GEOMETRY.
There is certainly no royal road to geometry, but the way may be
rendered easy and pleasant by timely preparations for the journey.
Without any previous knowledge of the country, or of its peculiar
language, how can we expect that our young traveller should advance
with facility or pleasure? We are anxious that our pupil should
acquire a taste for accurate reasoning, and we resort to Geometry, as
the most perfect, and the purest series of ratiocination which has
been invented. Let us, then, sedulously avoid whatever may disgust
him; let his first steps be easy, and successful; let them be
frequently repeated until he can trace them without a guide.
We have recommended in the chapter upon Toys, that children should,
from their earliest years, be accustomed to the shape of what are
commonly called regular solids; they should also be accustomed to the
figures in mathematical diagrams. To these should be added their
respective names, and the whole language of the science should be
rendered as familiar as possible.
Mr. Donne, an ingenious mathematician of Bristol, has published a
prospectus of an Essay on Mechanical Geometry: he has executed, and
employed with success, models in wood and metal for demonstrating
propositions in geometry in a _palpable_ manner. We have endeavoured,
in vain, to procure a set of these models for our own pupils, but we
have no doubt of their entire utility.
What has been acquired in childhood, should not be suffered to escape
the memory. Dionysius[19] had mathematical diagrams described upon
the floors of his apartments, and thus recalled their demonstrations
to his memory. The slightest addition that can be conceived, if it be
continued daily, will imperceptibly, not only preserve what has been
already acquired, but will, in a few years, amount to as large a stock
of mathematical knowledge as we could wish. It is not our object to
make mathematicians, but to make it easy to our pupil to become a
mathematician, if his interest, or his ambition, make it desirable;
and, above all, to habituate him to clear reasoning, and close
attention. And we may here remark, that an early acquaintance with the
accuracy of mathematical demonstration, does not, within our
experience, contract the powers of the imagination. On the contrary,
we think that a young lady of twelve years old, who is now no more,
and who had an uncommon propensity to mathematical reasoning, had an
imagination remarkably vivid and inventive.[20]
We have accustomed our pupils to form in their minds the conception of
figures generated from points and lines, and surfaces supposed to move
in different directions, and with different velocities. It may be
thought, that this would be a difficult occupation for young minds;
but, upon trial, it will be found not only easy to them, but
entertaining. In their subsequent studies, it will be of material
advantage; it will facilitate their progress not only in pure
mathematics, but in mechanics and astronomy, and in every operation of
the mind which requires exact reflection.
To demand steady thought from a person who has not been trained to it,
is one of the most unprofitable and dangerous requisitions that can be
made in education.
"Full in the midst of Euclid dip at once,
And petrify a genius to a dunce."
In the usual commencement of mathematical studies, the learner is
required to admit that a point, of which he sees the prototype, a dot
before him, has neither length, breadth, nor thickness. This, surely,
is a degree of faith not absolutely necessary for the neophyte in
science. It is an absurdity which has, with much success, been
attacked in "Observations on the Nature of Demonstrative Evidence," by
Doctor Beddoes.
We agree with the doctor as to the impropriety of calling a visible
dot, a point without dimensions. But, notwithstanding the high respect
which the author commands by a steady pursuit of truth on all subjects
of human knowledge, we cannot avoid protesting against part of the
doctrine which he has endeavoured to inculcate. That the names point,
radius, &c. are derived from sensible objects, need not be disputed;
but surely the word centre can be understood by the human mind without
the presence of any visible or tangible substance.
Where two lines meet, their junction cannot have dimensions;
where two radii of a circle meet, they constitute the centre,
and the name centre may be used for ever without any relation
to a tangible or visible point. The word boundary, in like manner,
means the extreme limit we call a line; but to assert that it has
thickness, would, from the very terms which are used to describe it,
be a direct contradiction. Bishop Berkely, Mr. Walton, Philathetes
Cantabrigiensis, and Mr. Benjamin Robins, published several pamphlets
upon this subject about half a century ago. No man had a more
penetrating mind than Berkely; but we apprehend that Mr. Robins closed
the dispute against him. This is not meant as an appeal to authority,
but to apprize such of our readers as wish to consider the argument,
where they may meet an accurate investigation of the subject. It is
sufficient for our purpose, to warn preceptors not to insist upon
their pupils' acquiescence in the dogma, that a point, represented by
a dot, is without dimensions; and at the same time to profess, that we
understand distinctly what is meant by mathematicians when they speak
of length without breadth, and of a superfices without depth;
expressions which, to our minds, convey a meaning as distinct as the
name of any visible or tangible substance in nature, whose varieties
from shade, distance, colour, smoothness, heat, &c. are infinite, and
not to be comprehended in any definition.
In fact, this is a dispute merely about words, and as the extension of
the art of printing puts it in the power of every man to propose and
to defend his opinions at length, and at leisure, the best friends may
support different sides of a question with mutual regard, and the most
violent enemies with civility and decorum. Can we believe that Tycho
Brahe lost half his nose in a dispute with a Danish nobleman about a
mathematical demonstration?
FOOTNOTES:
[19] Plutarch.--Life of Dion.
[20] V. Rivuletta, a little story written _entirely_ by her in 1786.