Septemher I, 1 910] 



NATURE 



291 



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 tnat 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 

 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 

 alwavs 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 deductive, they 

 are branches of Pure Mathematics. The older traditiona' 

 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 mi.xture 

 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 observa- 

 tional 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, .\ttempts 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 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 standardising the results of examinations. That 

 this is true may well be believed, and it was indeed foreseen 

 by many of those who tool; part in bringing about the 

 dethronement 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 subject 

 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 university 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 dis- 

 couraging 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. .'\n 

 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 repro- 

 duce, 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 objection really implies a demand for a purely deduc- 

 tive treatment of the subject. I think some of those who 

 raise it hardly realise 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 exploration 

 of the foundations of geometry we possess at the present 

 time tolerably satisfactory methods of purely deductive 

 treatment of the subject ; in regard to mechanics, notwith- 



NO. 2 13 1, VOL. 84] 



standing the valuable w-ork of Mach, Hcrz, 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 appre- 

 ciated 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 training 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 univer- 

 sities, 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 teaching which I have 

 advocated be allowed, it will be open to those teachers who 

 find it advisable in the interests of their students to 

 emphasise the logical side of their teaching to do so ;' and 

 it is certainly of value in all cases to direct the attention 

 of students to those points in a proof where the intuitional 

 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 integral calculus for students of en- 

 gineering and other branches of physical science has led to 

 the publication during the last few years of a considerable 

 number 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 skilful insistence on the principles 

 of the subject, 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 

 engineering 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 to regard Mathematics as consisting merely of 

 formula and of rules which provide the means of per- 

 forming the numerical computations necessary for solving 

 certain categories of problems which occur in the practical 

 sciences. Apart from the deplorable effect, on the educa- 

 tional side, of degrading Mathematics to this level, the 

 practical effect of reducing it to a number of rule-of-thumb 

 processes can only be to make those who learn it in so 

 unintelligent a manner incapable of applying mathematical 

 methods to any practical problem in which the data diffei 

 oven slightly from those in the model problems which they 

 have studied. Only a firm grasp of the principles will give 

 the necessary freedom in handling the methods of Mathe- 

 matics required for the various practical problems in the 

 solution of which they are essential. 



UNIVERSITY AND EDUCATIONAL 

 INTELLIGENCE. 



A Merch.int Venturers' research scholarship of the 

 value of 50/., tenable for one year in the faculty of 

 engineering of the University of Bristol, which is provided 

 and maintained in the Merchant Venturers' Technical 

 College, has been awarded to Mr. Harold Heaton Emsley. 



The Child, a new monthly journal devoted to child 

 welfare, "will appear in the early autumn, under the 

 general editorship of Dr. T. N. Kelynack. The jourtial 

 will be suited to the requirements of all engaged in child 

 study or working for the betterment of child life. The 

 publishers will be Messrs. John Bale, Sons and Daniels- 

 son, Ltd., S3-91 Great Titchfield Street, Oxford Street, 

 London, W. 



