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 Geometrie," 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.