Three Absurdities of Certain Modern Theories of Education [pp. 265-292]

Absurdities of Modern Education.
(we use this term not as excluding reason, but as implying a
demi-conviction of some ratio or reason not yet clearly traced)
then assumed as a stand-point, from whence to reason back,
until by reaching in this way some previously known and fixed
position, the guess at last is verified.
It may have been that in the mind of the first geometrical
explorer, the greatest part of the previous propositions in the
first book of the Elements, had been thus, one after another,
brought in to make a bridge from this great suspected truth to
the elementary axioms and definitions from which it seemed to
stand at so great a distance; and thus this theorem, so important in all other applications of the mathematics, may have been
the suggestive mother of that whole system of geometry, which
has come down to us under the name of Euclid.
This, or something like this, may have been the order of geometrical discovery; and yet, except as a problem of interest in
mental philosophy, it would be very unwise to attempt to teach
geometry in the way in which Pythagoras, and Euclid, and
Archimedes, and Descartes, may have discovered it, and not in
the way in which they afterwards scientifically arranged it,
making of it a logical harmony far more important and interesting than the mere amount of mathematical knowledge conveyed,
and presenting the eternal truths of space and figure in that
perfect order in which they are ever seen by the perfect mind,
instead of that in which they were reached by the groping
human intellect.
And here, although it may seem a digression from the main
subject, we cannot refrain from expressing regret, that this
great excellence of the old Greek geometry, its perfect logical
harmony, has been so much sacrificed in methods aiming rather
at what is comparatively of secondary importance, the mere
amount of mathematical knowledge, and the shortest methods
by which it might be reached. We may smile at the apparent
simplicity of some of the more elementary propositions in
Euclid; teacher and scholar, in their impatience, may pass
them by with affected contempt; the rigid adherence to a certain order may seem an unnecessary restraint; and yet we may
well doubt, whether some of the profoundest modern develop ments of the analytical calculus required higher powers of mind,
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