Septembee 23, 1910] 



SCIENCE 



401 



complete accord. It is a sound educational 

 principle that instruction sliou^ld 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 I think there are some serious dangers 

 connected with the movement towards ma- 

 king 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 al- 

 ways been successfully avoided. 



Geometry and mechanics are both sub- 

 jects with two sides: on the one side, the 

 observational, they are physical sciences; 

 on the other side, the abstract and deduct- 

 ive, they are branches of pure mathe- 

 matics. The older traditional treatment of 

 these subjects has been of a mixed char- 

 acter, in which deduction and induction 

 occurred side by side throughout, but far 

 too much stress was laid upon the deduct- 

 ive 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 instruc- 

 tion of the newer school of teachers. In 

 the earliest teaching of the subjects they 

 should, I believe, be treated wholly as ob- 

 servational studies. At a later stage a 

 mixed treatment must be emploj^ed, ob- 

 servation and deduction going hand in 

 hand, more stress being, however, laid on 

 the obsei'vational 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 experi- 



ence. Attempts to fix too rigidly any par- 

 ticular order of treatment of these subjects 

 are much to be deprecated, and, unfortu- 

 nately, 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 pre- 

 scribed scheme. 



There are at the present time some signs 

 of reaction against the recent movement 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 

 standardizing 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 de- 

 thronement of Euclid as a text-book. 

 From the point of view of the examiner it 

 is without doubt an enormous simplifica- 

 tion if all the students have learned the 

 subject in the same order, and have 

 studied the same text-book. But, admit- 

 ting 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 unhesita- 

 tingly 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 university teaching into moulds 

 .determined not by the true interests of 

 education, but by the mechanical exigencies 

 of the examination S3dlabus. The exam- 

 iner has thus exercised a potent influence 

 in discouraging initiative and individual- 

 ity of method on the part of the teacher; 

 he has robbed the teacher of that free- 



