ATTACKS UPON THE STUDY OF MATHEMATICS 371 



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 intro- 

 duced 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 mathe- 

 matical study throw upon the educational problems of to-day? Hamil- 

 ton gathered a cloud of witnesses which, in so far as the testimony ad- 

 duced 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 Hamiltouian 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 under- 

 standing the mode of proof called the reductio ad ahsurdum 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 observa- 

 tion, 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 



