PRESIDENTIAL ADDRESS. 521 



my disposal I propose to devote to a few words about some matters connected 

 with the teaching of the more elementary parts of Mathematics. Of late years a 

 new spirit has come over the mathematical teaching in many of our institutions, 

 due in no small measure to the reforming zeal of our General Treasurer, Professor 

 John Perry. The changes that have been made followed a recognition of the fact 

 that the abstract mode of treatment of the subject that had been traditional was 

 not only wholly unsuitable as a training for physicists and engineers, but was also 

 to a large extent a failure in relation to general education, because it neglected to 

 bring out clearly the bearing of the subject on the concrete side of things. With 

 the general principle that a much less abstract mode of treatment than was 

 formerly customary is desirable for a variety of reasons. I am in complete accord. 

 It is a sound educational principle that instruction should begin with the concrete 

 side, and should only gradually introduce the more general and abstract aspects 

 of the subject; an abstract treatment on a purely logical basis being reserved only 

 for that highest and latest stage which will be reached only by a small minority 

 of students. At the same time T think there are some serious dangers connected 

 with the movement towards making the teaching of Mathematics more practical 

 than formerly, and I do not think that, in making the recent changes in the modes 

 of teaching, these dangers have always been successfully avoided. 



Geometry and mechanics are both subjects with two sides : on the one side, the 

 observational, they are physical sciences ; on the other side, the abstract and deduc- 

 tive, they are branches of Pure Mathematics. The older traditional treatment^ of 

 these subjects has been of a mixed character, in which deduction and induction 

 occurred side by side throughout, but far too much stress was laid upon the 

 deductive side, especially in the earlier stages of instruction. It is the proportion 

 of the two elements in the mixture that has been altered by the changed methods 

 of instruction of the newer school of teachers. In the earliest teaching of the 

 subjects they should, I believe, be treated wholly as observational studies. At a 

 later stage a mixed treatment must be employed, observation and deduction going 

 hand in hand, more stress being, however, laid on the observational side than 

 was formerly customary. This mixed treatment leaves much opening for variety 

 of method; its character must depend to a large extent on the age and general 

 mental development of the pupils ; it should allow free scope for the individual 

 methods of various teachers as suggested to those teachers by experience. 

 Attempts to fix too rigidly any particular order of treatment of these subjects are 

 much to be deprecated, and. unfortunately, such attempts are now being made. 

 To have escaped from the thraldom of Euclid will avail little if the study of 

 geometry in all the schools is to fall under the domination of some other rigidly 

 prescribed scheme. 



There are at the present time some signs of reaction against the recent move- 

 ment of reform in the teaching of geometry. It is found that the lack of a 

 regular order in the sequence of propositions increases the difficulty of the 

 examiner in appraising the performance of the candidates, and in standardising 

 the results of examinations. That this is true may well be believed, and it was 

 indeed foreseen by many of those who took part in bringing about the dethrone- 

 ment of Euclid as a text-book. From the point of view of the examiner it is 

 without doubt an enormous simplification if all the students have learned the sub- 

 ject in the same order, and have studied the same text-book. But, admitting 

 this fact, ought decisive weight to be allowed to it? I am decidedly of opinion 

 that it ought not. I think the convenience of the examiner, and even precision 

 in the results of examinations, ought unhesitatingly to be sacrificed when they 

 are in conflict — as I believe they are in this case — with the vastly more important 

 interests of education. Of the many evils which our examination system^ has 

 inflicted upon us, the central one has consisted in forcing our school and univer- 

 sity teaching into moulds determined not by the true interests of education, but 

 by the mechanical exigencies of the examination syllabus. The examiner has 

 thus exercised a potent influence in discouraging initiative and individuality of 

 method on the part of the teacher ; he has robbed the teacher of that freedom 

 which is essential for any high degree of efficiency. An objection of a different 

 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 



