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ATTACKS UPON THE STUDY OF MATHEMATICS

nothing of observation, nothing of induction, nothing of experiment, nothing of causation."

I, of course, am not so absurd as to maintain that the habit of observation of external nature will be best or in any degree cultivated by the study of mathematics, at all events as that study is at present conducted, and no one can desire more earnestly than myself to see natural and experimental science introduced into our schools as a primary and indispensable branch of education: I think that that study and mathematical culture should go on hand in hand together, and that they would greatly influence each other for their mutual good. I would rejoice to see mathematics taught with that life and animation which the presence and example of her young and buoyant sister could not fail to impart, short roads preferred to long ones, Euclid honourably shelved or buried "deeper than e'er plummet sounded" out of the schoolboy's reach, morphology introduced into the elements of Algebra—projection, correlation, and motion accepted as aids to geometry—the mind of the student quickened and elevated and his faith awakened by early initiation into the ruling ideas of polarity, continuity, infinity, and familiarization with the doctrine of the imaginary and inconceivable.

What light, if any, do these attacks and these defenses of mathematical study throw upon the educational problems of to-day? Hamilton gathered a cloud of witnesses which, in so far as the testimony adduced was sincere, proved that mathematical study alone is not the proper education for life. That mathematical study is pernicious Hamilton did not succeed in proving. It would seem, therefore, as if the Hamiltonian controversy was somewhat barren in useful results. Probably no one to-day advocates the well-nigh exclusive study of mathematics or of any other science as the best education obtainable.

Schopenhauer attacked mainly the logic of mathematics as found in Euclid. As a critique of the logic as used by Euclid the attack is childish and has no value for us. From the standpoint of educational method it points out the difficulty experienced by children in understanding the mode of proof called the reductio ad absurdum and emphasizes the constant need of appeal to the intuition in the teaching of mathematics.

The attack made by Huxley touches questions which are more subtle. Sylvester, in his rejoinder, proved conclusively that the mathematician engaged in original research does exercise powers of internal observation, of induction, of experimentation and even of causation. Are these powers exercised by the pupil in the class room? That depends. When English teachers required several books of Euclid to be memorized, even including the lettering of figures, no original exercises being demanded, then indeed such teaching knew nothing of observation, induction, experiment, and causation, except that a good memory as a cause was seen to bring about a pass mark as an effect. But when attention is paid to the solution of original exercises, and to the heuristic or genetic development of certain parts of the subject, then surely the young pupil