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THE TEACHING OF
GEOMETRY
BY
DAVID EUGENE SMITH
GINN AND COMPANY
BOSTON . NEW YORK . CHICAGO . LONDON
COPYRIGHT, 1911, BY DAVID EUGENE SMITH
ALL RIGHTS RESERVED
911.6
The Athenaeum Press
GINN AND COMPANY . PROPRIETORS
BOSTON . U.S.A.
PREFACE
A book upon the teaching of geometry may be planned in divers ways. It
may be written to exploit a new theory of geometry, or a new method of
presenting the science as we already have it. On the other hand, it may
be ultraconservative, making a plea for the ancient teaching and the
ancient geometry. It may be prepared for the purpose of setting forth
the work as it now is, or with the tempting but dangerous idea of
prophecy. It may appeal to the iconoclast by its spirit of destruction,
or to the disciples of _laissez faire_ by its spirit of conserving what
the past has bequeathed. It may be written for the few who always lead,
or think they lead, or for the many who are ranked by the few as
followers. And in view of these varied pathways into the joint domain of
geometry and education, a writer may well afford to pause before he sets
his pen to paper, and to decide with care the route that he will take.
At present in America we have a fairly well-defined body of matter in
geometry, and this occupies a fairly well-defined place in the
curriculum. There are not wanting many earnest teachers who would change
both the matter and the place in a very radical fashion. There are not
wanting others, also many in number, who are content with things as they
find them. But by far the largest part of the teaching body is of a mind
to welcome the natural and gradual evolution of geometry toward better
things, contributing to this evolution as much as it can, glad to know
the best that others have to offer, receptive of ideas that make for
better teaching, but out of sympathy with either the extreme of
revolution or the extreme of stagnation.
It is for this larger class, the great body of progressive teachers,
that this book is written. It stands for vitalizing geometry in every
legitimate way; for improving the subject matter in such manner as not
to destroy the pupil's interest; for so teaching geometry as to make it
appeal to pupils as strongly as any other subject in the curriculum; but
for the recognition of geometry for geometry's sake and not for the sake
of a fancied utility that hardly exists. Expressing full appreciation of
the desirability of establishing a motive for all studies, so as to have
the work proceed with interest and vigor, it does not hesitate to
express doubt as to certain motives that have been exploited, nor to
stand for such a genuine, thought-compelling development of the science
as is in harmony with the mental powers of the pupils in the American
high school.
For this class of teachers the author hopes that the book will prove of
service, and that through its perusal they will come to admire the
subject more and more, and to teach it with greater interest. It offers
no panacea, it champions no single method, but it seeks to set forth
plainly the reasons for teaching a geometry of the kind that we have
inherited, and for hoping for a gradual but definite improvement in the
science and in the methods of its presentation.
DAVID EUGENE SMITH
CONTENTS
CHAPTER PAGE
I. CERTAIN QUESTIONS NOW AT ISSUE 1
II. WHY GEOMETRY IS STUDIED 7
III. A BRIEF HISTORY OF GEOMETRY 26
IV. DEVELOPMENT OF THE TEACHING OF GEOMETRY 40
V. EUCLID 47
VI. EFFORTS AT IMPROVING EUCLID 57
VII. THE TEXTBOOK IN GEOMETRY 70
VIII. THE RELATION OF ALGEBRA TO GEOMETRY 84
IX. THE INTRODUCTION TO GEOMETRY 93
X. THE CONDUCT OF A CLASS IN GEOMETRY 108
XI. THE AXIOMS AND POSTULATES 116
XII. THE DEFINITIONS OF GEOMETRY 132
XIII. HOW TO ATTACK THE EXERCISES 160
XIV. BOOK I AND ITS PROPOSITIONS 165
XV. THE LEADING PROPOSITIONS OF BOOK II 201
XVI. THE LEADING PROPOSITIONS OF BOOK III 227
XVII. THE LEADING PROPOSITIONS OF BOOK IV 252
XVIII. THE LEADING PROPOSITIONS OF BOOK V 269
XIX. THE LEADING PROPOSITIONS OF BOOK VI 289
XX. THE LEADING PROPOSITIONS OF BOOK VII 303
XXI. THE LEADING PROPOSITIONS OF BOOK VIII 321
INDEX 335
THE TEACHING OF GEOMETRY
CHAPTER I
CERTAIN QUESTIONS NOW AT ISSUE
It is commonly said at the present time that the opening of the
twentieth century is a period of unusual advancement in all that has to
do with the school. It would be pleasant to feel that we are living in
such an age, but it is doubtful if the future historian of education
will find this to be the case, or that biographers will rank the leaders
of our generation relatively as high as many who have passed away, or
that any great movements of the present will be found that measure up to
certain ones that the world now recognizes as epoch-making. Every
generation since the invention of printing has been a period of
agitation in educational matters, but out of all the noise and
self-assertion, out of all the pretense of the chronic revolutionist,
out of all the sham that leads to dogmatism, so little is remembered
that we are apt to feel that the past had no problems and was content
simply to accept its inheritance. In one sense it is not a misfortune
thus to be blinded by the dust of present agitation and to be deafened
by the noisy clamor of the agitator, since it stirs us to action at
finding ourselves in the midst of the skirmish; but in another sense it
is detrimental to our progress, since we thereby tend to lose the idea
of perspective, and the coin comes to appear to our vision as large as
the moon.
In considering a question like the teaching of geometry, we at once find
ourselves in the midst of a skirmish of this nature. If we join
thoughtlessly in the noise, we may easily persuade ourselves that we are
waging a mighty battle, fighting for some stupendous principle, doing
deeds of great valor and of personal sacrifice. If, on the other hand,
we stand aloof and think of the present movement as merely a chronic
effervescence, fostered by the professional educator at the expense of
the practical teacher, we are equally shortsighted. Sir Conan Doyle
expressed this sentiment most delightfully in these words:
The dead are such good company that one may come to think too
little of the living. It is a real and pressing danger with
many of us that we should never find our own thoughts and our
own souls, but be ever obsessed by the dead.
In every generation it behooves the open-minded, earnest, progressive
teacher to seek for the best in the way of improvement, to endeavor to
sift the few grains of gold out of the common dust, to weigh the values
of proposed reforms, and to put forth his efforts to know and to use the
best that the science of education has to offer. This has been the
attitude of mind of the real leaders in the school life of the past, and
it will be that of the leaders of the future.
With these remarks to guide us, it is now proposed to take up the issues
of the present day in the teaching of geometry, in order that we may
consider them calmly and dispassionately, and may see where the
opportunities for improvement lie.
At the present time, in the educational circles of the United States,
questions of the following type are causing the chief discussion among
teachers of geometry:
1. Shall geometry continue to be taught as an application of logic, or
shall it be treated solely with reference to its applications?
2. If the latter is the purpose in view, shall the propositions of
geometry be limited to those that offer an opportunity for real
application, thus contracting the whole subject to very narrow
dimensions?
3. Shall a subject called geometry be extended over several years, as is
the case in Europe,[1] or shall the name be applied only to serious
demonstrative geometry[2] as given in the second year of the four-year
high school course in the United States at present?
4. Shall geometry be taught by itself, or shall it be either mixed with
algebra (say a day of one subject followed by a day of the other) or
fused with it in the form of a combined mathematics?
5. Shall a textbook be used in which the basal propositions are proved
in full, the exercises furnishing the opportunity for original work and
being looked upon as the most important feature, or shall one be
employed in which the pupil is expected to invent the proofs for the
basal propositions as well as for the exercises?
6. Shall the terminology and the spirit of a modified Euclid and
Legendre prevail in the future as they have in the past, or shall there
be a revolution in the use of terms and in the general statements of the
propositions?
7. Shall geometry be made a strong elective subject, to be taken only by
those whose minds are capable of serious work? Shall it be a required
subject, diluted to the comprehension of the weakest minds? Or is it
now, by proper teaching, as suitable for all pupils as is any other
required subject in the school curriculum? And in any case, will the
various distinct types of high schools now arising call for distinct
types of geometry?
This brief list might easily be amplified, but it is sufficiently
extended to set forth the trend of thought at the present time, and to
show that the questions before the teachers of geometry are neither
particularly novel nor particularly serious. These questions and others
of similar nature are really side issues of two larger questions of far
greater significance: (1) Are the reasons for teaching demonstrative
geometry such that it should be a required subject, or at least a
subject that is strongly recommended to all, whatever the type of high
school? (2) If so, how can it be made interesting?
The present work is written with these two larger questions in mind,
although it considers from time to time the minor ones already
mentioned, together with others of a similar nature. It recognizes that
the recent growth in popular education has brought into the high school
a less carefully selected type of mind than was formerly the case, and
that for this type a different kind of mathematical training will
naturally be developed. It proceeds upon the theory, however, that for
the normal mind,--for the boy or girl who is preparing to win out in the
long run,--geometry will continue to be taught as demonstrative
geometry, as a vigorous thought-compelling subject, and along the
general lines that the experience of the world has shown to be the best.
Soft mathematics is not interesting to this normal mind, and a sham
treatment will never appeal to the pupil; and this book is written for
teachers who believe in this principle, who believe in geometry for the
sake of geometry, and who earnestly seek to make the subject so
interesting that pupils will wish to study it whether it is required or
elective. The work stands for the great basal propositions that have
come down to us, as logically arranged and as scientifically proved as
the powers of the pupils in the American high school will permit; and it
seeks to tell the story of these propositions and to show their possible
and their probable applications in such a way as to furnish teachers
with a fund of interesting material with which to supplement the book
work of their classes.
After all, the problem of teaching any subject comes down to this: Get a
subject worth teaching and then make every minute of it interesting.
Pupils do not object to work if they like a subject, but they do object
to aimless and uninteresting tasks. Geometry is particularly fortunate
in that the feeling of accomplishment comes with every proposition
proved; and, given a class of fair intelligence, a teacher must be
lacking in knowledge and enthusiasm who cannot foster an interest that
will make geometry stand forth as the subject that brings the most
pleasure, and that seems the most profitable of all that are studied in
the first years of the high school.
Continually to advance, continually to attempt to make mathematics
fascinating, always to conserve the best of the old and to sift out and
use the best of the new, to believe that "mankind is better served by
nature's quiet and progressive changes than by earthquakes,"[3] to
believe that geometry as geometry is so valuable and so interesting that
the normal mind may rightly demand it,--this is to ally ourselves with
progress. Continually to destroy, continually to follow strange gods,
always to decry the best of the old, and to have no well-considered aim
in the teaching of a subject,--this is to join the forces of reaction,
to waste our time, to be recreant to our trust, to blind ourselves to
the failures of the past, and to confess our weakness as teachers. It is
with the desire to aid in the progressive movement, to assist those who
believe that real geometry should be recommended to all, and to show
that geometry is both attractive and valuable that this book is written.
FOOTNOTES:
[1] And really, though not nominally, in the United States, where the
first concepts are found in the kindergarten, and where an excellent
course in mensuration is given in any of our better class of
arithmetics. That we are wise in not attempting serious demonstrative
geometry much earlier seems to be generally conceded.
[2] The third stage of geometry as defined in the recent circular (No.
711) of the British Board of Education, London, 1909.
[3] The closing words of a sensible review of the British Board of
Education circular (No. 711), on "The Teaching of Geometry" (London,
1909), by H. S. Hall in the _School World_, 1909, p. 222.
CHAPTER II
WHY GEOMETRY IS STUDIED
With geometry, as with other subjects, it is easier to set forth what
are not the reasons for studying it than to proceed positively and
enumerate the advantages. Although such a negative course is not
satisfying to the mind as a finality, it possesses definite advantages
in the beginning of such a discussion as this. Whenever false prophets
arise, and with an attitude of pained superiority proclaim unworthy aims
in human life, it is well to show the fallacy of their position before
proceeding to a constructive philosophy. Taking for a moment this
negative course, let us inquire as to what are not the reasons for
studying geometry, or, to be more emphatic, as to what are not the
worthy reasons.
In view of a periodic activity in favor of the utilities of geometry, it
is well to understand, in the first place, that geometry is not studied,
and never has been studied, because of its positive utility in
commercial life or even in the workshop. In America we commonly allow at
least a year to plane geometry and a half year to solid geometry; but
all of the facts that a skilled mechanic or an engineer would ever need
could be taught in a few lessons. All the rest is either obvious or is
commercially and technically useless. We prove, for example, that the
angles opposite the equal sides of a triangle are equal, a fact that is
probably quite as obvious as the postulate that but one line can be
drawn through a given point parallel to a given line. We then prove,
sometimes by the unsatisfactory process of _reductio ad absurdum_, the
converse of this proposition,--a fact that is as obvious as most other
facts that come to our consciousness, at least after the preceding
proposition has been proved. And these two theorems are perfectly fair
types of upwards of one hundred sixty or seventy propositions comprising
Euclid's books on plane geometry. They are generally not useful in daily
life, and they were never intended to be so. There is an oft-repeated
but not well-authenticated story of Euclid that illustrates the feeling
of the founders of geometry as well as of its most worthy teachers. A
Greek writer, Stobaeus, 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 aesthetic, 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'etre_ 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.--ABBE 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 Rebiere, Mathematiques et
Mathematiciens, Paris, 1893.
FOOTNOTES:
[4] In an address in London, June 15, 1909, at a dinner to Sir Ernest
Shackelton.
[5] Governor Hughes, now Justice Hughes, of New York, at the Peary
testimonial on February 8, 1910, at New York City.
[6] The first work upon this subject, and indeed the first printed
treatise on curves in general, was written by the famous artist of
Nuernberg, Albrecht Duerer.
[7] Several of these writers are mentioned in Chapter IV.
[8] If any reader chances upon George Birkbeck's English translation of
Charles Dupin's "Mathematics Practically Applied," Halifax, 1854, he
will find that Dupin gave more good applications of geometry than all of
our American advocates of practical geometry combined.
[9] See, for example, Henrici's "Congruent Figures," London, 1879, and
the review of Borel's "Elements of Mathematics," by Professor Sisam in
the _Bulletin of the American Mathematical Society_, July, 1910, a
matter discussed later in this work.
[10] T. J. McCormack, "Why do we study Mathematics: a Philosophical and
Historical Retrospect," p. 9, Cedar Rapids, Iowa, 1910.
[11] Of the fair and candid arguments against the culture value of
mathematics, one of the best of the recent ones is that by G. F. Swain,
in the _Atti del IV Congresso Internazionale dei Matematici_, Rome,
1909, Vol. III, p. 361. The literature of this school is quite
extensive, but Perry's "England's Neglect of Science," London, 1900, and
"Discussion on the Teaching of Mathematics," London, 1901, are typical.
[12] In his novel, "The Morals of Marcus Ordeyne."
[13] G. W. L. Carson, "The Functions of Geometry as a Subject of
Education," p. 3, Tonbridge, 1910.
[14] It may well be, however, that the growing curriculum may justify
some reduction in the time formerly assigned to geometry, and any
reasonable proposition of this nature should be fairly met by teachers
of mathematics.
[15] Professor Muensterberg, in the _Metropolitan Magazine_ for July,
1910.
CHAPTER III
A BRIEF HISTORY OF GEOMETRY
The geometry of very ancient peoples was largely the mensuration of
simple areas and solids, such as is taught to children in elementary
arithmetic to-day. They early learned how to find the area of a
rectangle, and in the oldest mathematical records that have come down to
us there is some discussion of the area of triangles and the volume of
solids.
The earliest documents that we have relating to geometry come to us from
Babylon and Egypt. Those from Babylon are written on small clay tablets,
some of them about the size of the hand, these tablets afterwards having
been baked in the sun. They show that the Babylonians of that period
knew something of land measures, and perhaps had advanced far enough to
compute the area of a trapezoid. For the mensuration of the circle they
later used, as did the early Hebrews, the value [pi] = 3. A tablet in
the British Museum shows that they also used such geometric forms as
triangles and circular segments in astrology or as talismans.
The Egyptians must have had a fair knowledge of practical geometry long
before the date of any mathematical treatise that has come down to us,
for the building of the pyramids, between 3000 and 2400 B.C., required
the application of several geometric principles. Some knowledge of
surveying must also have been necessary to carry out the extensive
plans for irrigation that were executed under Amenemhat III, about 2200
B.C.
The first definite knowledge that we have of Egyptian mathematics comes
to us from a manuscript copied on papyrus, a kind of paper used about
the Mediterranean in early times. This copy was made by one Aah-mesu
(The Moon-born), commonly called Ahmes, who probably flourished about
1700 B.C. The original from which he copied, written about 2300 B.C.,
has been lost, but the papyrus of Ahmes, written nearly four thousand
years ago, is still preserved, and is now in the British Museum. In this
manuscript, which is devoted chiefly to fractions and to a crude
algebra, is found some work on mensuration. Among the curious rules are
the incorrect ones that the area of an isosceles triangle equals half
the product of the base and one of the equal sides; and that the area of
a trapezoid having bases _b_, _b'_, and the nonparallel sides each equal
to _a_, is 1/2_a_(_b_ + _b'_). One noteworthy advance appears, however.
Ahmes gives a rule for finding the area of a circle, substantially as
follows: Multiply the square on the radius by (16/9)^2, which is
equivalent to taking for [pi] the value 3.1605. This papyrus also
contains some treatment of the mensuration of solids, particularly with
reference to the capacity of granaries. There is also some slight
mention of similar figures, and an extensive treatment of unit
fractions,--fractions that were quite universal among the ancients. In
the line of algebra it contains a brief treatment of the equation of the
first degree with one unknown, and of progressions.[16]
Herodotus tells us that Sesostris, king of Egypt,[17] divided the land
among his people and marked out the boundaries after the overflow of the
Nile, so that surveying must have been well known in his day. Indeed,
the _harpedonaptae_, 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 deg., but that no other
regular polygons can be so placed. To his school is also due the proof
for the general case that the sum of the angles of a triangle equals two
right angles, the first knowledge of the size of each angle of a regular
polygon, and the construction of at least one star-polygon, the
star-pentagon, which became the badge of his fraternity. The brotherhood
founded by Pythagoras proved so offensive to the government that it was
dispersed before the death of the master. Pythagoras fled to Megapontum,
a seaport lying to the north of Crotona, and there he died about 501
B.C.[19]
[Illustration: FANCIFUL PORTRAIT OF PYTHAGORAS Calandri's Arithmetic,
1491]
For two centuries after Pythagoras geometry passed through a period of
discovery of propositions. The state of the science may be seen from
the fact that Oenopides of Chios, who flourished about 465 B.C., and who
had studied in Egypt, was celebrated because he showed how to let fall a
perpendicular to a line, and how to make an angle equal to a given
angle. A few years later, about 440 B.C., Hippocrates of Chios wrote the
first Greek textbook on mathematics. He knew that the areas of circles
are proportional to the squares on their radii, but was ignorant of the
fact that equal central angles or equal inscribed angles intercept equal
arcs.
Antiphon and Bryson, two Greek scholars, flourished about 430 B.C. The
former attempted to find the area of a circle by doubling the number of
sides of a regular inscribed polygon, and the latter by doing the same
for both inscribed and circumscribed polygons. They thus approximately
exhausted the area between the polygon and the circle, and hence this
method is known as the method of exhaustions.
About 420 B.C. Hippias of Elis invented a certain curve called the
quadratrix, by means of which he could square the circle and trisect any
angle. This curve cannot be constructed by the unmarked straightedge and
the compasses, and when we say that it is impossible to square the
circle or to trisect any angle, we mean that it is impossible by the
help of these two instruments alone.
During this period the great philosophic school of Plato (429-348 B.C.)
flourished at Athens, and to this school is due the first systematic
attempt to create exact definitions, axioms, and postulates, and to
distinguish between elementary and higher geometry. It was at this time
that elementary geometry became limited to the use of the compasses and
the unmarked straightedge, which took from this domain the possibility
of constructing a square equivalent to a given circle ("squaring the
circle"), of trisecting any given angle, and of constructing a cube that
should have twice the volume of a given cube ("duplicating the cube"),
these being the three famous problems of antiquity. Plato and his school
interested themselves with the so-called Pythagorean numbers, that is,
with numbers that would represent the three sides of a right triangle
and hence fulfill the condition that _a_^2 + _b_^2 = _c_^2. Pythagoras
had already given a rule that would be expressed in modern form, as
1/4(_m_^2 + 1)^2 = _m_^2 + 1/4(_m_^2 - 1)^2. The school of Plato found
that [(1/2_m_)^2 + 1]^2 = _m_^2 + [(1/2_m_)^2 - 1]^2. By giving various
values to _m_, different Pythagorean numbers may be found. Plato's
nephew, Speusippus (about 350 B.C.), wrote upon this subject. Such
numbers were known, however, both in India and in Egypt, long before
this time.
One of Plato's pupils was Philippus of Mende, in Egypt, who flourished
about 380 B.C. It is said that he discovered the proposition relating to
the exterior angle of a triangle. His interest, however, was chiefly in
astronomy.
Another of Plato's pupils was Eudoxus of Cnidus (408-355 B.C.). He
elaborated the theory of proportion, placing it upon a thoroughly
scientific foundation. It is probable that Book V of Euclid, which is
devoted to proportion, is essentially the work of Eudoxus. By means of
the method of exhaustions of Antiphon and Bryson he proved that the
pyramid is one third of a prism, and the cone is one third of a
cylinder, each of the same base and the same altitude. He wrote the
first textbook known on solid geometry.
The subject of conic sections starts with another pupil of Plato's,
Menaechmus, who lived about 350 B.C. He cut the three forms of conics
(the ellipse, parabola, and hyperbola) out of three different forms of
cone,--the acute-angled, right-angled, and obtuse-angled,--not noticing
that he could have obtained all three from any form of right circular
cone. It is interesting to see the far-reaching influence of Plato.
While primarily interested in philosophy, he laid the first scientific
foundations for a system of mathematics, and his pupils were the leaders
in this science in the generation following his greatest activity.
The great successor of Plato at Athens was Aristotle, the teacher of
Alexander the Great. He also was more interested in philosophy than in
mathematics, but in natural rather than mental philosophy. With him
comes the first application of mathematics to physics in the hands of a
great man, and with noteworthy results. He seems to have been the first
to represent an unknown quantity by letters. He set forth the theory of
the parallelogram of forces, using only rectangular components, however.
To one of his pupils, Eudemus of Rhodes, we are indebted for a history
of ancient geometry, some fragments of which have come down to us.
The first great textbook on geometry, and the greatest one that has ever
appeared, was written by Euclid, who taught mathematics in the great
university at Alexandria, Egypt, about 300 B.C. Alexandria was then
practically a Greek city, having been named in honor of Alexander the
Great, and being ruled by the Greeks.
In his work Euclid placed all of the leading propositions of plane
geometry then known, and arranged them in a logical order. Most
geometries of any importance written since his time have been based upon
Euclid, improving the sequence, symbols, and wording as occasion
demanded. He also wrote upon other branches of mathematics besides
elementary geometry, including a work on optics. He was not a great
creator of mathematics, but was rather a compiler of the work of others,
an office quite as difficult to fill and quite as honorable.
Euclid did not give much solid geometry because not much was known then.
It was to Archimedes (287-212 B.C.), a famous mathematician of Syracuse,
on the island of Sicily, that some of the most important propositions of
solid geometry are due, particularly those relating to the sphere and
cylinder. He also showed how to find the approximate value of [pi] by a
method similar to the one we teach to-day, proving that the real value
lay between 3 1/7 and 3 10/71. The story goes that the sphere and
cylinder were engraved upon his tomb, and Cicero, visiting Syracuse many
years after his death, found the tomb by looking for these symbols.
Archimedes was the greatest mathematical physicist of ancient times.
The Greeks contributed little more to elementary geometry, although
Apollonius of Perga, who taught at Alexandria between 250 and 200 B.C.,
wrote extensively on conic sections, and Hypsicles of Alexandria, about
190 B.C., wrote on regular polyhedrons. Hypsicles was the first Greek
writer who is known to have used sexagesimal fractions,--the degrees,
minutes, and seconds of our angle measure. Zenodorus (180 B.C.) wrote on
isoperimetric figures, and his contemporary, Nicomedes of Gerasa,
invented a curve known as the conchoid, by means of which he could
trisect any angle. Another contemporary, Diocles, invented the cissoid,
or ivy-shaped curve, by means of which he solved the famous problem of
duplicating the cube, that is, constructing a cube that should have
twice the volume of a given cube.
The greatest of the Greek astronomers, Hipparchus (180-125 B.C.), lived
about this period, and with him begins spherical trigonometry as a
definite science. A kind of plane trigonometry had been known to the
ancient Egyptians. The Greeks usually employed the chord of an angle
instead of the half chord (sine), the latter having been preferred by
the later Arab writers.
The most celebrated of the later Greek physicists was Heron of
Alexandria, formerly supposed to have lived about 100 B.C., but now
assigned to the first century A.D. His contribution to geometry was the
formula for the area of a triangle in terms of its sides a, b, and c,
with s standing for the semiperimeter 1/2(_a_ + _b_ + _c_). The formula
is [sqrt](_s_(_s_-_a_)(_s_-_b_)(_s_-_c_)).
Probably nearly contemporary with Heron was Menelaus of Alexandria, who
wrote a spherical trigonometry. He gave an interesting proposition
relating to plane and spherical triangles, their sides being cut by a
transversal. For the plane triangle _ABC_, the sides _a_, _b_, and _c_
being cut respectively in _X_, _Y_, and _Z_, the theorem asserts
substantially that
(_AZ_/_BZ_) . (_BX_/_CX_) . (_CY_/_AY_) = 1.
The most popular writer on astronomy among the Greeks was Ptolemy
(Claudius Ptolemaeus, 87-165 A.D.), who lived at Alexandria. He wrote a
work entitled "Megale Syntaxis" (The Great Collection), which his
followers designated as _Megistos_ (greatest), on which account the Arab
translators gave it the name "Almagest" (_al_ meaning "the"). He
advanced the science of trigonometry, but did not contribute to
geometry.
At the close of the third century Pappus of Alexandria (295 A.D.) wrote
on geometry, and one of his theorems, a generalized form of the
Pythagorean proposition, is mentioned in Chapter XVI of this work. Only
two other Greek writers on geometry need be mentioned. Theon of
Alexandria (370 A.D.), the father of the Hypatia who is the heroine of
Charles Kingsley's well-known novel, wrote a commentary on Euclid to
which we are indebted for some historical information. Proclus (410-485
A.D.) also wrote a commentary on Euclid, and much of our information
concerning the first Book of Euclid is due to him.
The East did little for geometry, although contributing considerably to
algebra. The first great Hindu writer was Aryabhatta, who was born in
476 A.D. He gave the very close approximation for [pi], expressed in
modern notation as 3.1416. He also gave rules for finding the volume of
the pyramid and sphere, but they were incorrect, showing that the Greek
mathematics had not yet reached the Ganges. Another Hindu writer,
Brahmagupta (born in 598 A.D.), wrote an encyclopedia of mathematics. He
gave a rule for finding Pythagorean numbers, expressed in modern symbols
as follows:
1/4((_p_^2/_q_) + _q_)^2 = 1/4((_p_^2/_q_) - _q_)^2 + _p_^2.
He also generalized Heron's formula by asserting that the area of an
inscribed quadrilateral of sides _a_, _b_, _c_, _d_, and semiperimeter
_s_, is [sqrt]((_s_ - _a_)(_s_ - _b_)(_s_ - _c_)(_s_ - _d_)).
The Arabs, about the time of the "Arabian Nights Tales" (800 A.D.), did
much for mathematics, translating the Greek authors into their language
and also bringing learning from India. Indeed, it is to them that modern
Europe owed its first knowledge of Euclid. They contributed nothing of
importance to elementary geometry, however.
The greatest of the Arab writers was Mohammed ibn Musa al-Khowarazmi
(820 A.D.). He lived at Bagdad and Damascus. Although chiefly interested
in astronomy, he wrote the first book bearing the name "algebra"
("Al-jabr wa'l-muq[=a]balah," Restoration and Equation), composed an
arithmetic using the Hindu numerals,[20] and paid much attention to
geometry and trigonometry.
Euclid was translated from the Arabic into Latin in the twelfth century,
Greek manuscripts not being then at hand, or being neglected because of
ignorance of the language. The leading translators were Athelhard of
Bath (1120), an English monk; Gherard of Cremona (1160), an Italian
monk; and Johannes Campanus (1250), chaplain to Pope Urban IV.
The greatest European mathematician of the Middle Ages was Leonardo of
Pisa[21] (_ca._ 1170-1250). He was very influential in making the
Hindu-Arabic numerals known in Europe, wrote extensively on algebra, and
was the author of one book on geometry. He contributed nothing to the
elementary theory, however. The first edition of Euclid was printed in
Latin in 1482, the first one in English appearing in 1570.
Our symbols are modern, + and - first appearing in a German work in
1489; = in Recorde's "Whetstone of Witte" in 1557; > and < in the works
of Harriot (1560-1621); and x 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 Geometrie" in 1637. From this springs the modern analytic geometry,
a subject that has revolutionized the methods of all mathematics. Most
of the subsequent discoveries in mathematics have been in higher
branches. To the great Swiss mathematician Euler (1707-1783) is due,
however, one proposition that has found its way into elementary
geometry, the one showing the relation between the number of edges,
vertices, and faces of a polyhedron.
There has of late arisen a modern elementary geometry devoted chiefly to
special points and lines relating to the triangle and the circle, and
many interesting propositions have been discovered. The subject is so
extensive that it cannot find any place in our crowded curriculum, and
must necessarily be left to the specialist.[22] Some idea of the nature
of the work may be obtained from a mention of a few propositions:
The medians of a triangle are concurrent in the centroid, or center of
gravity of the triangle.
The bisectors of the various interior and exterior angles of a triangle
are concurrent by threes in the incenter or in one of the three
excenters of the triangle.
The common chord of two intersecting circles is a special case of their
radical axis, and tangents to the circles from any point on the radical
axis are equal.
If _O_ is the orthocenter of the triangle _ABC_, and _X_, _Y_, _Z_ are
the feet of the perpendiculars from _A_, _B_, _C_ respectively, and _P_,
_Q_, _R_ are the mid-points of _a_, _b_, _c_ respectively, and _L_, _M_,
_N_ are the mid-points of _OA_, _OB_, _OC_ respectively; then the points
_L_, _M_, _N_; _P_, _Q_, _R_; _X_, _Y_, _Z_ all lie on a circle, the
"nine points circle."
In the teaching of geometry it adds a human interest to the subject to
mention occasionally some of the historical facts connected with it. For
this reason this brief sketch will be supplemented by many notes upon
the various important propositions as they occur in the several books
described in the later chapters of this work.
FOOTNOTES:
[16] It was published in German translation by A. Eisenlohr, "Ein
mathematisches Handbuch der alten Aegypter," Leipzig, 1877, and in
facsimile by the British Museum, under the title, "The Rhind Papyrus,"
in 1898.
[17] Generally known as Rameses II. He reigned in Egypt about 1350 B.C.
[18] Two excellent works on Thales and his successors, and indeed the
best in English, are the following: G. J. Allman, "Greek Geometry from
Thales to Euclid," Dublin, 1889; J. Gow, "A History of Greek
Mathematics," Cambridge, 1884. On all mathematical subjects the best
general history is that of M. Cantor, "Geschichte der Mathematik," 4
vols, Leipzig, 1880-1908.
[19] Another good work on Greek geometry, with considerable material on
Pythagoras, is by C. A. Bretschneider, "Die Geometrie und die Geometer
vor Eukleides," Leipzig, 1870.
[20] Smith and Karpinski, "The Hindu-Arabic Numerals," Boston, 1911.
[21] For a sketch of his life see Smith and Karpinski, loc. cit.
[22] Those who care for a brief description of this phase of the subject
may consult J. Casey, "A Sequel to Euclid," Dublin, fifth edition, 1888;
W. J. M'Clelland, "A Treatise on the Geometry of the Circle," New York,
1891; M. Simon, "Ueber die Entwicklung der Elementar-Geometrie im XIX.
Jahrhundert," Leipzig, 1906.
CHAPTER IV
DEVELOPMENT OF THE TEACHING OF GEOMETRY
We know little of the teaching of geometry in very ancient times, but we
can infer its nature from the teaching that is still seen in the native
schools of the East. Here a man, learned in any science, will have a
group of voluntary students sitting about him, and to them he will
expound the truth. Such schools may still be seen in India, Persia, and
China, the master sitting on a mat placed on the ground or on the floor
of a veranda, and the pupils reading aloud or listening to his words of
exposition.
In Egypt geometry seems to have been in early times mere mensuration,
confined largely to the priestly caste. It was taught to novices who
gave promise of success in this subject, and not to others, the idea of
general culture, of training in logic, of the cultivation of exact
expression, and of coming in contact with truth being wholly wanting.
In Greece it was taught in the schools of philosophy, often as a general
preparation for philosophic study. Thus Thales introduced it into his
Ionic school, Pythagoras made it very prominent in his great school at
Crotona in southern Italy (Magna Graecia), 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 Finaeus (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 Nuernberg,
founded in 1526, and the Cathedral school at Wuerttemberg (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 _lycees_ 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 Theaetetus, and also demonstrating
with perfect certainty what his predecessors had but
insufficiently proved. He flourished in the time of the first
Ptolemy, for Archimedes, who closely followed this ruler,[25]
speaks of Euclid. Furthermore it is related that Ptolemy one
time demanded of him if there was in geometry no shorter way
than that of the "Elements," to whom he replied that there was
no royal road to geometry.[26] He was therefore younger than
the pupils of Plato, but older than Eratosthenes and
Archimedes; for the latter were contemporary with one another,
as Eratosthenes somewhere says.[27]
Thus we have in a few lines, from one who lived perhaps seven or eight
hundred years after Euclid, nearly all that is known of the most famous
teacher of geometry that ever lived. Nevertheless, even this little
tells us about when he flourished, for Hermotimus and Philippus were
pupils of Plato, who died in 347 B.C., whereas Archimedes was born about
287 B.C. and was writing about 250 B.C. Furthermore, since Ptolemy I
reigned from 306 to 283 B.C., Euclid must have been teaching about 300
B.C., and this is the date that is generally assigned to him.
Euclid probably studied at Athens, for until he himself assisted in
transferring the center of mathematical culture to Alexandria, it had
long been in the Grecian capital, indeed since the time of Pythagoras.
Moreover, numerous attempts had been made at Athens to do exactly what
Euclid succeeded in doing,--to construct a logical sequence of
propositions; in other words, to write a textbook on plane geometry. It
was at Athens, therefore, that he could best have received the
inspiration to compose his "Elements."[28] After finishing his education
at Athens it is quite probable that he, like other savants of the
period, was called to Alexandria by Ptolemy Soter, the king, to assist
in establishing the great school which made that city the center of the
world's learning for several centuries. In this school he taught, and
here he wrote the "Elements" and numerous other works, perhaps ten in
all.
Although the Greek writers who may have known something of the life of
Euclid have little to say of him, the Arab writers, who could have known
nothing save from Greek sources, have allowed their imaginations the
usual latitude in speaking of him and of his labors. Thus
Al-Qif[t.][=i], who wrote in the thirteenth century, has this to say in
his biographical treatise "Ta'r[=i]kh al-[H.]ukam[=a]":
Euclid, son of Naucrates, grandson of Zenarchus, called the
author of geometry, a Greek by nationality, domiciled at
Damascus, born at Tyre, most learned in the science of
geometry, published a most excellent and most useful work
entitled "The Foundation or Elements of Geometry," a subject in
which no more general treatise existed before among the Greeks;
nay, there was no one even of later date who did not walk in
his footsteps and frankly profess his doctrine.
This is rather a specimen of the Arab tendency to manufacture history
than a serious contribution to the biography of Euclid, of whose
personal history we have only the information given by Proclus.
[Illustration: EUCLID
From an old print]
Euclid's works at once took high rank, and they are mentioned by various
classical authors. Cicero knew of them, and Capella (_ca._ 470 A.D.),
Cassiodorius (_ca._ 515 A.D.), and Boethius (_ca._ 480-524 A.D.) were
all more or less familiar with the "Elements." With the advance of the
Dark Ages, however, learning was held in less and less esteem, so that
Euclid was finally forgotten, and manuscripts of his works were either
destroyed or buried in some remote cloister. The Arabs, however, whose
civilization assumed prominence from about 750 A.D. to about 1500,
translated the most important treatises of the Greeks, and Euclid's
"Elements" among the rest. One of these Arabic editions an English monk
of the twelfth century, one Athelhard (AEthelhard) of Bath, found and
translated into Latin (_ca._ 1120 A.D.). A little later Gherard of
Cremona (1114-1187) made a new translation from the Arabic, differing in
essential features from that of Athelhard, and about 1260 Johannes
Campanus made still a third translation, also from Arabic into
Latin.[29] There is reason to believe that Athelhard, Campanus, and
Gherard may all have had access to an earlier Latin translation, since
all are quite alike in some particulars while diverging noticeably in
others. Indeed, there is an old English verse that relates:
The clerk Euclide on this wyse hit fonde
Thys craft of gemetry yn Egypte londe ...
Thys craft com into England, as y yow say,
Yn tyme of good Kyng Adelstone's day.
If this be true, Euclid was known in England as early as 924-940 A.D.
Without going into particulars further, it suffices to say that the
modern knowledge of Euclid came first through the Arabic into the Latin,
and the first printed edition of the "Elements" (Venice, 1482) was the
Campanus translation. Greek manuscripts now began to appear, and at the
present time several are known. There is a manuscript of the ninth
century in the Bodleian library at Oxford, one of the tenth century in
the Vatican, another of the tenth century in Florence, one of the
eleventh century at Bologna, and two of the twelfth century at Paris.
There are also fragments containing bits of Euclid in Greek, and going
back as far as the second and third century A.D. The first modern
translation from the Greek into the Latin was made by Zamberti (or
Zamberto),[30] and was printed at Venice in 1513. The first translation
into English was made by Sir Henry Billingsley and was printed in 1570,
sixteen years before he became Lord Mayor of London.
Proclus, in his commentary upon Euclid's work, remarks:
In the whole of geometry there are certain leading theorems,
bearing to those which follow the relation of a principle,
all-pervading, and furnishing proofs of many properties. Such
theorems are called by the name of _elements_, and their
function may be compared to that of the letters of the alphabet
in relation to language, letters being indeed called by the
same name in Greek [[Greek: stoicheia], stoicheia].[31]
This characterizes the work of Euclid, a collection of the basic
propositions of geometry, and chiefly of plane geometry, arranged in
logical sequence, the proof of each depending upon some preceding
proposition, definition, or assumption (axiom or postulate). The number
of the propositions of plane geometry included in the "Elements" is not
entirely certain, owing to some disagreement in the manuscripts, but it
was between one hundred sixty and one hundred seventy-five. It is
possible to reduce this number by about thirty or forty, because Euclid
included a certain amount of geometric algebra; but beyond this we
cannot safely go in the way of elimination, since from the very nature
of the "Elements" these propositions are basic. The efforts at revising
Euclid have been generally confined, therefore, to rearranging his
material, to rendering more modern his phraseology, and to making a book
that is more usable with beginners if not more logical in its
presentation of the subject. While there has been an improvement upon
Euclid in the art of bookmaking, and in minor matters of phraseology and
sequence, the educational gain has not been commensurate with the effort
put forth. With a little modification of Euclid's semi-algebraic Book II
and of his treatment of proportion, with some scattering of the
definitions and the inclusion of well-graded exercises at proper places,
and with attention to the modern science of bookmaking, the "Elements"
would answer quite as well for a textbook to-day as most of our modern
substitutes, and much better than some of them. It would, moreover, have
the advantage of being a classic,--somewhat the same advantage that
comes from reading Homer in the original instead of from Pope's metrical
translation. This is not a plea for a return to the Euclid text, but for
a recognition of the excellence of Euclid's work.
The distinctive feature of Euclid's "Elements," compared with the modern
American textbook, is perhaps this: Euclid begins a book with what seems
to him the easiest proposition, be it theorem or problem; upon this he
builds another; upon these a third, and so on, concerning himself but
little with the classification of propositions. Furthermore, he arranges
his propositions so as to construct his figures before using them. We,
on the other hand, make some little attempt to classify our propositions
within each book, and we make no attempt to construct our figures before
using them, or at least to prove that the constructions are correct.
Indeed, we go so far as to study the properties of figures that we
cannot construct, as when we ask for the size of the angle of a regular
heptagon. Thus Euclid begins Book I by a problem, to construct an
equilateral triangle on a given line. His object is to follow this by
problems on drawing a straight line equal to a given straight line, and
cutting off from the greater of two straight lines a line equal to the
less. He now introduces a theorem, which might equally well have been
his first proposition, namely, the case of the congruence of two
triangles, having given two sides and the included angle. By means of
his third and fourth propositions he is now able to prove the _pons
asinorum_, that the angles at the base of an isosceles triangle are
equal. We, on the other hand, seek to group our propositions where this
can conveniently be done, putting the congruent propositions together,
those about inequalities by themselves, and the propositions about
parallels in one set. The results of the two arrangements are not
radically different, and the effect of either upon the pupil's mind does
not seem particularly better than that of the other. Teachers who have
used both plans quite commonly feel that, apart from Books II and V,
Euclid is nearly as easily understood as our modern texts, if presented
in as satisfactory dress.
The topics treated and the number of propositions in the plane geometry
of the "Elements" are as follows:
Book I. Rectilinear figures 48
Book II. Geometric algebra 14
Book III. Circles 37
Book IV. Problems about circles 16
Book V. Proportion 25
Book VI. Applications of proportion 33
----
173
Of these we now omit Euclid's Book II, because we have an algebraic
symbolism that was unknown in his time, although he would not have used
it in geometry even had it been known. Thus his first proposition in
Book II is as follows:
If there be two straight lines, and one of them be cut into any
number of segments whatever, the rectangle contained by the two
straight lines is equal to the rectangles contained by the
uncut straight line and each of the segments.
This amounts to saying that if _x_ = _p_ + _q_ + _r_ + ..., then
_ax_ = _ap_ + _aq_ + _ar_ + .... We also materially simplify Euclid's
Book V. He, for example, proves that "If four magnitudes be
proportional, they will also be proportional alternately." This he
proves generally for any kind of magnitude, while we merely prove it for
numbers having a common measure. We say that we may substitute for the
older form of proportion, namely,
_a_ : _b_ = _c_ : _d_,
the fractional form _a_/_b_ = _c_/_d_.
From this we have _ad_ = _bc_.
Whence _a_/_c_ = _b_/_d_.
In this work we assume that we may multiply equals by _b_ and _d_. But
suppose _b_ and _d_ are cubes, of which, indeed, we do not even know the
approximate numerical measure; what shall we do? To Euclid the
multiplication by a cube or a polygon or a sphere would have been
entirely meaningless, as it always is from the standpoint of pure
geometry. Hence it is that our treatment of proportion has no serious
standing in geometry as compared with Euclid's, and our only
justification for it lies in the fact that it is easier. Euclid's
treatment is much more rigorous than ours, but it is adapted to the
comprehension of only advanced students, while ours is merely a
confession, and it should be a frank confession, of the weakness of our
pupils, and possibly, at times, of ourselves.
If we should take Euclid's Books II and V for granted, or as
sufficiently evident from our study of algebra, we should have remaining
only one hundred thirty-four propositions, most of which may be
designated as basal propositions of plane geometry. Revise Euclid as we
will, we shall not be able to eliminate any large number of his
fundamental truths, while we might do much worse than to adopt these one
hundred thirty-four propositions _in toto_ as the bases, and indeed as
the definition, of elementary plane geometry.
=Bibliography.= Heath, The Thirteen Books of Euclid's Elements,
3 vols., Cambridge, 1908; Frankland, The First Book of Euclid,
Cambridge, 1906; Smith, Dictionary of Greek and Roman
Biography, article Eukleides; Simon, Euclid und die sechs
planimetrischen Buecher, Leipzig, 1901; Gow, History of Greek
Mathematics, Cambridge, 1884, and any of the standard histories
of mathematics. Both Heath and Simon give extensive
bibliographies. The latest standard Greek and Latin texts are
Heiberg's, published by Teubner of Leipzig.
FOOTNOTES:
[23] Riccardi, Saggio di una bibliografia Euclidea, Part I, p. 3,
Bologna, 1887. Riccardi lists well towards two thousand editions.
[24] Hermotimus of Colophon and Philippus of Mende.
[25] Literally, "Who closely followed the first," i.e. the first
Ptolemy.
[26] Menaechmus is said to have replied to a similar question of
Alexander the Great: "O King, through the country there are royal roads
and roads for common citizens, but in geometry there is one road for
all."
[27] This is also shown in a letter from Archimedes to Eratosthenes,
recently discovered by Heiberg.
[28] On this phase of the subject, and indeed upon Euclid and his
propositions and works in general, consult T. L. Heath, "The Thirteen
Books of Euclid's Elements," 3 vols., Cambridge, 1908, a masterly
treatise of which frequent use has been made in preparing this work.
[29] A contemporary copy of this translation is now in the library of
George A. Plimpton, Esq., of New York. See the author's "Rara
Arithmetica," p. 433, Boston, 1909.
[30] A beautiful vellum manuscript of this translation is in the library
of George A. Plimpton, Esq., of New York. See the author's "Rara
Arithmetica," p. 481, Boston, 1909.
[31] Heath, loc. cit., Vol. I, p. 114.
CHAPTER VI
EFFORTS AT IMPROVING EUCLID
From time to time an effort is made by some teacher, or association of
teachers, animated by a serious desire to improve the instruction in
geometry, to prepare a new syllabus that shall mark out some "royal
road," and it therefore becomes those who are interested in teaching to
consider with care the results of similar efforts in recent years. There
are many questions which such an attempt suggests: What is the real
purpose of the movement? What will the teaching world say of the result?
Shall a reckless, ill-considered radicalism dominate the effort,
bringing in a distasteful terminology and symbolism merely for its
novelty, insisting upon an ultralogical treatment that is beyond the
powers of the learner, rearranging the subject matter to fit some narrow
notion of the projectors, seeking to emasculate mathematics by looking
only to the applications, riding some little hobby in the way of some
particular class of exercises, and cutting the number of propositions to
a minimum that will satisfy the mere demands of the artisan? Such are
some of the questions that naturally arise in the mind of every one who
wishes well for the ancient science of geometry.
It is not proposed in this chapter to attempt to answer these questions,
but rather to assist in understanding the problem by considering the
results of similar attempts. If it shall be found that syllabi have
been prepared under circumstances quite as favorable as those that
obtain at present, and if these syllabi have had little or no real
influence, then it becomes our duty to see if new plans may be worked
out so as to be more successful than their predecessors. If the older
attempts have led to some good, it is well to know what is the nature of
this good, to the end that new efforts may also result in something of
benefit to the schools.
It is proposed in this chapter to call attention to four important
syllabi, setting forth briefly their distinguishing features and drawing
some conclusions that may be helpful in other efforts of this nature.
In England two noteworthy attempts have been made within a century,
looking to a more satisfactory sequence and selection of propositions
than is found in Euclid. Each began with a list of propositions arranged
in proper sequence, and each was thereafter elaborated into a textbook.
Neither accomplished fully the purpose intended, but each was
instrumental in provoking healthy discussion and in improving the texts
from which geometry is studied.
The first of these attempts was made by Professor Augustus de Morgan,
under the auspices of the Society for the Diffusion of Useful Knowledge,
and it resulted in a textbook, including "plane, solid, and spherical"
geometry, in six books. According to De Morgan's plan, plane geometry
consisted of three books, the number of propositions being as follows:
Book I. Rectilinear figures 60
Book II. Ratio, proportion, applications 69
Book III. The circle 65
----
Total for plane geometry 194
Of the 194 propositions De Morgan selected 114 with their corollaries as
necessary for a beginner who is teaching himself.
In solid geometry the plan was as follows:
Book IV. Lines in different planes, solids
contained by planes 52
Book V. Cylinder, cone, sphere 25
Book VI. Figures on a sphere 42
----
Total for solid geometry 119
Of these 119 propositions De Morgan selected 76 with their corollaries
as necessary for a beginner, thus making 190 necessary propositions out
of 305 desirable ones, besides the corollaries in plane and solid
geometry. In other words, of the desirable propositions he considered
that about two thirds are absolutely necessary.
It is interesting to note, however, that he summed up the results of his
labors by saying:
It will be found that the course just laid down, excepting the
sixth book of it only, is not of much greater extent, nor very
different in point of matter from that of Euclid, whose
"Elements" have at all times been justly esteemed a model not
only of easy and progressive instruction in geometry, but of
accuracy and perspicuity in reasoning.
De Morgan's effort, essentially that of a syllabus-maker rather than a
textbook writer, although it was published under the patronage of a
prominent society with which were associated the names of men like Henry
Hallam, Rowland Hill, Lord John Russell, and George Peacock, had no
apparent influence on geometry either in England or abroad. Nevertheless
the syllabus was in many respects excellent; it rearranged the matter,
it classified the propositions, it improved some of the terminology, and
it reduced the number of essential propositions; it had the assistance
of De Morgan's enthusiasm and of the society with which he was so
prominently connected, and it was circulated with considerable
generosity throughout the English-speaking world; but in spite of all
this it is to-day practically unknown.
A second noteworthy attempt in England was made about a quarter of a
century ago by a society that was organized practically for this very
purpose, the Association for the Improvement of Geometrical Teaching.
This society was composed of many of the most progressive teachers in
England, and it included in its membership men of high standing in
mathematics in the universities. As a result of their labors a syllabus
was prepared, which was elaborated into a textbook, and in 1889 a
revised syllabus was issued.
As to the arrangement of matter, the syllabus departs from Euclid
chiefly by separating the problems from the theorems, as is the case in
our American textbooks, and in improving the phraseology. The course is
preceded by some simple exercises in the use of the compasses and ruler,
a valuable plan that is followed by many of the best teachers
everywhere. Considerable attention is paid to logical processes before
beginning the work, such terms as "contrapositive" and "obverse," and
such rules as the "rule of conversion" and the "rule of identity" being
introduced before any propositions are considered.
The arrangement of the work and the number of propositions in plane
geometry are as follows:
Book I. The straight line 51
Book II. Equality of areas 19
Book III. The circle 42
Book IV. Ratio and proportion 32
Book V. Proportion 24
----
Total for plane geometry 168
Here, then, is the result of several years of labor by a somewhat
radical organization, fostered by excellent mathematicians, and carried
on in a country where elementary geometry is held in highest esteem, and
where Euclid was thought unsuited to the needs of the beginner. The
number of propositions remains substantially the same as in Euclid, and
the introduction of some unusable logic tends to counterbalance the
improvement in sequence of the propositions. The report provoked
thought; it shook the Euclid stronghold; it was probably instrumental in
bringing about the present upheaval in geometry in England, but as a
working syllabus it has not appealed to the world as the great
improvement upon Euclid's "Elements" that was hoped by many of its early
advocates.
The same association published later, and republished in 1905, a "Report
on the Teaching of Geometry," in which it returned to Euclid, modifying
the "Elements" by omitting certain propositions, changing the order and
proof of others, and introducing a few new theorems. It seems to reduce
the propositions to be proved in plane geometry to about one hundred
fifteen, and it recommends the omission of the incommensurable case.
This number is, however, somewhat misleading, for Euclid frequently puts
in one proposition what we in America, for educational reasons, find it
better to treat in two, or even three, propositions. This report,
therefore, reaches about the same conclusion as to the geometric facts
to be mastered as is reached by our later textbook writers in America.
It is not extreme, and it stands for good mathematics.
In the United States the influence of our early wars with England, and
the sympathy of France at that time, turned the attention of our
scholars of a century ago from Cambridge to Paris as a mathematical
center. The influx of French mathematics brought with it such works as
Legendre's geometry (1794) and Bourdon's algebra, and made known the
texts of Lacroix, Bertrand, and Bezout. Legendre's geometry was the
result of the efforts of a great mathematician at syllabus-making, a
natural thing in a country that had early broken away from Euclid.
Legendre changed the Greek sequence, sought to select only propositions
that are necessary to a good understanding of the subject, and added a
good course in solid geometry. His arrangement, with the number of
propositions as given in the Davies translation, is as follows:
Book I. Rectilinear figures 31
Book II. Ratio and proportion 14
Book III. The circle 48
Book IV. Proportions of figures and areas 51
Book V. Polygons and circles 17
----
Total for plane geometry 161
Legendre made, therefore, practically no reduction in the number of
Euclid's propositions, and his improvement on Euclid consisted chiefly
in his separation of problems and theorems, and in a less rigorous
treatment of proportion which boys and girls could comprehend.
D'Alembert had demanded that the sequence of propositions should be
determined by the order in which they had been discovered, but Legendre
wisely ignored such an extreme and gave the world a very usable book.
The principal effect of Legendre's geometry in America was to make every
textbook writer his own syllabus-maker, and to put solid geometry on a
more satisfactory footing. The minute we depart from a standard text
like Euclid's, and have no recognized examining body, every one is free
to set up his own standard, always within the somewhat uncertain
boundary prescribed by public opinion and by the colleges. The efforts
of the past few years at syllabus-making have been merely attempts to
define this boundary more clearly.
Of these attempts two are especially worthy of consideration as having
been very carefully planned and having brought forth such definite
results as to appeal to a large number of teachers. Other syllabi have
been made and are familiar to many teachers, but in point of clearness
of purpose, conciseness of expression, and form of publication they have
not been such as to compare with the two in question.
The first of these is the Harvard syllabus, which is placed in the hands
of students for reference when trying the entrance examinations of that
university, a plan not followed elsewhere. It sets forth the basal
propositions that should form the essential part of the student's
preparation, and that are necessary and sufficient for proving any
"original proposition" (to take the common expression) that may be set
on the examination. The propositions are arranged by books as follows:
Book I. Angles, triangles, parallels 25