremities as poles," and appears in substantially
this form in the works of Leibnitz and Gauss. The definition of a
straight line as "such a line as, with another straight line, does not
inclose space," is only a modification of this one. The other definition
of Heron states that in a straight line "all its parts fit on all in all
ways," and this in its modern form is perhaps the most satisfactory of
all. In this modern form it may be stated, "A line such that any part,
placed with its ends on any other part, must lie wholly in the line, is
called a straight line," in which the force of the word "must" should be
noted. This whole historical discussion goes to show how futile it is to
attempt to define a straight line. What is needed is that we should
explain what is meant by a straight line, that we should illustrate it,
and that pupils should then read the definition understandingly.
5. SURFACE. _A surface is that which has length and breadth._ This is
substantially the common definition of our modern textbooks. As with
line, so with surface, the definition is not entirely satisfactory, and
the chief consideration is that the meaning of the term should be made
clear by explanations and illustrations. The shadow cast on a table top
is a good illustration, since all idea of thickness is wanting. It adds
to the understanding of the concept to introduce Aristotle's statement
that a surface is generated by a moving line, modified by saying that
it _may_ be so generated, since the line might slide along its own
trace, or, as is commonly said in mathematics, along itself.
6. _The extremities of a surface are lines._ This is open to the same
explanation and objection as definition 3, and is not usually given in
modern textbooks. Proclus calls attention to the fact that the statement
is hardly true for a complete spherical surface.
7. PLANE. _A plane surface is a surface which lies evenly with the
straight lines on itself._ Euclid here follows his definition of
straight line, with a result that is equally unsatisfactory. For
teaching purposes the translation from the Greek is not clear to a
beginner, since "lies evenly" is a term not simpler than the one
defined. As with the definition of a straight line, so with that of a
plane, numerous efforts at improvement have been made. Proclus,
following a hint of Heron's, defines it as "the surface which is
stretched to the utmost," and also, this time influenced by Archimedes's
assumption concerning a straight line, as "the least surface among all
those which have the same extremities." Heron gave one of the best
definitions, "A surface all the parts of which have the property of
fitting on [each other]." The definition that has met with the widest
acceptance, however, is a modification of one due to Proclus, "A surface
such that a straight line fits on all parts of it." Proclus elsewhere
says, "[A plane surface is] such that the straight line fits on it all
ways," and Heron gives it in this form, "[A plane surface is] such that,
if a straight line pass through two points on it, the line coincides
with it at every spot, all ways." In modern form this appears as
follows: "A surface such that a straight line joining any two of its
points lies wholly in the surface is called a plane," and for teaching
purposes we have no better definition. It is often known as Simson's
definition, having been given by Robert Simson in 1756.
The French mathematician, Fourier, proposed to define a plane as formed
by the aggregate of all the straight lines which, passing through one
point on a straight line in space, are perpendicular to that line. This
is clear, but it is not so usable for beginners as Simson's definition.
It appears as a theorem in many recent geometries. The German
mathematician, Crelle, defined a plane as a surface containing all the
straight lines (throughout their whole length) passing through a fixed
point and also intersecting a straight line in space, but of course this
intersected straight line must not pass through the fixed point.
Crelle's definition is occasionally seen in modern textbooks, but it is
not so clear to the pupil as Simson's. Of the various ultrascientific
definitions of a plane that have been suggested of late it is hardly of
use to speak in a book concerned primarily with practical teaching. No
one of them is adapted to the needs and the comprehension of the
beginner, and it seems that we are not likely to improve upon the
so-called Simson form.
8. PLANE ANGLE. _A plane angle is the inclination to each other of two
lines in a plane which meet each other and do not lie in a straight
line._ This definition, it will be noticed, includes curvilinear angles,
and the expression "and do not lie in a straight line" states that the
lines must not be continuous one with the other, that is, that zero and
straight angles are excluded. Since Euclid does not use the curvilinear
angle, and it is only the rectilinear angle with which we are concerned,
we will pass to the next definition and consider this one in connection
therewith.
9. RECTILINEAR ANGLE. _When the lines containing the angle are straight,
the angle is called rectilinear._ This definition, taken with the
preceding one, has always been a subject of criticism. In the first
place it expressly excludes the straight angle, and, indeed, the angles
of Euclid are always less than 180°, contrary to our modern concept. In
the second place it defines angle by means of the word "inclination,"
which is itself as difficult to define as angle. To remedy these defects
many substitutes have been proposed. Apollonius defined angle as "a
contracting of a surface or a solid at one point under a broken line or
surface." Another of the Greeks defined it as "a quantity, namely, a
distance between the lines or surfaces containing it." Schotten[56] says
that the definitions of angle generally fall into three groups:
_a._ An angle is the difference of direction between two lines that
meet. This is no better than Euclid's, since "difference of direction"
is as difficult to define as "inclination."
_b._ An angle is the amount of turning necessary to bring one side to
the position of the other side.
_c._ An angle is the portion of the plane included between its sides.
Of these, _b_ is given by way of explanation in most modern textbooks.
Indeed, we cannot do better than simply to define an angle as the
opening between two lines which meet, and then explain what is meant by
size, through the bringing in of the idea of rotation. This is a simple
presentation, it is easily understood, and it is sufficiently accurate
for the real purpose in mind, namely, the grasping of the concept. We
should frankly acknowledge that the concept of angle is such a simple
one that a satisfactory definition is impossible, and we should
therefore confine our attention to having the concept understood.
10. _When a straight line set up on a straight line makes the adjacent
angles equal to one another, each of the equal angles is right, and the
straight line standing on the other is called a perpendicular to that on
which it stands._ We at present separate these definitions and simplify
the language.
11. _An obtuse angle is an angle greater than a right angle._
12. _An acute angle is an angle less than a right angle._
The question sometimes asked as to whether an angle of 200° is obtuse,
and whether a negative angle, say -90°, is acute, is answered by saying
that Euclid did not conceive of angles equal to or greater than 180° and
had no notion of negative quantities. Generally to-day we define an
obtuse angle as "greater than one and less than two right angles." An
acute angle is defined as "an angle less than a right angle," and is
considered as positive under the general understanding that all
geometric magnitudes are positive unless the contrary is stated.
13. _A boundary is that which is an extremity of anything._ The
definition is not exactly satisfactory, for a circle is the boundary of
the space inclosed, but we hardly consider it as the extremity of that
space. Euclid wishes the definition before No. 14.
14. _A figure is that which is contained by any boundary or boundaries._
The definition is not satisfactory, since it excludes the unlimited
straight line, the angle, an assemblage of points, and other
combinations of lines and points which we should now consider as
figures.
15. _A circle is a plane figure contained by one line such that all the
straight lines falling upon it from one point among those lying within
the figure are equal to one another._
16. _And the point is called the center of the circle._
Some commentators add after "one line," definition 15, the words "which
is called the circumference," but these are not in the oldest
manuscripts. The Greek idea of a circle was usually that of part of a
plane which is bounded by a line called in modern times the
circumference, although Aristotle used "circle" as synonymous with "the
bounding line." With the growth of modern mathematics, however, and
particularly as a result of the development of analytic geometry, the
word "circle" has come to mean the bounding line, as it did with
Aristotle, a century before Euclid's time. This has grown out of the
equations of the various curves, _x_^2 + _y_^2 = _r_^2 representing the
circle-_line_, _a_^2_y_^2 + _b_^2_x_^2 = _a_^2_b_^2 representing the
ellipse-_line_, and so on. It is natural, therefore, that circle,
ellipse, parabola, and hyperbola should all be looked upon as lines.
Since this is the modern use of "circle" in English, it has naturally
found its way into elementary geometry, in order that students should
not have to form an entirely different idea of circle on beginning
analytic geometry. The general body of American teachers, therefore, at
present favors using "circle" to mean the bounding line and
"circumference" to mean the length of that line. This requires
redefining "area of a circle," and this is done by saying that it is the
area of the plane space inclosed. The matter is not of greatest
consequence, but teachers will probably prefer to join in the modern
American usage of the term.
17. DIAMETER. _A diameter of the circle is any straight line drawn
through the center and terminated in both directions by the
circumference of the circle, and such a straight line also bisects the
circle._ The word "diameter" is from two Greek words meaning a "through
measurer," and it was also used by Euclid for the diagonal of a square,
and more generally for the diagonal of any parallelogram. The word
"diagonal" is a later term and means the "through angle." It will be
noticed that Euclid adds to the usual definition the statement that a
diameter bisects the circle. He does this apparently to justify his
definition (18), of a semicircle (a half circle).
Thales is said to have been the first to prove that a diameter bisects
the circle, this being one of three or four propositions definitely
attributed to him, and it is sometimes given as a proposition to be
proved. As a proposition, however, it is unsatisfactory, since the proof
of what is so evident usually instills more doubt than certainty in the
minds of beginners.
18. SEMICIRCLE. _A semicircle is the figure contained by the diameter
and the circumference cut off by it. And the center of the semicircle is
the same as that of the circle._ Proclus remarked that the semicircle is
the only plane figure that has its center on its perimeter. Some writers
object to defining a circle as a line and then speaking of the area of a
circle, showing minds that have at least one characteristic of that of
Proclus. The modern definition of semicircle is "half of a circle," that
is, an arc of 180°, although the term is commonly used to mean both the
arc and the segment.
19. RECTILINEAR FIGURES. _Rectilinear figures are those which are
contained by straight lines, trilateral figures being those contained by
three, quadrilateral those contained by four, and multilateral those
contained by more than four, straight lines._
20. _Of trilateral figures, an equilateral triangle is that which has
its three sides equal, an isosceles triangle that which has two of its
sides alone equal, and a scalene triangle that which has its three sides
unequal._
21. _Further, of trilateral figures, a right-angled triangle is that
which has a right angle, an obtuse-angled triangle that which has an
obtuse angle, and an acute-angled triangle that which has its three
angles acute._
These three definitions may properly be considered together.
"Rectilinear" is from the Latin translation of the Greek _euthygrammos_,
and means "right-lined," or "straight-lined." Euclid's idea of such a
figure is that of the space inclosed, while the modern idea is tending
to become that of the inclosing lines. In elementary geometry, however,
the Euclidean idea is still held. "Trilateral" is from the Latin
translation of the Greek _tripleuros_ (three-sided). In elementary
geometry the word "triangle" is more commonly used, although
"quadrilateral" is more common than "quadrangle." The use of these two
different forms is eccentric and is merely a matter of fashion. Thus we
speak of a pentagon but not of a tetragon or a trigon, although both
words are correct in form. The word "multilateral" (many-sided) is a
translation of the Greek _polypleuros_. Fashion has changed this to
"polygonal" (many-angled), the word "multilateral" rarely being seen.
Of the triangles, "equilateral" means "equal-sided"; "isosceles" is from
the Greek _isoskeles_, meaning "with equal legs," and "scalene" from
_skalenos_, possibly from _skazo_ (to limp), or from _skolios_
(crooked). Euclid's limitation of isosceles to a triangle with two, and
only two, equal sides would not now be accepted. We are at present more
given to generalizing than he was, and when we have proved a proposition
relating to the isosceles triangle, we wish to say that we have thereby
proved it for the equilateral triangle. We therefore say that an
isosceles triangle has two sides equal, leaving it possible that all
three sides should be equal. The expression "equal legs" is now being
discarded on the score of inelegance. In place of "right-angled
triangle" modern writers speak of "right triangle," and so for the
obtuse and acute triangles. The terms are briefer and are as readily
understood. It may add a little interest to the subject to know that
Plutarch tells us that the ancients thought that "the power of the
triangle is expressive of the nature of Pluto, Bacchus, and Mars." He
also states that the Pythagoreans called "the equilateral triangle the
head-born Minerva and Tritogeneia (born of Triton) because it may be
equally divided by the perpendicular lines drawn from each of its
angles."
22. _Of quadrilateral figures a square is that which is both equilateral
and right-angled; an oblong that which is right-angled but not
equilateral; a rhombus that which is equilateral and not right-angled;
and a rhomboid that which has its opposite sides and angles equal to one
another, but is neither equilateral nor right-angled. And let all
quadrilaterals other than these be called trapezia._ In this definition
Euclid also specializes in a manner not now generally approved. Thus we
are more apt to-day to omit the oblong and rhomboid as unnecessary, and
to define "rhombus" in such a manner as to include a square. We use
"parallelogram" to cover "rhomboid," "rhombus," "oblong," and "square."
For "oblong" we use "rectangle," letting it include square. Euclid's
definition of "square" illustrates his freedom in stating more
attributes than are necessary, in order to make sure that the concept is
clear; for he might have said that it "is that which is equilateral and
has one right angle." We may profit by his method, sacrificing logic to
educational necessity. Euclid does not use "oblong," "rhombus,"
"rhomboid," and "trapezium" (_plural_, "trapezia") in his proofs, so
that he might well have omitted the definitions, as we often do.
23. PARALLELS. _Parallel straight lines are straight lines which, being
in the same plane and being produced indefinitely in both directions, do
not meet one another in either direction._ This definition of parallels,
simplified in its language, is the one commonly used to-day. Other
definitions have been suggested, but none has been so generally used.
Proclus states that Posidonius gave the definition based upon the lines
always being at the same distance apart. Geminus has the same idea in
his definition. There are, as Schotten has pointed out, three general
types of definitions of parallels, namely:
_a._ They have no point in common. This may be expressed by saying that
(1) they do not intersect, (2) they meet at infinity.
_b._ They are equidistant from one another.
_c._ They have the same direction.
Of these, the first is Euclid's, the idea of the point at infinity being
suggested by Kepler (1604). The second part of this definition is, of
course, unusable for beginners. Dr. (now Sir Thomas) Heath says, "It
seems best, therefore, to leave to higher geometry the conception of
infinitely distant points on a line and of two straight lines meeting at
infinity, like imaginary points of intersection, and, for the purposes
of elementary geometry, to rely on the plain distinction between
'parallel' and 'cutting,' which average human intelligence can readily
grasp."
The direction definition seems to have originated with Leibnitz. It is
open to the serious objection that "direction" is not easy of
definition, and that it is used very loosely. If two people on different
meridians travel due north, do they travel in the same direction? on
parallel lines? The definition is as objectionable as that of angle as
the "difference of direction" of two intersecting lines.
From these definitions of the first book of Euclid we see (1) what a
small number Euclid considered as basal; (2) what a change has taken
place in the generalization of concepts; (3) how the language has
varied. Nevertheless we are not to be commended if we adhere to Euclid's
small number, because geometry is now taught to pupils whose vocabulary
is limited. It is necessary to define more terms, and to scatter the
definitions through the work for use as they are needed, instead of
massing them at the beginning, as in a dictionary. The most important
lesson to be learned from Euclid's definitions is that only the basal
ones, relatively few in number, need to be learned, and these because
they are used as the foundations upon which proofs are built. It should
also be noticed that Euclid explains nothing in these definitions; they
are hard statements of fact, massed at the beginning of his treatise.
Not always as statements, and not at all in their arrangement, are they
suited to the needs of our boys and girls at present.
Having considered Euclid's definitions of Book I, it is proper to turn
to some of those terms that have been added from time to time to his
list, and are now usually incorporated in American textbooks. It will be
seen that most of these were assumed by Euclid to be known by his
mature readers. They need to be defined for young people, but most of
them are not basal, that is, they are not used in the proofs of
propositions. Some of these terms, such as magnitudes, curve line,
broken line, curvilinear figure, bisector, adjacent angles, reflex
angles, oblique angles and lines, and vertical angles, need merely a
word of explanation so that they may be used intelligently. If they were
numerous enough to make it worth the while, they could be classified in
our textbooks as of minor importance, but such a course would cause more
trouble than it is worth.
Other terms have come into use in modern times that are not common
expressions with which students are familiar. Such a term is "straight
angle," a concept not used by Euclid, but one that adds so materially to
the interest and value of geometry as now to be generally recognized.
There is also the word "perigon," meaning the whole angular space about
a point. This was excluded by the Greeks because their idea of angle
required it to be less than a straight angle. The word means "around
angle," and is the best one that has been coined for the purpose. "Flat
angle" and "whole angle" are among the names suggested for these two
modern concepts. The terms "complement," "supplement," and "conjugate,"
meaning the difference between a given angle and a right angle, straight
angle, and perigon respectively, have also entered our vocabulary and
need defining.
There are also certain terms expressing relationship which Euclid does
not define, and which have been so changed in recent times as to require
careful definition at present. Chief among these are the words "equal,"
"congruent," and "equivalent." Euclid used the single word "equal" for
all three concepts, although some of his recent editors have changed it
to "identically equal" in the case of congruence. In modern speech we
use the word "equal" commonly to mean "like-valued," "having the same
measure," as when we say the circumference of a circle "equals" a
straight line whose length is 2[pi]_r_, although it could not coincide
with it. Of late, therefore, in Europe and America, and wherever
European influence reaches, the word "congruent" is coming into use to
mean "identically equal" in the sense of superposable. We therefore
speak of congruent triangles and congruent parallelograms as being those
that are superposable.
It is a little unfortunate that "equal" has come to be so loosely used
in ordinary conversation that we cannot keep it to mean "congruent"; but
our language will not permit it, and we are forced to use the newer
word. Whenever it can be used without misunderstanding, however, it
should be retained, as in the case of "equal straight lines," "equal
angles," and "equal arcs of the same circle." The mathematical and
educational world will never consent to use "congruent straight lines,"
or "congruent angles," for the reason that the terms are unnecessarily
long, no misunderstanding being possible when "equal" is used.
The word "equivalent" was introduced by Legendre at the close of the
eighteenth century to indicate equality of length, or of area, or of
volume. Euclid had said, "Parallelograms which are on the same base and
in the same parallels are equal to one another," while Legendre and his
followers would modify the wording somewhat and introduce "equivalent"
for "equal." This usage has been retained. Congruent polygons are
therefore necessarily equivalent, but equivalent polygons are not in
general congruent. Congruent polygons have mutually equal sides and
mutually equal angles, while equivalent polygons have no equality save
that of area.
In general, as already stated, these and other terms should be defined
just before they are used instead of at the beginning of geometry. The
reason for this, from the educational standpoint and considering the
present position of geometry in the curriculum, is apparent.
We shall now consider the definitions of Euclid's Book III, which is
usually taken as Book II in America.
1. EQUAL CIRCLES. _Equal circles are those the diameters of which are
equal, or the radii of which are equal._
Manifestly this is a theorem, for it asserts that if the radii of two
circles are equal, the circles may be made to coincide. In some
textbooks a proof is given by superposition, and the proof is
legitimate, but Euclid usually avoided superposition if possible.
Nevertheless he might as well have proved this as that two triangles are
congruent if two sides and the included angle of the one are
respectively equal to the corresponding parts of the other, and he might
as well have postulated the latter as to have substantially postulated
this fact. For in reality this definition is a postulate, and it was so
considered by the great Italian mathematician Tartaglia (_ca._
1500-_ca._ 1557). The plan usually followed in America to-day is to
consider this as one of many unproved propositions, too evident, indeed,
for proof, accepted by intuition. The result is a loss in the logic of
Euclid, but the method is thought to be better adapted to the mind of
the youthful learner. It is interesting to note in this connection that
the Greeks had no word for "radius," and were therefore compelled to use
some such phrase as "the straight line from the center," or, briefly,
"the from the center," as if "from the center" were one word.
2. TANGENT. _A straight line is said to touch a circle which, meeting
the circle and being produced, does not cut the circle._
Teachers who prefer to use "circumference" instead of "circle" for the
line should notice how often such phrases as "cut the circle" and
"intersecting circle" are used,--phrases that signify nothing unless
"circle" is taken to mean the line. So Aristotle uses an expression
meaning that the locus of a certain point is a circle, and he speaks of
a circle as passing through "all the angles." Our word "touch" is from
the Latin _tangere_, from which comes "tangent," and also "tag," an old
touching game.
3. TANGENT CIRCLES. _Circles are said to touch one another which,
meeting one another, do not cut one another._
The definition has not been looked upon as entirely satisfactory, even
aside from its unfortunate phraseology. It is not certain, for instance,
whether Euclid meant that the circles could not cut at some other point
than that of tangency. Furthermore, no distinction is made between
external and internal contact, although both forms are used in the
propositions. Modern textbook makers find it convenient to define
tangent circles as those that are tangent to the same straight line at
the same point, and to define external and internal tangency by
reference to their position with respect to the line, although this may
be characterized as open to about the same objection as Euclid's.
4. DISTANCE. _In a circle straight lines are said to be equally distant
from the center, when the perpendiculars drawn to them from the center
are equal._
It is now customary to define "distance" from a point to a line as the
length of the perpendicular from the point to the line, and to do this
in Book I. In higher mathematics it is found that distance is not a
satisfactory term to use, but the objections to it have no particular
significance in elementary geometry.
5. GREATER DISTANCE. _And that straight line is said to be at a greater
distance on which the greater perpendicular falls._
Such a definition is not thought essential at the present time.
6. SEGMENT. _A segment of a circle is the figure contained by a straight
line and the circumference of a circle._
The word "segment" is from the Latin root _sect_, meaning "cut." So we
have "sector" (a cutter), "section" (a cut), "intersect," and so on. The
word is not limited to a circle; we have long spoken of a spherical
segment, and it is common to-day to speak of a line segment, to which
some would apply a new name "sect." There is little confusion in the
matter, however, for the context shows what kind of a segment is to be
understood, so that the word "sect" is rather pedantic than important.
It will be noticed that Euclid here uses "circumference" to mean "arc."
7. ANGLE OF A SEGMENT. _An angle of a segment is that contained by a
straight line and a circumference of a circle._
This term has entirely dropped out of geometry, and few teachers would
know what it meant if they should hear it used. Proclus called such
angles "mixed."
8. ANGLE IN A SEGMENT. _An angle in a segment is the angle which, when a
point is taken on the circumference of the segment and straight lines
are joined from it to the extremities of the straight line which is the
base of the segment, is contained by the straight lines so joined._
Such an involved definition would not be usable to-day. Moreover, the
words "circumference of the segment" would not be used.
9. _And when the straight lines containing the angle cut off a
circumference, the angle is said to stand upon that circumference._
10. SECTOR. _A sector of a circle is the figure which, when an angle is
constructed at the center of the circle, is contained by the straight
lines containing the angle and the circumference cut off by them._
There is no reason for such an extended definition, our modern
phraseology being both more exact (as seen in the above use of
"circumference" for "arc") and more intelligible. The Greek word for
"sector" is "knife" (_tomeus_), "sector" being the Latin translation. A
sector is supposed to resemble a shoemaker's knife, and hence the
significance of the term. Euclid followed this by a definition of
similar sectors, a term now generally abandoned as unnecessary.
It will be noticed that Euclid did not use or define the word "polygon."
He uses "rectilinear figure" instead. Polygon may be defined to be a
bounding line, as a circle is now defined, or as the space inclosed by a
broken line, or as a figure formed by a broken line, thus including both
the limited plane and its boundary. It is not of any great consequence
geometrically which of these ideas is adopted, so that the usual
definition of a portion of a plane bounded by a broken line may be taken
as sufficient for elementary purposes. It is proper to call attention,
however, to the fact that we may have cross polygons of various types,
and that the line that "bounds" the polygon must be continuous, as the
definition states. That is, in the second of these figures the shaded
portion is not considered a polygon. Such special cases are not liable
to arise, but if questions relating to them are suggested, the teacher
should be prepared to answer them. If suggested to a class, a note of
this kind should come out only incidentally as a bit of interest, and
should not occupy much time nor be unduly emphasized.
[Illustration]
It may also be mentioned to a class at some convenient time that the old
idea of a polygon was that of a convex figure, and that the modern idea,
which is met in higher mathematics, leads to a modification of earlier
concepts. For example, here is a quadrilateral with one of its
diagonals, _BD_, _outside_ the figure. Furthermore, if we consider a
quadrilateral as a figure formed by four intersecting lines, _AC_, _CF_,
_BE_, and _EA_, it is apparent that this _general quadrilateral_ has six
vertices, _A_, _B_, _C_, _D_, _E_, _F_, and three diagonals, _AD_, _BF_,
and _CE_. Such broader ideas of geometry form the basis of what is
called modern elementary geometry.
[Illustration]
The other definitions of plane geometry need not be discussed, since all
that have any historical interest have been considered. On the whole it
may be said that our definitions to-day are not in general so carefully
considered as those of Euclid, who weighed each word with greatest
skill, but they are more teachable to beginners, and are, on the whole,
more satisfactory from the educational standpoint. The greatest lesson
to be learned from this discussion is that the number of basal
definitions to be learned for subsequent use is very small.
Since teachers are occasionally disturbed over the form in which
definitions are stated, it is well to say a few words upon this subject.
There are several standard types that may be used. (1) We may use the
dictionary form, putting the word defined first, thus: "_Right
triangle_. A triangle that has one of its angles a right angle." This is
scientifically correct, but it is not a complete sentence, and hence it
is not easily repeated when it has to be quoted as an authority. (2) We
may put the word defined at the end, thus: "A triangle that has one of
its angles a right angle is called a right triangle." This is more
satisfactory. (3) We may combine (1) and (2), thus: "_Right triangle_. A
triangle that has one of its angles a right angle is called a right
triangle." This is still better, for it has the catchword at the
beginning of the paragraph.
There is occasionally some mental agitation over the trivial things of a
definition, such as the use of the words "is called." It would not be a
very serious matter if they were omitted, but it is better to have them
there. The reason is that they mark the statement at once as a
definition. For example, suppose we say that "a triangle that has one of
its angles a right angle is a right triangle." We have also the fact
that "a triangle whose base is the diameter of a semicircle and whose
vertex lies on the semicircle is a right triangle." The style of
statement is the same, and we have nothing in the phraseology to show
that the first is a definition and the second a theorem. This may
happen with most of the definitions, and hence the most careful writers
have not consented to omit the distinctive words in question.
Apropos of the definitions of geometry, the great French philosopher and
mathematician, Pascal, set forth certain rules relating to this subject,
as also to the axioms employed, and these may properly sum up this
chapter.
1. Do not attempt to define terms so well known in themselves that there
are no simpler terms by which to express them.
2. Admit no obscure or equivocal terms without defining them.
3. Use in the definitions only terms that are perfectly understood or
are there explained.
4. Omit no necessary principles without general agreement, however clear
and evident they may be.
5. Set forth in the axioms only those things that are in themselves
perfectly evident.
6. Do not attempt to demonstrate anything that is so evident in itself
that there is nothing more simple by which to prove it.
7. Prove whatever is in the least obscure, using in the demonstration
only axioms that are perfectly evident in themselves, or propositions
already demonstrated or allowed.
8. In case of any uncertainty arising from a term employed, always
substitute mentally the definition for the term itself.
=Bibliography.= Heath, Euclid, as cited; Frankland, The First
Book of Euclid, as cited; Smith, Teaching of Elementary
Mathematics, p. 257, New York, 1900; Young, Teaching of
Mathematics, p. 189, New York, 1907; Veblen, On Definitions, in
the _Monist_, 1903, p. 303.
FOOTNOTES:
[53] Free use has been made of W. B. Frankland, "The First Book of
Euclid's 'Elements,'" Cambridge, 1905; T. L. Heath, "The Thirteen Books
of Euclid's 'Elements,'" Cambridge, 1908; H. Schotten, "Inhalt und
Methode des planimetrischen Unterrichts," Leipzig, 1893; M. Simon,
"Euclid und die sechs planimetrischen Bücher," Leipzig, 1901.
[54] For a facsimile of a thirteenth-century MS. containing this
definition, see the author's "Rara Arithmetica," Plate IV, Boston, 1909.
[55] Our slang expression "The cart before the horse" is suggestive of
this procedure.
[56] Loc. cit., Vol. II, p. 94.
CHAPTER XIII
HOW TO ATTACK THE EXERCISES
The old geometry, say of a century ago, usually consisted, as has been
stated, of a series of theorems fully proved and of problems fully
solved. During the nineteenth century exercises were gradually
introduced, thus developing geometry from a science in which one learned
by seeing things done, into one in which he gained power by actually
doing things. Of the nature of these exercises ("originals," "riders"),
and of their gradual change in the past few years, mention has been made
in Chapter VII. It now remains to consider the methods of attacking
these exercises.
It is evident that there is no single method, and this is a fortunate
fact, since if it were not so, the attack would be too mechanical to be
interesting. There is no one rule for solving every problem nor even for
seeing how to begin. On the other hand, a pupil is saved some time by
having his attention called to a few rather definite lines of attack,
and he will undoubtedly fare the better by not wasting his energies over
attempts that are in advance doomed to failure.
There are two general questions to be considered: first, as to the
discovery of new truths, and second, as to the proof. With the first the
pupil will have little to do, not having as yet arrived at this stage in
his progress. A bright student may take a little interest in seeing
what he can find out that is new (at least to him), and if so, he may
be told that many new propositions have been discovered by the accurate
drawing of figures; that some have been found by actually weighing
pieces of sheet metal of certain sizes; and that still others have made
themselves known through paper folding. In all of these cases, however,
the supposed proposition must be proved before it can be accepted.
As to the proof, the pupil usually wanders about more or less until he
strikes the right line, and then he follows this to the conclusion. He
should not be blamed for doing this, for he is pursuing the method that
the world followed in the earliest times, and one that has always been
common and always will be. This is the synthetic method, the building up
of the proof from propositions previously proved. If the proposition is
a theorem, it is usually not difficult to recall propositions that may
lead to the demonstration, and to select the ones that are really
needed. If it is a problem, it is usually easy to look ahead and see
what is necessary for the solution and to select the preceding
propositions accordingly.
But pupils should be told that if they do not rather easily find the
necessary propositions for the construction or the proof, they should
not delay in resorting to another and more systematic method. This is
known as the method of analysis, and it is applicable both to theorems
and to problems. It has several forms, but it is of little service to a
pupil to have these differentiated, and it suffices that he be given the
essential feature of all these forms, a feature that goes back to Plato
and his school in the fifth century B.C.
For a theorem, the method of analysis consists in reasoning as follows:
"I can prove this proposition if I can prove this thing; I can prove
this thing if I can prove that; I can prove that if I can prove a third
thing," and so the reasoning runs until the pupil comes to the point
where he is able to add, "but I _can_ prove that." This does not prove
the proposition, but it enables him to reverse the process, beginning
with the thing he can prove and going back, step by step, to the thing
that he is to prove. Analysis is, therefore, his method of discovery of
the way in which he may arrange his synthetic proof. Pupils often wonder
how any one ever came to know how to arrange the proofs of geometry, and
this answers the question. Some one guessed that a statement was true;
he applied analysis and found that he _could_ prove it; he then applied
synthesis and _did_ prove it.
For a problem, the method of analysis is much the same as in the case of
a theorem. Two things are involved, however, instead of one, for here we
must make the construction and then prove that this construction is
correct. The pupil, therefore, first supposes the problem solved, and
sees what results follow. He then reverses the process and sees if he
can attain these results and thus effect the required construction. If
so, he states the process and gives the resulting proof. For example:
In a triangle _ABC_, to draw _PQ_ parallel to the base _AB_,
cutting the sides in _P_ and _Q_, so that _PQ_ shall equal
_AP_ + _BQ_.
[Illustration]
=Analysis.= Assume the problem solved.
Then _AP_ must equal some part of _PQ_ as _PX_, and _BQ_ must
equal _QX_.
But if _AP_ = _PX_, what must [L]_PXA_ equal?
[because] _PQ_ is || _AB_, what does [L]_PXA_ equal?
Then why must [L]_BAX_ = [L]_XAP_?
Similarly, what about [L]_QBX_ and [L]_XBA_?
=Construction.= Now reverse the process. What may we do to [Ls]
_A_ and _B_ in order to fix _X_? Then how shall _PQ_ be drawn?
Now give the proof.
[Illustration]
[Illustration]
The third general method of attack applies chiefly to problems where
some point is to be determined. This is the method of the intersection
of loci. Thus, to locate an electric light at a point eighteen feet from
the point of intersection of two streets and equidistant from them,
evidently one locus is a circle with a radius eighteen feet and the
center at the vertex of the angle made by the streets, and the other
locus is the bisector of the angle. The method is also occasionally
applicable to theorems. For example, to prove that the perpendicular
bisectors of the sides of a triangle are concurrent. Here the locus of
points equidistant from _A_ and _B_ is _PP'_, and the locus of points
equidistant from _B_ and _C_ is _QQ'_. These can easily be shown to
intersect, as at _O_. Then _O_, being equidistant from _A_, _B_, and
_C_, is also on the perpendicular bisector of _AC_. Therefore these
bisectors are concurrent in _O_.
These are the chief methods of attack, and are all that should be given
to an average class for practical use.
Besides the methods of attack, there are a few general directions that
should be given to pupils.
1. In attacking either a theorem or a problem, take the most general
figure possible. Thus, if a proposition relates to a quadrilateral, take
one with unequal sides and unequal angles rather than a square or even a
rectangle. The simpler figures often deceive a pupil into feeling that
he has a proof, when in reality he has one only for a special case.
2. Set forth very exactly the thing that is given, using letters
relating to the figure that has been drawn. Then set forth with the same
exactness the thing that is to be proved. The neglect to do this is the
cause of a large per cent of the failures. The knowing of exactly what
we have to do and exactly what we have with which to do it is half the
battle.
3. If the proposition seems hazy, the difficulty is probably with the
wording. In this case try substituting the definition for the name of
the thing defined. Thus instead of thinking too long about proving that
the median to the base of an isosceles triangle is perpendicular to the
base, draw the figure and think that there is given
_AC_ = _BC_,
_AD_ = _BD_,
and that there is to be proved that
[L]_CDA_ = [L]_BDC_.
[Illustration]
Here we have replaced "median," "isosceles," and "perpendicular" by
statements that express the same idea in simpler language.
=Bibliography.= Petersen, Methods and Theories for the Solution
of Geometric Problems of Construction, Copenhagen, 1879, a
curious piece of English and an extreme view of the subject,
but well worth consulting; Alexandroff, Problèmes de géométrie
élémentaire, Paris, 1899, with a German translation in 1903;
Loomis, Original Investigation; or, How to attack an Exercise
in Geometry, Boston, 1901; Sauvage, Les Lieux géométriques en
géométrie élémentaire, Paris, 1893; Hadamard, Leçons de
géométrie, p. 261, Paris, 1898; Duhamel, Des Méthodes dans les
sciences de raisonnement, 3^e éd., Paris, 1885; Henrici and
Treutlein, Lehrbuch der Elementar-Geometrie, Leipzig, 3. Aufl.,
1897; Henrici, Congruent Figures, London, 1879.
CHAPTER XIV
BOOK I AND ITS PROPOSITIONS
Having considered the nature of the geometry that we have inherited, and
some of the opportunities for improving upon the methods of presenting
it, the next question that arises is the all-important one of the
subject matter, What shall geometry be in detail? Shall it be the text
or the sequence of Euclid? Few teachers have any such idea at the
present time. Shall it be a mere dabbling with forms that are seen in
mechanics or architecture, with no serious logical sequence? This is an
equally dangerous extreme. Shall it be an entirely new style of geometry
based upon groups of motions? This may sometime be developed, but as yet
it exists in the future if it exists at all, since the recent efforts in
this respect are generally quite as ill suited to a young pupil as is
Euclid's "Elements" itself.
No one can deny the truth of M. Bourlet's recent assertion that
"Industry, daughter of the science of the nineteenth century, reigns
to-day the mistress of the world; she has transformed all ancient
methods, and she has absorbed in herself almost all human activity."[57]
Neither can one deny the justice of his comparison of Euclid with a
noble piece of Gothic architecture and of his assertion that as modern
life demands another type of building, so it demands another type of
geometry.
But what does this mean? That geometry is to exist merely as it touches
industry, or that bad architecture is to replace the good? By no means.
A building should to-day have steam heat and elevators and electric
lights, but it should be constructed of just as enduring materials as
the Parthenon, and it should have lines as pleasing as those of a Gothic
façade. Architecture should still be artistic and construction should
still be substantial, else a building can never endure. So geometry must
still exemplify good logic and must still bring to the pupil a feeling
of exaltation, or it will perish and become a mere relic in the museum
of human culture.
What, then, shall the propositions of geometry be, and in what manner
shall they answer to the challenge of the industrial epoch in which we
live? In reply, they must be better adapted to young minds and to all
young minds than Euclid ever intended his own propositions to be.
Furthermore, they must have a richness of application to pure geometry,
in the way of carefully chosen exercises, that Euclid never attempted.
And finally, they must have application to this same life of industry of
which we have spoken, whenever this can really be found, but there must
be no sham and pretense about it, else the very honesty that permeated
the ancient geometry will seem to the pupil to be wanting in the whole
subject.[58]
Until some geometry on a radically different basis shall appear, and of
this there is no very hopeful sign at present, the propositions will be
the essential ones of Euclid, excluding those that may be considered
merely intuitive, and excluding all that are too difficult for the pupil
who to-day takes up their study. The number will be limited in a
reasonable way, and every genuine type of application will be placed
before the teacher to be used as necessity requires. But a fair amount
of logic will be retained, and the effort to make of geometry an empty
bauble of a listless mind will be rejected by every worthy teacher. What
the propositions should be is a matter upon which opinions may justly
differ; but in this chapter there is set forth a reasonable list for
Book I, arranged in a workable sequence, and this list may fairly be
taken as typical of what the American school will probably use for many
years to come. With the list is given a set of typical applications, and
some of the general information that will add to the interest in the
work and that should form part of the equipment of the teacher.
An ancient treatise was usually written on a kind of paper called
papyrus, made from the pith of a large reed formerly common in Egypt,
but now growing luxuriantly only above Khartum in Upper Egypt, and near
Syracuse in Sicily; or else it was written on parchment, so called from
Pergamos in Asia Minor, where skins were first prepared in parchment
form; or occasionally they were written on ordinary leather. In any case
they were generally written on long strips of the material used, and
these were rolled up and tied. Hence we have such an expression as
"keeping the roll" in school, and such a word as "volume," which has in
it the same root as "involve" (to roll in), and "evolve" (to roll out).
Several of these rolls were often necessary for a single treatise, in
which case each was tied, and all were kept together in a receptacle
resembling a pail, or in a compartment on a shelf. The Greeks called
each of the separate parts of a treatise _biblion_ ([Greek: biblion]), a
word meaning "book." Hence we have the books of the Bible, the books of
Homer, and the books of Euclid. From the same root, indeed, comes Bible,
bibliophile (booklover), bibliography (list of books), and kindred
words. Thus the books of geometry are the large chapters of the subject,
"chapter" being from the Latin _caput_ (head), a section under a new
heading. There have been efforts to change "books" to "chapters," but
they have not succeeded, and there is no reason why they should succeed,
for the term is clear and has the sanction of long usage.
THEOREM. _If two lines intersect, the vertical angles are equal._
This was Euclid's Proposition 15, being put so late because he based the
proof upon his Proposition 13, now thought to be best taken without
proof, namely, "If a straight line set upon a straight line makes
angles, it will make either two right angles or angles equal to two
right angles." It is found to be better pedagogy to assume that this
follows from the definition of straight angle, with reference, if
necessary, to the meaning of the sum of two angles. This proposition on
vertical angles is probably the best one with which to begin geometry,
since it is not so evident as to seem to need no proof, although some
prefer to rank it as semiobvious, while the proof is so simple as easily
to be understood. Eudemus, a Greek who wrote not long before Euclid,
attributed the discovery of this proposition to Thales of Miletus (_ca._
640-548 B.C.), one of the Seven Wise Men of Greece, of whom Proclus
wrote: "Thales it was who visited Egypt and first transferred to
Hellenic soil this theory of geometry. He himself, indeed, discovered
much, but still more did he introduce to his successors the principles
of the science."
The proposition is the only basal one relating to the intersection of
two lines, and hence there are no others with which it is necessarily
grouped. This is the reason for placing it by itself, followed by the
congruence theorems.
There are many familiar illustrations of this theorem. Indeed, any two
crossed lines, as in a pair of shears or the legs of a camp stool, bring
it to mind. The word "straight" is here omitted before "lines" in
accordance with the modern convention that the word "line" unmodified
means a straight line. Of course in cases of special emphasis the
adjective should be used.
THEOREM. _Two triangles are congruent if two sides and the included
angle of the one are equal respectively to two sides and the included
angle of the other._
This is Euclid's Proposition 4, his first three propositions being
problems of construction. This would therefore have been his first
proposition if he had placed his problems later, as we do to-day. The
words "congruent" and "equal" are not used as in Euclid, for reasons
already set forth on page 151. There have been many attempts to
rearrange the propositions of Book I, putting in separate groups those
concerning angles, those concerning triangles, and those concerning
parallels, but they have all failed, and for the cogent reason that such
a scheme destroys the logical sequence. This proposition may properly
follow the one on vertical angles simply because the latter is easier
and does not involve superposition.
As far as possible, Euclid and all other good geometers avoid the proof
by superposition. As a practical test superposition is valuable, but as
a theoretical one it is open to numerous objections. As Peletier pointed
out in his (1557) edition of Euclid, if the superposition of lines and
figures could freely be assumed as a method of demonstration, geometry
would be full of such proofs. There would be no reason, for example, why
an angle should not be constructed equal to a given angle by superposing
the given angle on another part of the plane. Indeed, it is possible
that we might then assume to bisect an angle by imagining the plane
folded like a piece of paper. Heath (1908) has pointed out a subtle
defect in Euclid's proof, in that it is said that because two lines are
equal, they can be made to coincide. Euclid says, practically, that if
two lines can be made to coincide, they are equal, but he does not say
that if two straight lines are equal, they can be made to coincide. For
the purposes of elementary geometry the matter is hardly worth bringing
to the attention of a pupil, but it shows that even Euclid did not cover
every point.
Applications of this proposition are easily found, but they are all very
much alike. There are dozens of measurements that can be made by simply
constructing a triangle that shall be congruent to another triangle. It
seems hardly worth the while at this time to do more than mention one
typical case,[59] leaving it to teachers who may find it desirable to
suggest others to their pupils.
[Illustration]
Wishing to measure the distance across a river, some boys
sighted from _A_ to a point _P_. They then turned and measured
_AB_ at right angles to _AP_. They placed a stake at _O_,
halfway from _A_ to _B_, and drew a perpendicular to _AB_ at
_B_. They placed a stake at _C_, on this perpendicular, and in
line with _O_ and _P_. They then found the width by measuring
_BC_. Prove that they were right.
This involves the ranging of a line, and the running of a line at right
angles to a given line, both of which have been described in Chapter IX.
It is also fairly accurate to run a line at any angle to a given line by
sighting along two pins stuck in a protractor.
THEOREM. _Two triangles are congruent if two angles and the included
side of the one are equal respectively to two angles and the included
side of the other._
Euclid combines this with his Proposition 26:
If two triangles have the two angles equal to two angles
respectively, and one side equal to one side, namely, either
the side adjoining the equal angles, or that subtending one of
the equal angles, they will also have the remaining sides equal
to the remaining sides, and the remaining angle to the
remaining angle.
He proves this cumbersome statement without superposition, desiring to
avoid this method, as already stated, whenever possible. The proof by
superposition is old, however, for Al-Nair[=i]z[=i][60] gives it and
ascribes it to some earlier author whose name he did not know. Proclus
tells us that "Eudemus in his geometrical history refers this theorem to
Thales. For he says that in the method by which they say that Thales
proved the distance of ships in the sea, it was necessary to make use of
this theorem." How Thales did this is purely a matter of conjecture, but
he might have stood on the top of a tower rising from the level shore,
or of such headlands as abound near Miletus, and by some simple
instrument sighted to the ship. Then, turning, he might have sighted
along the shore to a point having the same angle of declination, and
then have measured the distance from the tower to this point. This
seems more reasonable than any of the various plans suggested, and it is
found in so many practical geometries of the first century of printing
that it seems to have long been a common expedient. The stone astrolabe
from Mesopotamia, now preserved in the British Museum, shows that such
instruments for the measuring of angles are very old, and for the
purposes of Thales even a pair of large compasses would have answered
very well. An illustration of the method is seen in Belli's work of
1569, as here shown. At the top of the picture a man is getting the
angle by means of the visor of his cap; at the bottom of the picture a
man is using a ruler screwed to a staff.[61] The story goes that one of
Napoleon's engineers won the imperial favor by quickly measuring the
width of a stream that blocked the progress of the army, using this very
method.
[Illustration: SIXTEENTH-CENTURY MENSURATION
Belli's "Del Misurar con la Vista," Venice, 1569]
This proposition is the reciprocal or dual of the preceding one. The
relation between the two may be seen from the following arrangement:
Two triangles are congruent if two _sides_ and the included
_angle_ of the one are equal respectively to two _sides_ and
the included _angle_ of the other.
Two triangles are congruent if two _angles_ and the included
_side_ of the one are equal respectively to two _angles_ and
the included _side_ of the other.
In general, to every proposition involving _points_ and _lines_ there is
a reciprocal proposition involving _lines_ and _points_ respectively
that is often true,--indeed, that is always true in a certain line of
propositions. This relation is known as the Principle of Reciprocity or
of Duality. Instead of points and lines we have here angles (suggested
by the vertex points) and lines. It is interesting to a class to have
attention called to such relations, but it is not of sufficient
importance in elementary geometry to justify more than a reference here
and there. There are other dual features that are seen in geometry
besides those given above.
THEOREM. _In an isosceles triangle the angles opposite the equal sides
are equal._
This is Euclid's Proposition 5, the second of his theorems, but he adds,
"and if the equal straight lines be produced further, the angles under
the base will be equal to one another." Since, however, he does not use
this second part, its genuineness is doubted. He would not admit the
common proof of to-day of supposing the vertical angle bisected, because
the problem about bisecting an angle does not precede this proposition,
and therefore his proof is much more involved than ours. He makes
_CX_ = _CY_, and proves [triangles]_XBC_ and _YAC_ congruent,[62] and
also [triangles]_XBA_ and _YAB_ congruent. Then from [L]_YAC_ he takes
[L]_YAB_, leaving [L]_BAC_, and so on the other side, leaving [L]_CBA_,
these therefore being equal.
[Illustration]
This proposition has long been called the _pons asinorum_, or bridge of
asses, but no one knows where or when the name arose. It is usually
stated that it came from the fact that fools could not cross this
bridge, and it is a fact that in the Middle Ages this was often the
limit of the student's progress in geometry. It has however been
suggested that the name came from Euclid's figure, which resembles the
simplest type of a wooden truss bridge. The name is applied by the
French to the Pythagorean Theorem.
Proclus attributes the discovery of this proposition to Thales. He also
says that Pappus (third century A.D.), a Greek commentator on Euclid,
proved the proposition as follows:
Let _ABC_ be the triangle, with _AB_ = _AC_. Conceive of this
as two triangles; then _AB_ = _AC_, _AC_ = _AB_, and [L]_A_ is
common; hence the [triangles]_ABC_ and _ACB_ are congruent, and
[L]_B_ of the one equals [L]_C_ of the other.
This is a better plan than that followed by some textbook writers of
imagining [triangle]_ABC_ taken up and laid down on _itself_. Even to
lay it down on its "trace" is more objectionable than the plan of
Pappus.
THEOREM. _If two angles of a triangle are equal, the sides opposite the
equal angles are equal, and the triangle is isosceles._
The statement is, of course, tautological, the last five words being
unnecessary from the mathematical standpoint, but of value at this stage
of the student's progress as emphasizing the nature of the triangle.
Euclid stated the proposition thus, "If in a triangle two angles be
equal to one another, the sides which subtend the equal angles will also
be equal to one another." He did not define "subtend," supposing such
words to be already understood. This is the first case of a converse
proposition in geometry. Heath distinguishes the logical from the
geometric converse. The logical converse of Euclid I, 5, would be that
"_some_ triangles with two angles equal are isosceles," while the
geometric converse is the proposition as stated. Proclus called
attention to two forms of converse (and in the course of the work, but
not at this time, the teacher may have to do the same): (1) the complete
converse, in which that which is given in one becomes that which is to
be proved in the other, and vice versa, as in this and the preceding
proposition; (2) the partial converse, in which two (or even more)
things may be given, and a certain thing is to be proved, the converse
being that one (or more) of the preceding things is now given, together
with what was to be proved, and the other given thing is now to be
proved. Symbolically, if it is given that _a_ = _b_ and _c_ = _d_, to
prove that _x_ = _y_, the partial converse would have given _a_ = _b_
and _x_ = _y_, to prove that _c_ = _d_.
Several proofs for the proposition have been suggested, but a careful
examination of all of them shows that the one given below is, all things
considered, the best one for pupils beginning geometry and following
the sequence laid down in this chapter. It has the sanction of some of
the most eminent mathematicians, and while not as satisfactory in some
respects as the _reductio ad absurdum_, mentioned below, it is more
satisfactory in most particulars. The proof is as follows:
[Illustration:
=Given the triangle ABC, with the angle A equal to the angle B.=]
_To prove that_ _AC_ = _BC_.
=Proof.= Suppose the second triangle _A'B'C'_ to be an exact
reproduction of the given triangle _ABC_.
Turn the triangle _A'B'C'_ over and place it upon _ABC_ so that _B'_
shall fall on _A_ and _A'_ shall fall on _B_.
Then _B'A'_ will coincide with _AB_.
Since [L]_A'_ = [L]_B'_, Given
and [L]_A_ = [L]_A'_, Hyp.
[therefore][L]_A_ = [L]_B'_.
[therefore]_B'C'_ will lie along _AC_.
Similarly, _A'C'_ will lie along _BC_.
Therefore _C'_ will fall on both _AC_ and _BC_, and hence at their
intersection.
[therefore]_B'C'_ = _AC_.
But _B'C'_ was made equal to _BC_.
[therefore]_AC_ = _BC_. Q.E.D.
If the proposition should be postponed until after the one on the sum of
the angles of a triangle, the proof would be simpler, but it is
advantageous to couple it with its immediate predecessor. This simpler
proof consists in bisecting the vertical angle, and then proving the
two triangles congruent. Among the other proofs is that of the _reductio
ad absurdum_, which the student might now meet, but which may better be
postponed. The phrase _reductio ad absurdum_ seems likely to continue in
spite of the efforts to find another one that is simpler. Such a proof
is also called an indirect proof, but this term is not altogether
satisfactory. Probably both names should be used, the Latin to explain
the nature of the English. The Latin name is merely a translation of one
of several Greek names used by Aristotle, a second being in English
"proof by the impossible," and a third being "proof leading to the
impossible." If teachers desire to introduce this form of proof here, it
must be borne in mind that only one supposition can be made if such a
proof is to be valid, for if two are made, then an absurd conclusion
simply shows that either or both must be false, but we do not know which
is false, or if only one is false.
THEOREM. _Two triangles are congruent if the three sides of the one are
equal respectively to the three sides of the other._
It would be desirable to place this after the fourth proposition
mentioned in this list if it could be done, so as to get the triangles
in a group, but we need the fourth one for proving this, so that the
arrangement cannot be made, at least with this method of proof.
This proposition is a "partial converse" of the second
proposition in this list; for if the triangles are _ABC_ and
_A'B'C'_, with sides _a_, _b_, _c_ and _a'_, _b'_, _c'_, then
the second proposition asserts that if _b_ = _b'_, _c_ = _c'_,
and [L]_A_ = [L]_A'_, then _a_ = _a'_ and the triangles are
congruent, while this proposition asserts that if _a_ = _a'_,
_b_ = _b'_, and _c_ = _c'_, then [L]_A_ = [L]_A'_ and the
triangles are congruent.
The proposition was known at least as early as Aristotle's time. Euclid
proved it by inserting a preliminary proposition to the effect that it
is impossible to have on the same base _AB_ and the same side of it two
different triangles _ABC_ and _ABC'_, with _AC_ = _AC'_, and
_BC_ = _BC'_. The proof ordinarily given to-day, wherein the two
triangles are constructed on opposite sides of the base, is due to Philo
of Byzantium, who lived after Euclid's time but before the Christian
era, and it is also given by Proclus. There are really three cases, if
one wishes to be overparticular, corresponding to the three pairs of
equal sides. But if we are allowed to take the longest side for the
common base, only one case need be considered.
Of the applications of the proposition one of the most important relates
to making a figure rigid by means of diagonals. For example, how many
diagonals must be drawn in order to make a quadrilateral rigid? to make
a pentagon rigid? a hexagon? a polygon of _n_ sides. In particular, the
following questions may be asked of a class:
[Illustration]
1. Three iron rods are hinged at the extremities, as shown in
this figure. Is the figure rigid? Why?
2. Four iron rods are hinged, as shown in this figure. Is the
figure rigid? If not, where would you put in the fifth rod to
make it rigid? Prove that this would accomplish the result.
[Illustration]
Another interesting application relates to the most ancient form of
leveling instrument known to us. This kind of level is pictured on very
ancient monuments, and it is still used in many parts of the world.
Pupils in manual training may make such an instrument, and indeed one is
easily made out of cardboard. If the plumb line passes through the
mid-point of the base, the two triangles are congruent and the plumb
line is then perpendicular to the base. In other words, the base is
level. With such simple primitive instruments, easily made by pupils, a
good deal of practical mathematical work can be performed. The
interesting old illustration here given shows how this form of level was
used three hundred years ago.
[Illustration: EARLY METHODS OF LEVELING
Pomodoro's "La geometria prattica," Rome, 1624]
[Illustration]
Teachers who seek for geometric figures in practical mechanics will find
this proposition illustrated in the ordinary hoisting apparatus of the
kind here shown. From the study of such forms and of simple roof and
bridge trusses, a number of the usual properties of the isosceles
triangle may be derived.
THEOREM. _The sum of two lines drawn from a given point to the
extremities of a given line is greater than the sum of two other lines
similarly drawn, but included by them._
It should be noted that the words "the extremities of" are necessary,
for it is possible to draw from a certain point within a certain
triangle two lines to the base such that their sum is greater than the
sum of the other two sides.
[Illustration]
Thus, in the right triangle _ABC_ draw any line _CX_ from _C_
to the base. Make _XY_ = _AC_, and _CP_ = _PY_. Then it is
easily shown that _PB_ + _PX_ > _CB_ + _CA_.
[Illustration]
It is interesting to a class to have a teacher point out that,
in this figure, _AP_ + _PB_ < _AC_ + _CB_, and _AP'_ + _P'B_ <
_AP_ + _PB_, and that the nearer _P_ gets to _AB_, the shorter
_AP_ + _PB_ becomes, the limit being the line _AB_. From this
we may _infer_ (although we have not proved) that "a straight
line (_AB_) is the shortest path between two points."
THEOREM. _Only one perpendicular can be drawn to a given line from a
given external point._
THEOREM. _Two lines drawn from a point in a perpendicular to a given
line, cutting off on the given line equal segments from the foot of the
perpendicular, are equal and make equal angles with the perpendicular._
THEOREM. _Of two lines drawn from the same point in a perpendicular to a
given line, cutting off on the line unequal segments from the foot of
the perpendicular, the more remote is the greater._
THEOREM. _The perpendicular is the shortest line that can be drawn to a
straight line from a given external point._
These four propositions, while known to the ancients and incidentally
used, are not explicitly stated by Euclid. The reason seems to be that
he interspersed his problems with his theorems, and in his Propositions
11 and 12, which treat of drawing a perpendicular to a line, the
essential features of these theorems are proved. Further mention will be
made of them when we come to consider the problems in question. Many
textbook writers put the second and third of the four before the first,
forgetting that the first is assumed in the other two, and hence should
precede them.
THEOREM. _Two right triangles are congruent if the hypotenuse and a side
of the one are equal respectively to the hypotenuse and a side of the
other._
THEOREM. _Two right triangles are congruent if the hypotenuse and an
adjacent angle of the one are equal respectively to the hypotenuse and
an adjacent angle of the other._
As stated in the notes on the third proposition in this sequence,
Euclid's cumbersome Proposition 26 covers several cases, and these two
among them. Of course this present proposition could more easily be
proved after the one concerning the sum of the angles of a triangle, but
the proof is so simple that it is better to leave the proposition here
in connection with others concerning triangles.
THEOREM. _Two lines in the same plane perpendicular to the same line
cannot meet, however far they are produced._
This proposition is not in Euclid, and it is introduced for educational
rather than for mathematical reasons. Euclid introduced the subject by
the proposition that, if alternate angles are equal, the lines are
parallel. It is, however, simpler to begin with this proposition, and
there is some advantage in stating it in such a way as to prove that
parallels exist before they are defined. The proposition is properly
followed by the definition of parallels and by the postulate that has
been discussed on page 127.
A good application of this proposition is the one concerning a method of
drawing parallel lines by the use of a carpenter's square. Here two
lines are drawn perpendicular to the edge of a board or a ruler, and
these are parallel.
THEOREM. _If a line is perpendicular to one of two parallel lines, it is
perpendicular to the other also._
This, like the preceding proposition, is a special case under a later
theorem. It simplifies the treatment of parallels, however, and the
beginner finds it easier to approach the difficulties gradually, through
these two cases of perpendiculars. It should be noticed that this is an
example of a partial converse, as explained on page 175. The preceding
proposition may be stated thus: If _a_ is [perp] to _x_ and _b_ is
[perp] to _x_, then _a_ is || to _b_. This proposition may be stated
thus: If _a_ is [perp] to _x_ and _a_ is || to _b_, then _b_ is [perp]
to _x_. This is, therefore, a partial converse.
These two propositions having been proved, the usual definitions of the
angles made by a transversal of two parallels may be given. It is
unfortunate that we have no name for each of the two groups of four
equal angles, and the name of "transverse angles" has been suggested.
This would simplify the statements of certain other propositions; thus:
"If two parallel lines are cut by a transversal, the transverse angles
are equal," and this includes two propositions as usually given. There
is not as yet, however, any general sanction for the term.
THEOREM. _If two parallel lines are cut by a transversal, the
alternate-interior angles are equal._
Euclid gave this as half of his Proposition 29. Indeed, he gives only
four theorems on parallels, as against five propositions and several
corollaries in most of our American textbooks. The reason for increasing
the number is that each proposition may be less involved. Thus, instead
of having one proposition for both exterior and interior angles, modern
authors usually have one for the exterior and one for the interior, so
as to make the difficult subject of parallels easier for beginners.
THEOREM. _When two straight lines in the same plane are cut by a
transversal, if the alternate-interior angles are equal, the two
straight lines are parallel._
This is the converse of the preceding theorem, and is half of Euclid I,
28, his theorem being divided for the reason above stated. There are
several typical pairs of equal or supplemental angles that would lead to
parallel lines, of which Euclid uses only part, leaving the other cases
to be inferred. This accounts for the number of corollaries in this
connection in later textbooks.
Surveyors make use of this proposition when they wish, without using a
transit instrument, to run one line parallel to another.
[Illustration]
For example, suppose two boys are laying out a tennis court and
they wish to run a line through _P_ parallel to _AB_. Take a
60-foot tape and swing it around _P_ until the other end rests
on _AB_, as at _M_. Put a stake at _O_, 30 feet from _P_ and
_M_. Then take any convenient point _N_ on _AB_, and measure
_ON_. Suppose it equals 20 feet. Then sight from _N_ through
_O_, and put a stake at _Q_ just 20 feet from _O_. Then _P_ and
_Q_ determine the parallel, according to the proposition just
mentioned.
THEOREM. _If two parallel lines are cut by a transversal, the
exterior-interior angles are equal._
This is also a part of Euclid I, 29. It is usually followed by several
corollaries, covering the minor and obvious cases omitted by the older
writers. While it would be possible to dispense with these corollaries,
they are helpful for definite reference in later propositions.
THEOREM. _The sum of the three angles of a triangle is equal to two
right angles._
Euclid stated this as follows: "In any triangle, if one of the sides be
produced, the exterior angle is equal to the two interior and opposite
angles, and the three interior angles of the triangle are equal to two
right angles." This states more than is necessary for the basal fact of
the proposition, which is the constancy of the sum of the angles.
The theorem is one of the three most important propositions in plane
geometry, the other two being the so-called Pythagorean Theorem, and a
proposition relating to the proportionality of the sides of two
triangles. These three form the foundation of trigonometry and of the
mensuration of plane figures.
The history of the proposition is extensive. Eutocius (_ca._ 510 A.D.),
in his commentary on Apollonius, says that Geminus (first century B.C.)
testified that "the ancients investigated the theorem of the two right
angles in each individual species of triangle, first in the equilateral,
again in the isosceles, and afterwards in the scalene triangle." This,
indeed, was the ancient plan, to proceed from the particular to the
general. It is the natural order, it is the world's order, and it is
well to follow it in all cases of difficulty in the classroom.
Proclus (410-485 A.D.) tells us that Eudemus, who lived just before
Euclid (or probably about 325 B.C.), affirmed that the theorem was due
to the Pythagoreans, although this does not necessarily mean to the
actual pupils of Pythagoras. The proof as he gives it consists in
showing that _a_ = _a_´, _b_ = _b_´, and _a_´ + _c_ + _b_´ = two right
angles. Since the proposition about the exterior angle of a triangle is
attributed to Philippus of Mende (_ca._ 380 B.C.), the figure given by
Eudemus is probably the one used by the Pythagoreans.
[Illustration]
There is also some reason for believing that Thales (_ca._ 600 B.C.)
knew the theorem, for Diogenes Laertius (_ca._ 200 A.D.) quotes
Pamphilius (first century A.D.) as saying that "he, having learned
geometry from the Egyptians, was the first to inscribe a right triangle
in a circle, and sacrificed an ox." The proof of this proposition
requires the knowledge that the sum of the angles, at least in a right
triangle, is two right angles. The proposition is frequently referred to
by Aristotle.
There have been numerous attempts to prove the proposition without the
use of parallel lines. Of these a German one, first given by Thibaut in
the early part of the eighteenth century, is among the most interesting.
This, in simplified form, is as follows:
[Illustration]
Suppose an indefinite line _XY_ to lie on _AB_. Let it swing
about _A_, counterclockwise, through [L]_A_, so as to lie on
_AC_, as _X'Y'_. Then let it swing about _C_, through [L]_C_,
so as to lie on _CB_, as _X''Y''_. Then let it swing about _B_,
through [L]_B_, so as to lie on _BA_, as _X'''Y'''_. It now
lies on _AB_, but it is turned over, _X'''_ being where _Y_
was, and _Y'''_ where _X_ was. In turning through [Ls]_A_, _B_,
and _C_ it has therefore turned through two right angles.
One trouble with the proof is that the rotation has not been about the
same point, so that it has never been looked upon as other than an
interesting illustration.
Proclus tried to prove the theorem by saying that, if we have two
perpendiculars to the same line, and suppose them to revolve about their
feet so as to make a triangle, then the amount taken from the right
angles is added to the vertical angle of the triangle, and therefore the
sum of the angles continues to be two right angles. But, of course, to
prove his statement requires a perpendicular to be drawn from the vertex
to the base, and the theorem of parallels to be applied.
Pupils will find it interesting to cut off the corners of a paper
triangle and fit the angles together so as to make a straight angle.
This theorem furnishes an opportunity for many interesting exercises,
and in particular for determining the third angle when two angles of a
triangle are given, or the second acute angle of a right triangle when
one acute angle is given.
Of the simple outdoor applications of the proposition, one of the best
is illustrated in this figure.
[Illustration]
To ascertain the height of a tree or of the school building,
fold a piece of paper so as to make an angle of 45°. Then walk
back from the tree until the top is seen at an angle of 45°
with the ground (being therefore careful to have the base of
the triangle level). Then the height _AC_ will equal the base
_AB_, since _ABC_ is isosceles. A paper protractor may be used
for the same purpose.
Distances can easily be measured by constructing a large equilateral
triangle of heavy pasteboard, and standing pins at the vertices for the
purpose of sighting.
[Illustration]
To measure _PC_, stand at some convenient point _A_ and sight
along _APC_ and also along _AB_. Then walk along _AB_ until a
point _B_ is reached from which _BC_ makes with _BA_ an angle
of the triangle (60°). Then _AC_ = _AB_, and since _AP_ can be
measured, we can find _PC_.
Another simple method of measuring a distance _AC_ across a stream is
shown in this figure.
[Illustration]
Measure the angle _CAX_, either in degrees, with a protractor,
or by sighting along a piece of paper and marking down the
angle. Then go along _XA_ produced until a point _B_ is reached
from which _BC_ makes with _A_ an angle equal to half of angle
_CAX_. Then it is easily shown that _AB_ = _AC_.
A navigator uses the same principle when he "doubles the angle on the
bow" to find his distance from a lighthouse or other object.
[Illustration]
If he is sailing on the course _ABC_ and notes a lighthouse _L_
when he is at _A_, and takes the angle _A_, and if he notices
when the angle that the lighthouse makes with his course is
just twice the angle noted at _A_, then _BL_ = _AB_. He has
_AB_ from his log (an instrument that tells how far a ship goes
in a given time), so he knows _BL_. He has "doubled the angle
on the bow" to get this distance.
It would have been possible for Thales, if he knew this proposition, to
have measured the distance of the ship at sea by some such device as
this:
[Illustration]
Make a large isosceles triangle out of wood, and, standing at
_T_, sight to the ship and along the shore on a line _TA_,
using the vertical angle of the triangle. Then go along _TA_
until a point _P_ is reached, from which _T_ and _S_ can be
seen along the sides of a base angle of the triangle. Then
_TP_ = _TS_. By measuring _TB_, _BS_ can then be found.
THEOREM. _The sum of two sides of a triangle is greater than the third
side, and their difference is less than the third side_.
If the postulate is assumed that a straight line is the shortest path
between two points, then the first part of this theorem requires no
further proof, and the second part follows at once from the axiom of
inequalities. This seems the better plan for beginners, and the
proposition may be considered as semiobvious. Euclid proved the first
part, not having assumed the postulate. Proclus tells us that the
Epicureans (the followers of Epicurus, the Greek philosopher, 342-270
B.C.) used to ridicule this theorem, saying that even an ass knew it,
for if he wished to get food, he walked in a straight line and not along
two sides of a triangle. Proclus replied that it was one thing to know
the truth and another thing to prove it, meaning that the value of
geometry lay in the proof rather than in the mere facts, a thing that
all who seek to reform the teaching of geometry would do well to keep in
mind. The theorem might simply appear as a corollary under the postulate
if it were of any importance to reduce the number of propositions one
more.
If the proposition is postponed until after those concerning the
inequalities of angles and sides of a triangle, there are several good
proofs.
[Illustration]
For example, produce _AC_ to _X_,
making
_CX_ = _CB_.
Then [L]_X_ = [L]_XBC_.
[therefore] [L]_XBA_ > [L]_X_.
[therefore] _AX_ > _AB_.
[therefore] _AC_ + _CB_ > _AB_.
The above proof is due to Euclid. Heron of Alexandria (first century
A.D.) is said by Proclus to have given the following:
[Illustration]
Let _CX_ bisect [L]_C_.
Then [L]_BXC_ > [L]_ACX_.
[therefore] [L]_BXC_ > [L]_XCB_.
[therefore] _CB_ > _XB_.
Similarly, _AC_ > _AX_.
Adding, _AC_ + _CB_ > _AB_.
THEOREM. _If two sides of a triangle are unequal, the angles opposite
these sides are unequal, and the angle opposite the greater side is the
greater._
Euclid stated this more briefly by saying, "In any triangle the greater
side subtends the greater angle." This is not so satisfactory, for there
may be no greater side.
THEOREM. _If two angles of a triangle are unequal, the sides opposite
these angles are unequal, and the side opposite the greater angle is the
greater._
Euclid also stated this more briefly, but less satisfactorily, thus, "In
any triangle the greater angle is subtended by the greater side."
Students should have their attention called to the fact that these two
theorems are reciprocal or dual theorems, the words "sides" and
"angles" of the one corresponding to the words "angles" and "sides"
respectively of the other.
It may also be noticed that the proof of this proposition
involves what is known as the Law of Converse; for
(1) if _b_ = _c_, then [L]_B_ = [L]_C_;
(2) if _b_ > _c_, then [L]_B_ > [L]_C_;
(3) if _b_ < _c_, then [L]_B_ < [L]_C_;
therefore the converses must necessarily be true as a matter of
logic; for
if [L]_B_ = [L]_C_, then _b_ cannot be greater than _c_ without
violating (2), and _b_ cannot be less than _c_ without
violating (3), therefore _b_ = _c_;
and if [L]_B_ > [L]_C_, then _b_ cannot equal _c_ without
violating (1), and _b_ cannot be less than _c_ without
violating (3), therefore _b_ > _c_;
similarly, if [L]_B_ < [L]_C_, then _b_ < _c_.
This Law of Converse may readily be taught to pupils, and it has several
applications in geometry.
THEOREM. _If two triangles have two sides of the one equal respectively
to two sides of the other, but the included angle of the first triangle
greater than the included angle of the second, then the third side of
the first is greater than the third side of the second, and conversely._
[Illustration]
In this proposition there are three possible cases: the point _Y_ may
fall below _AB_, as here shown, or on _AB_, or above _AB_. As an
exercise for pupils all three may be considered if desired. Following
Euclid and most early writers, however, only one case really need be
proved, provided that is the most difficult one, and is typical. Proclus
gave the proofs of the other two cases, and it is interesting to pupils
to work them out for themselves. In such work it constantly appears that
every proposition suggests abundant opportunity for originality, and
that the complete form of proof in a textbook is not a bar to
independent thought.
The Law of Converse, mentioned on page 190, may be applied to the
converse case if desired.
THEOREM. _Two angles whose sides are parallel, each to each, are either
equal or supplementary._
This is not an ancient proposition, although the Greeks were well aware
of the principle. It may be stated so as to include the case of the
sides being perpendicular, each to each, but this is better left as an
exercise. It is possible, by some circumlocution, to so state the
theorem as to tell in what cases the angles are equal and in what cases
supplementary. It cannot be tersely stated, however, and it seems better
to leave this point as a subject for questioning by the teacher.
THEOREM. _The opposite sides of a parallelogram are equal._
THEOREM. _If the opposite sides of a quadrilateral are equal, the figure
is a parallelogram._
[Illustration]
This proposition is a very simple test for a parallelogram. It is the
principle involved in the case of the common folding parallel ruler, an
instrument that has long been recognized as one of the valuable tools
of practical geometry. It will be of some interest to teachers to see
one of the early forms of this parallel ruler, as shown in the
illustration.[63] If such an instrument is not available in the school,
one suitable for illustrative purposes can easily be made from
cardboard.
[Illustration: PARALLEL RULER OF THE SEVENTEENTH CENTURY
San Giovanni's "Seconda squara mobile," Vicenza, 1686]
A somewhat more complicated form of this instrument may also be made by
pupils in manual training, as is shown in this illustration from Bion's
great treatise. The principle involved may be taken up in class, even if
the instrument is not used. It is evident that, unless the workmanship
is unusually good, this form of parallel ruler is not as accurate as the
common one illustrated above. The principle is sometimes used in iron
gates.
[Illustration: PARALLEL RULER OF THE EIGHTEENTH CENTURY
N. Bion's "Traité de la construction ... des instrumens de
mathématique," The Hague, 1723]
THEOREM. _Two parallelograms are congruent if two sides and the included
angle of the one are equal respectively to two sides and the included
angle of the other._
This proposition is discussed in connection with the one that follows.
THEOREM. _If three or more parallels intercept equal segments on one
transversal, they intercept equal segments on every transversal._
These two propositions are not given in Euclid, although generally
required by American syllabi of the present time. The last one is
particularly useful in subsequent work. Neither one offers any
difficulty, and neither has any interesting history. There are, however,
numerous interesting applications to the last one. One that is used in
mechanical drawing is here illustrated.
[Illustration]
If it is desired to divide a line _AB_ into five equal parts,
we may take a piece of ruled tracing paper and lay it over the
given line so that line 0 passes through _A_, and line 5
through _B_. We may then prick through the paper and thus
determine the points on _AB_. Similarly, we may divide _AB_
into any other number of equal parts.
Among the applications of these propositions is an interesting one due
to the Arab Al-Nair[=i]z[=i] (_ca._ 900 A.D.). The problem is to divide
a line into any number of equal parts, and he begins with the case of
trisecting _AB_. It may be given as a case of practical drawing even
before the problems are reached, particularly if some preliminary work
with the compasses and straightedge has been given.
Make _BQ_ and _AQ'_ perpendicular to _AB_, and make _BP_ = _PQ_
= _AP'_ = _P'Q'_. Then [triangle]_XYZ_ is congruent to
[triangle]_YBP_, and also to [triangle]_XAP'_. Therefore
_AX_ = _XY_ = _YB_. In the same way we might continue to produce
_BQ_ until it is made up of _n_ - 1 lengths _BP_, and so for _AQ'_,
and by properly joining points we could divide _AB_ into _n_
equal parts. In particular, if we join _P_ and _P'_, we bisect
the line _AB_.
[Illustration]
THEOREM. _If two sides of a quadrilateral are equal and parallel, then
the other two sides are equal and parallel, and the figure is a
parallelogram._
This was Euclid's first proposition on parallelograms, and Proclus
speaks of it as the connecting link between the theory of parallels and
that of parallelograms. The ancients, writing for mature students, did
not add the words "and the figure is a parallelogram," because that
follows at once from the first part and from the definition of
"parallelogram," but it is helpful to younger students because it
emphasizes the fact that here is a test for this kind of figure.
THEOREM. _The diagonals of a parallelogram bisect each other._
This proposition was not given in Euclid, but it is usually required in
American syllabi. There is often given in connection with it the
exercise in which it is proved that the diagonals of a rectangle are
equal. When this is taken, it is well to state to the class that
carpenters and builders find this one of the best checks in laying out
floors and other rectangles. It is frequently applied also in laying out
tennis courts. If the class is doing any work in mensuration, such as
finding the area of the school grounds, it is a good plan to check a few
rectangles by this method.
An interesting outdoor application of the theory of parallelograms is
the following:
[Illustration]
Suppose you are on the side of this stream opposite to _XY_,
and wish to measure the length of _XY_. Run a line _AB_ along
the bank. Then take a carpenter's square, or even a large book,
and walk along _AB_ until you reach _P_, a point from which you
can just see _X_ and _B_ along two sides of the square. Do the
same for _Y_, thus fixing _P_ and _Q_. Using the tape, bisect
_PQ_ at _M_. Then walk along _YM_ produced until you reach a
point _Y'_ that is exactly in line with _M_ and _Y_, and also
with _P_ and _X_. Then walk along _XM_ produced until you reach
a point _X'_ that is exactly in line with _M_ and _X_, and also
with _Q_ and _Y_. Then measure _Y'X'_ and you have the length
of _XY_. For since _YX'_ is [perp] to _PQ_, and _XY'_ is also
[perp] to _PQ_, _YX'_ is || to _XY'_. And since _PM_ = _MQ_,
therefore _XM_ = _MX'_ and _Y'M_ = _MY_. Therefore _Y'X'YX_ is
a parallelogram.
The properties of the parallelogram are often applied to proving figures
of various kinds congruent, or to constructing them so that they will be
congruent.
[Illustration]
For example, if we draw _A'B'_ equal and parallel to _AB_,
_B'C'_ equal and parallel to _BC_, and so on, it is easily
proved that _ABCD_ and _A'B'C'D'_ are congruent. This may be
done by ordinary superposition, or by sliding _ABCD_ along the
dotted parallels.
There are many applications of this principle of parallel translation in
practical construction work. The principle is more far-reaching than
here intimated, however, and a few words as to its significance will now
be in place.
The efforts usually made to improve the spirit of Euclid are trivial.
They ordinarily relate to some commonplace change of sequence, to some
slight change in language, or to some narrow line of applications. Such
attempts require no particular thought and yield no very noticeable
result. But there is a possibility, remote though it may be at present,
that a geometry will be developed that will be as serious as Euclid's
and as effective in the education of the thinking individual. If so, it
seems probable that it will not be based upon the congruence of
triangles, by which so many propositions of Euclid are proved, but upon
certain postulates of motion, of which one is involved in the above
illustration,--the postulate of parallel translation. If to this we join
the two postulates of rotation about an axis,[64] leading to axial
symmetry; and rotation about a point,[65] leading to symmetry with
respect to a center, we have a group of three motions upon which it is
possible to base an extensive and rigid geometry.[66] It will be through
some such effort as this, rather than through the weakening of the
Euclid-Legendre style of geometry, that any improvement is likely to
come. At present, in America, the important work for teachers is to
vitalize the geometry they have,--an effort in which there are great
possibilities,--seeing to it that geometry is not reduced to mere froth,
and recognizing the possibility of another geometry that may sometime
replace it,--a geometry as rigid, as thought-compelling, as logical,
and as truly educational.
THEOREM. _The sum of the interior angles of a polygon is equal to two
right angles, taken as many times less two as the figure has sides._
This interesting generalization of the proposition about the sum of the
angles of a triangle is given by Proclus. There are several proofs, but
all are based upon the possibility of dissecting the polygon into
triangles. The point from which lines are drawn to the vertices is
usually taken at a vertex, so that there are _n_ - 2 triangles. It may
however be taken within the figure, making _n_ triangles, from the sum
of the angles of which the four right angles about the point must be
subtracted. The point may even be taken on one side, or outside the
polygon, but the proof is not so simple. Teachers who desire to do so
may suggest to particularly good students the proving of the theorem for
a concave polygon, or even for a cross polygon, although the latter
requires negative angles.
Some schools have transit instruments for the use of their classes in
trigonometry. In such a case it is a good plan to measure the angles in
some piece of land so as to verify the proposition, as well as show the
care that must be taken in reading angles. In the absence of this
exercise it is well to take any irregular polygon and measure the angles
by the help of a protractor, and thus accomplish the same results.
THEOREM. _The sum of the exterior angles of a polygon, made by producing
each of its sides in succession, is equal to four right angles._
This is also a proposition not given by the ancient writers. We have,
however, no more valuable theorem for the purpose of showing the nature
and significance of the negative angle; and teachers may arouse a great
deal of interest in the negative quantity by showing to a class that
when an interior angle becomes 180° the exterior angle becomes 0, and
when the polygon becomes concave the exterior angle becomes negative,
the theorem holding for all these cases. We have few better
illustrations of the significance of the negative quantity, and few
better opportunities to use the knowledge of this kind of quantity
already acquired in algebra.
[Illustration]
In the hilly and mountainous parts of America, where irregular-shaped
fields are more common than in the more level portions, a common test
for a survey is that of finding the exterior angles when the transit
instrument is set at the corners. In this field these angles are given,
and it will be seen that the sum is 360°. In the absence of any outdoor
work a protractor may be used to measure the exterior angles of a
polygon drawn on paper. If there is an irregular piece of land near the
school, the exterior angles can be fairly well measured by an ordinary
paper protractor.
The idea of locus is usually introduced at the end of Book I. It is too
abstract to be introduced successfully any earlier, although authors
repeat the attempt from time to time, unmindful of the fact that all
experience is opposed to it. The loci propositions are not ancient. The
Greeks used the word "locus" (in Greek, _topos_), however. Proclus, for
example, says, "I call those locus theorems in which the same property
is found to exist on the whole of some locus." Teachers should be
careful to have the pupils recognize the necessity for proving two
things with respect to any locus: (1) that any point on the supposed
locus satisfies the condition; (2) that any point outside the supposed
locus does not satisfy the given condition. The first of these is called
the "sufficient condition," and the second the "necessary condition."
Thus in the case of the locus of points in a plane equidistant from two
given points, it is _sufficient_ that the point be on the perpendicular
bisector of the line joining the given points, and this is the first
part of the proof; it is also _necessary_ that it be on this line, i.e.
it cannot be outside this line, and this is the second part of the
proof. The proof of loci cases, therefore, involves a consideration of
"the necessary and sufficient condition" that is so often spoken of in
higher mathematics. This expression might well be incorporated into
elementary geometry, and when it becomes better understood by teachers,
it probably will be more often used.
In teaching loci it is helpful to call attention to loci in space
(meaning thereby the space of three dimensions), without stopping to
prove the proposition involved. Indeed, it is desirable all through
plane geometry to refer incidentally to solid geometry. In the
mensuration of plane figures, which may be boundaries of solid figures,
this is particularly true.
It is a great defect in most school courses in geometry that
they are entirely confined to two dimensions. Even if solid
geometry in the usual sense is not attempted, every occasion
should be taken to liberate boys' minds from what becomes the
tyranny of paper. Thus the questions: "What is the locus of a
point equidistant from two given points; at a constant distance
from a given straight line or from a given point?" should be
extended to space.[67]
The two loci problems usually given at this time, referring to a point
equidistant from the extremities of a given line, and to a point
equidistant from two intersecting lines, both permit of an interesting
extension to three dimensions without any formal proof. It is possible
to give other loci at this point, but it is preferable merely to
introduce the subject in Book I, reserving the further discussion until
after the circle has been studied.
It is well, in speaking of loci, to remember that it is entirely proper
to speak of the "locus of a point" or the "locus of points." Thus the
locus of a _point_ so moving in a plane as constantly to be at a given
distance from a fixed point in the plane is a circle. In analytic
geometry we usually speak of the locus of a _point_, thinking of the
point as being anywhere on the locus. Some teachers of elementary
geometry, however, prefer to speak of the locus of _points_, or the
locus of _all points_, thus tending to make the language of elementary
geometry differ from that of analytic geometry. Since it is a trivial
matter of phraseology, it is better to recognize both forms of
expression and to let pupils use the two interchangeably.
FOOTNOTES:
[57] Address at Brussels, August, 1910.
[58] For a recent discussion of this general subject, see Professor
Hobson on "The Tendencies of Modern Mathematics," in the _Educational
Review_, New York, 1910, Vol. XL, p. 524.
[59] A more extended list of applications is given later in this work.
[60] Ab[=u]'l-'Abb[=a]s al-Fadl ibn H[=a]tim al-Nair[=i]z[=i], so called
from his birthplace, Nair[=i]z, was a well-known Arab writer. He died
about 922 A.D. He wrote a commentary on Euclid.
[61] This illustration, taken from a book in the author's library,
appeared in a valuable monograph by W. E. Stark, "Measuring Instruments
of Long Ago," published in _School Science and Mathematics_, Vol. X, pp.
48, 126. With others of the same nature it is here reproduced by the
courtesy of Principal Stark and of the editors of the journal in which
it appeared.
[62] In speaking of two congruent triangles it is somewhat easier to
follow the congruence if the two are read in the same order, even though
the relatively unimportant counterclockwise reading is neglected. No one
should be a slave to such a formalism, but should follow the plan when
convenient.
[63] Stark, loc. cit.
[64] Of which so much was made by Professor Olaus Henrici in his
"Congruent Figures," London, 1879,--a book that every teacher of
geometry should own.
[65] Much is made of this in the excellent work by Henrici and
Treutlein, "Lehrbuch der Geometrie," Leipzig, 1881.
[66] Méray did much for this movement in France, and the recent works of
Bourlet and Borel have brought it to the front in that country.
[67] W. N. Bruce, "Teaching of Geometry and Graphic Algebra in Secondary
Schools," Board of Education circular (No. 711), p. 8, London, 1909.
CHAPTER XV
THE LEADING PROPOSITIONS OF BOOK II
Having taken up all of the propositions usually given in Book I, it
seems unnecessary to consider as specifically all those in subsequent
books. It is therefore proposed to select certain ones that have some
special interest, either from the standpoint of mathematics or from that
of history or application, and to discuss them as fully as the
circumstances seem to warrant.
THEOREMS. _In the same circle or in equal circles equal central angles
intercept equal arcs; and of two unequal central angles the greater
intercepts the greater arc_, and conversely for both of these cases.
Euclid made these the twenty-sixth and twenty-seventh propositions of
his Book III, but he limited them as follows: "In equal circles equal
angles stand on equal circumferences, whether they stand at the centers
or at the circumferences, and conversely." He therefore included two of
our present theorems in one, thus making the proposition doubly hard for
a beginner. After these two propositions the Law of Converse, already
mentioned on page 190, may properly be introduced.
THEOREMS. _In the same circle or in equal circles, if two arcs are
equal, they are subtended by equal chords; and if two arcs are unequal,
the greater is subtended by the greater chord_, and conversely.
Euclid dismisses all this with the simple theorem, "In equal circles
equal circumferences are subtended by equal straight lines." It will
therefore be noticed that he has no special word for "chord" and none
for "arc," and that the word "circumference," which some teachers are so
anxious to retain, is used to mean both the whole circle and any arc. It
cannot be doubted that later writers have greatly improved the language
of geometry by the use of these modern terms. The word "arc" is the
same, etymologically, as "arch," each being derived from the Latin
_arcus_ (a bow). "Chord" is from the Greek, meaning "the string of a
musical instrument." "Subtend" is from the Latin _sub_ (under), and
_tendere_ (to stretch).
It should be noticed that Euclid speaks of "equal circles," while we
speak of "the same circle or equal circles," confining our proofs to the
latter, on the supposition that this sufficiently covers the former.
THEOREM. _A line through the center of a circle perpendicular to a chord
bisects the chord and the arcs subtended by it._
This is an improvement on Euclid, III, 3: "If in a circle a straight
line through the center bisects a straight line not through the center,
it also cuts it at right angles; and if it cuts it at right angles, it
also bisects it." It is a very important proposition, theoretically and
practically, for it enables us to find the center of a circle if we know
any part of its arc. A civil engineer, for example, who wishes to find
the center of the circle of which some curve (like that on a running
track, on a railroad, or in a park) is an arc, takes two chords, say of
one hundred feet each, and erects perpendicular bisectors. It is well to
ask a class why, in practice, it is better to take these chords some
distance apart. Engineers often check their work by taking three chords,
the perpendicular bisectors of the three passing through a single
point. Illustrations of this kind of work are given later in this
chapter.
THEOREM. _In the same circle or in equal circles equal chords are
equidistant from the center, and chords equidistant from the center are
equal._
This proposition is practically used by engineers in locating points on
an arc of a circle that is too large to be described by a tape, or that
cannot easily be reached from the center on account of obstructions.
[Illustration]
If part of the curve _APB_ is known, take _P_ as the mid-point.
Then stretch the tape from _A_ to _B_ and draw _PM_
perpendicular to it. Then swing the length _AM_ about _P_, and
_PM_ about _B_, until they meet at _L_, and stretch the length
_AB_ along _PL_ to _Q_. This fixes the point _Q_. In the same
way fix the point _C_. Points on the curve can thus be fixed as
near together as we wish. The chords _AB_, _PQ_, _BC_, and so
on, are equal and are equally distant from the center.
THEOREM. _A line perpendicular to a radius at its extremity is tangent
to the circle._
The enunciation of this proposition by Euclid is very interesting. It is
as follows:
The straight line drawn at right angles to the diameter of a
circle at its extremity will fall outside the circle, and into
the space between the straight line and the circumference
another straight line cannot be interposed; further, the angle
of the semicircle is greater and the remaining angle less than
any acute rectilineal angle.
The first assertion is practically that of tangency,--"will fall outside
the circle." The second one states, substantially, that there is only
one such tangent, or, as we say in modern mathematics, the tangent is
unique. The third statement relates to the angle formed by the diameter
and the circumference,--a mixed angle, as Proclus called it, and a kind
of angle no longer used in elementary geometry. The fourth statement
practically asserts that the angle between the tangent and circumference
is less than any assignable quantity. This gives rise to a difficulty
that seems to have puzzled many of Euclid's commentators, and that will
interest a pupil: As the circle diminishes this angle apparently
increases, while as the circle increases the angle decreases, and yet
the angle is always stated to be zero. Vieta (1540-1603), who did much
to improve the science of algebra, attempted to explain away the
difficulty by adopting a notion of circle that was prevalent in his
time. He said that a circle was a polygon of an infinite number of sides
(which it cannot be, by definition), and that, a tangent simply
coincided with one of the sides, and therefore made no angle with it;
and this view was also held by Galileo (1564-1642), the great physicist
and mathematician who first stated the law of the pendulum.
THEOREM. _Parallel lines intercept equal arcs on a circle._
The converse of this proposition has an interesting application in
outdoor work.
[Illustration]
Suppose we wish to run a line through _P_ parallel to a given
line _AB_. With any convenient point _O_ as a center, and _OP_
as a radius, describe a circle cutting _AB_ in _X_ and _Y_.
Draw _PX_. Then with _Y_ as a center and _PX_ as a radius draw
an arc cutting the circle in _Q_. Then run the line from _P_ to
_Q_. _PQ_ is parallel to _AB_ by the converse of the above
theorem, which is easily shown to be true for this figure.
THEOREM. _If two circles are tangent to each other, the line of centers
passes through the point of contact._
There are many illustrations of this theorem in practical work, as in
the case of cogwheels. An interesting application to engineering is seen
in the case of two parallel streets or lines of track which are to be
connected by a "reversed curve."
[Illustration]
If the lines are _AB_ and _CD_, and the connection is to be
made, as shown, from _B_ to _C_, we may proceed as follows:
Draw _BC_ and bisect it at _M_. Erect _PO_, the perpendicular
bisector of _BM_; and _BO_, perpendicular to _AB_. Then _O_ is
one center of curvature. In the same way fix _O'_. Then to
check the work apply this theorem, _M_ being in the line of
centers _OO'_. The curves may now be drawn, and they will be
tangent to _AB_, to _CD_, and to each other.
At this point in the American textbooks it is the custom to insert a
brief treatment of measurement, explaining what is meant by ratio,
commensurable and incommensurable quantities, constant and variable, and
limit, and introducing one or more propositions relating to limits. The
object of this departure from the ancient sequence, which postponed this
subject to the book on ratio and proportion, is to treat the circle more
completely in Book III. It must be confessed that the treatment is not
as scientific as that of Euclid, as will be explained under Book III,
but it is far better suited to the mind of a boy or girl.
It begins by defining measurement in a practical way, as the finding of
the number of times a quantity of any kind contains a known quantity of
the same kind. Of course this gives a number, but this number may be a
surd, like [sqrt]2. In other words, the magnitude measured may be
incommensurable with the unit of measure, a seeming paradox. With this
difficulty, however, the pupil should not be called upon to contend at
this stage in his progress. The whole subject of incommensurables might
safely be postponed, although it may be treated in an elementary fashion
at this time. The fact that the measure of the diagonal of a square, of
which a side is unity, is [sqrt]2, and that this measure is an
incommensurable number, is not so paradoxical as it seems, the paradox
being verbal rather than actual.
It is then customary to define ratio as the quotient of the numerical
measures of two quantities in terms of a common unit. This brings all
ratios to the basis of numerical fractions, and while it is not
scientifically so satisfactory as the ancient concept which considered
the terms as lines, surfaces, angles, or solids, it is more practical,
and it suffices for the needs of elementary pupils.
"Commensurable," "incommensurable," "constant," and "variable" are then
defined, and these definitions are followed by a brief discussion of
limit. It simplifies the treatment of this subject to state at once that
there are two classes of limits,--those which the variable actually
reaches, and those which it can only approach indefinitely near. We find
the one as frequently as we find the other, although it is the latter
that is referred to in geometry. For example, the superior limit of a
chord is a diameter, and this limit the chord may reach. The inferior
limit is zero, but we do not consider the chord as reaching this limit.
It is also well to call the attention of pupils to the fact that a
quantity may decrease towards its limit as well as increase towards it.
Such further definitions as are needed in the theory of limits are now
introduced. Among these is "area of a circle." It might occur to some
pupil that since a circle is a line (as used in modern mathematics), it
can have no area. This is, however, a mere quibble over words. It is not
pretended that the line has area, but that "area of a circle" is merely
a shortened form of the expression "area inclosed by a circle."
The Principle of Limits is now usually given as follows: "If, while
approaching their respective limits, two variables are always equal,
their limits are equal." This was expressed by D'Alembert in the
eighteenth century as "Magnitudes which are the limits of equal
magnitudes are equal," or this in substance. It would easily be possible
to elaborate this theory, proving, for example, that if _x_ approaches
_y_ as its limit, then _ax_ approaches _ay_ as its limit, and _x/a_
approaches _y/a_ as its limit, and so on. Very much of this theory,
however, wearies a pupil so that the entire meaning of the subject is
lost, and at best the treatment in elementary geometry is not rigorous.
It is another case of having to sacrifice a strictly scientific
treatment to the educational abilities of the pupil. Teachers wishing to
find a scientific treatment of the subject should consult a good work on
the calculus.
THEOREM. _In the same circle or in equal circles two central angles have
the same ratio as their intercepted arcs._
This is usually proved first for the commensurable case and then for the
incommensurable one. The latter is rarely understood by all of the
class, and it may very properly be required only of those who show some
aptitude in geometry. It is better to have the others understand fully
the commensurable case and see the nature of its applications, possibly
reading the incommensurable proof with the teacher, than to stumble
about in the darkness of the incommensurable case and never reach the
goal. In Euclid there was no distinction between the two because his
definition of ratio covered both; but, as we shall see in Book III, this
definition is too difficult for our pupils. Theon of Alexandria (fourth
century A.D.), the father of the Hypatia who is the heroine of
Kingsley's well-known novel, wrote a commentary on Euclid, and he adds
that sectors also have the same ratio as the arcs, a fact very easily
proved. In propositions of this type, referring to the same circle or to
equal circles, it is not worth while to ask pupils to take up both
cases, the proof for either being obviously a proof for the other.
Many writers state this proposition so that it reads that "central
angles are _measured by_ their intercepted arcs." This, of course, is
not literally true, since we can measure anything only by some thing, of
the same kind. Thus we measure a volume by finding how many times it
contains another volume which we take as a unit, and we measure a length
by taking some other length as a unit; but we cannot measure a given
length in quarts nor a given weight in feet, and it is equally
impossible to measure an arc by an angle, and vice versa. Nevertheless
it is often found convenient to _define_ some brief expression that has
no meaning if taken literally, in such way that it shall acquire a
meaning. Thus we _define_ "area of a circle," even when we use "circle"
to mean a line; and so we may define the expression "central angles are
measured by their intercepted arcs" to mean that central angles have the
same numerical measure as these arcs. This is done by most writers, and
is legitimate as explaining an abbreviated expression.
THEOREM. _An inscribed angle is measured by half the intercepted arc._
In Euclid this proposition is combined with the preceding one in his
Book VI, Proposition 33. Such a procedure is not adapted to the needs of
students to-day. Euclid gave in Book III, however, the proposition (No.
20) that a central angle is twice an inscribed angle standing on the
same arc. Since Euclid never considered an angle greater than 180°, his
inscribed angle was necessarily less than a right angle. The first one
who is known to have given the general case, taking the central angle as
being also greater than 180°, was Heron of Alexandria, probably of the
first century A.D.[68] In this he was followed by various later
commentators, including Tartaglia and Clavius in the sixteenth century.
One of the many interesting exercises that may be derived from this
theorem is seen in the case of the "horizontal danger angle" observed by
ships.
[Illustration]
If some dangerous rocks lie off the shore, and _L_ and _L'_ are
two lighthouses, the angle _A_ is determined by observation, so
that _A_ will lie on a circle inclosing the dangerous area.
Angle _A_ is called the "horizontal danger angle." Ships
passing in sight of the two lighthouses _L_ and _L'_ must keep
out far enough so that the angle _L'SL_ shall be less than
angle _A_.
To this proposition there are several important corollaries, including
the following:
1. _An angle inscribed in a semicircle is a right angle._ This corollary
is mentioned by Aristotle and is attributed to Thales, being one of the
few propositions with which his name is connected. It enables us to
describe a circle by letting the arms of a carpenter's square slide
along two nails driven in a board, a pencil being held at the vertex.
[Illustration]
A more practical use for it is made by machinists to determine whether a
casting is a true semicircle. Taking a carpenter's square as here shown,
if the vertex touches the curve at every point as the square slides
around, it is a true semicircle. By a similar method a circle may be
described by sliding a draftsman's triangle so that two sides touch two
tacks driven in a board.
[Illustration]
Another interesting application of this corollary may be seen
by taking an ordinary paper protractor _ACB_, and fastening a
plumb line at _B_. If the protractor is so held that the plumb
line cuts the semicircle at _C_, then _AC_ is level because it
is perpendicular to the vertical line _BC_. Thus, if a class
wishes to determine the horizontal line _AC_, while sighting up
a hill in the direction _AB_, this is easily determined without
a spirit level.
It follows from this corollary, as the pupil has already found, that the
mid-point of the hypotenuse of a right triangle is equidistant from the
three vertices. This is useful in outdoor measuring, forming the basis
of one of the best methods of letting fall a perpendicular from an
external point to a line.
[Illustration]
Suppose _XY_ to be the edge of a sidewalk, and _P_ a point in
the street from which we wish to lay a gas pipe perpendicular
to the walk. From _P_ swing a cord or tape, say 60 feet long,
until it meets _XY_ at _A_. Then take _M_, the mid-point of
_PA_, and swing _MP_ about _M_, to meet _XY_ at _B_. Then _B_
is the foot of the perpendicular, since [L]_PBA_ can be
inscribed in a semicircle.
2. _Angles inscribed in the same segment are equal._
[Illustration]
By driving two nails in a board, at _A_ and _B_, and taking an
angle _P_ made of rigid material (in particular, as already
stated, a carpenter's square), a pencil placed at _P_ will
generate an arc of a circle if the arms slide along _A_ and
_B_. This is an interesting exercise for pupils.
THEOREM. _An angle formed by two chords intersecting within the circle
is measured by half the sum of the intercepted arcs._
THEOREM. _An angle formed by a tangent and a chord drawn from the point
of tangency is measured by half the intercepted arc._
THEOREM. _An angle formed by two secants, a secant and a tangent, or two
tangents, drawn to a circle from an external point, is measured by half
the difference of the intercepted arcs._
These three theorems are all special cases of the general proposition
that the angle included between two lines that cut (or touch) a circle
is measured by half the sum of the intercepted arcs. If the point passes
from within the circle to the circle itself, one arc becomes zero and
the angle becomes an inscribed angle. If the point passes outside the
circle, the smaller arc becomes negative, having passed through zero.
The point may even "go to infinity," as is said in higher mathematics,
the lines then becoming parallel, and the angle becoming zero, being
measured by half the sum of one arc and a negative arc of the same
absolute value. This is one of the best illustrations of the Principle
of Continuity to be found in geometry.
PROBLEM. _To let fall a perpendicular upon a given line from a given
external point._
This is the first problem that a student meets in most American
geometries. The reason for treating the problems by themselves instead
of mingling them with the theorems has already been discussed.[69] The
student now has a sufficient body of theorems, by which he can prove
that his constructions are correct, and the advantage of treating these
constructions together is greater than that of following Euclid's plan
of introducing them whenever needed.
Proclus tells us that "this problem was first investigated by
Oenopides,[70] who thought it useful for astronomy." Proclus speaks of
such a line as a gnomon, a common name for the perpendicular on a
sundial, which casts the shadow by which the time of day is known. He
also speaks of two kinds of perpendiculars, the plane and solid, the
former being a line perpendicular to a line, and the latter a line
perpendicular to a plane.
It is interesting to notice that the solution tacitly assumes that a
certain arc is going to cut the given line in two points, and only two.
Strictly speaking, why may it not cut it in only one point, or even in
three points? We really assume that if a straight line is drawn through
a point within a circle, this line must get out of the circle on each
of two sides of the given point, and in getting out it must cut the
circle twice. Proclus noticed this assumption and endeavored to prove
it. It is better, however, not to raise the question with beginners,
since it seems to them like hair-splitting to no purpose.
The problem is of much value in surveying, and teachers would do well to
ask a class to let fall a perpendicular to the edge of a sidewalk from a
point 20 feet from the walk, using an ordinary 66-foot or 50-foot tape.
Practically, the best plan is to swing 30 feet of the tape about the
point and mark the two points of intersection with the edge of the walk.
Then measure the distance between the points and take half of this
distance, thus fixing the foot of the perpendicular.
PROBLEM. _At a given point in a line, to erect a perpendicular to that
line._
This might be postponed until after the problem to bisect an angle,
since it merely requires the bisection of a straight angle; but
considering the immaturity of the average pupil, it is better given
independently. The usual case considers the point not at the extremity
of the line, and the solution is essentially that of Euclid. In
practice, however, as for example in surveying, the point may be at the
extremity, and it may not be convenient to produce the line.
[Illustration]
Surveyors sometimes measure _PB_ = 3 ft., and then take 9 ft.
of tape, the ends being held at _B_ and _P_, and the tape being
stretched to _A_, so that _PA_ = 4 ft. and _AB_ = 5 ft. Then
_P_ is a right angle by the Pythagorean Theorem. This theorem
not having yet been proved, it cannot be used at this time.
A solution for the problem of erecting a perpendicular from the
extremity of a line that cannot be produced, depending, however, on the
problem of bisecting an angle, and therefore to be given after that
problem, is attributed by Al-Nair[=i]z[=i] (tenth century A.D.) to Heron
of Alexandria. It is also given by Proclus.
[Illustration]
Required to draw from _P_ a perpendicular to _AP_. Take _X_
anywhere on the line and erect _XY_ [perp] to _AP_ in the usual
manner. Bisect [L]_PXY_ by the line _XM_. On _XY_ take _XN_ = _XP_,
and draw _NM_ [perp] to _XY_. Then draw _PM_. The proof
is evident.
These may at the proper time be given as interesting variants of the
usual solution.
PROBLEM. _To bisect a given line._
Euclid said "finite straight line," but this wording is not commonly
followed, because it will be inferred that the line is finite if it is
to be bisected, and we use "line" alone to mean a straight line.
Euclid's plan was to construct an equilateral triangle (by his
Proposition 1 of Book I) on the line as a base, and then to bisect the
vertical angle. Proclus tells us that Apollonius of Perga, who wrote the
first great work on conic sections, used a plan which is substantially
that which is commonly found in textbooks to-day,--constructing two
isosceles triangles upon the line as a common base, and connecting their
vertices.
PROBLEM. _To bisect a given angle._
It should be noticed that in the usual solution two arcs intersect, and
the point thus determined is connected with the vertex. Now these two
arcs intersect twice, and since one of the points of intersection may be
the vertex itself, the other point of intersection must be taken. It is
not, however, worth while to make much of this matter with pupils.
Proclus calls attention to the possible suggestion that the point of
intersection may be imagined to lie outside the angle, and he proceeds
to show the absurdity; but here, again, the subject is not one of value
to beginners. He also contributes to the history of the trisection of an
angle. Any angle is easily trisected by means of certain higher curves,
such as the conchoid of Nicomedes (_ca._ 180 B.C.), the quadratrix of
Hippias of Elis (_ca._ 420 B.C.), or the spiral of Archimedes (_ca._ 250
B.C.). But since this problem, stated algebraically, requires the
solution of a cubic equation, and this involves, geometrically, finding
three points, we cannot solve the problem by means of straight lines and
circles alone. In other words, the trisection of _any_ angle, by the use
of the straightedge and compasses alone, is impossible. Special angles
may however be trisected. Thus, to trisect an angle of 90° we need only
to construct an angle of 60°, and this can be done by constructing an
equilateral triangle. But while we cannot trisect the angle, we may
easily approximate trisection. For since, in the infinite geometric
series 1/2 + 1/8 + 1/32 + 1/128 + ..., _s_ = _a_ ÷ (1 - _r_), we have
_s_ = 1/2 ÷ 3/4 = 2/3. In other words, if we add 1/2 of the angle, 1/8
of the angle, 1/32 of the angle, and so on, we approach as a limit 2/3
of the angle; but all of these fractions can be obtained by repeated
bisections, and hence by bisections we may approximate the trisection.
The approximate bisection (or any other division) of an angle may of
course be effected by the help of the protractor and a straightedge. The
geometric method is, however, usually more accurate, and it is
advantageous to have the pupils try both plans, say for bisecting an
angle of about 49 1/2°.
[Illustration]
Applications of this problem are numerous. It may be desired, for
example, to set a lamp-post on a line bisecting the angle formed by two
streets that come together a little unsymmetrically, as here shown, in
which case the bisecting line can easily be run by the use of a
measuring tape, or even of a stout cord.
A more interesting illustration is, however, the following:
[Illustration]
Let the pupils set a stake, say about 5 feet high, at a point
_N_ on the school grounds about 9 A.M., and carefully measure
the length of the shadow, _NW_, placing a small wooden pin at
_W_. Then about 3 P.M. let them watch until the shadow _NE_ is
exactly the same length that it was when _W_ was fixed, and
then place a small wooden pin at _E_. If the work has been very
carefully done, and they take the tape and bisect the line
_WE_, thus fixing the line _NS_, they will have a north and
south line. If this is marked out for a short distance from
_N_, then when the shadow falls on _NS_, it will be noon by sun
time (not standard time) at the school.
PROBLEM. _From a given point in a given line, to draw a line making an
angle equal to a given angle._
Proclus says that Eudemus attributed to Oenopides the discovery of the
solution which Euclid gave, and which is substantially the one now
commonly seen in textbooks. The problem was probably solved in some
fashion before the time of Oenopides, however. The object of the problem
is primarily to enable us to draw a line parallel to a given line.
Practically, the drawing of one line parallel to another is usually
effected by means of a parallel ruler (see page 191), or by the use of
draftsmen's triangles, as here shown, or even more commonly by the use
of a T-square, such as is here seen. This illustration shows two
T-squares used for drawing lines parallel to the sides of a board upon
which the drawing paper is fastened.[71]
[Illustration]
[Illustration]
An ingenious instrument described by Baron Dupin is illustrated below.
[Illustration]
To the bar _A_ is fastened the sliding check _B_. A movable
check _D_ may be fastened by a screw _C_. A sharp point is
fixed in _B_, so that as _D_ slides along the edge of a board,
the point marks a line parallel to the edge. Moreover, _F_ and
_G_ are two brass arms of equal length joined by a pointed
screw _H_ that marks a line midway between _B_ and _D_.
Furthermore, it is evident that _H_ will draw a line bisecting
any irregular board if the checks _B_ and _D_ are kept in
contact with the irregular edges.
Book II offers two general lines of application that may be introduced
to advantage, preferably as additions to the textbook work. One of these
has reference to topographical drawing and related subjects, and the
other to geometric design. As long as these can be introduced to the
pupil with an air of reality, they serve a good purpose, but if made a
part of textbook work, they soon come to have less interest than the
exercises of a more abstract character. If a teacher can relate the
problems in topographical drawing to the pupil's home town, and can
occasionally set some outdoor work of the nature here suggested, the
results are usually salutary; but if he reiterates only a half-dozen
simple propositions time after time, with only slight changes in the
nature of the application, then the results will not lead to a
cultivation of power in geometry,--a point which the writers on applied
geometry usually fail to recognize.
[Illustration]
One of the simple applications of this book relates to the rounding of
corners in laying out streets in some of our modern towns where there is
a desire to depart from the conventional square corner. It is also used
in laying out park walks and drives.
[Illustration]
The figure in the middle of the page represents two streets,
_AP_ and _BQ_, that would, if prolonged, intersect at _C_. It
is required to construct an arc so that they shall begin to
curve at _P_ and _Q_, where _CP_ = _CQ_, and hence the "center
of curvature" _O_ must be found.
The problem is a common one in railroad work, only here _AP_ is
usually oblique to _BQ_ if they are produced to meet at _C_, as
in the second figure on page 218. It is required to construct
an arc so that the tracks shall begin to curve at _P_ and _Q_,
where _CP_ = _CQ_.
[Illustration]
The problem becomes a little more complicated, and correspondingly more
interesting, when we have to find the center of curvature for a street
railway track that must turn a corner in such a way as to allow, say,
exactly 5 feet from the point _P_, on account of a sidewalk.
[Illustration]
The problem becomes still more difficult if we have two roads of
different widths that we wish to join on a curve. Here the two centers
of curvature are not the same, and the one road narrows to the other on
the curve. The solutions will be understood from a study of the figures.
The number of problems of this kind that can easily be made is
limitless, and it is well to avoid the danger of hobby riding on this
or any similar topic. Therefore a single one will suffice to close this
group.
[Illustration]
If a road _AB_ on an arc described about _O_, is to be joined
to road _CD_, described about _O'_, the arc _BC_ should
evidently be internally tangent to _AB_ and externally tangent
to _CD_. Hence the center is on _BOX_ and _O'CY_, and is
therefore at _P_. The problem becomes more real if we give some
width to the roads in making the drawing, and imagine them in a
park that is being laid out with drives.
It will be noticed that the above problems require the erecting of
perpendiculars, the bisecting of angles, and the application of the
propositions on tangents.
A somewhat different line of problems is that relating to the passing of
a circle through three given points. It is very easy to manufacture
problems of this kind that have a semblance of reality.
[Illustration]
For example, let it be required to plan a driveway from the
gate _G_ to the porch _P_ so as to avoid a mass of rocks _R_,
an arc of a circle to be taken. Of course, if we allow pupils
to use the Pythagorean Theorem at this time (and for metrical
purposes this is entirely proper, because they have long been
familiar with it), then we may ask not only for the drawing,
but we may, for example, give the length from _G_ to the point
on _R_ (which we may also call _R_), and the angle _RGO_ as
60°, to find the radius.
A second general line of exercises adapted to Book II is a continuation
of the geometric drawing recommended as a preliminary to the work in
demonstrative geometry. The copying or the making of designs requiring
the describing of circles, their inscription in or circumscription about
triangles, and their construction in various positions of tangency, has
some value as applying the various problems studied in this book. For a
number of years past, several enthusiastic teachers have made much of
the designs found in Gothic windows, having their pupils make the
outline drawings by the help of compasses and straightedge. While such
work has its value, it is liable soon to degenerate into purposeless
formalism, and hence to lose interest by taking the vigorous mind of
youth from the strong study of geometry to the weak manipulation of
instruments. Nevertheless its value should be appreciated and conserved,
and a few illustrations of these forms are given in order that the
teacher may have examples from which to select. The best way of using
this material is to offer it as supplementary work, using much or
little, as may seem best, thus giving to it a freshness and interest
that some have trouble in imparting to the regular book work.
The best plan is to sketch rapidly the outline of a window on the
blackboard, asking the pupils to make a rough drawing, and to bring in a
mathematical drawing on the following day.
[Illustration]
It might be said, for example, that in planning a Gothic window
this drawing is needed. The arc _BC_ is drawn with _A_ as a
center and _AB_ as a radius. The small arches are described
with _A_, _D_, and _B_ as centers and _AD_ as a radius. The
center _P_ is found by taking _A_ and _B_ as centers and _AE_ as
a radius. How may the points _D_, _E_, and _F_ be found? Draw
the figure. From the study of the rectilinear figures suggested
by such a simple pattern the properties of the equilateral
triangle may be inferred.
The Gothic window also offers some interesting possibilities in
connection with the study of the square. For example, the illustration
given on page 223 shows a number of traceries involving the construction
of a square, the bisecting of angles, and the describing of circles.[72]
[Illustration]
The properties of the square, a figure now easily constructed by the
pupils, are not numerous. What few there are may be brought out through
the study of art forms, if desired. In case these forms are shown to a
class, it is important that they should be selected from good designs.
We have enough poor art in the world, so that geometry should not
contribute any more. This illustration is a type of the best medieval
Gothic parquetry.[73]
[Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]
Even simple designs of a semipuzzling nature have their advantage in
this connection. In the following example the inner square contains all
of the triangles, the letters showing where they may be fitted.[74]
Still more elaborate designs, based chiefly upon the square and circle,
are shown in the window traceries on page 225, and others will be given
in connection with the study of the regular polygons.
[Illustration]
Designs like the figure below are typical of the simple forms, based on
the square and circle, that pupils may profitably incorporate in any
work in art design that they may be doing at the time they are studying
the circle and the problems relating to perpendiculars and squares.
[Illustration]
Among the applications of the problem to draw a tangent to a given
circle is the case of the common tangents to two given circles. Some
authors give this as a basal problem, although it is more commonly given
as an exercise or a corollary. One of the most obvious applications of
the idea is that relating to the transmission of circular motion by
means of a band over two wheels,[75] _A_ and _B_, as shown on page 226.
[Illustration: GOTHIC DESIGNS EMPLOYING CIRCLES AND BISECTED ANGLES]
The band may either not be crossed (the case of the two exterior
tangents), or be crossed (the interior tangents), the latter allowing
the wheels to turn in opposite directions. In case the band is liable to
change its length, on account of stretching or variation in heat or
moisture, a third wheel, _D_, is used. We then have the case of tangents
to three pairs of circles. Illustrations of this nature make the
exercise on the drawing of common tangents to two circles assume an
appearance of genuine reality that is of advantage to the work.
[Illustration]
FOOTNOTES:
[68] This is the latest opinion. He is usually assigned to the first
century B.C.
[69] See page 54.
[70] A Greek philosopher and mathematician of the fifth century B.C.
[71] This illustration and the following two are from C. Dupin,
"Mathematics Practically Applied," translated from the French by G.
Birkbeck, Halifax, 1854. This is probably the most scholarly attempt
ever made at constructing a "practical geometry."
[72] This illustration and others of the same type used in this work are
from the excellent drawings by R. W. Billings, in "The Infinity of
Geometric Design Exemplified," London, 1849.
[73] From H. Kolb, "Der Ornamentenschatz ... aus allen Kunst-Epochen,"
Stuttgart, 1883. The original is in the Church of Saint Anastasia in
Verona.
[74] From J. Bennett, "The Arcanum ... A Concise Theory of Practicable
Geometry," London, 1838, one of the many books that have assumed to
revolutionize geometry by making it practical.
[75] The figures are from Dupin, loc. cit.
CHAPTER XVI
THE LEADING PROPOSITIONS OF BOOK III
In the American textbooks Book III is usually assigned to proportion. It
is therefore necessary at the beginning of this discussion to consider
what is meant by ratio and proportion, and to compare the ancient and
the modern theories. The subject is treated by Euclid in his Book V, and
an anonymous commentator has told us that it "is the discovery of
Eudoxus, the teacher of Plato." Now proportion had been known long
before the time of Eudoxus (408-355 B.C.), but it was numerical
proportion, and as such it had been studied by the Pythagoreans. They
were also the first to study seriously the incommensurable number, and
with this study the treatment of proportion from the standpoint of
rational numbers lost its scientific position with respect to geometry.
It was because of this that Eudoxus worked out a theory of geometric
proportion that was independent of number as an expression of ratio.
The following four definitions from Euclid are the basal ones of the
ancient theory:
A ratio is a sort of relation in respect of size between two
magnitudes of the same kind.
Magnitudes are said to have a ratio to one another which are
capable, when multiplied, of exceeding one another.
Magnitudes are said to be in the same ratio, the first to the
second and the third to the fourth, when, if any equimultiples
whatever be taken of the first and third, and any
equimultiples whatever of the second and fourth, the former
equimultiples alike exceed, are alike equal to, or alike fall
short of, the latter equimultiples respectively taken in
corresponding order.
Let magnitudes which have the same ratio be called
proportional.[76]
Of these, the first is so loose in statement as often to have been
thought to be an interpolation of some later writer. It was probably,
however, put into the original for the sake of completeness, to have
some kind of statement concerning ratio as a preliminary to the
important definition of quantities in the same ratio. Like the
definition of "straight line," it was not intended to be taken seriously
as a mathematical statement.
The second definition is intended to exclude zero and infinite
magnitudes, and to show that incommensurable magnitudes are included.
The third definition is the essential one of the ancient theory. It
defines what is meant by saying that magnitudes are in the same ratio;
in other words, it defines a proportion. Into the merits of the
definition it is not proposed to enter, for the reason that it is no
longer met in teaching in America, and is practically abandoned even
where the rest of Euclid's work is in use. It should be said, however,
that it is scientifically correct, that it covers the case of
incommensurable magnitudes as well as that of commensurable ones, and
that it is the Greek forerunner of the modern theories of irrational
numbers.
As compared with the above treatment, the one now given in textbooks is
unscientific. We define ratio as "the quotient of the numerical measures
of two quantities of the same kind," and proportion as "an equality of
ratios."
But what do we mean by the quotient, say of [sqrt]2 by [sqrt]3? And when
we multiply a ratio by [sqrt]5, what is the meaning of this operation?
If we say that [sqrt]2 : [sqrt]3 means a quotient, what meaning shall we
assign to "quotient"? If it is the number that shows how many times one
number is contained in another, how many _times_ is [sqrt]3 contained in
[sqrt]2? If to multiply is to take a number a certain number of times,
how many times do we take it when we multiply by [sqrt]5? We certainly
take it more than 2 times and less than 3 times, but what meaning can we
assign to [sqrt]5 times? It will thus be seen that our treatment of
proportion assumes that we already know the theory of irrationals and
can apply it to geometric magnitudes, while the ancient treatment is
independent of this theory.
Educationally, however, we are forced to proceed as we do. Just as
Dedekind's theory of numbers is a simple one for college students, so is
the ancient theory of proportion; but as the former is not suited to
pupils in the high school, so the latter must be relegated to the
college classes. And in this we merely harmonize educational progress
with world progress, for the numerical theory of proportion long
preceded the theory of Eudoxus.
The ancients made much of such terms as duplicate, triplicate,
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° | .087 | .996 | .087 || 50° | .766 | .643 | 1.192
------+------+------+-------++-------+------+------+--------
10° | .174 | .985 | .176 || 55° | .819 | .574 | 1.428
------+------+------+-------++-------+------+------+--------
15° | .259 | .966 | .268 || 60° | .866 | .500 | 1.732
------+------+------+-------++-------+------+------+--------
20° | .342 | .940 | .364 || 65° | .906 | .423 | 2.145
------+------+------+-------++-------+------+------+--------
25° | .423 | .906 | .466 || 70° | .940 | .342 | 2.748
------+------+------+-------++-------+------+------+--------
30° | .500 | .866 | .577 || 75° | .966 | .259 | 3.732
------+------+------+-------++-------+------+------+--------
35° | .574 | .819 | .700 || 80° | .985 | .174 | 5.671
------+------+------+-------++-------+------+------+--------
40° | .643 | .766 | .839 || 85° | .996 | .087 |11.430
------+------+------+-------++-------+------+------+--------
45° | .707 | .707 | 1.000 || 90° | 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° is
0.99619, and the sine of 85° 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°, then since _y_ = _r_ sin _O_, we have
_y_ = 30 × 0.643 = 19.29. If we know that _x_ = 60 and
[L]_O_ = 35°, then since _y_ = _x_ tan _O_, we have
_y_ = 60 × 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 "Traité de la construction ... des instrumens de
mathématique," 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," Nürnberg, 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 × BC will equal CD × 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_ × _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 æsthetic 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 Athenæus (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_ × _AP_ = (_AC_)^2.
Similarly, _AB_ × _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°, until it comes into the position
_A'BC'_. Then because it has been turned through 90°, _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°, 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," §
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° 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° = 60°),
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° and the vertical angle 36°, 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 Hêng (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 Méray 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 × 2 × 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._
By changing "plane" to "line," and by making other obvious changes to
correspond, this reduces to the analogous proposition of plane geometry.
The figure formed by the plane perpendicular to the edge is also the
figure of that analogous proposition. This at once suggests that there
are two planes in the locus, provided the planes of the dihedral angle
are taken as indefinite in extent, and that these planes are
perpendicular to each other. It may interest some of the pupils to draw
this general figure, analogous to the one in plane geometry.
THEOREM. _The projection of a straight line not perpendicular to a plane
upon that plane is a straight line._
In higher mathematics it would simply be said that the projection is a
straight line, the special case of the projection of a perpendicular
being considered as a line-segment of zero length. There is no
advantage, however, of bringing in zero and infinity in the course in
elementary geometry. The legitimate reason for the modern use of these
terms is seldom understood by beginners.
This subject of projection (Latin _pro-_, "forth," and _jacere_, "to
throw") is extensively used in modern mathematics and also in the
elementary work of the draftsman, and it will be referred to a little
later. At this time, however, it is well to call attention to the fact
that the projection of a straight line on a plane is a straight line or
a point; the projection of a curve may be a curve or it may be straight;
the projection of a point is a point; and the projection of a plane
(which is easily understood without defining it) may be a surface or it
may be a straight line. An artisan represents a solid by drawing its
projection upon two planes at right angles to each other, and a map
maker (cartographer) represents the surface of the earth by projecting
it upon a plane. A photograph of the class is merely the projection of
the class upon a photographic plate (plane), and when we draw a figure
in solid geometry, we merely project the solid upon the plane of the
paper.
There are other projections than those formed by lines that are
perpendicular to the plane. The lines may be oblique to the plane, and
this is the case with most projections. A photograph, for example, is
not formed by lines perpendicular to a plane, for they all converge in
the camera. If the lines of projection are all perpendicular to the
plane, the projection is said to be orthographic, from the Greek
_ortho-_ (straight) and _graphein_ (to draw). A good example of
orthographic projection may be seen in the shadow cast by an object upon
a piece of paper that is held perpendicular to the sun's rays. A good
example of oblique projection is a shadow on the floor of the
schoolroom.
THEOREM. _Between two straight lines not in the same plane there can be
one common perpendicular, and only one._
The usual corollary states that this perpendicular is the shortest line
joining them. It is interesting to compare this with the case of two
lines in the same plane. If they are parallel, there may be any number
of common perpendiculars. If they intersect, there is still a common
perpendicular, but this can hardly be said to be between them, except
for its zero segment.
There are many simple illustrations of this case. For example, what is
the shortest line between any given edge of the ceiling and the various
edges of the floor of the schoolroom? If two galleries in a mine are to
be connected by an air shaft, how shall it be planned so as to save
labor? Make a drawing of the plan.
At this point the polyhedral angle is introduced. The word is from the
Greek _polys_ (many) and _hedra_ (seat). Students have more difficulty
in grasping the meaning of the size of a polyhedral angle than is the
case with dihedral and plane angles. For this reason it is not good
policy to dwell much upon this subject unless the question arises, since
it is better understood when the relation of the polyhedral angle and
the spherical polygon is met. Teachers will naturally see that just as
we may measure the plane angle by taking the ratio of an arc to the
whole circle, and of a dihedral angle by taking the ratio of that part
of the cylindric surface that is cut out by the planes to the whole
surface, so we may measure a polyhedral angle by taking the ratio of the
spherical polygon to the whole spherical surface. It should also be
observed that just as we may have cross polygons in a plane, so we may
have spherical polygons that are similarly tangled, and that to these
will correspond polyhedral angles that are also cross, their
representation by drawings being too complicated for class use.
The idea of symmetric solids may be illustrated by a pair of gloves, all
their parts being mutually equal but arranged in opposite order. Our
hands, feet, and ears afford other illustrations of symmetric solids.
THEOREM. _The sum of the face angles of any convex polyhedral angle is
less than four right angles._
There are several interesting points of discussion in connection with
this proposition. For example, suppose the vertex _V_ to approach the
plane that cuts the edges in _A_, _B_, _C_, _D_, ..., the edges
continuing to pass through these as fixed points. The sum of the angles
about _V_ approaches what limit? On the other hand, suppose _V_ recedes
indefinitely; then the sum approaches what limit? Then what are the two
limits of this sum? Suppose the polyhedral angle were concave, why would
the proof not hold?
FOOTNOTES:
[87] These may be purchased through the Leipziger Lehrmittelanstalt,
Leipzig, Germany, which will send catalogues to intending buyers.
[88] An excellent set of stereoscopic views of the figures of solid
geometry, prepared by E. M. Langley of Bedford, England, is published by
Underwood & Underwood, New York. Such a set may properly have place in a
school library or in a classroom in geometry, to be used when it seems
advantageous.
CHAPTER XX
THE LEADING PROPOSITIONS OF BOOK VII
Book VII relates to polyhedrons, cylinders, and cones. It opens with the
necessary definitions relating to polyhedrons, the etymology of the
terms often proving interesting and valuable when brought into the work
incidentally by the teacher. "Polyhedron" is from the Greek _polys_
(many) and _hedra_ (seat). The Greek plural, _polyhedra_, is used in
early English works, but "polyhedrons" is the form now more commonly
seen in America. "Prism" is from the Greek _prisma_ (something sawed,
like a piece of wood sawed from a beam). "Lateral" is from the Latin
_latus_ (side). "Parallelepiped" is from the Greek _parallelos_
(parallel) and _epipedon_ (a plane surface), from _epi_ (on) and _pedon_
(ground). By analogy to "parallelogram" the word is often spelled
"parallelopiped," but the best mathematical works now adopt the
etymological spelling above given. "Truncate" is from the Latin
_truncare_ (to cut off).
A few of the leading propositions are now considered.
THEOREM. _The lateral area of a prism is equal to the product of a
lateral edge by the perimeter of the right section._
It should be noted that although some syllabi do not give the
proposition that parallel sections are congruent, this is necessary for
this proposition, because it shows that the right sections are all
congruent and hence that any one of them may be taken.
It is, of course, possible to construct a prism so oblique and so low
that a right section, that is, a section cutting all the lateral edges
at right angles, is impossible. In this case the lateral faces must be
extended, thus forming what is called a _prismatic space_. This term may
or may not be introduced, depending upon the nature of the class.
This proposition is one of the most important in Book VII, because it is
the basis of the mensuration of the cylinder as well as the prism.
Practical applications are easily suggested in connection with beams,
corridors, and prismatic columns, such as are often seen in school
buildings. Most geometries supply sufficient material in this line,
however.
THEOREM. _An oblique prism is equivalent to a right prism whose base is
equal to a right section of the oblique prism, and whose altitude is
equal to a lateral edge of the oblique prism._
This is a fundamental theorem leading up to the mensuration of the
prism. Attention should be called to the analogous proposition in plane
geometry relating to the area of the parallelogram and rectangle, and to
the fact that if we cut through the solid figure by a plane parallel to
one of the lateral edges, the resulting figure will be that of the
proposition mentioned. As in the preceding proposition, so in this case,
there may be a question raised that will make it helpful to introduce
the idea of prismatic space.
THEOREM. _The opposite lateral faces of a parallelepiped are congruent
and parallel._
It is desirable to refer to the corresponding case in plane geometry,
and to note again that the figure is obtained by passing a plane
through the parallelepiped parallel to a lateral edge. The same may be
said for the proposition about the diagonal plane of a parallelepiped.
These two propositions are fundamental in the mensuration of the prism.
THEOREM. _Two rectangular parallelepipeds are to each other as the
products of their three dimensions._
This leads at once to the corollary that the volume of a rectangular
parallelepiped equals the product of its three dimensions, the
fundamental law in the mensuration of all solids. It is preceded by the
proposition asserting that rectangular parallelepipeds having congruent
bases are proportional to their altitudes. This includes the
incommensurable case, but this case may be omitted.
The number of simple applications of this proposition is practically
unlimited. In all such cases it is advisable to take a considerable
number of numerical exercises in order to fix in mind the real nature of
the proposition. Any good geometry furnishes a certain number of these
exercises.
The following is an interesting property of the rectangular
parallelepiped, often called the rectangular solid:
If the edges are _a_, _b_, and _c_, and the diagonal is _d_,
then (_a_/_d_)^2 + (_b_/_d_)^2 + (_c_/_d_)^2 = 1. This property
is easily proved by the Pythagorean Theorem, for
_d_^2 = _a_^2 + _b_^2 + _c_^2, whence
(_a_^2 + _b_^2 + _c_^2) / _d_^2 = 1.
In case _c_ = 0, this reduces to the Pythagorean Theorem. The
property is the fundamental one of solid analytic geometry.
THEOREM. _The volume of any parallelepiped is equal to the product of
its base by its altitude._
This is one of the few propositions in Book VII where a model is of any
advantage. It is easy to make one out of pasteboard, or to cut one from
wood. If a wooden one is made, it is advisable to take an oblique
parallelepiped and, by properly sawing it, to transform it into a
rectangular one instead of using three different solids.
On account of its awkward form, this figure is sometimes called the
Devil's Coffin, but it is a name that it would be well not to
perpetuate.
THEOREM. _The volume of any prism is equal to the product of its base by
its altitude._
This is also one of the basal propositions of solid geometry, and it has
many applications in practical mensuration. A first-class textbook will
give a sufficient list of problems involving numerical measurement, to
fix the law in mind. For outdoor work, involving measurements near the
school or within the knowledge of the pupils, the following problem is a
type:
[Illustration]
If this represents the cross section of a railway embankment
that is _l_ feet long, _h_ feet high, _b_ feet wide at the
bottom, and _b'_ feet wide at the top, find the number of cubic
feet in the embankment. Find the volume if _l_ = 300, _h_ = 8,
_b_ = 60, and _b'_ = 28.
The mensuration of the volume of the prism, including the rectangular
parallelepiped and cube, was known to the ancients. Euclid was not
concerned with practical measurement, so that none of this part of
geometry appears in his "Elements." We find, however, in the papyrus of
Ahmes, directions for the measuring of bins, and the Egyptian builders,
long before his time, must have known the mensuration of the rectangular
parallelepiped. Among the Hindus, long before the Christian era, rules
were known for the construction of altars, and among the Greeks the
problem of constructing a cube with twice the volume of a given cube
(the "duplication of the cube") was attacked by many mathematicians. The
solution of this problem is impossible by elementary geometry.
If _e_ equals the edge of the given cube, then _e_^3 is its
volume and 2_e_^3 is the volume of the required cube. Therefore
the edge of the required cube is _e_[3root]2. Now if _e_ is
given, it is not possible with the straightedge and compasses
to construct a line equal to _e_[3root]2, although it is easy
to construct one equal to _e_[sqrt]2.
The study of the pyramid begins at this point. In practical measurement
we usually meet the regular pyramid. It is, however, a simple matter to
consider the oblique pyramid as well, and in measuring volumes we
sometimes find these forms.
THEOREM. _The lateral area of a regular pyramid is equal to half the
product of its slant height by the perimeter of its base._
This leads to the corollary concerning the lateral area of the frustum
of a regular pyramid. It should be noticed that the regular pyramid may
be considered as a frustum with the upper base zero, and the proposition
as a special case under the corollary. It is also possible, if we
choose, to let the upper base of the frustum pass through the vertex and
cut the lateral edges above that point, although this is too complicated
for most pupils. If this case is considered, it is well to bring in the
general idea of _pyramidal space_, the infinite space bounded on several
sides by the lateral faces, of the pyramid. This pyramidal space is
double, extending on two sides of the vertex.
THEOREM. _If a pyramid is cut by a plane parallel to the base:_
1. _The edges and altitude are divided proportionally._
2. _The section is a polygon similar to the base._
To get the analogous proposition of plane geometry, pass a plane through
the vertex so as to cut the base. We shall then have the sides and
altitude of the triangle divided proportionally, and of course the
section will merely be a line-segment, and therefore it is similar to
the base line.
The cutting plane may pass through the vertex, or it may cut the
pyramidal space above the vertex. In either case the proof is
essentially the same.
THEOREM. _The volume of a triangular pyramid is equal to one third of
the product of its base by its altitude, and this is also true of any
pyramid._
This is stated as two theorems in all textbooks, and properly so. It is
explained to children who are studying arithmetic by means of a hollow
pyramid and a hollow prism of equal base and equal altitude. The pyramid
is filled with sand or grain, and the contents is poured into the prism.
This is repeated, and again repeated, showing that the volume of the
prism is three times the volume of the pyramid. It sometimes varies the
work to show this to a class in geometry.
This proposition was first proved, so Archimedes asserts, by Eudoxus of
Cnidus, famous as an astronomer, geometer, physician, and lawgiver, born
in humble circumstances about 407 B.C. He studied at Athens and in
Egypt, and founded a famous school of geometry at Cyzicus. His discovery
also extended to the volume of the cone, and it was his work that gave
the beginning to the science of stereometry, the mensuration part of
solid geometry.
THEOREM. _The volume of the frustum of any pyramid is equal to the sum
of the volumes of three pyramids whose common altitude is the altitude
of the frustum, and whose bases are the lower base, the upper base,
and the mean proportional between the bases of the frustum._
Attention should be called to the fact that this formula _v_ = 1/3
_a_(_b_ + _b'_ + [sqrt](_bb'_)) applies to the pyramid by letting
_b'_ = 0, to the prism by letting _b_ = _b'_, and also to the
parallelepiped and cube, these being special forms of the prism. This
formula is, therefore, a very general one, relating to all the
polyhedrons that are commonly met in mensuration.
THEOREM. _There cannot be more than five regular convex polyhedrons._
Eudemus of Rhodes, one of the principal pupils of Aristotle, in his
history of geometry of which Proclus preserves some fragments, tells us
that Pythagoras discovered the construction of the "mundane figures,"
meaning the five regular polyhedrons. Iamblichus speaks of the discovery
of the dodecahedron in these words:
As to Hippasus, who was a Pythagorean, they say that he
perished in the sea on account of his impiety, inasmuch as he
boasted that he first divulged the knowledge of the sphere with
the twelve pentagons. Hippasus assumed the glory of the
discovery to himself, whereas everything belongs to Him, for
thus they designate Pythagoras, and do not call Him by name.
Iamblichus here refers to the dodecahedron inscribed in the sphere. The
Pythagoreans looked upon these five solids as fundamental forms in the
structure of the universe. In particular Plato tells us that they
asserted that the four elements of the real world were the tetrahedron,
octahedron, icosahedron, and cube, and Plutarch ascribes this doctrine
to Pythagoras himself. Philolaus, who lived in the fifth century B.C.,
held that the elementary nature of bodies depended on their form. The
tetrahedron was assigned to fire, the octahedron to air, the icosahedron
to water, and the cube to earth, it being asserted that the smallest
constituent part of each of these substances had the form here assigned
to it. Although Eudemus attributes all five to Pythagoras, it is certain
that the tetrahedron, cube, and octahedron were known to the Egyptians,
since they appear in their architectural decorations. These solids were
studied so extensively in the school of Plato that Proclus also speaks
of them as the Platonic bodies, saying that Euclid "proposed to himself
the construction of the so-called Platonic bodies as the final aim of
his arrangement of the 'Elements.'" Aristæus, probably a little older
than Euclid, wrote a book upon these solids.
As an interesting amplification of this proposition, the centers of the
faces (squares) of a cube may be connected, an inscribed octahedron
being thereby formed. Furthermore, if the vertices of the cube are _A_,
_B_, _C_, _D_, _A'_, _B'_, _C'_, _D'_, then by drawing _AC_, _CD'_,
_D'A_, _D'B'_, _B'A_, and _B'C_, a regular tetrahedron will be formed.
Since the construction of the cube is a simple matter, this shows how
three of the five regular solids may be constructed. The actual
construction of the solids is not suited to elementary geometry.[89]
It is not difficult for a class to find the relative areas of the cube
and the inscribed tetrahedron and octahedron. If _s_ is the side of the
cube, these areas are 6_s_^2, (1/2)_s_^2[sqrt]3, and _s_^2[sqrt]3; that
is, the area of the octahedron is twice that of the tetrahedron
inscribed in the cube.
Somewhat related to the preceding paragraph is the fact that the edges
of the five regular solids are incommensurable with the radius of the
circumscribed sphere. This fact seems to have been known to the Greeks,
perhaps to Theætetus (_ca._ 400 B.C.) and Aristæus (_ca._ 300 B.C.),
both of whom wrote on incommensurables.
Just as we may produce the sides of a regular polygon and form a regular
cross polygon or stellar polygon, so we may have stellar polyhedrons.
Kepler, the great astronomer, constructed some of these solids in 1619,
and Poinsot, a French mathematician, carried the constructions so far in
1801 that several of these stellar polyhedrons are known as Poinsot
solids. There is a very extensive literature upon this subject.
The following table may be of some service in assigning problems in
mensuration in connection with the regular polyhedrons, although some of
the formulas are too difficult for beginners to prove. In the table _e_ =
edge of the polyhedron, _r_ = radius of circumscribed sphere, _r'_ =
radius of inscribed sphere, _a_ = total area, _v_ = volume.
==========================================================
NUMBER | | |
OF FACES| 4 | 6 | 8
--------+-----------------+--------------+----------------
_r_ | _e_[sqrt](3/8) |(_e_/2)[sqrt]3| _e_[sqrt](1/2)
| | |
_r'_ | _e_[sqrt](1/24) | _e_/2 | _e_[sqrt](1/6)
| | |
_a_ | _e_^2[sqrt]3 | 6_e_^2 | 2_e_^2[sqrt]3
| | |
_v_ |(_e_^3/12)[sqrt]2| _e_^3 |(_e_^3/3)[sqrt]2
----------------------------------------------------------
========================================================================
NUMBER | |
OF FACES| 12 | 20
--------+----------------------------------+----------------------------
_r_ |(_e_/4)[sqrt]3([sqrt]5 + 1) |_e_[sqrt]((5 + [sqrt]5)/8)
| |
_r'_ |(_e_/2)[sqrt]((25 + 11[sqrt]5)/10)|(_e_[sqrt]3)/12([sqrt]5 + 3)
| |
_a_ |3_e_^2[sqrt](5(5 + 2[sqrt]5)) | (5_e_^2)[sqrt]3
| |
_v_ |((_e_^3)/4)(15 + 7[sqrt]5) |((5_e_^3)/12)([sqrt]5 + 3)
------------------------------------------------------------------------
Some interest is added to the study of polyhedrons by calling attention
to their occurrence in nature, in the form of crystals. The computation
of the surfaces and volumes of these forms offers an opportunity for
applying the rules of mensuration, and the construction of the solids
by paper folding or by the cutting of crayon or some other substance
often arouses a considerable interest. The following are forms of
crystals that are occasionally found:
[Illustration]
They show how the cube is modified by having its corners cut off. A cube
may be inscribed in an octahedron, its vertices being at the centers of
the faces of the octahedron. If we think of the cube as expanding, the
faces of the octahedron will cut off the corners of the cube as seen in
the first figure, leaving the cube as shown in the second figure. If the
corners are cut off still more, we have the third figure.
Similarly, an octahedron may be inscribed in a cube, and by letting it
expand a little, the faces of the cube will cut off the corners of the
octahedron. This is seen in the following figures:
[Illustration]
This is a form that is found in crystals, and the computation of the
surface and volume is an interesting exercise. The quartz crystal, an
hexagonal pyramid on an hexagonal prism, is found in many parts of the
country, or is to be seen in the school museum, and this also forms an
interesting object of study in this connection.
The properties of the cylinder are next studied. The word is from the
Greek _kylindros_, from _kyliein_ (to roll). In ancient mathematics
circular cylinders were the only ones studied, but since some of the
properties are as easily proved for the case of a noncircular directrix,
it is not now customary to limit them in this way. It is convenient to
begin by a study of the cylindric surface, and a piece of paper may be
curved or rolled up to illustrate this concept. If the paper is brought
around so that the edges meet, whatever curve may form a cross section
the surface is said to inclose a _cylindric space_. This concept is
sometimes convenient, but it need be introduced only as necessity for
using it arises. The other definitions concerning the cylinder are so
simple as to require no comment.
The mensuration of the volume of a cylinder depends upon the assumption
that the cylinder is the limit of a certain inscribed or circumscribed
prism as the number of sides of the base is indefinitely increased. It
is possible to give a fairly satisfactory and simple proof of this fact,
but for pupils of the age of beginners in geometry in America it is
better to make the assumption outright. This is one of several cases in
geometry where a proof is less convincing than the assumed statement.
THEOREM. _The lateral area of a circular cylinder is equal to the
product of the perimeter of a right section of the cylinder by an
element._
For practical purposes the cylinder of revolution (right circular
cylinder) is the one most frequently used, and the important formula is
therefore _l_ = 2[pi]_rh_ where _l_ = the lateral area, _r_ = the
radius, and _h_ = the altitude. Applications of this formula are easily
found.
THEOREM. _The volume of a circular cylinder is equal to the product of
its base by its altitude._
Here again the important case is that of the cylinder of revolution,
where _v_ = [pi]_r_^2_h_.
The number of applications of this proposition is, of course, very
great. In architecture and in mechanics the cylinder is constantly seen,
and the mensuration of the surface and the volume is important. A single
illustration of this type of problem will suffice.
A machinist is making a crank pin (a kind of bolt) for an
engine, according to this drawing. He considers it as weighing
the same as three steel cylinders having the diameters and
lengths in inches as here shown, where 7 3/4" means 7 3/4
inches. He has this formula for the weight (_w_) of a steel
cylinder where _d_ is the diameter and _l_ is the length:
_w_ = 0.07[pi]_d_^2_l_. Taking [pi] = 3 1/7, find the weight of
the pin.
The most elaborate study of the cylinder, cone, and sphere (the "three
round bodies") in the Greek literature is that of Archimedes of Syracuse
(on the island of Sicily), who lived in the third century B.C.
Archimedes tells us, however, that Eudoxus (born _ca._ 407 B.C.)
discovered that any cone is one third of a cylinder of the same base and
the same altitude. Tradition says that Archimedes requested that a
sphere and a cylinder be carved upon his tomb, and that this was done.
Cicero relates that he discovered the tomb by means of these symbols.
The tomb now shown to visitors in ancient Syracuse as that of
Archimedes cannot be his, for it bears no such figures, and is not
"outside the gate of Agrigentum," as Cicero describes.
The cone is now introduced. A conic surface is easily illustrated to a
class by taking a piece of paper and rolling it up into a cornucopia,
the space inclosed being a _conic space_, a term that is sometimes
convenient. The generation of a conic surface may be shown by taking a
blackboard pointer and swinging it around by its tip so that the other
end moves in a curve. If we consider a straight line as the limit of a
curve, then the pointer may swing in a plane, and so a plane is the
limit of a conic surface. If we swing the pointer about a point in the
middle, we shall generate the two nappes of the cone, the conic space
now being double.
In practice the right circular cone, or cone of revolution, is the
important type, and special attention should be given to this form.
THEOREM. _Every section of a cone made by a plane passing through its
vertex is a triangle._
At this time, or in speaking of the preliminary definitions, reference
should be made to the conic sections. Of these there are three great
types: (1) the ellipse, where the cutting plane intersects all the
elements on one side of the vertex; a circle is a special form of the
ellipse; (2) the parabola, where the plane is parallel to an element;
(3) the hyperbola, where the plane cuts some of the elements on one side
of the vertex, and the rest on the other side; that is, where it cuts
both nappes. It is to be observed that the ellipse may vary greatly in
shape, from a circle to a very long ellipse, as the cutting plane
changes from being perpendicular to the axis to being nearly parallel to
an element. The instant it becomes parallel to an element the ellipse
changes suddenly to a parabola. If the plane tips the slightest amount
more, the section becomes an hyperbola.
While these conic sections are not studied in elementary geometry, the
terms should be known for general information, particularly the ellipse
and parabola. The study of the conic sections forms a large part of the
work of analytic geometry, a subject in which the figures resemble the
graphic work in algebra, this having been taken from "analytics," as the
higher subject is commonly called. The planets move about the sun in
elliptic orbits, and Halley's comet that returned to view in 1909-1910
has for its path an enormous ellipse. Most comets seem to move in
parabolas, and a body thrown into the air would take a parabolic path if
it were not for the resistance of the atmosphere. Two of the sides of
the triangle in this proposition constitute a special form of the
hyperbola.
The study of conic sections was brought to a high state by the Greeks.
They were not known to the Pythagoreans, but were discovered by
Menæchmus in the fourth century B.C. This discovery is mentioned by
Proclus, who says, "Further, as to these sections, the conics were
conceived by Menæchmus."
Since if the cutting plane is perpendicular to the axis the section is a
circle, and if oblique it is an ellipse, a parabola, or an hyperbola, it
follows that if light proceeds from a point, the shadow of a circle is a
circle, an ellipse, a parabola, or an hyperbola, depending on the
position of the plane on which the shadow falls. It is interesting and
instructive to a class to see these shadows, but of course not much time
can be allowed for such work. At this point the chief thing is to have
the names "ellipse" and "parabola," so often met in reading, understood.
It is also of interest to pupils to see at this time the method of
drawing an ellipse by means of a pencil stretching a string band that
moves about two pins fastened in the paper. This is a practical method,
and is familiar to all teachers who have studied analytic geometry. In
designing elliptic arches, however, three circular arcs are often
joined, as here shown, the result being approximately an elliptic arc.
[Illustration]
Here _O_ is the center of arc _BC_, _M_ of arc _AB_, and _N_ of
arc _CD_. Since _XY_ is perpendicular to _BM_ and _BO_, it is
tangent to arcs _AB_ and _BC_, so there is no abrupt turning at
_B_, and similarly for _C_.[90]
THEOREM. _The volume of a circular cone is equal to one third the
product of its base by its altitude._
It is easy to prove this for noncircular cones as well, but since they
are not met commonly in practice, they may be omitted in elementary
geometry. The important formula at this time is _v_ = 1/3[pi]_r_^2_h_.
As already stated, this proposition was discovered by Eudoxus of Cnidus
(born _ca._ 407 B.C., died _ca._ 354 B.C.), a man who, as already
stated, was born poor, but who became one of the most illustrious and
most highly esteemed of all the Greeks of his time.
THEOREM. _The lateral area of a frustum of a cone of revolution is equal
to half the sum of the circumferences of its bases multiplied by the
slant height._
An interesting case for a class to notice is that in which the upper
base becomes zero and the frustum becomes a cone, the proposition being
still true. If the upper base is equal to the lower base, the frustum
becomes a cylinder, and still the proposition remains true. The
proposition thus offers an excellent illustration of the elementary
Principle of Continuity.
Then follows, in most textbooks, a theorem relating to the volume of a
frustum.
In the case of a cone of revolution
_v_ = (1/3)[pi]_h_(_r_^2 + _r'_^2 + _rr'_). Here if _r'_ = 0, we
have _v_ = (1/3)[pi]_r_^2_h_, the volume of a cone. If _r'_ = _r_,
we have _v_ = (1/3)[pi]_h_(_r_^2 + _r_^2 + _r_^2) = [pi]_hr_^2, the
volume of a cylinder.
If one needs examples in mensuration beyond those given in a first-class
textbook, they are easily found. The monument to Sir Christopher Wren,
the professor of geometry in Cambridge University, who became the great
architect of St. Paul's Cathedral in London, has a Latin inscription
which means, "Reader, if you would see his monument, look about you." So
it is with practical examples in Book VII.
Appended to this Book, or more often to the course in solid geometry, is
frequently found a proposition known as Euler's Theorem. This is often
considered too difficult for the average pupil and is therefore omitted.
On account of its importance, however, in the theory of polyhedrons,
some reference to it at this time may be helpful to the teacher. The
theorem asserts that in any convex polyhedron the number of edges
increased by two is equal to the number of vertices increased by the
number of faces. In other words, that _e_ + 2 = _v_ + _f_. On account of
its importance a proof will be given that differs from the one
ordinarily found in textbooks.
Let _s__{1}, _s__{2}, ···, _s__{_n_} be the number of sides of
the various faces, and _f_ the number of faces. Now since the
sum of the angles of a polygon of _s_ sides is (_s_ - 2)180°,
therefore the sum of the angles of all the faces is
(_s__{1} + _s__{2} + _s__{3} + ··· + _s__{_n_} - 2_f_)180°.
But _s__{1} + _s__{2} + _s__{3} + ··· + _s__{_n_} is twice the
number of edges, because each edge belongs to two faces.
[therefore] the sum of the angles of all the faces is
(2_e_ - 2_f_)180°, or (_e_ - _f_)360°.
Since the polyhedron is convex, it is possible to find some
outside point of view, _P_, from which some face, as _ABCDE_,
covers up the whole figure, as in this illustration. If we
think of all the vertices projected on _ABCDE_, by lines
through _P_, the sum of the angles of all the faces will be the
same as the sum of the angles of all their projections on
_ABCDE_. Calling _ABCDE_ _s__{1}, and thinking of the
projections as traced by dotted lines on the opposite side of
_s__{1}, this sum is evidently equal to
(1) the sum of the angles in _s__{1}, or (_s__{1} - 2) 180°,
plus
(2) the sum of the angles on the other side of _s__{1}, or
(_s__{1} - 2)180°, plus
(3) the sum of the angles about the various points shown as
inside of _s__{1}, of which there are _v_ - _s__{1} points,
about each of which the sum of the angles is 360°, making
(_v_ - _s__{1})360° in all.
[Illustration]
Adding, we have
(_s__{1} - 2)180° + (_s__{1} - 2)180° + (_v_ - _s__{1})360°
= [(_s__{1} - 2) + (_v_ - _s__{1})]360°
= (_v_ - 2)360°.
Equating the two sums already found, we have
(_e_ - _f_)360° = (_v_ - 2)360°,
or _e_ - _f_ = _v_ - 2,
or _e_ + 2 = _v_ + _f_.
This proof is too abstract for most pupils in the high school, but it is
more scientific than those found in any of the elementary textbooks, and
teachers will find it of service in relieving their own minds of any
question as to the legitimacy of the theorem.
Although this proposition is generally attributed to Euler, and was,
indeed, rediscovered by him and published in 1752, it was known to the
great French geometer Descartes, a fact that Leibnitz mentions.[91]
This theorem has a very practical application in the study of crystals,
since it offers a convenient check on the count of faces, edges, and
vertices. Some use of crystals, or even of polyhedrons cut from a piece
of crayon, is desirable when studying Euler's proposition. The following
illustrations of common forms of crystals may be used in this
connection:
[Illustration]
The first represents two truncated pyramids placed base to base. Here
_e_ = 20, _f_ = 10, _v_ = 12, so that _e_ + 2 = _f_ + _v_. The second
represents a crystal formed by replacing each edge of a cube by a plane,
with the result that _e_ = 40, _f_ = 18, and _v_ = 24. The third
represents a crystal formed by replacing each edge of an octahedron by a
plane, it being easy to see that Euler's law still holds true.
FOOTNOTES:
[89] The actual construction of these solids is given by Pappus. See his
"Mathematicae Collectiones," p. 48, Bologna, 1660.
[90] The illustration is from Dupin, loc. cit.
[91] For the historical bibliography consult G. Holzmüller, _Elemente
der Stereometrie_, Vol. I, p. 181, Leipzig, 1900.
CHAPTER XXI
THE LEADING PROPOSITIONS OF BOOK VIII
Book VIII treats of the sphere. Just as the circle may be defined either
as a plane surface or as the bounding line which is the locus of a point
in a plane at a given distance from a fixed point, so a sphere may be
defined either as a solid or as the bounding surface which is the locus
of a point in space at a given distance from a fixed point. In higher
mathematics the circle is defined as the bounding line and the sphere as
the bounding surface; that is, each is defined as a locus. This view of
the circle as a line is becoming quite general in elementary geometry,
it being the desire that students may not have to change definitions in
passing from elementary to higher mathematics. The sphere is less
frequently looked upon in geometry as a surface, and in popular usage it
is always taken as a solid.
Analogous to the postulate that a circle may be described with any given
point as a center and any given line as a radius, is the postulate for
constructing a sphere with any given center and any given radius. This
postulate is not so essential, however, as the one about the circle,
because we are not so concerned with constructions here as we are in
plane geometry.
A good opportunity is offered for illustrating several of the
definitions connected with the study of the sphere, such as great
circle, axis, small circle, and pole, by referring to geography.
Indeed, the first three propositions usually given in Book VIII have a
direct bearing upon the study of the earth.
THEOREM. _A plane perpendicular to a radius at its extremity is tangent
to the sphere._
The student should always have his attention called to the analogue in
plane geometry, where there is one. If here we pass a plane through the
radius in question, the figure formed on the plane will be that of a
line tangent to a circle. If we revolve this about the line of the
radius in question, as an axis, the circle will generate the sphere
again, and the tangent line will generate the tangent plane.
THEOREM. _A sphere may be inscribed in any given tetrahedron._
Here again we may form a corresponding proposition of plane geometry by
passing a plane through any three points of contact of the sphere and
the tetrahedron. We shall then form the figure of a circle inscribed in
a triangle. And just as in the case of the triangle we may have escribed
circles by producing the sides, so in the case of the tetrahedron we may
have escribed spheres by producing the planes indefinitely and
proceeding in the same way as for the inscribed sphere. The figure is
difficult to draw, but it is not difficult to understand, particularly
if we construct the tetrahedron out of pasteboard.
THEOREM. _A sphere may be circumscribed about any given tetrahedron._
By producing one of the faces indefinitely it will cut the sphere in a
circle, and the resulting figure, on the plane, will be that of the
analogous proposition of plane geometry, the circle circumscribed about
a triangle. It is easily proved from the proposition that the four
perpendiculars erected at the centers of the faces of a tetrahedron meet
in a point (are concurrent), the analogue of the proposition about the
perpendicular bisectors of the sides of a triangle.
THEOREM. _The intersection of two spherical surfaces is a circle whose
plane is perpendicular to the line joining the centers of the surfaces
and whose center is in that line._
The figure suggests the case of two circles in plane geometry. In the
case of two circles that do not intersect or touch, one not being within
the other, there are four common tangents. If the circles touch, two
close up into one. If one circle is wholly within the other, this last
tangent disappears. The same thing exists in relation to two spheres,
and the analogous cases are formed by revolving the circles and tangents
about the line through their centers.
In plane geometry it is easily proved that if two circles intersect, the
tangents from any point on their common chord produced are equal. For if
the common chord is _AB_ and the point _P_ is taken on _AB_ produced,
then the square on any tangent from _P_ is equal to _PB_ × _PA_. The
line _PBA_ is sometimes called the _radical axis_.
Similarly in this proposition concerning spheres, if from any point in
the plane of the circle formed by the intersection of the two spherical
surfaces lines are drawn tangent to either sphere, these tangents are
equal. For it is easily proved that all tangents to the same sphere from
an external point are equal, and it can be proved as in plane geometry
that two tangents to the two spheres are equal.
Among the interesting analogies between plane and solid geometry is the
one relating to the four common tangents to two circles. If the figure
be revolved about the line of centers, the circles generate spheres and
the tangents generate conical surfaces. To study this case for various
sizes and positions of the two spheres is one of the most interesting
generalizations of solid geometry.
An application of the proposition is seen in the case of an
eclipse, where the sphere _O'_ represents the moon, _O_ the
earth, and _S_ the sun. It is also seen in the case of the full
moon, when _S_ is on the other side of the earth. In this case
the part _MIN_ is fully illuminated by the moon, but the zone
_ABNM_ is only partly illuminated, as the figure shows.[92]
[Illustration]
THEOREM. _The sum of the sides of a spherical polygon is less than
360°._
In all such cases the relation to the polyhedral angle should be made
clear. This is done in the proofs usually given in the textbooks. It is
easily seen that this is true only with the limitation set forth in most
textbooks, that the spherical polygons considered are convex. Thus we
might have a spherical triangle that is concave, with its base 359°, and
its other two sides each 90°, the sum of the sides being 539°.
THEOREM. _The sum of the angles of a spherical triangle is greater than
180° and less than 540°._
It is for the purpose of proving this important fact that polar
triangles are introduced. This proposition shows the relation of the
spherical to the plane triangle. If our planes were in reality slightly
curved, being small portions of enormous spherical surfaces, then the
sum of the angles of a triangle would not be exactly 180°, but would
exceed 180° by some amount depending on the curvature of the surface.
Just as a being may be imagined as having only two dimensions, and
living always on a plane surface (in a space of two dimensions), and
having no conception of a space of three dimensions, so we may think of
ourselves as living in a space of three dimensions but surrounded by a
space of four dimensions. The flat being could not point to a third
dimension because he could not get out of his plane, and we cannot point
to the fourth dimension because we cannot get out of our space. Now what
the flat being thinks is his plane may be the surface of an enormous
sphere in our three dimensions; in other words, the space he lives in
may curve through some higher space without his being conscious of it.
So our space may also curve through some higher space without our being
conscious of it. If our planes have really some curvature, then the sum
of the angles of our triangles has a slight excess over 180°. All this
is mere speculation, but it may interest some student to know that the
idea of fourth and higher dimensions enters largely into mathematical
investigation to-day.
THEOREM. _Two symmetric spherical triangles are equivalent._
While it is not a subject that has any place in a school, save perhaps
for incidental conversation with some group of enthusiastic students, it
may interest the teacher to consider this proposition in connection with
the fourth dimension just mentioned. Consider these triangles, where
[L]_A_ = [L]_A'_, _AB_ = _A'B'_, _AC_ = _A'C'_. We prove them congruent
by superposition, turning one over and placing it upon the other. But
suppose we were beings in Flatland, beings with only two dimensions and
without the power to point in any direction except in the plane we lived
in. We should then be unable to turn [triangle]_A'B'C'_ over so that it
could coincide with [triangle]_ABC_, and we should have to prove these
triangles equivalent in some other way, probably by dividing them into
isosceles triangles that could be superposed.
[Illustration]
[Illustration]
Now it is the same thing with symmetric spherical triangles; we cannot
superpose them. But might it not be possible to do so if we could turn
them through the fourth dimension exactly as we turn the Flatlander's
triangle through our third dimension? It is interesting to think about
this possibility even though we carry it no further, and in these side
lights on mathematics lies much of the fascination of the subject.
THEOREM. _The shortest line that can be drawn on the surface of a sphere
between two points is the minor arc of a great circle joining the two
points._
It is always interesting to a class to apply this practically. By taking
a terrestrial globe and drawing a great circle between the southern
point of Ireland and New York City, we represent the shortest route for
ships crossing to England. Now if we notice where this great-circle arc
cuts the various meridians and mark this on an ordinary Mercator's
projection map, such as is found in any schoolroom, we shall find that
the path of the ship does not make a straight line. Passengers at sea
often do not understand why the ship's course on the map is not a
straight line; but the chief reason is that the ship is taking a
great-circle arc, and this is not, in general, a straight line on a
Mercator projection. The small circles of latitude are straight lines,
and so are the meridians and the equator, but other great circles are
represented by curved lines.
THEOREM. _The area of the surface of a sphere is equal to the product of
its diameter by the circumference of a great circle._
This leads to the remarkable formula, _a_ = 4[pi]_r_^2. That the area of
the sphere, a curved surface, should exactly equal the sum of the areas
of four great circles, plane surfaces, is the remarkable feature. This
was one of the greatest discoveries of Archimedes (_ca._ 287-212 B.C.),
who gives it as the thirty-fifth proposition of his treatise on the
"Sphere and the Cylinder," and who mentions it specially in a letter to
his friend Dositheus, a mathematician of some prominence. Archimedes
also states that the surface of a sphere is two thirds that of the
circumscribed cylinder, or the same as the curved surface of this
cylinder. This is evident, since the cylindric surface of the cylinder
is 2[pi]_r_ × 2_r_, or 4[pi]_r_^2, and the two bases have an area
[pi]_r_^2 + [pi]_r_^2, making the total area 6[pi]_r_^2.
THEOREM. _The area of a spherical triangle is equal to the area of a
lune whose angle is half the triangle's spherical excess._
This theorem, so important in finding areas on the earth's surface,
should be followed by a considerable amount of computation of triangular
areas, else it will be rather meaningless. Students tend to memorize a
proof of this character, and in order to have the proposition mean what
it should to them, they should at once apply it. The same is true of the
following proposition on the area of a spherical polygon. It is probable
that neither of these propositions is very old; at any rate, they do not
seem to have been known to the writers on elementary mathematics among
the Greeks.
THEOREM. _The volume of a sphere is equal to the product of the area of
its surface by one third of its radius._
This gives the formula _v_ = (4/3)[pi]_r_^3. This is one of the greatest
discoveries of Archimedes. He also found as a result that the volume of
a sphere is two thirds the volume of the circumscribed cylinder. This is
easily seen, since the volume of the cylinder is [pi]_r_^2 × 2_r_, or
2[pi]_r_^3, and (4/3)[pi]_r_^3 is 2/3 of 2[pi]_r_^3. It was because of
these discoveries on the sphere and cylinder that Archimedes wished
these figures engraved upon his tomb, as has already been stated. The
Roman general Marcellus conquered Syracuse in 212 B.C., and at the sack
of the city Archimedes was killed by an ignorant soldier. Marcellus
carried out the wishes of Archimedes with respect to the figures on his
tomb.
The volume of a sphere can also be very elegantly found by means of a
proposition known as Cavalieri's Theorem. This asserts that if two
solids lie between parallel planes, and are such that the two sections
made by any plane parallel to the given planes are equal in area, the
solids are themselves equal in volume. Thus, if these solids have the
same altitude, _a_, and if _S_ and _S'_ are equal sections made by a
plane parallel to _MN_, then the solids have the same volume. The proof
is simple, since prisms of the same altitude, say _a_/_n_, and on the
bases _S_ and _S'_ are equivalent, and the sums of _n_ such prisms are
the given solids; and as _n_ increases, the sums of the prisms approach
the solids as their limits; hence the volumes are equal.
[Illustration]
This proposition, which will now be applied to finding the volume of the
sphere, was discovered by Bonaventura Cavalieri (1591 or 1598-1647). He
was a Jesuit professor in the University of Bologna, and his best known
work is his "Geometria Indivisilibus," which he wrote in 1626, at least
in part, and published in 1635 (second edition, 1647). By means of the
proposition it is also possible to prove several other theorems, as that
the volumes of triangular pyramids of equivalent bases and equal
altitudes are equal.
[Illustration]
To find the volume of a sphere, take the quadrant _OPQ_, in the
square _OPRQ_. Then if this figure is revolved about _OP_,
_OPQ_ will generate a hemisphere, _OPR_ will generate a cone of
volume (1/3)[pi]_r_^3, and _OPRQ_ will generate a cylinder of
volume [pi]_r_^3. Hence the figure generated by _ORQ_ will have
a volume [pi]_r_^3 - (1/3)[pi]_r_^3, or (2/3)[pi]_r_^3, which
we will call _x_.
Now _OA_ = _AB_, and _OC_ = _AD_; also (_OC_)^2 - (_OA_)^2 = (_AC_)^2,
so that (_AD_)^2 - (_AB_)^2 = (_AC_)^2,
and [pi](_AD_)^2 - [pi](_AB_)^2 = [pi](_AC_)^2.
But [pi](_AD_)^2 - [pi](_AB_)^2 is the area of the ring
generated by _BD_, a section of _x_, and [pi](_AC_)^2 is the
corresponding section of the hemisphere. Hence, by Cavalieri's
Theorem,
(2/3)[pi]_r_^3 = the volume of the hemisphere.
[therefore] (4/3)[pi]_r_^3 = the volume of the sphere.
In connection with the sphere some easy work in quadratics may be
introduced even if the class has had only a year in algebra.
For example, suppose a cube is inscribed in a hemisphere of
radius _r_ and we wish to find its edge, and thereby its
surface and its volume.
If _x_ = the edge of the cube, the diagonal of the base must be
_x_[sqrt]2, and the projection of _r_ (drawn from the center of
the base to one of the vertices) on the base is half of this
diagonal, or (_x_[sqrt]2)/2.
Hence, by the Pythagorean Theorem,
_r_^2 = _x_^2 + ((_x_[sqrt]2)/2)^2 = (3/2)_x_^2
[therefore] _x_ = _r_[sqrt](2/3),
and the total surface is 6_x_^2 = 4_r_^2,
and the volume is _x_^3 = (2/3)_r_^3[sqrt](2/3).
FOOTNOTES:
[92] The illustration is from Dupin, loc. cit.
L'ENVOI
In the Valley of Youth, through which all wayfarers must pass on their
journey from the Land of Mystery to the Land of the Infinite, there is a
village where the pilgrim rests and indulges in various excursions for
which the valley is celebrated. There also gather many guides in this
spot, some of whom show the stranger all the various points of common
interest, and others of whom take visitors to special points from which
the views are of peculiar significance. As time has gone on new paths
have opened, and new resting places have been made from which these
views are best obtained. Some of the mountain peaks have been neglected
in the past, but of late they too have been scaled, and paths have been
hewn out that approach the summits, and many pilgrims ascend them and
find that the result is abundantly worth the effort and the time.
The effect of these several improvements has been a natural and usually
friendly rivalry in the body of guides that show the way. The mountains
have not changed, and the views are what they have always been. But
there are not wanting those who say, "My mountain may not be as lofty as
yours, but it is easier to ascend"; or "There are quarries on my peak,
and points of view from which a building may be seen in process of
erection, or a mill in operation, or a canal, while your mountain shows
only a stretch of hills and valleys, and thus you will see that mine is
the more profitable to visit." Then there are guides who are themselves
often weak of limb, and who are attached to numerous sand dunes, and
these say to the weaker pilgrims, "Why tire yourselves climbing a rocky
mountain when here are peaks whose summits you can reach with ease and
from which the view is just as good as that from the most famous
precipice?" The result is not wholly disadvantageous, for many who pass
through the valley are able to approach the summits of the sand dunes
only, and would make progress with greatest difficulty should they
attempt to scale a real mountain, although even for them it would be
better to climb a little way where it is really worth the effort instead
of spending all their efforts on the dunes.
Then, too, there have of late come guides who have shown much ingenuity
by digging tunnels into some of the greatest mountains. These they have
paved with smooth concrete, and have arranged for rubber-tired cars that
run without jar to the heart of some mountain. Arrived there the pilgrim
has a glance, as the car swiftly turns in a blaze of electric light, at
a roughly painted panorama of the view from the summit, and he is
assured by the guide that he has accomplished all that he would have
done, had he laboriously climbed the peak itself.
In the midst of all the advocacy of sand-dune climbing, and of
rubber-tired cars to see a painted view, the great body of guides still
climb their mountains with their little groups of followers, and the
vigor of the ascent and the magnificence of the view still attract all
who are strong and earnest, during their sojourn in the Valley of Youth.
Among the mountains that have for ages attracted the pilgrims is Mons
Latinus, usually called in the valley by the more pleasing name Latina.
Mathematica, and Rhetorica, and Grammatica are also among the best
known. A group known as Montes Naturales comprises Physica, Biologica,
and Chemica, and one great peak with minor peaks about it is called by
the people Philosophia. There are those who claim that these great
masses of rock are too old to be climbed, as if that affected the view;
while others claim that the ascent is too difficult and that all who do
not favor the sand dunes are reactionary. But this affects only a few
who belong to the real mountains, and the others labor diligently to
improve the paths and to lessen unnecessary toil, but they seek not to
tear off the summits nor do they attend to the amusing attempts of those
who sit by the hillocks and throw pebbles at the rocky sides of the
mountains upon which they work.
* * * * *
Geometry is a mountain. Vigor is needed for its ascent. The views all
along the paths are magnificent. The effort of climbing is stimulating.
A guide who points out the beauties, the grandeur, and the special
places of interest commands the admiration of his group of pilgrims.
One who fails to do this, who does not know the paths, who puts
unnecessary burdens upon the pilgrim, or who blindfolds him in his
progress, is unworthy of his position. The pretended guide who says that
the painted panorama, seen from the rubber-tired car, is as good as the
view from the summit is simply a fakir and is generally recognized as
such. The mountain will stand; it will not be used as a mere commercial
quarry for building stone; it will not be affected by pellets thrown
from the little hillocks about; but its paths will be freed from
unnecessary flints, they will be straightened where this can
advantageously be done, and new paths on entirely novel plans will be
made as time goes on, but these paths will be hewed out of rock, not
made out of the dreams of a day. Every worthy guide will assist in all
these efforts at betterment, and will urge the pilgrim at least to
ascend a little way because of the fact that the same view cannot be
obtained from other peaks; but he will not take seriously the efforts of
the fakir, nor will he listen with more than passing interest to him who
proclaims the sand heap to be a Matterhorn.
INDEX
Ahmes, 27, 254, 278, 306
Alexandroff, 164
Algebra, 37, 84
Al-Khowarazmi, 37
Allman, G. J., 29
Almagest, 35
Al-Nair[=i]z[=i], 171, 193, 214, 264
Al-Qif[t.][=i], 49
Analysis, 41, 161
Angle, 142, 155;
trisection of, 31, 215
Anthonisz, Adriaen, 279
Antiphon, 31, 32, 276
Apollodotus (Apollodorus), 259
Apollonius, 34, 214, 231
Applied problems, 75, 103, 178, 186, 192, 195, 203, 204, 209, 215, 217,
242, 267, 295, 317
Appreciation of geometry, 19
Arab geometry, 37, 51
Archimedes, 34, 42, 48, 139, 141, 215, 276, 278, 314, 327, 328
Aristæus, 310
Aristotle, 33, 42, 134, 135, 137, 145, 154, 177, 209
Aryabhatta, 36, 279
Associations, syllabi of, 58, 60, 64
Assumptions, 116
Astrolabe, 172
Athelhard of Bath, 37, 51
Athenæus, 259
Axioms, 31, 41, 116
Babylon, 26, 272
Bartoli, 10, 44, 238
Belli, 10, 44, 172
Beltinus, 239, 241
Beltrami, 127
Bennett, J., 224
Bernoulli, 280
Bertrand, 62
Betz, 131
Bezout, 62
Bhaskara, 232, 268
Billings, R. W., 222
Billingsley, 52
Bion, 192, 239
Boethius, 43, 50
Bolyai, 128
Bonola, 128
Books of geometry, 165, 167, 201, 227, 252, 269, 289, 303, 321
Bordas-Demoulin, 24
Borel, 11, 67, 196
Bosanquet, 272
Bossut, 23
Bourdon, 62
Bourlet, 67, 165, 196
Brahmagupta, 36, 268, 279
Bretschneider, C. A., 30
Brouncker, 280
Bruce, W. N., 199
Bryson, 31, 32, 276
Cajori, 46
Calandri, 30
Campanus, 37, 51, 135
Cantor, M., 29, 46
Capella, 50, 135
Capra, 44
Carson, G. W. L., 18, 96, 114
Casey, J., 38
Cassiodorius, 50
Cataneo, 10, 44
Cavalieri, 136, 329
Chinese values of [pi], 279
Church schools, 43
Cicero, 34, 50, 259, 314
Circle, 145, 201, 270, 287;
squaring the, 31, 32, 277
Circumference, 145
Cissoid, 34
Class in geometry, 108
Clavius, 121
Colleges, geometry in the, 46
Collet, 24
Commensurable magnitudes, 206, 207
Conchoid, 34
Condorcet, 23
Cone, 315
Congruent, 151
Conic sections, 33, 315
Continuity, 212
Converse proposition, 175, 190, 191
Crelle, 142
Cube, duplicating the, 32, 307
Cylinder, 313
D'Alembert, 24, 67
Dase, 279
Decagon, 273
Definitions, 41, 132
De Judaeis, 239, 241
De Morgan, A., 58
De Paolis, 67
Descartes, 38, 84, 320
Diameter, 146
Dihedral, 298
Diocles, 34
Diogenes Laertius, 259
Diorismus, 41
Direction, 150
Distance, 154
Doyle, Conan, 8
Drawing, 95, 221, 281
Duality, 173
Duhamel, 164
Dupin, 11, 217
Duplication problem, 32, 307
Dürer, 10
Educational problems, 1
Egypt, 26, 40
Eisenlohr, 27
Engel and Stäckel, 128
England, 14, 46, 58, 60
Epicureans, 188
Equal, 151, 153
Equilateral, 147
Equivalent, 151
Eratosthenes, 48
Euclid, 33, 42, 43, 44, 119, 125, 135, 156, 165, 167 ff., 201 ff.,
et passim;
editions of, 47, 52;
efforts at improving, 57;
life of, 47;
nature of his "Elements," 52, 55;
opinions of, 8
Eudemus, 33, 168, 171, 185, 216, 309
Eudoxus, 32, 41, 48, 227, 308, 314, 317
Euler, 38, 280, 318
Eutocius, 184
Exercises, nature of, 74, 103;
how to attack, 160
Exhaustions, method of, 31
Extreme and mean ratio, 250
Figures in geometry, 104, 107, 113
Finaeus, 44, 239, 240, 243
Fourier, 142
Fourth dimension, 326
Frankland, 56, 117, 127, 135, 159
Fusion, of algebra and geometry, 84;
of geometry and trigonometry, 91
Gargioli, 44
Gauss, 140, 274
Geminus, 126, 128, 149
Geometry, books of, 165, 167, 201, 227, 252, 269, 289, 303, 321;
compared with other subjects, 14;
introduction to, 93;
modern, 38;
of motion, 68, 196;
reasons for teaching, 7, 15, 20;
related to algebra, 84;
textbooks in, 70
Gerbert, 43
Gherard of Cremona, 37, 51
Gnomon, 212
Golden section, 250
Gothic windows, 75, 221 ff., 274, 282
Gow, J., 29, 56
Greece, 28, 40
Gregoire de St. Vincent, 267
Gregory, 280
Grévy, 67
Gymnasia, geometry in the, 45
Hadamard, 164
Hamilton, W., 14
Harmonic division, 231
Harpedonaptae, 28
Harriot, 37
Harvard syllabus, 63
Heath, T. L., 49, 56, 119, 126, 127, 135, 149, 159, 170, 175, 228, 261
Hebrews, 26
Henrici, O., 11, 14, 25, 164, 196
Henrici and Treutlein, 68, 164, 196
Hermite, 281
Herodotus, 28
Heron, 35, 137, 139, 141, 209, 259, 267
Hexagon, regular, 272
High schools, geometry in the, 45
Hilbert, 119, 131
Hipparchus, 35
Hippasus, 273, 309
Hippias, 31, 215
Hippocrates, 31, 41, 281
History of geometry, 26
Hobson, 166
Hoffmann, 242
Holzmüller, 320
Hughes, Justice, 9
Hypatia, 36
Hypsicles, 34
Hypsometer, 245
Iamblichus, 273, 309
Illusions, optical, 100
Ingrami, 128
Instruments, 96, 178, 236
Introduction to geometry, 93
Ionic school, 28
Jackson, C. S., 12
Jones, W., 271
Junge, 259
Karagiannides, 128
Karpinski, 37
Kaye, G. B., 232
Kepler, 24, 149
Kingsley, C., 36
Klein, F., 68, 89
Kolb, 222
Lacroix, 24, 46, 62, 66
Langley, E. M., 291
Laplace, 101
Legendre, 10, 45, 62, 127, 128, 152
Leibnitz, 140, 150
Leon, 41
Leonardo da Vinci, 264
Leonardo of Pisa, 37, 43, 279
Lettering figures, 105
Limits, 207
Lindemann, 278, 281
Line defined, 137
Lobachevsky, 128
Loci, 163, 198
Locke, W. J., 13
Lodge, A., 14
Logic, 17, 104
Loomis, 164
Ludolph van Ceulen, 279
Lycées, geometry in, 45
M'Clelland, 38
McCormack, T. J., 11
Measured by, 208
Memorizing, 12, 79
Menæchmus, 33, 316
Menelaus, 35
Méray, 67, 68, 196, 289
Methods, 41, 42, 115
Metius, 279
Mikami, 264
Minchin, 14
Models, 93, 290
Modern geometry, 38
Mohammed ibn Musa, 37
Moore, E. H., 131
Mosaics, 274
Müller, H., 68
Münsterberg, 22
Napoleon, 24, 287
Newton, 24
Nicomedes, 34, 215
Octant, 242
Oenopides, 31, 212, 216
Optical illusions, 100
Oughtred, 37
Paciuolo, 86
Pamphilius, 185
Pappus, 36, 230, 263
Parallelepiped, 303
Parallels, 149, 181
Parquetry, 222, 274
Pascal, 24, 159
Peletier, 169
Perigon, 151
Perry, J., 13, 14
Petersen, 164
Philippus of Mende, 32, 185
Philo, 178
Philolaus, 309
[pi], 26, 27, 34, 36, 271, 278, 280
Plane, 140
Plato, 25, 31, 41, 48, 129, 136, 137, 309, 310
Playfair, 128
Pleasure of geometry, 16
Plimpton, G. A., 51, 52
Plutarch, 259
Poinsot, 311
Point, 135
Polygons, 156, 252, 269, 274
Polyhedrons, 301, 303, 310
Pomodoro, 179
_Pons asinorum_, 174, 265
Posidonius, 128, 149
Postulates, 31, 41, 116, 125, 292
Practical geometry, 3, 7, 9, 44
Printing, effect of, 44
Prism, 303
Problems, applied, 75, 103, 178, 186, 192, 195, 203, 204, 209,
215, 217, 242, 267, 295, 317
Proclus, 36, 47, 48, 52, 71, 127, 128, 136, 137, 139, 140, 149,
155, 186, 188, 197, 212, 214, 253, 258, 310, 311
Projections, 300
Proofs in full, 79
Proportion, 32, 227
Psychology, 12, 20
Ptolemy, C., 35, 278;
king, 48, 49
Pyramid, 307
Pythagoras, 29, 40, 258, 272, 273, 310
Pythagorean Theorem, 28, 36, 258
Pythagorean numbers, 32, 36, 263, 266
Pythagoreans, 136, 137, 185, 227, 269, 309
Quadrant, 236
Quadratrix, 31, 215
Quadrilaterals, 148, 157
Quadrivium, 42
Questions at issue, 3
Rabelais, 24
Radius, 153
Ratio, 205, 227
Real problem defined, 75, 103
Reasons for studying geometry, 7, 100
Rebière, 25
Reciprocal propositions, 173
Recitation in geometry, 113
Recorde, 37
Rectilinear figures, 146
Reductio ad absurdum, 41, 177
Regular polygons, 269
"Rhind Papyrus," 27
Rhombus and rhomboid, 148
Riccardi, 47
Richter, 279
Roman surveyors, 247
Saccheri, 127
Sacrobosco, 43
San Giovanni, 192
Sauvage, 164
Sayre, 272
Scalene, 147
Schlegel, 68
Schopenhauer, 121, 265
Schotten, 46, 135, 149
Sector, 154, 156
Segment, 154
Semicircle, 146
Shanks, 279
Similar figures, 232
Simon, 38, 56, 135
Simson, 142
Sisam, 11
Smith, D. E., 25, 37, 51, 52, 119, 131, 135, 159
Solid geometry, 289
Speusippus, 32
Sphere, 321
Square on a line, 257
Squaring the circle, 31, 32, 277
Stäckel, 128
Stamper, 46, 83
Stark, W. E., 172, 238
Stereoscopic slides, 291
Stobæus, 8
Straight angle, 151
Straight line, 138
Suggested proofs, 81
Sulvasutras, 232
Superposition, 169
Surface, 140
Swain, G. F., 13
Syllabi, 58, 60, 63, 64, 66, 67, 80, 82
Sylvester II, 43
Synthetic method, 161
Tangent, 154
Tartaglia, 153
Tatius, Achilles, 272
Teaching geometry, reasons for, 7, 15, 20;
development of, 40
Textbooks, 32, 33, 41, 70, 80, 82
Thales, 28, 168, 171, 185, 210
Theætetus, 48, 310
Theon of Alexandria, 36
Thibaut, 185
Thoreau, 246
Trapezium, 148
Treutlein, 68, 164
Triangle, 147
Trigonometry, 234
Trisection problem, 31, 215
Trivium, 42
Universities, geometry in the, 43
Uselessness of mathematics, 13
Veblen, 131, 159
Vega, 279
Veronese, 68
Vieta, 279
Vogt, 259
Wallis, 127, 280
Young, J. W. A., 25, 131, 159, 277
Zamberti, 52
Zenodorus, 34, 253
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