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SCIENCE 



[N. o. Vol. XXXII. No. 821 



dom which is essential for any high de- 

 gree of efficiency. An objection of a dif- 

 ferent character to the newer modes of 

 teaching geometry has been frequently 

 made of late. It is said that the students 

 are induced to accept and reproduce, as 

 proofs of theorems, arguments which are 

 not really proofs, and thus that the logical 

 training which should be imparted by a 

 study of geometry is vitiated. If this ob- 

 jection really implies a demand for a 

 purely deductive treatment of the subject, 

 I think some of those who raise it hardly 

 realize all that would be involved in the 

 complete satisfaction of their requirement. 

 I have already remarked that Euclid's 

 treatment of the subject is not rigorous as 

 regards logic. Owing to the recent ex- 

 ploration of the foundations of geometry 

 we possess at the • present time tolerably 

 satisfactory methods of purely deductive 

 treatment of the subject; in regard to me- 

 chanics, notwithstanding the valuable 

 work of Mach, Herz and others, this is not 

 yet the case. But, in the schemes of purely 

 deductive geometry, the systems of axioms 

 and postulates are far from being of a 

 very simple character; their real nature, 

 and the necessity for many of them, can 

 only be appreciated at a much later stage 

 in mathematical education than the one of 

 which I am speaking. A purely logical 

 treatment is the highest stage in the train- 

 ing of the mathematician, and is wholly 

 unsuitable — and, indeed, quite impossible 

 — in those stages beyond which the great 

 majority of students never pass. It can 

 then, in the case of all students, except a 

 few advanced ones in the universities, only 

 be a question of degree how far the purely 

 logical factor in the proofs of propositions 

 shall be modified by the introduction of 

 elements derived from observation or 

 spatial intuition. If the freedom of teach- 

 ing which I have advocated be allowed, it 



will be open to those teachers who find it 

 advisable in the interests of their students 

 to emphasize the logical side of their teach- 

 ing to do so ; and it is certainly of value in 

 all cases to draw the attention of students 

 to those points in a proof where the intui- 

 tional element enters. I draw, then, the 

 conclusion that a mixed treatment of 

 geometry, as of mechanics, must prevail in 

 the future, as it has done in the past, but 

 that the proportion of the observational or 

 intuitional factor to the logical one must 

 vary in accordance with the needs and 

 intellectual attainments of the students, 

 and that a large measure of freedom of 

 judgment in this regard should be left to 

 the teacher. 



The great and increasing importance of 

 a knowledge of the differential and inte- 

 gral calculus for students of engineering 

 and other branches of physical science has 

 led to the publication during the last few 

 years of a considerable niimber of text- 

 books on this subject intended for the use 

 of such students. Some of these text-books 

 are excellent, and their authors, by a skil- 

 ful insistence on the principles of the sub- 

 ject, have done their utmost to guard 

 against the very real dangers which attend 

 attempts to adapt such a subject to the 

 practical needs of engineers and others. 

 It is quite true that a great mass of detail 

 which has gradually come to form part — 

 often much too large a part — of the 

 material of the student of mathematics, 

 may with great advantage be ignored by 

 those whose main study is to be engineer- 

 ing science or physics. Yet it cannot be 

 too strongly insisted on that a firm grasp 

 of the principles, as distinct from the mere 

 processes of calculation, is essential if 

 mathematics is to be a tool really useful to 

 the engineer and the physicist. There is a 

 danger, which experience has shown to be 

 only too real, that such students may learn 



