eighty, because he several times combined two or more in one. These we
may reduce to about one hundred thirty with perfect safety, or less than
one a day for a school year, but to reduce still further is undesirable
as well as unnecessary. It would not be difficult to dispense with a few
more; indeed, we might dispense with thirty more if we should set about
it, although we must never forget that a goodly number in addition to
those needed for the logical sequence are necessary for the wide range
of exercises that are offered. But let it be clear that if we teach 100
instead of 130, our results are liable to be about 100/130 as
satisfactory. We may theorize on pedagogy as we please, but geometry
will pay us about in proportion to what we give.

And as to the exercises, what is the basis of selection? In general, let
it be said that any exercise that pretends to be real should be so, and
that words taken from science or measurements do not necessarily make
the problem genuine. To take a proposition and apply it in a manner that
the world never sanctions is to indulge in deceit. On the other hand,
wholly to neglect the common applications of geometry to handwork of
various kinds is to miss one of our great opportunities to make the
subject vital to the pupil, to arouse new interest, and to give a
meaning to it that is otherwise wanting. It should always be remembered
that mental discipline, whatever the phrase may mean, can as readily be
obtained from a genuine application of a theorem as from a mere
geometric puzzle. On the other hand, it is evident that not more than 25
per cent of propositions have any genuine applications outside of
geometry, and that if we are to attempt any applications at all, these
must be sought mainly in the field of pure geometry. In the exercises,
therefore, we seek to-day a sane and a balanced book, giving equal
weight to theory and to practice, to the demands of the artisan and to
those of the mathematician, to the applications of concrete science and
to those of pure geometry, thus making a fusion of pure and applied
mathematics, with the latter as prominent as the supply of genuine
problems permits. The old is not all bad and the new is not all good,
and a textbook is a success in so far as it selects boldly the good that
is in the old and rejects with equal boldness the bad that is in the
new.

Lest the nature of the exercises of geometry may be misunderstood, it is
well that we consider for a moment what constitutes a genuine
application of the subject. It is the ephemeral fashion just at present
in America to call these genuine applications by the name of "real
problems." The name is an unfortunate importation, but that is not a
matter of serious moment. The important thing is that we should know
what makes a problem "real" to the pupil of geometry, especially as the
whole thing is coming rapidly into disrepute through the mistaken zeal
of some of its supporters.

A real problem is a problem that the average citizen may sometime be
called upon to solve; that, if so called upon, he will solve in the
manner indicated; and that is expressed in terms that are familiar to
the pupil.

This definition, which seems fairly to state the conditions under which
a problem can be called "real" in the schoolroom, involves three points:
(1) people must be liable to meet such a problem; (2) in that case they
will solve it in the way suggested by the book; (3) it must be clothed
in language familiar to the pupil. For example, let the problem be to
find the dimensions of a rectangular field, the data being the area of
the field and the area of a road four rods wide that is cut from three
sides of the field. As a real problem this is ridiculous, since no one
would ever meet such a case outside the puzzle department of a
schoolroom. Again, if by any stretch of a vigorous imagination any human
being should care to find the area of a piece of glass, bounded by the
arcs of circles, in a Gothic window in York Minster, it is fairly
certain that he would not go about it in the way suggested in some of
the earnest attempts that have been made by several successful teachers
to add interest to geometry. And for the third point, a problem is not
real to a pupil simply because it relates to moments of inertia or the
tensile strength of a steel bar. Indeed, it is unreal precisely because
it does talk of these things at a time when they are unfamiliar, and
properly so, to the pupil.

It must not be thought that puzzle problems, and unreal problems
generally, have no value. All that is insisted upon is that such
problems as the above are not "real," and that about 90 per cent of
problems that go by this name are equally lacking in the elements that
make for reality in this sense of the word. For the other 10 per cent of
such problems we should be thankful, and we should endeavor to add to
the number. As for the great mass, however, they are no better than
those that have stood the test of generations, and by their pretense
they are distinctly worse.

It is proper, however, to consider whether a teacher is not justified in
relating his work to those geometric forms that are found in art, let us
say in floor patterns, in domes of buildings, in oilcloth designs, and
the like, for the purpose of arousing interest, if for no other reason.
The answer is apparent to any teacher: It is certainly justifiable to
arouse the pupil's interest in his subject, and to call his attention to
the fact that geometric design plays an important part in art; but we
must see to it that our efforts accomplish this purpose. To make a
course in geometry one on oilcloth design would be absurd, and nothing
more unprofitable or depressing could be imagined in connection with
this subject. Of course no one would advocate such an extreme, but it
sometimes seems as if we are getting painfully near it in certain
schools.

A pupil has a passing interest in geometric design. He should learn to
use the instruments of geometry, and he learns this most easily by
drawing a few such patterns. But to keep him week after week on
questions relating to such designs of however great variety, and
especially to keep him upon designs relating to only one or two types,
is neither sound educational policy nor even common sense. That this
enthusiastic teacher or that one succeeds by such a plan is of no
significance; it is the enthusiasm that succeeds, not the plan.

The experience of the world is that pupils of geometry like to use the
subject practically, but that they are more interested in the pure
theory than in any fictitious applications, and this is why pure
geometry has endured, while the great mass of applied geometry that was
brought forward some three hundred years ago has long since been
forgotten. The question of the real applications of the subject is
considered in subsequent chapters.

In Chapter VI we considered the question of the number of regular
propositions to be expected in the text, and we have just considered the
nature of the exercises which should follow those propositions. It is
well to turn our attention next to the nature of the proofs of the basal
theorems. Shall they appear in full? Shall they be merely suggested
demonstrations? Shall they be only a series of questions that lead to
the proof? Shall the proofs be omitted entirely? Or shall there be some
combination of these plans?

The natural temptation in the nervous atmosphere of America is to listen
to the voice of the mob and to proceed at once to lynch Euclid and every
one who stands for that for which the "Elements" has stood these two
thousand years. This is what some who wish to be considered as educators
tend to do; in the language of the mob, to "smash things"; to call
reactionary that which does not conform to their ephemeral views. It is
so easy to be an iconoclast, to think that _cui bono_ is a conclusive
argument, to say so glibly that Raphael was not a great painter,--to do
anything but construct. A few years ago every one must take up with the
heuristic method developed in Germany half a century back and containing
much that was commendable. A little later one who did not believe that
the Culture Epoch Theory was vital in education was looked upon with
pity by a considerable number of serious educators. A little later the
man who did not think that the principle of Concentration in education
was a _regula aurea_ was thought to be hopeless. A little later it may
have been that Correlation was the saving factor, to be looked upon in
geometry teaching as a guiding beacon, even as the fusion of all
mathematics is the temporary view of a few enthusiasts to-day.[35]

And just now it is vocational training that is the catch phrase, and to
many this phrase seems to sound the funeral knell of the standard
textbook in geometry. But does it do so? Does this present cry of the
pedagogical circle really mean that we are no longer to have geometry
for geometry's sake? Does it mean that a panacea has been found for the
ills of memorizing without understanding a proof in the class of a
teacher who is so inefficient as to allow this kind of work to go on?
Does it mean that a teacher who does not see the human side of
geometry, who does not know the real uses of geometry, and who has no
faculty of making pupils enthusiastic over geometry,--that this teacher
is to succeed with some scrappy, weak, pretending apology for a real
work on the subject?

No one believes in stupid teaching, in memorizing a textbook, in having
a book that does all the work for a pupil, or in any of the other ills
of inefficient instruction. On the other hand, no fair-minded person can
condemn a type of book that has stood for generations until something
besides the mere transient experiments of the moment has been suggested
to replace it. Let us, for example, consider the question of having the
basal propositions proved in full, a feature that is so easy to condemn
as leading to memorizing.

The argument in favor of a book with every basal proposition proved in
full, or with most of them so proved, the rest having only suggestions
for the proof, is that the pupil has before him standard forms
exhibiting the best, most succinct, most clearly stated demonstrations
that geometry contains. The demonstrations stand for the same thing that
the type problems stand for in algebra, and are generally given in full
in the same way. The argument against the plan is that it takes away the
pupil's originality by doing all the work for him, allowing him to
merely memorize the work. Now if all there is to geometry were in the
basal propositions, this argument might hold, just as it would hold in
algebra in case there were only those exercises that are solved in full.
But just as this is not the case in algebra, the solved exercises
standing as types or as bases for the pupil's real work, so the
demonstrated proposition forms a relatively small part of geometry,
standing as a type, a basis for the more important part of the work.
Moreover, a pupil who uses a syllabus is exposed to a danger that should
be considered, namely, that of dishonesty. Any textbook in geometry will
furnish the proofs of most of the propositions in a syllabus, whatever
changes there may be in the sequence, and it is not a healthy condition
of mind that is induced by getting the proofs surreptitiously. Unless a
teacher has more time for the course than is usually allowed, he cannot
develop the new work as much as is necessary with only a syllabus, and
the result is that a pupil gets more of his work from other books and
has less time for exercises. The question therefore comes to this: Is it
better to use a book containing standard forms of proof for the basal
propositions, and have time for solving a large number of original
exercises and for seeking the applications of geometry? Or is it better
to use a book that requires more time on the basal propositions, with
the danger of dishonesty, and allows less time for solving originals? To
these questions the great majority of teachers answer in favor of the
textbook with most of the basal propositions fully demonstrated. In
general, therefore, it is a good rule to use the proofs of the basal
propositions as models, and to get the original work from the exercises.
Unless we preserve these model proofs, or unless we supply them with a
syllabus, the habit of correct, succinct self-expression, which is one
of the chief assets of geometry, will tend to become atrophied. So
important is this habit that "no system of education in which its
performance is neglected can hope or profess to evolve men and women who
are competent in the full sense of the word. So long as teachers of
geometry neglect the possibilities of the subject in this respect, so
long will the time devoted to it be in large part wasted, and so long
will their pupils continue to imbibe the vicious idea that it is much
more important to be able to do a thing than to say how it can be
done."[36]

It is here that the chief danger of syllabus-teaching lies, and it is
because of this patent fact that a syllabus without a carefully selected
set of model proofs, or without the unnecessary expenditure of time by
the class, is a dangerous kind of textbook.

What shall then be said of those books that merely suggest the proofs,
or that give a series of questions that lead to the demonstrations?
There is a certain plausibility about such a plan at first sight. But it
is easily seen to have only a fictitious claim to educational value. In
the first place, it is merely an attempt on the part of the book to take
the place of the teacher and to "develop" every lesson by the heuristic
method. The questions are so framed as to admit, in most cases, of only
a single answer, so that this answer might just as well be given instead
of the question. The pupil has therefore a proof requiring no more
effort than is the case in the standard form of textbook, but not given
in the clear language of a careful writer. Furthermore, the pupil is
losing here, as when he uses only a syllabus, one of the very things
that he should be acquiring, namely, the habit of reading mathematics.
If he met only syllabi without proofs, or "suggestive" geometries, or
books that endeavored to question every proof out of him, he would be in
a sorry plight when he tried to read higher mathematics, or even other
elementary treatises. It is for reasons such as these that the heuristic
textbook has never succeeded for any great length of time or in any wide
territory.

And finally, upon this point, shall the demonstrations be omitted
entirely, leaving only the list of propositions,--in other words, a pure
syllabus? This has been sufficiently answered above. But there is a
modification of the pure syllabus that has much to commend itself to
teachers of exceptional strength and with more confidence in themselves
than is usually found. This is an arrangement that begins like the
ordinary textbook and, after the pupil has acquired the form of proof,
gradually merges into a syllabus, so that there is no temptation to go
surreptitiously to other books for help. Such a book, if worked out with
skill, would appeal to an enthusiastic teacher, and would accomplish the
results claimed for the cruder forms of manual already described. It
would not be in general as safe a book as the standard form, but with
the right teacher it would bring good results.

In conclusion, there are two types of textbook that have any hope of
success. The first is the one with all or a large part of the basal
propositions demonstrated in full, and with these propositions not
unduly reduced in number. Such a book should give a large number of
simple exercises scattered through the work, with a relatively small
number of difficult ones. It should be modern in its spirit, with
figures systematically lettered, with each page a unit as far as
possible, and with every proof a model of clearness of statement and
neatness of form. Above all, it should not yield to the demand of a few
who are always looking merely for something to change, nor should it in
a reactionary spirit return to the old essay form of proof, which
hinders the pupil at this stage.

The second type is the semisyllabus, otherwise with all the spirit of
the first type. In both there should be an honest fusion of pure and
applied geometry, with no exercises that pretend to be practical
without being so, with no forced applications that lead the pupil to
measure things in a way that would appeal to no practical man, with no
merely narrow range of applications, and with no array of difficult
terms from physics and engineering that submerge all thought of
mathematics in the slough of despond of an unknown technical vocabulary.
Outdoor exercises, even if somewhat primitive, may be introduced, but it
should be perfectly understood that such exercises are given for the
purpose of increasing the interest in geometry, and they should be
abandoned if they fail of this purpose.

     =Bibliography.= For a list of standard textbooks issued prior
     to the present generation, consult the bibliography in Stamper,
     History of the Teaching of Geometry, New York, 1908.

FOOTNOTES:

[35] For some classes of schools and under certain circumstances courses
in combined mathematics are very desirable. All that is here insisted
upon is that any general fusion all along the line would result in weak,
insipid, and uninteresting mathematics. A beginning, inspirational
course in combined mathematics has a good reason for being in many high
schools in spite of its manifest disadvantages, and such a course may be
developed to cover all of the required mathematics given in certain
schools.

[36] Carson, loc. cit., p. 15.

CHAPTER VIII

THE RELATION OF ALGEBRA TO GEOMETRY

From the standpoint of theory there is or need be no relation whatever
between algebra and geometry. Algebra was originally the science of the
equation, as its name[37] indicates. This means that it was the science
of finding the value of an unknown quantity in a statement of equality.
Later it came to mean much more than this, and Newton spoke of it as
universal arithmetic, and wrote an algebra with this title. At present
the term is applied to the elements of a science in which numbers are
represented by letters and in which certain functions are studied,
functions which it is not necessary to specify at this time. The work
relates chiefly to functions involving the idea of number. In geometry,
on the other hand, the work relates chiefly to form. Indeed, in pure
geometry number plays practically no part, while in pure algebra form
plays practically no part.

In 1687 the great French philosopher, Descartes, wishing to picture
certain algebraic functions, wrote a work of about a hundred pages,
entitled "La Géométrie," and in this he showed a correspondence between
the numbers of algebra (which may be expressed by letters) and the
concepts of geometry. This was the first great step in the analytic
geometry that finally gave us the graph in algebra. Since then there
have been brought out from time to time other analogies between algebra
and geometry, always to the advantage of each science. This has led to a
desire on the part of some teachers to unite algebra and geometry into
one science, having simply a class in mathematics without these special
names.

It is well to consider the advantages and the disadvantages of such a
plan, and to decide as to the rational attitude to be taken by teachers
concerning the question at issue. On the side of advantages it is
claimed that there is economy of time and of energy. If a pupil is
studying formulas, let the formulas of geometry be studied; if he is
taking up ratio and proportion; let him do so for algebra and geometry
at the same time; if he is solving quadratics, let him apply them at
once to certain propositions concerning secants; and if he is proving
that (_a_ + _b_)^2 equals _a_^2 + 2_ab_ + _b_^2, let him do so by
algebra and by geometry simultaneously. It is claimed that not only is
there economy in this arrangement, but that the pupil sees mathematics
as a whole, and thus acquires more of a mastery than comes by our
present "tandem arrangement."

On the side of disadvantages it may be asked if the same arguments would
not lead us to teach Latin and Greek together, or Latin and French, or
all three simultaneously? If pupils should decline nouns in all three
languages at the same time, learn to count in all at the same time, and
begin to translate in all simultaneously, would there not be an economy
of time and effort, and would there not be developed a much broader view
of language? Now the fusionist of algebra and geometry does not like
this argument, and he says that the cases are not parallel, and he tries
to tell why they are not. He demands that his opponent abandon argument
by analogy and advance some positive reason why algebra and geometry
should not be fused. Then his opponent says that it is not for him to
advance any reason for what already exists, the teaching of the two
separately; that he has only to refute the fusionist's arguments, and
that he has done so. He asserts that algebra and geometry are as
distinct as chemistry and biology; that they have a few common points,
but not enough to require teaching them together. He claims that to
begin Latin and Greek at the same time has always proved to be
confusing, and that the same is true of algebra and geometry. He grants
that unified knowledge is desirable, but he argues that when the fine
arts of music and color work fuse, and when the natural sciences of
chemistry and physics are taught in the same class, and when we follow
the declension of a German noun by that of a French noun and a Latin
noun, and when we teach drawing and penmanship together, then it is well
to talk of mixing algebra and geometry.

It is well, before deciding such a question for ourselves (for evidently
we cannot decide it for the world), to consider what has been the result
of experience. Algebra and geometry were always taught together in early
times, as were trigonometry and astronomy. The Ahmes papyrus contains
both primitive algebra and primitive geometry. Euclid's "Elements"
contains not only pure geometry, but also a geometric algebra and the
theory of numbers. The early works of the Hindus often fused geometry
and arithmetic, or geometry and algebra. Even the first great printed
compendium of mathematics, the "S[=u]ma" of Paciuolo (1494) contained
all of the branches of mathematics. Much of this later attempt was not,
however, an example of perfect fusion, but rather of assigning one set
of chapters to algebra, another to geometry, and another to arithmetic.
So fusion, more or less perfect, has been tried over long periods, and
abandoned as each subject grew more complete in itself, with its own
language and its peculiar symbols.

But it is asserted that fusion is being carried on successfully to-day
by more than one enthusiastic teacher, and that this proves the
contention that the plan is a good one. Books are cited to show that the
arrangement is feasible, and classes are indicated where the work is
progressing along this line.

What, then, is the conclusion? That is a question for the teacher to
settle, but it is one upon which a writer on the teaching of mathematics
should not fear to express his candid opinion.

It is a fact that the Greek and Latin fusion is a fair analogy. There
are reasons for it, but there are many more against it, the chief one
being the confusion of beginning two languages at once, and the learning
simultaneously of two vocabularies that must be kept separate. It is
also a fact that algebra and geometry are fully as distinct as physics
and chemistry, or chemistry and biology. Life may be electricity, and a
brief cessation of oxidization in the lungs brings death, but these
facts are no reasons for fusing the sciences of physics, biology, and
chemistry. Algebra is primarily a theory of certain elementary
functions, a generalized arithmetic, while geometry is primarily a
theory of form with a highly refined logic to be used in its mastery.
They have a few things in common, as many other subjects have, but they
have very many more features that are peculiar to the one or the other.
The experience of the world has led it away from a simultaneous
treatment, and the contrary experience of a few enthusiastic teachers of
to-day proves only their own powers to succeed with any method. It is
easy to teach logarithms in the seventh school year, but it is not good
policy to do so under present conditions. So the experience of the world
is against the plan of strict fusion, and no arguments have as yet been
advanced that are likely to change the world's view. No one has written
a book combining algebra and geometry in this fashion that has helped
the cause of fusion a particle; on the contrary, every such work that
has appeared has damaged that cause by showing how unscientific a result
has come from the labor of an enthusiastic supporter of the movement.

But there is one feature that has not been considered above, and that is
a serious handicap to any effort at combining the two sciences in the
high school, and this is the question of relative difficulty. It is
sometimes said, in a doctrinaire fashion, that geometry is easier than
algebra, since form is easier to grasp than function, and that therefore
geometry should precede algebra. But every teacher of mathematics knows
better than this. He knows that the simplest form is easier to grasp
than the simplest function, but nevertheless that plane geometry, as we
understand the term to-day, is much more difficult than elementary
algebra for a pupil of fourteen. The child studies form in the
kindergarten before he studies number, and this is sound educational
policy. He studies form, in mensuration, throughout his course in
arithmetic, and this, too, is good educational policy. This kind of
geometry very properly precedes algebra. But the demonstrations of
geometry, the study by pupils of fourteen years of a geometry that was
written for college students and always studied by them until about
fifty years ago,--that is by no means as easy as the study of a simple
algebraic symbolism and its application to easy equations. If geometry
is to be taught for the same reasons as at present, it cannot
advantageously be taught earlier than now without much simplification,
and it cannot successfully be fused with algebra save by some teacher
who is willing to sacrifice an undue amount of energy to no really
worthy purpose. When great mathematicians like Professor Klein speak of
the fusion of all mathematics, they speak from the standpoint of
advanced students, not for the teacher of elementary geometry.

It is therefore probable that simple mensuration will continue, as a
part of arithmetic, to precede algebra, as at present; and that algebra
into or through quadratics will precede geometry,[38] drawing upon the
mensuration of arithmetic as may be needed; and that geometry will
follow this part of algebra, using its principles as far as possible to
assist in the demonstrations and to express and manipulate its formulas.
Plane geometry, or else a year of plane and solid geometry, will
probably, in this country, be followed by algebra, completing quadratics
and studying progressions; and by solid geometry, or a supplementary
course in plane and solid geometry, this work being elective in many, if
not all, schools.[39] It is also probable that a general review of
mathematics, where the fusion idea may be carried out, will prove to be
a feature of the last year of the high school, and one that will grow
in popularity as time goes on. Such a plan will keep algebra and
geometry separate, but it will allow each to use all of the other that
has preceded it, and will encourage every effort in this direction. It
will accomplish all that a more complete fusion really hopes to
accomplish, and it will give encouragement to all who seek to modernize
the spirit of each of these great branches of mathematics.

There is, however, a chance for fusion in two classes of school, neither
of which is as yet well developed in this country. The first is the
technical high school that is at present coming into some prominence. It
is not probable even here that the best results can be secured by
eliminating all mathematics save only what is applicable in the shop,
but if this view should prevail for a time, there would be so little
left of either algebra or geometry that each could readily be joined to
the other. The actual amount of algebra needed by a foreman in a machine
shop can be taught in about four lessons, and the geometry or
mensuration that he needs can be taught in eight lessons at the most.
The necessary trigonometry may take eight more, so that it is entirely
feasible to unite these three subjects. The boy who takes such a course
would know as much about mathematics as a child who had read ten pages
in a primer would know about literature, but he would have enough for
his immediate needs, even though he had no appreciation of mathematics
as a science. If any one asks if this is not all that the school should
give him, it might be well to ask if the school should give only the
ability to read, without the knowledge of any good literature; if it
should give only the ability to sing, without the knowledge of good
music; if it should give only the ability to speak, without any training
in the use of good language; and if it should give a knowledge of home
geography, without any intimation that the world is round,--an atom in
the unfathomable universe about us.

The second opportunity for fusion is possibly (for it is by no means
certain) to be found in a type of school in which the only required
courses are the initial ones. These schools have some strong advocates,
it being claimed that every pupil should be introduced to the large
branches of knowledge and then allowed to elect the ones in which he
finds himself the most interested. Whether or not this is sound
educational policy need not be discussed at this time; but if such a
plan were developed, it might be well to offer a somewhat superficial
(in the sense of abridged) course that should embody a little of
algebra, a little of geometry, and a little of trigonometry. This would
unconsciously become a bait for students, and the result would probably
be some good teaching in the class in question. It is to be hoped that
we may have some strong, well-considered textbooks upon this phase of
the work.

As to the fusion of trigonometry and plane geometry little need be said,
because the subject is in the doctrinaire stage. Trigonometry naturally
follows the chapter on similar triangles, but to put it there means, in
our crowded curriculum, to eliminate something from geometry. Which,
then, is better,--to give up the latter portion of geometry, or part of
it at least, or to give up trigonometry? Some advocates have entered a
plea for two or three lessons in trigonometry at this point, and this is
a feature that any teacher may introduce as a bit of interest, as is
suggested in Chapter XVI, just as he may give a popular talk to his
class upon the fourth dimension or the non-Euclidean geometry. The
lasting impression upon the pupil will be exactly the same as that of
four lessons in Sanskrit while he is studying Latin. He might remember
each with pleasure, Latin being related, as it is, to Sanskrit, and
trigonometry being an outcome of the theory of similar triangles. But
that either of these departures from the regular sequence is of any
serious mathematical or linguistic significance no one would feel like
asserting. Each is allowable on the score of interest, but neither will
add to the pupil's power in any essential feature.

Each of these subjects is better taught by itself, each using the other
as far as possible and being followed by a review that shall make use of
all. It is not improbable that we may in due time have high schools that
give less extended courses in algebra and geometry, adding brief
practical courses in trigonometry and the elements of the calculus; but
even in such schools it is likely to be found that geometry is best
taught by itself, making use of all the mathematics that has preceded
it.

It will of course be understood that the fusion of algebra and geometry
as here understood has nothing to do with the question of teaching the
two subjects simultaneously, say two days in the week for one and three
days for the other. This plan has many advocates, although on the whole
it has not been well received in this country. But what is meant here is
the actual fusing of algebra and geometry day after day,--a plan that
has as yet met with only a sporadic success, but which may be developed
for beginning classes in due time.

FOOTNOTES:

[37] _Al-jabr wa'l-muq[=a]balah_: "restoration and equation" is a fairly
good translation of the Arabic.

[38] Or be carried along at the same time as a distinct topic.

[39] With a single year for required geometry it would be better from
every point of view to cut the plane geometry enough to admit a fair
course in solid geometry.

CHAPTER IX

THE INTRODUCTION TO GEOMETRY

There are two difficult crises in the geometry course, both for the
pupil and for the teacher. These crises are met at the beginning of the
subject and at the beginning of solid geometry. Once a class has fairly
got into Book I, if the interest in the subject can be maintained, there
are only the incidental difficulties of logical advance throughout the
plane geometry. When the pupil who has been seeing figures in one plane
for a year attempts to visualize solids from a flat drawing, the second
difficult place is reached. Teachers going over solid geometry from year
to year often forget this difficulty, but most of them can easily place
themselves in the pupil's position by looking at the working drawings of
any artisan,--usually simple cases in the so-called descriptive
geometry. They will then realize how difficult it is to visualize a
solid from an unfamiliar kind of picture. The trouble is usually avoided
by the help of a couple of pieces of heavy cardboard or box board, and a
few knitting needles with which to represent lines in space. If these
are judiciously used in class for a few days, until the figures are
understood, the second crisis is easily passed. The continued use of
such material, however, or the daily use of either models or
photographs, weakens the pupil, even as a child is weakened by being
kept too long in a perambulator. Such devices have their place; they are
useful when needed, but they are pernicious when unnecessary. Just as
the mechanic must be able to make and to visualize his working drawings,
so the student of solid geometry must be able to get on with pencil and
paper, representing his solid figures in the flat.

But the introduction to plane geometry is not so easily disposed of. The
pupil at that time is entering a field that is entirely unfamiliar. He
is only fourteen or fifteen years of age, and his thoughts are
distinctly not on geometry. Of logic he knows little and cares less. He
is not interested in a subject of which he knows nothing, not even the
meaning of its name. He asks, naturally and properly, what it all
signifies, what possible use there is for studying geometry, and why he
should have to prove what seems to him evident without proof. To pass
him successfully through this stage has taxed the ingenuity of every
real teacher from the time of Euclid to the present; and just as Euclid
remarked to King Ptolemy, his patron, that there is no royal road to
geometry, so we may affirm that there is no royal road to the teaching
of geometry.

Nevertheless the experience of teachers counts for a great deal, and
this experience has shown that, aside from the matter of technic in
handling the class, certain suggestions are of value, and a few of these
will now be set forth.

First, as to why geometry is studied, it is manifestly impossible
successfully to explain to a boy of fourteen or fifteen the larger
reasons for studying anything whatever. When we confess ourselves
honestly we find that these reasons, whether in mathematics, the natural
sciences, handwork, letters, the vocations, or the fine arts, are none
too clear in our own minds, in spite of any pretentious language that we
may use. It is therefore most satisfactory to anticipate the question at
once, and to set the pupils, for a few days, at using the compasses and
ruler in the drawing of geometric designs and of the most common figures
that they will use. This serves several purposes: it excites their
interest, it guards against the slovenly figures that so often lead them
to erroneous conclusions, it has a genuine value for the future artisan,
and it shows that geometry is something besides mere theory. Whether the
textbook provides for it or not, the teacher will find a few days of
such work well spent, it being a simple matter to supplement the book in
this respect. There was a time when some form of mechanical drawing was
generally taught in the schools, but this has given place to more
genuine art work, leaving it to the teacher of geometry to impart such
knowledge of drawing as is a necessary preliminary to the regular study
of the subject.

Such work in drawing should go so far, and only so far, as to arouse an
interest in geometric form without becoming wearisome, and to
familiarize the pupil with the use of the instruments. He should be
counseled about making fine lines, about being careful in setting the
point of his compasses on the exact center that he wishes to use, and
about representing a point by a very fine dot, or, preferably at first,
by two crossed lines. Unless these details are carefully considered, the
pupil will soon find that the lines of his drawings do not fit together,
and that the result is not pleasing to the eye. The figures here given
are good ones upon which to begin, the dotted construction lines being
erased after the work is completed. They may be constructed with the
compasses and ruler alone, or the draftsman's T-square, triangle, and
protractor may be used, although these latter instruments are not
necessary. We should constantly remember that there is a danger in the
slavish use of instruments and of such helps as squared paper.

     Just as Euclid rode roughshod over the growing intellects of
     boys and girls, so may instruments ride roughshod over their
     growing perceptions by interfering with natural and healthy
     intuitions, and making them the subject of laborious
     measurement.[40]

The pupil who cannot see the equality of vertical angles intuitively
better than by the use of the protractor is abnormal. Nevertheless it is
the pupil's interest that is at stake, together with his ability to use
the instruments of daily life. If, therefore, he can readily be
supplied with draftsmen's materials, and is not compelled to use them in
a foolish manner, so much the better. They will not hurt his geometry if
the teacher does not interfere, and they will help his practical
drawing; but for obvious reasons we cannot demand that the pupil
purchase what is not really essential to his study of the subject. The
most valuable single instrument of the three just mentioned is the
protractor, and since a paper one costs only a few cents and is often
helpful in the drawing of figures, it should be recommended to pupils.

There is also another line of work that often arouses a good deal of
interest, namely, the simple field measures that can easily be made
about the school grounds. Guarding against the ever-present danger of
doing too much of such work, of doing work that has no interest for the
pupil, of requiring it done in a way that seems unreal to a class, and
of neglecting the essence of geometry by a line of work that involves no
new principles,--such outdoor exercises in measurement have a positive
value, and a plentiful supply of suggestions in this line is given in
the subsequent chapters. The object is chiefly to furnish a motive for
geometry, and for many pupils this is quite unnecessary. For some,
however, and particularly for the energetic, restless boy, such work has
been successfully offered by various teachers as an alternative to some
of the book work. Because of this value a considerable amount of such
work will be suggested for teachers who may care to use it, the textbook
being manifestly not the place for occasional topics of this nature.

For the purposes of an introduction only a tape line need be purchased.
Wooden pins and a plumb line can easily be made. Even before he comes
to the propositions in mensuration in geometry the pupil knows, from his
arithmetic, how to find ordinary areas and volumes, and he may therefore
be set at work to find the area of the school ground, or of a field, or
of a city block. The following are among the simple exercises for a
beginner:

1. Drive stakes at two corners, _A_ and _B_, of the school grounds,
putting a cross on top of each; or make the crosses on the sidewalk, so
as to get two points between which to measure. Measure from _A_ to _B_
by holding the tape taut and level, dropping perpendiculars when
necessary by means of the plumb line, as shown in the figure. Check the
work by measuring from _B_ back to _A_ in the same way. Pupils will find
that their work should always be checked, and they will be surprised to
see how the results will vary in such a simple measurement as this,
unless very great care is taken. If they learn the lesson of accuracy
thus early, they will have gained much.

2. Take two stakes, _X_, _Y_, in a field, preferably two or three
hundred feet apart, always marked on top with crosses so as to have
exact points from which to work. Let it then be required to stake out or
"range" the line from _X_ to _Y_ by placing stakes at specified
distances. One boy stands at _Y_ and another at _X_, each with a plumb
line. A third one takes a plumb line and stands at _P_, the observer at
_X_ motioning to him to move his plumb line to the right or the left
until it is exactly in line with _X_ and _Y_. A stake is then driven at
_P_, and the pupil at _X_ moves on to the stake _P_. Then _Q_ is
located in the same way, and then _R_, and so on. The work is checked by
ranging back from _Y_ to _X_. In some of the simple exercises suggested
later it is necessary to range a line so that this work is useful in
making measurements. The geometric principle involved is that two points
determine a straight line.

3. To test a perpendicular or to draw one line perpendicular to another
in a field, we may take a stout cord twelve feet long, having a knot at
the end of every foot. If this is laid along four feet, the ends of this
part being fixed, and it is stretched as here shown, so that the next
vertex is five feet from one of these ends and three feet from the other
end, a right angle will be formed. A right angle can also be run by
making a simple instrument, such as is described in Chapter XV. Still
another plan of drawing a line perpendicular to another line _AB_, from
a point _P_, consists in swinging a tape from _P_, cutting _AB_ at _X_
and _Y_, and then bisecting _XY_ by doubling the tape. This fixes the
foot of the perpendicular.

4. It is now possible to find the area of a field of irregular shape by
dividing it into triangles and trapezoids, as shown in the figure.
Pupils know from their work in arithmetic how to find the area of a
triangle or a trapezoid, so that the area of the field is easily found.
The work may be checked by comparing the results of different groups of
pupils, or by drawing another diagonal and dividing the field into other
triangles and trapezoids.

These are about as many types of field work as there is any advantage in
undertaking for the purpose of securing the interest of pupils as a
preliminary to the work in geometry. Whether any of it is necessary, and
for what pupils it is necessary, and how much it should trespass upon
the time of scientific geometry are matters that can be decided only by
the teacher of a particular class.

A second difficulty of the pupil is seen in his attitude of mind towards
proofs in general. He does not see why vertical angles should be proved
equal when he knows that they are so by looking at the figure. This
difficulty should also be anticipated by giving him some opportunity to
know the weakness of his judgment, and for this purpose figures like the
following should be placed before him. He should be asked which of these
lines is longer, _AB_ or _XY_. Two equal lines should then be arranged
in the form of a letter T, as here shown, and he should be asked which
is the longer, _AB_ or _CD_. A figure that is very deceptive,
particularly if drawn larger and with heavy cross lines, is this one in
which _AB_ and _CD_ are really parallel, but do not seem to be so. Other
interesting deceptions have to do with producing lines, as in these
figures, where it is quite difficult in advance to tell whether _AB_ and
_CD_ are in the same line, and similarly for _WX_ and _YZ_. Equally
deceptive is this figure, in which it is difficult to tell which line
_AB_ will lie along when produced. In the next figure _AB_ appears to be
curved when in reality it is straight, and _CD_ appears straight when in
reality it is curved. The first of the following circles seems to be
slightly flattened at the points _P_, _Q_, _R_, _S_, and in the second
one the distance _BD_ seems greater than the distance _AC_. There are
many equally deceptive figures, and a few of them will convince the
beginner that the proofs are necessary features of geometry.

It is interesting, in connection with the tendency to feel that a
statement is apparent without proof, to recall an anecdote related by
the French mathematician, Biot, concerning the great scientist, Laplace:

     Once Laplace, having been asked about a certain point in his
     "Celestial Mechanics," spent nearly an hour in trying to recall
     the chain of reasoning which he had carelessly concealed by the
     words "It is easy to see."

A third difficulty lies in the necessity for putting a considerable
number of definitions at the beginning of geometry, in order to get a
working vocabulary. Although practically all writers scatter the
definitions as much as possible, there must necessarily be some
vocabulary at the beginning. In order to minimize the difficulty of
remembering so many new terms, it is helpful to mingle with them a
considerable number of exercises in which these terms are employed, so
that they may become fixed in mind through actual use. Thus it is of
value to have a class find the complements of 27°, 32° 20', 41° 32' 48",
26.75°, 33 1/3°, and 0°. It is true that into the pure geometry of
Euclid the measuring of angles in degrees does not enter, but it has
place in the practical applications, and it serves at this juncture to
fix the meaning of a new term like "complement."

The teacher who thus anticipates the question as to the reason for
studying geometry, the mental opposition to proving statements, and the
forgetfulness of the meaning of common terms will find that much of the
initial difficulty is avoided. If, now, great care is given to the first
half dozen propositions, the pupil will be well on his way in geometry.
As to these propositions, two plans of selection are employed. The first
takes a few preliminary propositions, easily demonstrated, and seeks
thus to introduce the pupil to the nature of a proof. This has the
advantage of inspiring confidence and the disadvantage of appearing to
prove the obvious. The second plan discards all such apparently obvious
propositions as those about the equality of right angles, and the sum of
two adjacent angles formed by one line meeting another, and begins at
once on things that seem to the pupil as worth the proving. In this
latter plan the introduction is usually made with the proposition
concerning vertical angles, and the two simplest cases of congruent
triangles.

Whichever plan of selection is taken, it is important to introduce a
considerable number of one-step exercises immediately, that is,
exercises that require only one significant step in the proof. In this
way the pupil acquires confidence in his own powers, he finds that
geometry is not mere memorizing, and he sees that each proposition makes
him the master of a large field. To delay the exercises to the end of
each book, or even to delay them for several lessons, is to sow seeds
that will result in the attempt to master geometry by the sheer process
of memorizing.

As to the nature of these exercises, however, the mistake must not be
made of feeling that only those have any value that relate to football
or the laying out of a tennis court. Such exercises are valuable, but
such exercises alone are one-sided. Moreover, any one who examines the
hundreds of suggested exercises that are constantly appearing in various
journals, or who, in the preparation of teachers, looks through the
thousands of exercises that come to him in the papers of his students,
comes very soon to see how hollow is the pretense of most of them. As
has already been said, there are relatively few propositions in geometry
that have any practical applications, applications that are even honest
in their pretense. The principle that the writer has so often laid down
in other works, that whatever pretends to be practical should really be
so, applies with much force to these exercises. When we can find the
genuine application, if it is within reasonable grasp of the pupil, by
all means let us use it. But to put before a class of girls some
technicality of the steam engine that only a skilled mechanic would be
expected to know is not education,--it is mere sham. There is a noble
dignity to geometry, a dignity that a large majority of any class comes
to appreciate when guided by an earnest teacher; but the best way to
destroy this dignity, to take away the appreciation of pure mathematics,
and to furnish weaker candidates than now for advance in this field is
to deceive our pupils and ourselves into believing that the ultimate
purpose of mathematics is to measure things in a way in which no one
else measures them or has ever measured them.

In the proof of the early propositions of plane geometry, and again at
the beginning of solid geometry, there is a little advantage in using
colored crayon to bring out more distinctly the equal parts of two
figures, or the lines outside the plane, or to differentiate one plane
from another. This device, however, like that of models in solid
geometry, can easily be abused, and hence should be used sparingly, and
only until the purpose is accomplished. The student of mathematics must
learn to grasp the meaning of a figure drawn in black on white paper,
or, more rarely, in white on a blackboard, and the sooner he is able to
do this the better for him. The same thing may be said of the
constructing of models for any considerable number of figures in solid
geometry; enough work of this kind to enable a pupil clearly to
visualize the solids is valuable, but thereafter the value is usually
more than offset by the time consumed and the weakened power to grasp
the meaning of a geometric drawing.

There is often a tendency on the part of teachers in their first years
of work to overestimate the logical powers of their pupils and to
introduce forms of reasoning and technical terms that experience has
proved to be unsuited to one who is beginning geometry. Usually but
little harm is done, because the enthusiasm of any teacher who would use
this work would carry the pupils over the difficulties without much
waste of energy on their part. In the long run, however, the attempt is
usually abandoned as not worth the effort. Such a term as
"contrapositive," such distinctions as that between the logical and the
geometric converse, or between perfect and partial geometric conversion,
and such pronounced formalism as the "syllogistic method,"--all these
are happily unknown to most teachers and might profitably be unknown to
all pupils. The modern American textbook in geometry does not begin to
be as good a piece of logic as Euclid's "Elements," and yet it is to be
observed that none of these terms is found in this classic work, so that
they cannot be thought to be necessary to a logical treatment of the
subject. We need the word "converse," and some reference to the law of
converse is therefore permissible; the meaning of the _reductio ad
absurdum_, of a necessary and sufficient condition, and of the terms
"synthesis" and "analysis" may properly form part of the pupil's
equipment because of their universal use; but any extended incursion
into the domain of logic will be found unprofitable, and it is liable to
be positively harmful to a beginner in geometry.

A word should be said as to the lettering of the figures in the early
stages of geometry. In general, it is a great aid to the eye if this is
carried out with some system, and the following suggestions are given as
in accord with the best authors who have given any attention to the
subject:

1. In general, letter a figure counterclockwise, for the reason that we
read angles in this way in higher mathematics, and it is as easy to form
this habit now as to form one that may have to be changed. Where two
triangles are congruent, however, but have their sides arranged in
opposite order, it is better to letter them so that their corresponding
parts appear in the same order, although this makes one read clockwise.

2. For the same reason, read angles counterclockwise. Thus [L]_A_ is
read "_BAC_," the reflex angle on the outside of the triangle being read
"_CAB_." Of course this is not vital, and many authors pay no attention
to it; but it is convenient, and if the teacher habitually does it, the
pupils will also tend to do it. It is helpful in trigonometry, and it
saves confusion in the case of a reflex angle in a polygon. Designate an
angle by a single letter if this can conveniently be done.

3. Designate the sides opposite angles _A_, _B_, _C_, in a triangle, by
_a_, _b_, _c_, and use these letters in writing proofs.

4. In the case of two congruent triangles use the letters _A_, _B_, _C_
and _A'_, _B'_, _C'_, or _X_, _Y_, _Z_, instead of letters chosen at
random, like _D_, _K_, _L_. It is easier to follow a proof where some
system is shown in lettering the figures. Some teachers insist that a
pupil at the blackboard should not use the letters given in the
textbook, hoping thereby to avoid memorizing. While the danger is
probably exaggerated, it is easy to change with some system, using _P_,
_Q_, _R_ and _P'_, _Q'_, _R'_, for example.

5. Use small letters for lines, as above stated, and also place them
within angles, it being easier to speak of and to see [L]_m_ than
[L]_DEF_. The Germans have a convenient system that some American
teachers follow to advantage, but that a textbook has no right to
require. They use, as in the following figure, _A_ for the point, _a_
for the opposite side, and the Greek letter [alpha] (alpha) for the
angle. The learning of the first three Greek letters, alpha ([alpha]),
beta ([beta]), and gamma ([gamma]), is not a hardship, and they are
worth using, although Greek is so little known in this country to-day
that the alphabet cannot be demanded of teachers who do not care to use
it.

6. Also use small letters to represent numerical values. For example,
write _c_ = 2[pi]_r_ instead of _C_ = 2[pi]_R_. This is in accord with
the usage in algebra to which the pupil is accustomed.

7. Use initial letters whenever convenient, as in the case of _a_ for
area, _b_ for base, _c_ for circumference, _d_ for diameter, _h_ for
height (altitude), and so on.

Many of these suggestions seem of slight importance in themselves, and
some teachers will be disposed to object to any attempt at lettering a
figure with any regard to system. If, however, they will notice how a
class struggles to follow a demonstration given with reference to a
figure on the blackboard, they will see how helpful it is to have some
simple standards of lettering. It is hardly necessary to add that in
demonstrating from a figure on a blackboard it is usually better to say
"this line," or "the red line," than to say, without pointing to it,
"the line _AB_." It is by such simplicity of statement and by such
efforts to help the class to follow demonstrations that pupils are led
through many of the initial discouragements of the subject.

FOOTNOTES:

[40] Carson, loc. cit., p. 13.

CHAPTER X

THE CONDUCT OF A CLASS IN GEOMETRY

No definite rules can be given for the detailed conduct of a class in
any subject. If it were possible to formulate such rules, all the
personal magnetism of the teacher, all the enthusiasm, all the
originality, all the spirit of the class, would depart, and we should
have a dull, dry mechanism. There is no one best method of teaching
geometry or anything else. The experience of the schools has shown that
a few great principles stand out as generally accepted, but as to the
carrying out of these principles there can be no definite rules.

Let us first consider the general question of the employment of time in
a recitation in geometry. We might all agree on certain general
principles, and yet no two teachers ever would or ever should divide the
period even approximately in the same way. First, a class should have an
opportunity to ask questions. A teacher here shows his power at its
best, listening sympathetically to any good question, quickly seeing the
essential point, and either answering it or restating it in such a way
that the pupil can answer it for himself. Certain questions should be
answered by the teacher; he is there for that purpose. Others can at
once be put in such a light that the pupil can himself answer them.
Others may better be answered by the class. Occasionally, but more
rarely, a pupil may be told to "look that up for to-morrow," a plan
that is commonly considered by students as a confession of weakness on
the part of the teacher, as it probably is. Of course a class will waste
time in questioning a weak teacher, but a strong one need have no fear
on this account. Five minutes given at the opening of a recitation to
brisk, pointed questions by the class, with the same credit given to a
good question as to a good answer, will do a great deal to create a
spirit of comradeship, of frankness, and of honesty, and will reveal to
a sympathetic teacher the difficulties of a class much better than the
same amount of time devoted to blackboard work. But there must be no
dawdling, and the class must feel that it has only a limited time, say
five minutes at the most, to get the help it needs.

Next in order of the division of the time may be the teacher's report on
any papers that the class has handed in. It is impossible to tell how
much of this paper work should be demanded. The local school conditions,
the mental condition of the class, and the time at the disposal of the
teacher are all factors in the case. In general, it may be said that
enough of this kind of work is necessary to see that pupils are neat and
accurate in setting down their demonstrations. On the other hand, paper
work gives an opportunity for dishonesty, and it consumes a great deal
of the teacher's time that might better be given to reading good books
on the subject that he is teaching. If, however, any papers have been
submitted, about five minutes may well be given to a rapid review of the
failures and the successes. In general, it is good educational policy to
speak of the errors and failures impersonally, but occasionally to
mention by name any one who has done a piece of work that is worthy of
special comment. Pupils may better be praised in public and blamed in
private. There is such a thing, however, as praising too much, when
nothing worthy of note has been done, just as there is danger of blaming
too much, resulting in mere "nagging."

The third division of the recitation period may profitably go to
assigning the advance lesson. The class questions and the teacher's
report on written work have shown the mental status of the pupils, so
that the teacher now knows what he may expect for the next lesson. If he
assigns his lesson at the beginning of the period, he does not have this
information. If he waits to the end, he may be too hurried to give any
"development" that the new lesson may require. There can be no rule as
to how to assign a new lesson; it all depends upon what the lesson is,
upon the mental state of the class, and not a little upon the
idiosyncrasy of the teacher. The German educator, Herbart, laid down
certain formal steps in developing a new lesson, and his successors have
elaborated these somewhat as follows:

1. _Aim._ Always take a class into your confidence. Tell the members at
the outset the goal. No one likes to be led blindfolded.

2. _Preparation._ A few brief questions to bring the class to think of
what is to be considered.

3. _Presentation of the new._ Preferably this is done by questions, the
answers leading the members of the class to discover the new truth for
themselves.

4. _Apperception._ Calling attention to the fact that this new fact was
known before, in part, and that it relates to a number of things already
in the mind. The more the new can be tied up to the old the more
tenaciously it will be held.

5. _Generalization and application._

It is evident at once that a great deal of time may be wasted in always
following such a plan, perhaps in ever following it consciously. But, on
the other hand, probably every good teacher, whether he has heard of
Herbart or not, naturally covers these points in substantially this
order. For an inexperienced teacher it is helpful to be familiar with
them, that he may call to mind the steps, arranged in a psychological
sequence, that he would do well to follow. It must always be remembered
that there is quite as much danger in "developing" too much as in taking
the opposite extreme. A mechanical teacher may develop a new lesson
where there is need for only a question or two or a mere suggestion. It
should also be recognized that students need to learn to read
mathematics for themselves, and that always to take away every
difficulty by explanations given in advance is weakening to any one.

Therefore, in assigning the new lesson we may say "Take the next two
pages," and thus discourage most of the class. On the other hand, we may
spend an unnecessary amount of time and overdevelop the work of those
same pages, and have the whole lesson lose all its zest. It is here that
the genius of the teacher comes forth to find the happy mean.

The fourth division of the hour should be reached, in general, in about
ten minutes. This includes the recitation proper. But as to the nature
of this work no definite instructions should be attempted. To a good
teacher they would be unnecessary, to a poor one they would be harmful.
Part of the class may go to the board, and as they are working, the rest
may be reciting. Those at the board should be limited as to time, for
otherwise a premium is placed on mere dawdling. They should be so
arranged as to prevent copying, but the teacher's eye is the best
preventive of this annoying feature. Those at their seats may be called
upon one at a time to demonstrate at the blackboard, the rest being
called upon for quick responses, as occasion demands. The European plan
of having small blackboards is in many respects better than ours, since
pupils cannot so easily waste time. They have to work rapidly and talk
rapidly, or else take their seats.

What should be put on the board, whether the figure alone, or the figure
and the proof, depends upon the proposition. In general, there should be
a certain number of figures put on the board for the sake of rapid work
and as a basis for the proofs of the day. There should also be a certain
amount of written work for the sake of commending or of criticizing
adversely the proof used. There are some figures that are so complicated
as to warrant being put upon sheets of paper and hung before the class.
Thus there is no rule upon the subject, and the teacher must use his
judgment according to the circumstances and the propositions.

If the early "originals" are one-step exercises, and a pupil is required
to recite rapidly, a habit of quick expression is easily acquired that
leads to close attention on the part of all the class. Students as a
rule recite slower than they need to, from mere habit. Phlegmatic as we
think the German is, and nervous as is the American temperament, a
student in geometry in a German school will usually recite more quickly
and with more vigor than one with us. Our extensive blackboards have
something to do with this, allowing so many pupils to be working at the
board that a teacher cannot attend to them all. The result is a habit of
wasting the minutes that can only be overcome by the teacher setting a
definite but reasonable time limit, and holding the pupil responsible if
the work is not done in the time specified. If this matter is taken in
hand the first day, and special effort made in the early weeks of the
year, much of the difficulty can be overcome.

As to the nature of the recitation to be expected from the pupil, no
definite rule can be laid down, since it varies so much with the work of
the day. In general, however, a pupil should state the theorem quickly,
state exactly what is given and what is to be proved, with respect to
the figure, and then give the proof. At first it is desirable that he
should give the authorities in full, and later give only the essential
part in a few words. It is better to avoid the expression "by previous
proposition," for it soon comes to be abused, and of course the learning
of section numbers in a book is a barbarism. It is only by continually
stating the propositions used that a pupil comes to have well fixed in
his memory the basal theorems of geometry, and without these he cannot
make progress in his subsequent mathematics. In general, it is better to
allow a pupil to finish his proof before asking him any questions, the
constant interruptions indulged in by some teachers being the cause of
no little confusion and hesitancy on the part of pupils. Sometimes it is
well to have a figure drawn differently from the one in the book, or
lettered differently, so as to make sure that the pupil has not
memorized the proof, but in general such devices are unnecessary, for a
teacher can easily discover whether the proof is thoroughly understood,
either by the manner of the pupil or by some slight questioning. A good
textbook has the figures systematically lettered in some helpful way
that is easily followed by the class that is listening to the
recitation, and it is not advisable to abandon this for a random set of
letters arranged in no proper order.

It is good educational policy for the teacher to commend at least as
often as he finds fault when criticizing a recitation at the blackboard
and when discussing the pupils' papers. Optimism, encouragement,
sympathy, the genuine desire to help, the putting of one's self in the
pupil's place, the doing to the pupil as the teacher would that he
should do in return,--these are educational policies that make for
better geometry as they make for better life.

The prime failure in teaching geometry lies unquestionably in the lack
of interest on the part of the pupil, and this has been brought about by
the ancient plan of simply reading and memorizing proofs. It is to get
away from this that teachers resort to some such development of the
lesson in advance, as has been suggested above. It is usually a good
plan to give the easier propositions as exercises before they are
reached in the text, where this can be done. An English writer has
recently contributed this further idea:

     It might be more stimulating to encourage investigation than to
     demand proofs of stated facts; that is to say, "Here is a
     figure drawn in this way, find out anything you can about it."
     Some such exercises having been performed jointly by teachers
     and pupils, the lust of investigation and healthy competition
     which is present in every normal boy or girl might be awakened
     so far as to make such little researches really attractive;
     moreover, the training thus given is of far more value than
     that obtained by proving facts which are stated in advance, for
     it is seldom, if ever, that the problems of adult life present
     themselves in this manner. The spirit of the question, "What is
     true?" is positive and constructive, but that involved in "Is
     this true?" is negative and destructive.[41]

When the question is asked, "How shall I teach?" or "What is the
Method?" there is no answer such as the questioner expects. A Japanese
writer, Motowori, a great authority upon the Shinto faith of his people,
once wrote these words: "To have learned that there is no way to be
learned and practiced is really to have learned the way of the gods."

FOOTNOTES:

[41] Carson, loc. cit., p. 12.

CHAPTER XI

THE AXIOMS AND POSTULATES

The interest as well as the value of geometry lies chiefly in the fact
that from a small number of assumptions it is possible to deduce an
unlimited number of conclusions. With the truth of these assumptions we
are not so much concerned as with the reasoning by which we draw the
conclusions, although it is manifestly desirable that the assumptions
should not be false, and that they should be as few as possible.

It would be natural, and in some respects desirable, to call these
foundations of geometry by the name "assumptions," since they are simply
statements that are assumed to be true. The real foundation principles
cannot be proved; they are the means by which we prove other statements.
But as with most names of men or things, they have received certain
titles that are time-honored, and that it is not worth the while to
attempt to change. In English we call them axioms and postulates, and
there is no more reason for attempting to change these terms than there
is for attempting to change the names of geometry[42] and of
algebra.[43]

Since these terms are likely to continue, it is necessary to
distinguish between them more carefully than is often done, and to
consider what assumptions we are justified in including under each. In
the first place, these names do not go back to Euclid, as is ordinarily
supposed, although the ideas and the statements are his. "Postulate" is
a Latin form of the Greek [Greek: aitêma] (_aitema_), and appears only
in late translations. Euclid stated in substance, "Let the following be
assumed." "Axiom" ([Greek: axiôma], _axioma_) dates perhaps only from
Proclus (fifth century A.D.), Euclid using the words "common notions"
([Greek: koinai ennoiai], _koinai ennoiai_) for "axioms," as Aristotle
before him had used "common things," "common principles," and "common
opinions."

The distinction between axiom and postulate was not clearly made by
ancient writers. Probably what was in Euclid's mind was the Aristotelian
distinction that an axiom was a principle common to all sciences,
self-evident but incapable of proof, while the postulates were the
assumptions necessary for building up the particular science under
consideration, in this case geometry.[44]

We thus come to the modern distinction between axiom and postulate, and
say that a general statement admitted to be true without proof is an
axiom, while a postulate in geometry is a geometric statement admitted
to be true, without proof. For example, when we say "If equals are added
to equals, the sums are equal," we state an assumption that is taken
also as true in arithmetic, in algebra, and in elementary mathematics in
general. This is therefore an axiom. At one time such a statement was
defined as "a self-evident truth," but this has in recent years been
abandoned, since what is evident to one person is not necessarily
evident to another, and since all such statements are mere matters of
assumption in any case. On the other hand, when we say, "A circle may be
described with any given point as a center and any given line as a
radius," we state a special assumption of geometry, and this assumption
is therefore a geometric postulate. Some few writers have sought to
distinguish between axiom and postulate by saying that the former was an
assumed theorem and the latter an assumed problem, but there is no
standard authority for such a distinction, and indeed the difference
between a theorem and a problem is very slight. If we say, "A circle may
be passed through three points not in the same straight line," we state
a theorem; but if we say, "Required to pass a circle through three
points," we state a problem. The mental process of handling the two
propositions is, however, practically the same in spite of the minor
detail of wording. So with the statement, "A straight line may be
produced to any required length." This is stated in the form of a
theorem, but it might equally well be stated thus: "To produce a
straight line to any required length." It is unreasonable to call this
an axiom in one case and a postulate in the other. However stated, it is
a geometric postulate and should be so classed.

What, now, are the axioms and postulates that we are justified in
assuming, and what determines their number and character? It seems
reasonable to agree that they should be as few as possible, and that for
educational purposes they should be so clear as to be intelligible to
beginners. But here we encounter two conflicting ideas. To get the
"irreducible minimum" of assumptions is to get a set of statements quite
unintelligible to students beginning geometry or any other branch of
elementary mathematics. Such an effort is laudable when the results are
intended for advanced students in the university, but it is merely
suggestive to teachers rather than usable with pupils when it touches
upon the primary steps of any science. In recent years several such
attempts have been made. In particular, Professor Hilbert has given a
system[45] of congruence postulates, but they are rather for the
scientist than for the student of elementary geometry.

In view of these efforts it is well to go back to Euclid and see what
this great teacher of university men[46] had to suggest. The following
are the five "common notions" that Euclid deemed sufficient for the
purposes of elementary geometry.

1. _Things equal to the same thing are also equal to each other._ This
axiom has persisted in all elementary textbooks. Of course it is a
simple matter to attempt criticism,--to say that -2 is the square root
of 4, and +2 is also the square root of 4, whence -2 = +2; but it is
evident that the argument is not sound, and that it does not invalidate
the axiom. Proclus tells us that Apollonius attempted to prove the axiom
by saying, "Let _a_ equal _b_, and _b_ equal _c_. I say that _a_ equals
_c_. For, since _a_ equals _b_, _a_ occupies the same space as _b_.
Therefore _a_ occupies the same space as _c_. Therefore _a_ equals
_c_." The proof is of no value, however, save as a curiosity.

2. _And if to equals equals are added, the wholes are equal._

3. _If equals are subtracted from equals, the remainders are equal._

Axioms 2 and 3 are older than Euclid's time, and are the only ones given
by him relating to the solution of the equation. Certain other axioms
were added by later writers, as, "Things which are double of the same
thing are equal to one another," and "Things which are halves of the
same thing are equal to one another." These two illustrate the ancient
use of _duplatio_ (doubling) and _mediatio_ (halving), the primitive
forms of multiplication and division. Euclid would not admit the
multiplication axiom, since to him this meant merely repeated addition.
The partition (halving) axiom he did not need, and if needed, he would
have inferred its truth. There are also the axioms, "If equals are added
to unequals, the wholes are unequal," and "If equals are subtracted from
unequals, the remainders are unequal," neither of which Euclid would
have used because he did not define "unequals." The modern arrangement
of axioms, covering addition, subtraction, multiplication, division,
powers, and roots, sometimes of unequals as well as equals, comes from
the development of algebra. They are not all needed for geometry, but in
so far as they show the relation of arithmetic, algebra, and geometry,
they serve a useful purpose. There are also other axioms concerning
unequals that are of advantage to beginners, even though unnecessary
from the standpoint of strict logic.

4. _Things that coincide with one another are equal to one another._
This is no longer included in the list of axioms. It is rather a
definition of "equal," or of "congruent," to take the modern term. If
not a definition, it is certainly a postulate rather than an axiom,
being purely geometric in character. It is probable that Euclid included
it to show that superposition is to be considered a legitimate form of
proof, but why it was not placed among the postulates is not easily
seen. At any rate it is unfortunately worded, and modern writers
generally insert the postulate of motion instead,--that a figure may be
moved about in space without altering its size or shape. The German
philosopher, Schopenhauer (1844), criticized Euclid's axiom as follows:
"Coincidence is either mere tautology or something entirely empirical,
which belongs not to pure intuition but to external sensuous experience.
It presupposes, in fact, the mobility of figures."

5. _The whole is greater than the part._ To this Clavius (1574) added,
"The whole is equal to the sum of its parts," which may be taken to be a
definition of "whole," but which is helpful to beginners, even if not
logically necessary. Some writers doubt the genuineness of this axiom.

Having considered the axioms of Euclid, we shall now consider the axioms
that are needed in the study of elementary geometry. The following are
suggested, not from the standpoint of pure logic, but from that of the
needs of the teacher and pupil.

1. _If equals are added to equals, the sums are equal._ Instead of this
axiom, the one numbered 8 below is often given first. For convenience in
memorizing, however, it is better to give the axioms in the following
order: (1) addition, (2) subtraction, (3) multiplication, (4) division,
(5) powers and roots,--all of equal quantities.

2. _If equals are subtracted from equals, the remainders are equal._

3. _If equals are multiplied by equals, the products are equal._

4. _If equals are divided by equals, the quotients are equal._

5. _Like powers or like positive roots of equals are equal._ Formerly
students of geometry knew nothing of algebra, and in particular nothing
of negative quantities. Now, however, in American schools a pupil
usually studies algebra a year before he studies demonstrative geometry.
It is therefore better, in speaking of roots, to limit them to positive
numbers, since the two square roots of 4 (+2 and -2), for example, are
not equal. If the pupil had studied complex numbers before he began
geometry, it would have been advisable to limit the roots still further
to real roots, since the four fourth roots of 1 (+1, -1, +[sqrt](-1),
-[sqrt](-1)), for example, are not equal save in absolute value. It is
well, however, to eliminate these fine distinctions as far as possible,
since their presence only clouds the vision of the beginner.

It should also be noted that these five axioms might be combined in one,
namely, _If equals are operated on by equals in the same way, the
results are equal_. In Axiom 1 this operation is addition, in Axiom 2 it
is subtraction, and so on. Indeed, in order to reduce the number of
axioms two are already combined in Axiom 5. But there is a good reason
for not combining the first four with the fifth, and there is also a
good reason for combining two in Axiom 5. The reason is that these are
the axioms continually used in equations, and to combine them all in one
would be to encourage laxness of thought on the part of the pupil. He
would always say "by Axiom 1" whatever he did to an equation, and the
teacher would not be certain whether the pupil was thinking definitely
of dividing equals by equals, or had a hazy idea that he was
manipulating an equation in some other way that led to an answer. On the
other hand, Axiom 5 is not used as often as the preceding four, and the
interchange of integral and fractional exponents is relatively common,
so that the joining of these two axioms in one for the purpose of
reducing the total number is justifiable.

6. _If unequals are operated on by positive equals in the same way, the
results are unequal in the same order._ This includes in a single
statement the six operations mentioned in the preceding axioms; that is,
if _a_ > _b_ and if _x_ = _y_, then _a_ + _x_ > _b_ + _y_,
_a_ - _x_ > _b_ - _y_, _ax_ > _by_, etc. The reason for thus combining
six axioms in one in the case of inequalities is apparent. They are
rarely used in geometry, and if a teacher is in doubt as to the pupil's
knowledge, he can easily inquire in the few cases that arise, whereas it
would consume a great deal of time to do this for the many equations
that are met. The axiom is stated in such a way as to exclude
multiplying or dividing by negative numbers, this case not being needed.

7. _If unequals are added to unequals in the same order, the sums are
unequal in the same order; if unequals are subtracted from equals, the
remainders are unequal in the reverse order._ These are the only cases
in which unequals are necessarily combined with unequals, or operate
upon equals in geometry, and the axiom is easily explained to the class
by the use of numbers.

8. _Quantities that are equal to the same quantity or to equal
quantities are equal to each other._ In this axiom the word "quantity"
is used, in the common manner of the present time, to include number and
all geometric magnitudes (length, area, volume).

9. _A quantity may be substituted for its equal in an equation or in an
inequality._ This axiom is tacitly assumed by all writers, and is very
useful in the proofs of geometry. It is really the basis of several
other axioms, and if we were seeking the "irreducible minimum," it would
replace them. Since, however, we are seeking only a reasonably abridged
list of convenient assumptions that beginners will understand and use,
this axiom has much to commend it. If we consider the equations
(1) _a_ = _x_ and (2) _b_ = _x_, we see that for _x_ in equation (1) we
may substitute _b_ from equation (2) and have _a_ = _b_; in other words,
that Axiom 8 is included in Axiom 9. Furthermore, if (1) _a_ = _b_ and
(2) _x_ = _y_, then since _a_ + _x_ is the same as _a_ + _x_, we may, by
substituting, say that _a_ + _x_ = _a_ + _x_ = _b_ + _x_ = _b_ + _y_. In
other words, Axiom 1 is included in Axiom 9. Thus an axiom that includes
others has a legitimate place, because a beginner would be too much
confused by seeing its entire scope, and because he will make frequent
use of it in his mathematical work.

10. _If the first of three quantities is greater than the second, and
the second is greater than the third, then the first is greater than the
third._ This axiom is needed several times in geometry. The case in
which _a_ > _b_ and _b_ = _c_, therefore _a_ > _c_, is provided for in
Axiom 9.

11. _The whole is greater than any of its parts and is equal to the sum
of all its parts._ The latter part of this axiom is really only the
definition of "whole," and it would be legitimate to state a definition
accordingly and refer to it where the word is employed. Where, however,
we wish to speak of a polygon, for example, and wish to say that the
area is equal to the combined areas of the triangles composing it, it is
more satisfactory to have this axiom to which to refer. It will be
noticed that two related axioms are here combined in one, for a reason
similar to the one stated under Axiom 5.

In the case of the postulates we are met by a problem similar to the one
confronting us in connection with the axioms,--the problem of the
"irreducible minimum" as related to the question of teaching. Manifestly
Euclid used postulates that he did not state, and proved some statements
that he might have postulated.[47]

The postulates given by Euclid under the name [Greek:
aitêmata](_aitemata_) were requests made by the teacher to his pupil
that certain things be conceded. They were five in number, as follows:

1. _Let the following be conceded: to draw a straight line from any
point to any point._

Strictly speaking, Euclid might have been required to postulate that
points and straight lines exist, but he evidently considered this
statement sufficient. Aristotle had, however, already called attention
to the fact that a mere definition was sufficient only to show what a
concept is, and that this must be followed by a proof that the thing
exists. We might, for example, define _x_ as a line that bisects an
angle without meeting the vertex, but this would not show that an _x_
exists, and indeed it does not exist. Euclid evidently intended the
postulate to assert that this line joining two points is unique, which
is only another way of saying that two points determine a straight line,
and really includes the idea that two straight lines cannot inclose
space. For purposes of instruction, the postulate would be clearer if it
read, _One straight line, and only one, can be drawn through two given
points_.

2. _To produce a finite straight line continuously in a straight line._

In this postulate Euclid practically assumes that a straight line can be
produced only in a straight line; in other words, that two different
straight lines cannot have a common segment. Several attempts have been
made to prove this fact, but without any marked success.

3. _To describe a circle with any center and radius._

4. _That all right angles are equal to one another._

     While this postulate asserts the essential truth that a right
     angle is a _determinate magnitude_ so that it really serves as
     an invariable standard by which other (acute and obtuse) angles
     may be measured, much more than this is implied, as will easily
     be seen from the following consideration. If the statement is
     to be _proved_, it can only be proved by the method of applying
     one pair of right angles to another and so arguing their
     equality. But this method would not be valid unless on the
     assumption of the _invariability of figures_, which would have
     to be asserted as an antecedent postulate. Euclid preferred to
     assert as a postulate, directly, the fact that all right angles
     are equal; and hence his postulate must be taken as equivalent
     to the principle of _invariability of figures_, or its
     equivalent, the _homogeneity_ of space.[48]

It is better educational policy, however, to assert this fact more
definitely, and to state the additional assumption that figures may be
moved about in space without deformation. The fourth of Euclid's
postulates is often given as an axiom, following the idea of the Greek
philosopher Geminus (who flourished in the first century B.C.), but this
is because Euclid's distinction between axiom and postulate is not
always understood. Proclus (410-485 A.D.) endeavored to prove the
postulate, and a later and more scientific effort was made by the
Italian geometrician Saccheri (1667-1733). It is very commonly taken as
a postulate that all straight angles are equal, this being more evident
to the senses, and the equality of right angles is deduced as a
corollary. This method of procedure has the sanction of many of our best
modern scholars.

5. _That, if a straight line falling on two straight lines make the
interior angle on the same side less than two right angles, the two
straight lines, if produced indefinitely, meet on that side on which are
the angles less than the two right angles._

This famous postulate, long since abandoned in teaching the beginner in
geometry, is a remarkable evidence of the clear vision of Euclid. For
two thousand years mathematicians sought to prove it, only to
demonstrate the wisdom of its author in placing it among the
assumptions.[49] Every proof adduced contains some assumption that
practically conceals the postulate itself. Thus the great English
mathematician John Wallis (1616-1703) gave a proof based upon the
assumption that "given a figure, another figure is possible which is
similar to the given one, and of any size whatever." Legendre
(1752-1833) did substantially the same at one time, and offered several
other proofs, each depending upon some equally unprovable assumption.
The definite proof that the postulate cannot be demonstrated is due to
the Italian Beltrami (1868).

Of the alternative forms of the postulate, that of Proclus is generally
considered the best suited to beginners. As stated by Playfair (1795),
this is, "Through a given point only one parallel can be drawn to a
given straight line"; and as stated by Proclus, "If a straight line
intersect one of two parallels, it will intersect the other also."
Playfair's form is now the common "postulate of parallels," and is the
one that seems destined to endure.

Posidonius and Geminus, both Stoics of the first century B.C., gave as
their alternative, "There exist straight lines everywhere equidistant
from one another." One of Legendre's alternatives is, "There exists a
triangle in which the sum of the three angles is equal to two right
angles." One of the latest attempts to suggest a substitute is that of
the Italian Ingrami (1904), "Two parallel straight lines intercept, on
every transversal which passes through the mid-point of a segment
included between them, another segment the mid-point of which is the
mid-point of the first."

Of course it is entirely possible to assume that through a point more
than one line can be drawn parallel to a given straight line, in which
case another type of geometry can be built up, equally rigorous with
Euclid's. This was done at the close of the first quarter of the
nineteenth century by Lobachevsky (1793-1856) and Bolyai (1802-1860),
resulting in the first of several "non-Euclidean" geometries.[50]

Taking the problem to be that of stating a reasonably small number of
geometric assumptions that may form a basis to supplement the general
axioms, that shall cover the most important matters to which the student
must refer, and that shall be so simple as easily to be understood by a
beginner, the following are recommended:

1. _One straight line, and only one, can be drawn through two given
points._ This should also be stated for convenience in the form, _Two
points determine a straight line_. From it may also be drawn this
corollary, _Two straight lines can intersect in only one point_, since
two points would determine a straight line. Such obvious restatements of
or corollaries to a postulate are to be commended, since a beginner is
often discouraged by having to prove what is so obvious that a
demonstration fails to commend itself to his mind.

2. _A straight line may be produced to any required length._ This, like
Postulate 1, requires the use of a straightedge for drawing the physical
figure. The required length is attained by using the compasses to
measure the distance. The straightedge and the compasses are the only
two drawing instruments recognized in elementary geometry.[51] While
this involves more than Euclid's postulate, it is a better working
assumption for beginners.

3. _A straight line is the shortest path between two points._ This is
easily proved by the method of Euclid[52] for the case where the paths
are broken lines, but it is needed as a postulate for the case of curve
paths. It is a better statement than the common one that a straight line
is the shortest _distance_ between two points; for distance is measured
on a line, but it is not itself a line. Furthermore, there are
scientific objections to using the word "distance" any more than is
necessary.

4. _A circle may be described with any given point as a center and any
given line as a radius._ This involves the use of the second of the two
geometric instruments, the compasses.

5. _Any figure may be moved from one place to another without altering
the size or shape._ This is the postulate of the homogeneity of space,
and asserts that space is such that we may move a figure as we please
without deformation of any kind. It is the basis of all cases of
superposition.

6. _All straight angles are equal._ It is possible to prove this, and
therefore, from the standpoint of strict logic, it is unnecessary as a
postulate. On the other hand, it is poor educational policy for a
beginner to attempt to prove a thing that is so obvious. The attempt
leads to a loss of interest in the subject, the proposition being (to
state a paradox) hard because it is so easy. It is, of course, possible
to postulate that straight angles are equal, and to draw the conclusion
that their halves (right angles) are equal; or to proceed in the
opposite direction, and postulate that all right angles are equal, and
draw the conclusion that their doubles (straight angles) are equal. Of
the two the former has the advantage, since it is probably more obvious
that all straight angles are equal. It is well to state the following
definite corollaries to this postulate: (1) _All right angles are
equal_; (2) _From a point in a line only one perpendicular can be drawn
to the line_, since two perpendiculars would make the whole (right
angle) equal to its part; (3) _Equal angles have equal complements,
equal supplements, and equal conjugates_; (4) _The greater of two
angles has the less complement, the less supplement, and the less
conjugate._ All of these four might appear as propositions, but, as
already stated, they are so obvious as to be more harmful than useful to
beginners when given in such form.

The postulate of parallels may properly appear in connection with that
topic in Book I, and it is accordingly treated in Chapter XIV.

There is also another assumption that some writers are now trying to
formulate in a simple fashion. We take, for example, a line segment
_AB_, and describe circles with _A_ and _B_ respectively as centers, and
with a radius _AB_. We say that the circles will intersect as at _C_ and
_D_. But how do we know that they intersect? We assume it, just as we
assume that an indefinite straight line drawn from a point inclosed by a
circle will, if produced far enough, cut the circle twice. Of course a
pupil would not think of this if his attention was not called to it, and
the harm outweighs the good in doing this with one who is beginning the
study of geometry.

With axioms and with postulates, therefore, the conclusion is the same:
from the standpoint of scientific geometry there is an irreducible
minimum of assumptions, but from the standpoint of practical teaching
this list should give place to a working set of axioms and postulates
that meet the needs of the beginner.

     =Bibliography.= Smith, Teaching of Elementary Mathematics, New
     York, 1900; Young, The Teaching of Mathematics, New York, 1901;
     Moore, On the Foundations of Mathematics, _Bulletin of the
     American Mathematical Society_, 1903, p. 402; Betz, Intuition
     and Logic in Geometry, _The Mathematics Teacher_, Vol. II, p.
     3; Hilbert, The Foundations of Geometry, Chicago, 1902; Veblen,
     A System of Axioms for Geometry, _Transactions of the American
     Mathematical Society_, 1904, p. 343.

FOOTNOTES:

[42] From the Greek [Greek: gê], _ge_ (earth), + [Greek: metrein],
_metrein_ (to measure), although the science has not had to do directly
with the measure of the earth for over two thousand years.

[43] From the Arabic _al_ (the) + _jabr_ (restoration), referring to
taking a quantity from one side of an equation and then restoring the
balance by taking it from the other side (see page 37).

[44] One of the clearest discussions of the subject is in W. B.
Frankland, "The First Book of Euclid's 'Elements,'" p. 26, Cambridge,
1905.

[45] "Grundlagen der Geometrie," Leipzig, 1899. See Heath's "Euclid,"
Vol. I, p. 229, for an English version; also D. E. Smith, "Teaching of
Elementary Mathematics," p. 266, New York, 1900.

[46] We need frequently to recall the fact that Euclid's "Elements" was
intended for advanced students who went to Alexandria as young men now
go to college, and that the book was used only in university instruction
in the Middle Ages and indeed until recent times.

[47] For example, he moves figures without deformation, but states no
postulate on the subject; and he proves that one side of a triangle is
less than the sum of the other two sides, when he might have postulated
that a straight line is the shortest path between two points. Indeed,
his followers were laughed at for proving a fact so obvious as this one
concerning the triangle.

[48] T. L. Heath, "Euclid," Vol. I, p. 200.

[49] For a résumé of the best known attempts to prove this postulate,
see Heath, "Euclid," Vol. I, p. 202; W. B. Frankland, "Theories of
Parallelism," Cambridge, 1910.

[50] For the early history of this movement see Engel and Stäckel, "Die
Theorie der Parallellinien von Euklid bis auf Gauss," Leipzig, 1895;
Bonola, Sulla teoria delle parallele e sulle geometrie non-euclidee, in
his "Questioni riguardanti la geometria elementare," 1900;
Karagiannides, "Die nichteuklidische Geometrie vom Alterthum bis zur
Gegenwart," Berlin, 1893.

[51] This limitation upon elementary geometry was placed by Plato (died
347 B.C.), as already stated.

[52] Book I, Proposition 20.

CHAPTER XII

THE DEFINITIONS OF GEOMETRY

When we consider the nature of geometry it is evident that more
attention must be paid to accuracy of definitions than is the case in
most of the other sciences. The essence of all geometry worthy of
serious study is not the knowledge of some fact, but the proof of that
fact; and this proof is always based upon preceding proofs, assumptions
(axioms or postulates), or definitions. If we are to prove that one line
is perpendicular to another, it is essential that we have an exact
definition of "perpendicular," else we shall not know when we have
reached the conclusion of the proof.

The essential features of a definition are that the term defined shall
be described in terms that are simpler than, or at least better known
than, the thing itself; that this shall be done in such a way as to
limit the term to the thing defined; and that the description shall not
be redundant. It would not be a good definition to say that a right
angle is one fourth of a perigon and one half of a straight angle,
because the concept "perigon" is not so simple, and the term "perigon"
is not so well known, as the term and the concept "right angle," and
because the definition is redundant, containing more than is necessary.

It is evident that satisfactory definitions are not always possible; for
since the number of terms is limited, there must be at least one that is
at least as simple as any other, and this cannot be described in terms
simpler than itself. Such, for example, is the term "angle." We can
easily explain the meaning of this word, and we can make the concept
clear, but this must be done by a certain amount of circumlocution and
explanation, not by a concise and perfect definition. Unless a beginner
in geometry knows what an angle is before he reads the definition in a
textbook, he will not know from the definition. This fact of the
impossibility of defining some of the fundamental concepts will be
evident when we come to consider certain attempts that have been made in
this direction.

It should also be understood in this connection that a definition makes
no assertion as to the existence of the thing defined. If we say that a
tangent to a circle is an unlimited straight line that touches the
circle in one point, and only one, we do not assert that it is possible
to have such a line; that is a matter for proof. Not in all cases,
however, can this proof be given, as in the existence of the simplest
concepts. We cannot, for example, prove that a point or a straight line
exists after we have defined these concepts. We therefore tacitly or
explicitly assume (postulate) the existence of these fundamentals of
geometry. On the other hand, we can prove that a tangent exists, and
this may properly be considered a legitimate proposition or corollary of
elementary geometry. In relation to geometric proof it is necessary to
bear in mind, therefore, that we are permitted to define any term we
please; for example, "a seven-edged polyhedron" or Leibnitz's "ten-faced
regular polyhedron," neither of which exists; but, strictly speaking, we
have no right to make use of a definition in a proof until we have shown
or postulated that the thing defined has an existence. This is one of
the strong features of Euclid's textbook. Not being able to prove that a
point, a straight line, and a circle exists, he practically postulates
these facts; but he uses no other definition in a proof without showing
that the thing defined exists, and this is his reason for mingling his
problems with his theorems. At the present time we confessedly sacrifice
his logic in this respect for the reason that we teach geometry to
pupils who are too young to appreciate that logic.

It was pointed out by Aristotle, long before Euclid, that it is not a
satisfactory procedure to define a thing by means of terms that are
strictly not prior to it, as when we attempt to define something by
means of its opposite. Thus to define a curve as "a line, no part of
which is straight," would be a bad definition unless "straight" had
already been explicitly defined; and to define "bad" as "not good" is
unsatisfactory for the reason that "bad" and "good" are concepts that
are evolved simultaneously. But all this is only a detail under the
general principle that a definition must employ terms that are better
understood than the one defined.

It should be understood that some definitions are much more important
than others, considered from the point of view of the logic of geometry.
Those that enter into geometric proofs are basal; those that form part
of the conversational language of geometry are not. Euclid gave
twenty-three definitions in Book I, and did not make use of even all of
these terms. Other terms, those not employed in his proofs, he assumed
to be known, just as he assumed a knowledge of any other words in his
language. Such procedure would not be satisfactory under modern
conditions, but it is of great importance that the teacher should
recognize that certain definitions are basal, while others are merely
informational.

It is now proposed to consider the basal definitions of geometry, first,
that the teacher may know what ones are to be emphasized and learned;
and second, that he may know that the idea that the standard definitions
can easily be improved is incorrect. It is hoped that the result will be
the bringing into prominence of the basal concepts, and the discouraging
of attempts to change in unimportant respects the definitions in the
textbook used by the pupil.

In order to have a systematic basis for work, the definitions of two
books of Euclid will first be considered.[53]

1. POINT. _A point is that which has no part._ This was incorrectly
translated by Capella in the fifth century, "Punctum est cuius pars
nihil est" (a point is that of which a part is nothing), which is as
much as to say that the point itself is nothing. It generally appears,
however, as in the Campanus edition,[54] "Punctus est cuius pars non
est," which is substantially Euclid's wording. Aristotle tells of the
definitions of point, line, and surface that prevailed in his time,
saying that they all defined the prior by means of the posterior.[55]
Thus a point was defined as "an extremity of a line," a line as "the
extremity of a surface," and a surface as "the extremity of a
solid,"--definitions still in use and not without their value. For it
must not be assumed that scientific priority is necessarily priority in
fact; a child knows of "solid" before he knows of "point," so that it
may be a very good way to explain, if not to define, by beginning with
solid, passing thence to surface, thence to line, and thence to point.

The first definition of point of which Proclus could learn is attributed
by him to the Pythagoreans, namely, "a monad having position," the early
form of our present popular definition of a point as "position without
magnitude." Plato defined it as "the beginning of a line," thus
presupposing the definition of "line"; and, strangely enough, he
anticipated by two thousand years Cavalieri, the Italian geometer, by
speaking of points as "indivisible lines." To Aristotle, who protested
against Plato's definitions, is due the definition of a point as
"something indivisible but having position."

Euclid's definition is essentially that of Aristotle, and is followed by
most modern textbook writers, except as to its omission of the reference
to position. It has been criticized as being negative, "which has _no_
part"; but it is generally admitted that a negative definition is
admissible in the case of the most elementary concepts. For example,
"blind" must be defined in terms of a negation.

At present not much attention is given to the definition of "point,"
since the term is not used as the basis of a proof, but every effort is
made to have the concept clear. It is the custom to start from a small
solid, conceive it to decrease in size, and think of the point as the
limit to which it is approaching, using these terms in their usual sense
without further explanation.

2. LINE. _A line is breadthless length._ This is usually modified in
modern textbooks by saying that "a line is that which has length without
breadth or thickness," a statement that is better understood by
beginners. Euclid's definition is thought to be due to Plato, and is
only one of many definitions that have been suggested. The Pythagoreans
having spoken of the point as a monad naturally were led to speak of the
line as dyadic, or related to two. Proclus speaks of another definition,
"magnitude in one dimension," and he gives an excellent illustration of
line as "the edge of a shadow," thus making it real but not material.
Aristotle speaks of a line as a magnitude "divisible in one way only,"
as contrasted with a surface which is divisible in two ways, and with a
solid which is divisible in three ways. Proclus also gives another
definition as the "flux of a point," which is sometimes rendered as the
path of a moving point. Aristotle had suggested the idea when he wrote,
"They say that a line by its motion produces a surface, and a point by
its motion a line."

Euclid did not deem it necessary to attempt a classification of lines,
contenting himself with defining only a straight line and a circle, and
these are really the only lines needed in elementary geometry. His
commentators, however, made the attempt. For example. Heron (first
century A.D.) probably followed his definition of line by this
classification:

          { Straight
    Lines {              { Circular circumferences
          { Not straight { Spiral shaped
                         { Curved (generally)

Proclus relates that both Plato and Aristotle divided lines into
"straight," "circular," and "a mixture of the two," a statement which
is not quite exact, but which shows the origin of a classification not
infrequently found in recent textbooks. Geminus (_ca._ 50 B.C.) is said
by Proclus to have given two classifications, of which one will suffice
for our purposes:

          { Composite (broken line forming an angle)
          {
    Lines {             { Forming a figure, or determinate. (Circle,
          {             { ellipse, cissoid.)
          { Incomposite { Not forming a figure, or indeterminate and
                        { extending without a limit. (Straight
                        { line, parabola, hyperbola, conchoid.)

Of course his view of the cissoid, the curve represented by the equation
_y_^2(_a_ + _x_) = (_a_ - _x_)^3, is not the modern view.

3. _The extremities of a line are points._ This is not a definition in
the sense of its two predecessors. A modern writer would put it as a
note under the definition of line. Euclid did not wish to define a point
as the extremity of a line, for Aristotle had asserted that this was not
scientific; so he defined point and line, and then added this statement
to show the relation of one to the other. Aristotle had improved upon
this by stating that the "division" of a line, as well as an extremity,
is a point, as is also the intersection of two lines. These statements,
if they had been made by Euclid, would have avoided the objection made
by Proclus, that some lines have no extremities, as, for example, a
circle, and also a straight line extending infinitely in both
directions.

4. STRAIGHT LINE. _A straight line is that which lies evenly with
respect to the points on itself._ This is the least satisfactory of all
of the definitions of Euclid, and emphasizes the fact that the straight
line is the most difficult to define of the elementary concepts of
geometry. What is meant by "lies evenly"? Who would know what a
straight line is, from this definition, if he did not know in advance?

The ancients suggested many definitions of straight line, and it is well
to consider a few in order to appreciate the difficulties involved.
Plato spoke of it as "that of which the middle covers the ends," meaning
that if looked at endways, the middle would make it impossible to see
the remote end. This is often modified to read that "a straight line
when looked at endways appears as a point,"--an idea that involves the
postulate that our line of sight is straight. Archimedes made the
statement that "of all the lines which have the same extremities, the
straight line is the least," and this has been modified by later writers
into the statement that "a straight line is the shortest distance
between two points." This is open to two objections as a definition: (1)
a line is not distance, but distance is the _length_ of a line,--it is
measured on a line; (2) it is merely stating a property of a straight
line to say that "a straight line is the shortest path between two
points,"--a proper postulate but not a good definition. Equally
objectionable is one of the definitions suggested by both Heron and
Proclus, that "a straight line is a line that is stretched to its
uttermost"; for even then it is reasonable to think of it as a catenary,
although Proclus doubtless had in mind the Archimedes statement. He also
stated that "a straight line is a line such that if any part of it is in
a plane, the whole of it is in the plane,"--a definition that runs in a
circle, since plane is defined by means of straight line. Proclus also
defines it as "a uniform line, capable of sliding along itself," but
this is also true of a circle.

Of the various definitions two of the best go back to Heron, about the
beginning of our era. Proclus gives one of them in this form, "That line
which, when its ends remain fixed, itself remains fixed." Heron proposed
to add, "when it is, as it were, turned round in the same plane." This
has been modified into "that which does not change its position when it
is turned about its extremities as poles," and appears in substantially
this form in the works of Leibnitz and Gauss. The definition of a
straight line as "such a line as, with another straight line, does not
inclose space," is only a modification of this one. The other definition
of Heron states that in a straight line "all its parts fit on all in all
ways," and this in its modern form is perhaps the most satisfactory of
all. In this modern form it may be stated, "A line such that any part,
placed with its ends on any other part, must lie wholly in the line, is
called a straight line," in which the force of the word "must" should be
noted. This whole historical discussion goes to show how futile it is to
attempt to define a straight line. What is needed is that we should
explain what is meant by a straight line, that we should illustrate it,
and that pupils should then read the definition understandingly.

5. SURFACE. _A surface is that which has length and breadth._ This is
substantially the common definition of our modern textbooks. As with
line, so with surface, the definition is not entirely satisfactory, and
the chief consideration is that the meaning of the term should be made
clear by explanations and illustrations. The shadow cast on a table top
is a good illustration, since all idea of thickness is wanting. It adds
to the understanding of the concept to introduce Aristotle's statement
that a surface is generated by a moving line, modified by saying that
it _may_ be so generated, since the line might slide along its own
trace, or, as is commonly said in mathematics, along itself.

6. _The extremities of a surface are lines._ This is open to the same
explanation and objection as definition 3, and is not usually given in
modern textbooks. Proclus calls attention to the fact that the statement
is hardly true for a complete spherical surface.

7. PLANE. _A plane surface is a surface which lies evenly with the
straight lines on itself._ Euclid here follows his definition of
straight line, with a result that is equally unsatisfactory. For
teaching purposes the translation from the Greek is not clear to a
beginner, since "lies evenly" is a term not simpler than the one
defined. As with the definition of a straight line, so with that of a
plane, numerous efforts at improvement have been made. Proclus,
following a hint of Heron's, defines it as "the surface which is
stretched to the utmost," and also, this time influenced by Archimedes's
assumption concerning a straight line, as "the least surface among all
those which have the same extremities." Heron gave one of the best
definitions, "A surface all the parts of which have the property of
fitting on [each other]." The definition that has met with the widest
acceptance, however, is a modification of one due to Proclus, "A surface
such that a straight line fits on all parts of it." Proclus elsewhere
says, "[A plane surface is] such that the straight line fits on it all
ways," and Heron gives it in this form, "[A plane surface is] such that,
if a straight line pass through two points on it, the line coincides
with it at every spot, all ways." In modern form this appears as
follows: "A surface such that a straight line joining any two of its
points lies wholly in the surface is called a plane," and for teaching
purposes we have no better definition. It is often known as Simson's
definition, having been given by Robert Simson in 1756.

The French mathematician, Fourier, proposed to define a plane as formed
by the aggregate of all the straight lines which, passing through one
point on a straight line in space, are perpendicular to that line. This
is clear, but it is not so usable for beginners as Simson's definition.
It appears as a theorem in many recent geometries. The German
mathematician, Crelle, defined a plane as a surface containing all the
straight lines (throughout their whole length) passing through a fixed
point and also intersecting a straight line in space, but of course this
intersected straight line must not pass through the fixed point.
Crelle's definition is occasionally seen in modern textbooks, but it is
not so clear to the pupil as Simson's. Of the various ultrascientific
definitions of a plane that have been suggested of late it is hardly of
use to speak in a book concerned primarily with practical teaching. No
one of them is adapted to the needs and the comprehension of the
beginner, and it seems that we are not likely to improve upon the
so-called Simson form.

8. PLANE ANGLE. _A plane angle is the inclination to each other of two
lines in a plane which meet each other and do not lie in a straight
line._ This definition, it will be noticed, includes curvilinear angles,
and the expression "and do not lie in a straight line" states that the
lines must not be continuous one with the other, that is, that zero and
straight angles are excluded. Since Euclid does not use the curvilinear
angle, and it is only the rectilinear angle with which we are concerned,
we will pass to the next definition and consider this one in connection
therewith.

9. RECTILINEAR ANGLE. _When the lines containing the angle are straight,
the angle is called rectilinear._ This definition, taken with the
preceding one, has always been a subject of criticism. In the first
place it expressly excludes the straight angle, and, indeed, the angles
of Euclid are always less than 180°, contrary to our modern concept. In
the second place it defines angle by means of the word "inclination,"
which is itself as difficult to define as angle. To remedy these defects
many substitutes have been proposed. Apollonius defined angle as "a
contracting of a surface or a solid at one point under a broken line or
surface." Another of the Greeks defined it as "a quantity, namely, a
distance between the lines or surfaces containing it." Schotten[56] says
that the definitions of angle generally fall into three groups:

_a._ An angle is the difference of direction between two lines that
meet. This is no better than Euclid's, since "difference of direction"
is as difficult to define as "inclination."

_b._ An angle is the amount of turning necessary to bring one side to
the position of the other side.

_c._ An angle is the portion of the plane included between its sides.

Of these, _b_ is given by way of explanation in most modern textbooks.
Indeed, we cannot do better than simply to define an angle as the
opening between two lines which meet, and then explain what is meant by
size, through the bringing in of the idea of rotation. This is a simple
presentation, it is easily understood, and it is sufficiently accurate
for the real purpose in mind, namely, the grasping of the concept. We
should frankly acknowledge that the concept of angle is such a simple
one that a satisfactory definition is impossible, and we should
therefore confine our attention to having the concept understood.

10. _When a straight line set up on a straight line makes the adjacent
angles equal to one another, each of the equal angles is right, and the
straight line standing on the other is called a perpendicular to that on
which it stands._ We at present separate these definitions and simplify
the language.

11. _An obtuse angle is an angle greater than a right angle._

12. _An acute angle is an angle less than a right angle._

The question sometimes asked as to whether an angle of 200° is obtuse,
and whether a negative angle, say -90°, is acute, is answered by saying
that Euclid did not conceive of angles equal to or greater than 180° and
had no notion of negative quantities. Generally to-day we define an
obtuse angle as "greater than one and less than two right angles." An
acute angle is defined as "an angle less than a right angle," and is
considered as positive under the general understanding that all
geometric magnitudes are positive unless the contrary is stated.

13. _A boundary is that which is an extremity of anything._ The
definition is not exactly satisfactory, for a circle is the boundary of
the space inclosed, but we hardly consider it as the extremity of that
space. Euclid wishes the definition before No. 14.

14. _A figure is that which is contained by any boundary or boundaries._
The definition is not satisfactory, since it excludes the unlimited
straight line, the angle, an assemblage of points, and other
combinations of lines and points which we should now consider as
figures.

15. _A circle is a plane figure contained by one line such that all the
straight lines falling upon it from one point among those lying within
the figure are equal to one another._

16. _And the point is called the center of the circle._

Some commentators add after "one line," definition 15, the words "which
is called the circumference," but these are not in the oldest
manuscripts. The Greek idea of a circle was usually that of part of a
plane which is bounded by a line called in modern times the
circumference, although Aristotle used "circle" as synonymous with "the
bounding line." With the growth of modern mathematics, however, and
particularly as a result of the development of analytic geometry, the
word "circle" has come to mean the bounding line, as it did with
Aristotle, a century before Euclid's time. This has grown out of the
equations of the various curves, _x_^2 + _y_^2 = _r_^2 representing the
circle-_line_, _a_^2_y_^2 + _b_^2_x_^2 = _a_^2_b_^2 representing the
ellipse-_line_, and so on. It is natural, therefore, that circle,
ellipse, parabola, and hyperbola should all be looked upon as lines.
Since this is the modern use of "circle" in English, it has naturally
found its way into elementary geometry, in order that students should
not have to form an entirely different idea of circle on beginning
analytic geometry. The general body of American teachers, therefore, at
present favors using "circle" to mean the bounding line and
"circumference" to mean the length of that line. This requires
redefining "area of a circle," and this is done by saying that it is the
area of the plane space inclosed. The matter is not of greatest
consequence, but teachers will probably prefer to join in the modern
American usage of the term.

17. DIAMETER. _A diameter of the circle is any straight line drawn
through the center and terminated in both directions by the
circumference of the circle, and such a straight line also bisects the
circle._ The word "diameter" is from two Greek words meaning a "through
measurer," and it was also used by Euclid for the diagonal of a square,
and more generally for the diagonal of any parallelogram. The word
"diagonal" is a later term and means the "through angle." It will be
noticed that Euclid adds to the usual definition the statement that a
diameter bisects the circle. He does this apparently to justify his
definition (18), of a semicircle (a half circle).

Thales is said to have been the first to prove that a diameter bisects
the circle, this being one of three or four propositions definitely
attributed to him, and it is sometimes given as a proposition to be
proved. As a proposition, however, it is unsatisfactory, since the proof
of what is so evident usually instills more doubt than certainty in the
minds of beginners.

18. SEMICIRCLE. _A semicircle is the figure contained by the diameter
and the circumference cut off by it. And the center of the semicircle is
the same as that of the circle._ Proclus remarked that the semicircle is
the only plane figure that has its center on its perimeter. Some writers
object to defining a circle as a line and then speaking of the area of a
circle, showing minds that have at least one characteristic of that of
Proclus. The modern definition of semicircle is "half of a circle," that
is, an arc of 180°, although the term is commonly used to mean both the
arc and the segment.

19. RECTILINEAR FIGURES. _Rectilinear figures are those which are
contained by straight lines, trilateral figures being those contained by
three, quadrilateral those contained by four, and multilateral those
contained by more than four, straight lines._

20. _Of trilateral figures, an equilateral triangle is that which has
its three sides equal, an isosceles triangle that which has two of its
sides alone equal, and a scalene triangle that which has its three sides
unequal._

21. _Further, of trilateral figures, a right-angled triangle is that
which has a right angle, an obtuse-angled triangle that which has an
obtuse angle, and an acute-angled triangle that which has its three
angles acute._

These three definitions may properly be considered together.
"Rectilinear" is from the Latin translation of the Greek _euthygrammos_,
and means "right-lined," or "straight-lined." Euclid's idea of such a
figure is that of the space inclosed, while the modern idea is tending
to become that of the inclosing lines. In elementary geometry, however,
the Euclidean idea is still held. "Trilateral" is from the Latin
translation of the Greek _tripleuros_ (three-sided). In elementary
geometry the word "triangle" is more commonly used, although
"quadrilateral" is more common than "quadrangle." The use of these two
different forms is eccentric and is merely a matter of fashion. Thus we
speak of a pentagon but not of a tetragon or a trigon, although both
words are correct in form. The word "multilateral" (many-sided) is a
translation of the Greek _polypleuros_. Fashion has changed this to
"polygonal" (many-angled), the word "multilateral" rarely being seen.

Of the triangles, "equilateral" means "equal-sided"; "isosceles" is from
the Greek _isoskeles_, meaning "with equal legs," and "scalene" from
_skalenos_, possibly from _skazo_ (to limp), or from _skolios_
(crooked). Euclid's limitation of isosceles to a triangle with two, and
only two, equal sides would not now be accepted. We are at present more
given to generalizing than he was, and when we have proved a proposition
relating to the isosceles triangle, we wish to say that we have thereby
proved it for the equilateral triangle. We therefore say that an
isosceles triangle has two sides equal, leaving it possible that all
three sides should be equal. The expression "equal legs" is now being
discarded on the score of inelegance. In place of "right-angled
triangle" modern writers speak of "right triangle," and so for the
obtuse and acute triangles. The terms are briefer and are as readily
understood. It may add a little interest to the subject to know that
Plutarch tells us that the ancients thought that "the power of the
triangle is expressive of the nature of Pluto, Bacchus, and Mars." He
also states that the Pythagoreans called "the equilateral triangle the
head-born Minerva and Tritogeneia (born of Triton) because it may be
equally divided by the perpendicular lines drawn from each of its
angles."

22. _Of quadrilateral figures a square is that which is both equilateral
and right-angled; an oblong that which is right-angled but not
equilateral; a rhombus that which is equilateral and not right-angled;
and a rhomboid that which has its opposite sides and angles equal to one
another, but is neither equilateral nor right-angled. And let all
quadrilaterals other than these be called trapezia._ In this definition
Euclid also specializes in a manner not now generally approved. Thus we
are more apt to-day to omit the oblong and rhomboid as unnecessary, and
to define "rhombus" in such a manner as to include a square. We use
"parallelogram" to cover "rhomboid," "rhombus," "oblong," and "square."
For "oblong" we use "rectangle," letting it include square. Euclid's
definition of "square" illustrates his freedom in stating more
attributes than are necessary, in order to make sure that the concept is
clear; for he might have said that it "is that which is equilateral and
has one right angle." We may profit by his method, sacrificing logic to
educational necessity. Euclid does not use "oblong," "rhombus,"
"rhomboid," and "trapezium" (_plural_, "trapezia") in his proofs, so
that he might well have omitted the definitions, as we often do.