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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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*** START OF THE PROJECT GUTENBERG EBOOK THE TEACHING
OF GEOMETRY ***

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.

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.

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 π = 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 ½a(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 π 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]

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 Œnopides 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 a2 + b2
= c2. Pythagoras had already given a rule that would be expressed in
modern form, as ¼(m2 + 1)2 = m2 + ¼(m2 - 1)2. The school of Plato found
that ((½m)2 + 1)2 = m2 + ((½m)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 π
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 ½(a + b + c). The formula is

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 π, 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:

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

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-muqābalah," 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.

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-Qifṫī, who wrote in

the thirteenth century, has this to say in his biographical treatise "Ta'rīkh al-
Ḥukamā":
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.

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 [στοιχεια, 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.

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 Lines in different planes, solids contained by
52
IV. planes
Book V. Cylinder, cone, sphere 25
Book
Figures on a sphere 42
VI.
——
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.

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.

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 a2 + 2ab + b2, 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 "Sūma" 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.

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]

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:

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.

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.

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.

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.

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.

2. For the same reason, read angles counterclockwise. Thus ∠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 ∠m than ∠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 α (alpha) for the angle. The
learning of the first three Greek letters, alpha (α), beta (β), and gamma (γ), 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.

6. Also use small letters to represent numerical values. For example, write c = 2πr
instead of C = 2πR. 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.

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."

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 αιτημα (aitema), and appears only in late
translations. Euclid stated in substance, "Let the following be assumed." "Axiom"
(αξίωμα, axioma) dates perhaps only from Proclus (fifth century A.D.), Euclid
using the words "common notions" (κοιναὶ εννοιαι, 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, +√(-1), -√(-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 αἰτήματα (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.

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)
Forming a figure, or
Lines 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 y2(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, x2 + y2 = r2 representing the circle-line, a2y2 + b2x2 = a2b2 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 2πr, 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.

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.

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.

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.

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 ∠PXA equal?
∵ PQ is || AB, what does ∠PXA equal?
Then why must ∠BAX = ∠XAP?
Similarly, what about ∠QBX and ∠XBA?
Construction. Now reverse the process. What may we do to ⦞A and B in order to fix X?
Then how shall PQ be drawn? Now give the proof.

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
∠CDA = ∠BDC.

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, 3e 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 (βιβλίον), 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.

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-Nairīzī[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.

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
⧌XBC and YAC congruent,[62] and also ⧌XBA and YAB congruent. Then from ∠YAC he takes ∠YAB,
leaving ∠BAC, and so on the other side, leaving ∠CBA, these therefore being equal.

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 ∠A is common; hence the ⧌ABC and ACB are congruent, and ∠B of the
one equals ∠C of the other.
This is a better plan than that followed by some textbook writers of imagining ⧍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:

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 ∠A' = ∠B', Given
and ∠A = ∠A', Hyp.
∴∠A = ∠B'.
∴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.
∴B'C' = AC.

But B'C' was made equal to BC.

∴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 ∠A = ∠A', then a = a' and the triangles are congruent,
while this proposition asserts that if a = a', b = b', and c = c', then ∠A = ∠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:

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.

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.

Early Methods of Leveling Pomodoro's "La geometria prattica," Rome,
1624

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.

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.

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 ⊥ to x and b is ⊥ to x, then
a is || to b. This proposition may be stated thus: If a is ⊥ to x and a is || to b, then b is ⊥ 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.

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.

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:

Suppose an indefinite line XY to lie on AB. Let it swing about A, counterclockwise,
through ∠A, so as to lie on AC, as X'Y'. Then let it swing about C, through ∠C, so as to
lie on CB, as X''Y''. Then let it swing about B, through ∠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 ⦞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.

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.

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.

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.

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:

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.

For example, produce AC to X,
making

CX = CB.

Then ∠X = ∠XBC.

∴ ∠XBA > ∠X.

∴ AX > AB.

∴ 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:

Let CX bisect ∠C.

Then ∠BXC > ∠ACX.

∴∠BXC > ∠XCB.

∴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 ∠B = ∠C;
(2) if b > c, then ∠B > ∠C;
(3) if b < c, then ∠B < ∠C;
therefore the converses must necessarily be true as a matter of logic; for
if ∠B = ∠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 ∠B > ∠C, then b cannot equal c without violating (1), and b cannot be less than c
without violating (3), therefore b > c;
similarly, if ∠B < ∠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.

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.

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.

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.

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.

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-Nairīzī (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 ⧍XYZ is
congruent to ⧍YBP, and also to ⧍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.

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:

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 ⊥ to PQ, and XY' is also ⊥ 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.

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.

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.

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.

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.

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."

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 √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 √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.

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.

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.

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.

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 ∠PBA can be inscribed in a
semicircle.
2. Angles inscribed in the same segment are equal.

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 Œnopides,[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 througha 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.

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-Nairīzī (tenth century A.D.) to Heron of Alexandria. It is also given by
Proclus.

Required to draw from P a perpendicular to AP. Take X anywhere on the line and erect
XY ⊥ to AP in the usual manner. Bisect ∠PXY by the line XM. On XY take XN = XP, and
draw NM ⊥ 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°.

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:

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 Œnopides 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 Œnopides, 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]

An ingenious instrument described by Baron Dupin is illustrated below.

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.

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.

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.

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.

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.

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.
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.

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]

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]

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.

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.

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.

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.

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 √2 by √3? And when we multiply a ratio by √5, what is the
meaning of this operation? If we say that √2 : √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 √3 contained in √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 √5? We certainly take it more than 2 times and less than 3 times, but

what meaning can we assign to √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
P1 and P2.

∴AP1 : P1B = AP2 : P2B.

∴AP2 : P2B = AP2 - P1P2 : P1P2 - P2B,

and this is the criterion for the harmonic progression still seen in
many algebras. For, letting AP2 = a, P1P2 = b, P2B = 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 P1P2 is also called the harmonic mean
between AP2 and P2B, and the points A, P1, B, P2 are said to form
an harmonic range.

It may be noted that ∠P2CP1, 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 P1 and P2 are both fixed. Hence
C must lie on a semicircle with P1P2 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.

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, A1, A2, A3, ..., and B1, B2, B3, ..., as
such that A1A2, B1B2, C1C2, ... are concurrent in O, and A1O : A2O = B1O : B2O
= C1O : C2O = ... Here the constant ratio A1O : A2O 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.

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:

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 ∠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° 1.000 .000 ∞

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 ∠O = 40°, then
since y = r sin O, we have y = 30 × 0.643 = 19.29. If we know that x
= 60 and ∠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 ∠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.

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.

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]

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.

Surveying Instrument of the Eighteenth Century
N. Bion's "Traité de la construction ... des instrumens de

mathématique," The Hague, 1723

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.

The Quadrant Used for Depths Finaeus's
"Protomathesis," Paris, 1532

A Quadrant of the Sixteenth Century
De Judaeis's "De quadrante geometrico," Nürnberg, 1594

The Quadrant Used for Distances
Bettinus's "Apiaria universae philosophiae mathematieae,"
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.

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.

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.

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.

Another application of the same principle is seen in a method often taken for
measuring the height of a tree.

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.

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 ⊥ to AB.
Then the length of DC will be √6 in., and we may find √6
approximately by measuring DC.

Furthermore, if we introduce negative magnitudes into geometry,
and let DB = +3 and DA = -2, then DC will equal √(-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 √2, √3, √5, and √9, the last being integral. If,
as is not usually the case, the class has studied the complex number, the
absolute value of √(-6), √(-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.

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.

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 P3, 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 P4, then y = y' and we have xx' = y2, 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.

For, in this figure, let a = 3, b = 5, and c = 6.
Then ∵ x : y = b : a, and y = 6 - x,

we have x/(6 - x) = 5/3.

∴ 3x = 30 - 5x.

∴ x = 3 3/4, y = 2 1/4.

By the theorem, z2 = ab - xy
= 15 - (8 7/16) = 6 9/16.
∴ z = √(6 9/16) = 1/4 √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.

For if we designate the sides by a, b, and c, as usual, and let CD = d
and PB = x, then

(CP)2 = a2 - x2
= b2 - (c - x)2.
∴ a2 - x2 = b2 - c2 + 2cx - x2.
∴ x = (a2 - b2 + c2) / 2c.
∴ (CP)2 = a2 - ((a2 - b2 + c2) / 2c)2.

But CP · d = ab.
∴ d = 2abc / √(4a2c2 - (a2 - b2 + c2)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.

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 ∠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 ⧌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) = x2,
or x2 + ax - a2 = 0;
whence x = - a/2 ± a/2√5

= a(1.118 - 0.5)
= 0.618a,

the other root representing the external point.
That is, x = about 0.6a, and a - x = about 0.4a, 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.

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 ½
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.

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 ½(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.

This may be proved independently of the preceding proposition by
drawing the altitudes p and p'. Then
⧍ABC/⧍A'B'C' = cp/c'p'.

But c/c' = p/p',
by similar triangles.

∴ ⧍ABC/⧍A'B'C' = c2/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 32 + 42 = 52. 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.

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 12 + (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:

⧌ABC and APC are similar.

∴ AB × AP = (AC)2.

Similarly, AB × PB = (BC)2.

∴ 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:

Here Fig. 1 is evidently h2 + 4 ⧌.

Fig. 2 is evidently a2 + b2 + 4 ⧌.

∴ h2 + 4 ⧌ = a2 + b2 + 4 ⧌, the ⧌ all being congruent.

∴ h2 = a2 + b2.
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:

h2 = 4 · ab/2 + (b - a)2 = a2 + b2.
A somewhat similar proof can be based upon the following figure:

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.

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.

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-Nairīzī (died about 922 A.D.), attributes to Thābit ben Qurra
(826-901 A.D.) a proof substantially as follows:

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:

The construction of the following figure is evident. It is easily
shown that the four quadrilaterals ABMX, XNCA, SBCP, and SRQP
are congruent.
∴ 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 = ⧍ABA',
and 1/2(BC')2 = ⧍BC'C,

because BC = BC'.
∴ 1/2((AB)2 + (BC)2) = ⧍ABA' + ⧍BC'C.
∴ 1/2((AB)2 + (BC)2)
= ⧍AC'A' + ⧍A'C'C

(For ⧍ABA' + ⧍BC'A' + ⧍A'C'C is the second member of both
equations.)
= 1/2A'C' · AP
+ 1/2A'C' · PC
= 1/2A'C' · AC
= 1/2(AC)2.

∴ (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:

n2 + ((n2 - 1)/2)2 = ((n2 + 1)/2)2,

the sides being n, (n2 - 1)/2, and (n2 + 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, ···,

(n2 - 1)/2 = 0, 4, 12, 24, 40, ···,

(n2 + 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:

(2n)2 + (n2 - 1)2 = (n2 + 1)2.

If 2n = 2, 4, 6, 8, 10, ···,

then n2 - 1 = 0, 3, 8, 15, 24, ···,

and n2 + 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).

It should be observed that

a2 = b2 + c2 - 2b'c.
If ∠A = 90°, then b' = 0, and this becomes

a2 = b2 + c2.
If ∠A is obtuse, then b' passes through 0 and becomes negative, and
a2 = b2 + c2 + 2b'c.
Thus we have three propositions in one.

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 = √(s(s - a)(s - b)(s - c)),
where a, b, and c are the sides, and s is the semiperimeter ½(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 = ½bh.
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 √((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.

CHAPTER XVIII

THE LEADING PROPOSITIONS OF BOOK V

Book V treats of regular polygons and circles, and includes the computation of
the approximate value of π. 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 π. 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 π. 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 π, and of finding the two
formulas c = 2πr and a = πr2, 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 π.
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 = 2πr,
and a = πr2, 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 π. 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 2s sides, and then of 4s sides, and then of
8s sides, and in general of 2ns 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 2n · 3 sides;
From the square, regular polygons of 2n sides;
From the regular pentagon, regular polygons of 2n · 5 sides;
From the regular pentedecagon, regular polygons of 2n · 15 sides.
This gives us, for successive values of n, the following regular polygons of less
than 100 sides:

From 2n · 3, 3, 6, 12, 24, 48, 96;
From 2n, 2, 4, 8, 16, 32, 64;

From 2n · 5, 5, 10, 20, 40, 80;
From 2n · 15, 15, 30, 60.

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 2n · 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.

Mosaic from Damascus

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.

Arabic Pattern
The object of this proposition is, of course, to prepare the way for finding the
perimeter of a polygon of 2n 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 π, 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 SidePerimeter
s4 = 1.414... = 1.41 5.66
2
s8 = √(2-√(4-1.414 )) = 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 s12 and possibly s24, but aside
from this they are expected merely to know the process.
If it were possible to find the value of π exactly, we could find the
circumference exactly in terms of the radius, since c = 2πr. If we could find the
circumference exactly, we could find the area exactly, since a = πr2. If we could
find the area exactly in this form, π 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 π 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 √a by means of the straightedge and compasses, it
would be possible for us to square the circle if we could express π 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 π 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 π 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 π 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)2d2, which makes π = 256/81 or 3.1604....
Archimedes, using a plan somewhat similar to ours, found that π 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 √(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 π, 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 π 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 π, among them being the
following:
π/2 = 2/1 · 2/3 · 4/3 · 4/5 · 6/5 · 6/7 · 8/7 · 8/9 · .... (Wallis, 1616-1703)

4/π = 1 + 1/2
+ 9/2
+ 25/2
+ 49/2
+ .... (Brouncker, 1620-1684)

π/4 = 1 - 1/3 + 1/5 - 1/7 + .... (Gregory, 1638-1675)

π/6 = √(1/3) · (1 - 1/(3 · 3) + 1/(32 · 5) - 1/(33 · 7) + ...).

π/2 = (log i) / i. (Bernoulli)

π/(2√(3)) = 1 - 1/5 + 1/7 - 1/11 + 1/13 - 1/17 + 1/19...,
thus connecting the primes.

π2 / 16 = 1 - 1/22 - 1/32 + 1/42 - 1/52 + 1/62 - 1/72 - 1/82 + 1/92 + ....

π/2 = x/2 + sin x + (sin2 x) / 2 + (sin3 x) / 3 + .... (0 < x < 2π)

π/4 = 3/4 + 1/(2 · 3 · 4) - 1/(4 · 5 · 6) + 1/(6 · 7 · 8) - ....

2π2/3 = 7 - (1/(1 · 3) + 1/(3 · 6) + 1/(6 · 10) + ...).

π = 2n√(2 - √(2 + √(2 + √(2 + √(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 + eiπ = 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 = √(-1), and π. It was by means of this
relation that the transcendence of e was proved by the French mathematician
Hermite, and the transcendence of π by the German Lindemann.

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.

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:

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.

Choir of Lincoln Cathedral

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.

Gothic Designs employing Circles and
Bisected Angles

Gothic Designs employing Circles and
Squares

Gothic Designs employing Circles and the
Equilateral Triangle

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.

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.

An important application of the formula a = πr2 is seen in the area of the
annulus, or ring, the formula being a = πr2 - πr'2 = π(r2 - r'2) = π(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.

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 = πr2, and B = π(r/2)2. Hence B = 1/4A, 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.

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 nth 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.

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 a One plane does not determine a
plane. line.
Two intersecting lines Two intersecting planes
determine a plane. determine a line.
Two lines perpendicular to a Two planes perpendicular to a
plane are parallel. line are parallel.
If one of two parallel lines is If one of two parallel planes is
perpendicular to a plane, the 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?

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 d2 = a2 + b2 + c2, whence (a2 + b2 + c2) /
d2 = 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:

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 e3 is its volume and 2e3
is the volume of the required cube. Therefore the edge of the
required cube is e∛2. Now if e is given, it is not possible with the
straightedge and compasses to construct a line equal to e∛2,
although it is easy to construct one equal to e√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' + √(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
6s2, (1/2)s2√3, and s2√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
4 6 8 12 20
of Faces
r e√(3/8) (e/2)√3 e√(1/2) (e/4)√3(√5 + 1) e√((5 + √5)/8)
(e/2)√((25 +
r' e√(1/24) e/2 e√(1/6) (e√3)/12(√5 + 3)
11√5)/10)

a e2√3 6e2 2e2√3 3e2√(5(5 + 2√5)) (5e2)√3
v (e3/12)√2 e3 (e3/3)√2 ((e3)/4)(15 + 7√5) ((5e3)/12)(√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:

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:

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 = 2πrh 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 =
πr2h.
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 = 0.07πd2l. Taking π = 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.

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 = 1/3πr2h. 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)πh(r2 + r'2 + rr'). Here if
r' = 0, we have v = (1/3)πr2h, the volume of a cone. If r' = r, we
have v = (1/3)πh(r2 + r2 + r2) = πhr2, 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 s1, s2, ···, sn 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 (s1 + s2 + s3 + ··· + sn - 2f)180°.

But s1 + s2 + s3 + ··· + sn is twice the number of edges, because
each edge belongs to two faces.
∴ the sum of the angles of all the faces is
(2e - 2f)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 s1, and thinking of the
projections as traced by dotted lines on the opposite side of s1, this
sum is evidently equal to
(1) the sum of the angles in s1, or (s1 - 2) 180°, plus

(2) the sum of the angles on the other side of s1, or (s1 - 2)180°, plus

(3) the sum of the angles about the various points shown as inside
of s1, of which there are v - s1 points, about each of which the sum
of the angles is 360°, making (v - s1)360° in all.

Adding, we have
(s1 - 2)180° + (s1 - 2)180° + (v - s1)360° = ((s1 - 2) + (v - s1))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:

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.

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]

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 ∠A = ∠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 ⧍A'B'C' over so that it could
coincide with ⧍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.

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 = 4πr2. 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 2πr × 2r,
or 4πr2, and the two bases have an area πr2 + πr2, making the total area 6πr2.
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)πr3. 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 πr2 × 2r, or 2πr3, and (4/3)πr3 is 2/3 of 2πr3. 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.

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.

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)πr3, and OPRQ will generate a cylinder of volume πr3. Hence
the figure generated by ORQ will have a volume πr3 - (1/3)πr3, or
(2/3)πr3, 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 π(AD)2 - π(AB)2 = π(AC)2.

But π(AD)2 - π(AB)2 is the area of the ring generated by BD, a
section of x, and π(AC)2 is the corresponding section of the
hemisphere. Hence, by Cavalieri's Theorem,

(2/3)πr3 = the volume of the hemisphere.
∴ (4/3)πr3 = 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√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√2)/2.
Hence, by the Pythagorean Theorem,

r2 = x2 + ((x√2)/2)2 = (3/2)x2

∴ x = r√(2/3),

and the total surface is 6x2 = 4r2,

and the volume is x3 = (2/3)r3√(2/3).

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-Nairīzī, 171, 193, 214, 264
Al-Qifṫī, 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 π, 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