96 



NATURE 



[March 2;, 191; 



Even Chrystal blundered; he is the only 

 blunderer whose name is given by Mr. Barnard. 

 Chrystal 's was the last and greatest attempt to do 

 tor algebra what Euclid attempted for geometry, to 

 build up the whole structure on a few axioms the 

 truth of which was obvious. As the result of his 

 attempt Chrystal learned (and was always ready to 

 admit) how impossible of attainment this ideal is, a 

 conclusion which is to-day becoming generally 

 accepted. In the future, instead of trying to build 

 mathematics up on axioms which are absolutely funda- 

 mental and by reasoning which only a geniusis fit to 

 grasp, we shall use as the foundation properties which 

 are intelligible to every boy, we shall assume the truth 

 of these whether obvious or not, and upon these we 

 shall build the superstructure. The question of the 

 soundness of the foundation is not a question 

 for schoolboys, it is not even a question for the 

 average university student, it is a question of meta- 

 physics to be dealt with by the mathematical philo- 

 sopher. 



No. 30 is an account by Mr. L. M. Jones of the 

 work in a municipal secondary school. The course is 

 good, and ends with the calculus. It includes here 

 and there an item on the value of which all would 

 not agree, e.g. stocks and present value, solution of a 

 quadratic by guessing factors, and the use of the 

 straight line graph as introduction to graphs and the 

 calculus. A sound opinion of Mr. Jones's, which one 

 would like to see more widely adopted, is that the 

 time spent in arithmetic on contracted methods is out 

 of proportion to its value to the pupil, it being quicker 

 and surer in most natural questions to use all the 

 figures given than to contract. 



In No. 32 Mr. Garstang attempts to pile up a load 

 of wickedness on the Board of Education. He charges 

 Circular 711 with loose reasoning in the matter of 

 parallels, and quotes many authorities to show that a 

 rigorous development cannot be based on the method 

 of direction. But the withers of the Board are un- 

 wrung. It is the second and third stages of the 

 Circular which deal with the systematic development 

 of geometry; the first stage, containing the passage 

 which displeases Mr. Garstang, is not concerned with 

 rigorous development, but with the preliminary 

 acquisition of the concepts of the subject. 



At Oundle (paper No. 33) the data for practical 

 mathematics are supplied from " the school shops, 

 testing-rooms, and fields." This is admirable, and the 

 boys show a keenness about the results because of 

 their contact with reality, a keenness much greater 

 than is aroused by questions which are onlv of 

 academic interest to the pupils, however practical and 

 important thev mav be for men or for other boys. 

 A difficult problem for schools less fortunately situated 

 than Oundle is the invention of laboratory questions 

 which have real interest and importance for the boys 

 to whom they are set. 



No. 19 is a clear exposition by Prof. Gibson of 

 mathematics in Scotch schools, which must have been 

 of great value to members of the congress who were 

 investigating such matters. 



Preparatory Schools. 

 Paper No. 29 contains a pleasing sign of the times 

 in the cooperation of public and preparatory school- 

 masters. In former years a preparatory school had 

 to prepare boys for a great variety of scholarship 

 examinations, and a public school to continue the 

 education of boys taught on a great varietv of plans. 

 To obviate the consequent difficulties, representatives 

 of the Headmasters' Conference and the Association 

 of Preparatory Schools have drawn up a syllabus for 

 NO. 2265. VOL. 91] 



a boy's education in mathematics from nine to six- 

 teen. This syllabus is now pretty widely used ; it 

 also bears witness to the advance made in recent 

 years in the teaching of the subject. 



Training of Teachers. 



In No. 27 Dr. Nunn discusses the training of 

 teachers of mathematics. Perhaps the most interest- 

 ing part of his paper is his excellent syllabus of 

 mathematical studies. The first part of the svllabus 

 is compulsory, and includes numerical trigonometry 

 and the ideas of the calculus. It is arranged with 

 the object of giving a clear consciousness of mathe- 

 matical conceptions. The logical proofs of these con- 

 ceptions belongs to the second part, which is optional. 

 The introduction to the calculus is made on historical 

 lines, on which lines it is interesting to note that 

 integration preceded differentiation. 



One would like to see logarithms also follow tliB 

 historical order, and introduced in Napier's way, with- 

 out anv consideration of indices. Dr. Nunn's method 

 compels the treatment of negative and practical in- 

 dices in part i., for which they are too difficult. But 

 it is perhaps ungenerous to criticise a detail in a 

 scheme drawn on such broad and statesmanlike lines. 



Technical Institutions. 



No*. 24 and 26. — Most teachers of mathematics 

 have their pupils at their mercy. In evening technical 

 institutions we meet a new type, the youth who must 

 be persuaded to come in. It is interesting and im- 

 portant that while mathematics treated in an abstract 

 way deters him, the subject treated in connection with 

 (and arising out of) concrete problems related to the 

 boy's work not only persuades him to come in, but 

 often gives him such an interest that he goes on with 

 the abstract study. 



Mr. Abbott also contributes the valuable suggestion 

 that each locality should have an advisory committee 

 composed of teachers of elementary schools, evening 

 continuation schools, spcondarv schools, and technical 

 schools, for the coordination of the work of these 

 schools in regard to the preliminary training of tech- 

 nical students. 



Dr. Sumpner and Mr. Abbott agree in the statement 

 that students who come from elementary schools re- 

 quire murh training in accuracy. There is clearly still 

 room for rpform in the mathematical teaching of these 

 schools, when it is still necessary to recommend the 

 abandonment of "discount, stocks and shares, H.C.F. 

 and L.C.M., &c." 



Universities. 



In Nos. 21, 23, 25, 28, 31, 34, we have the views 

 of the universities. Various changes are advocated, 

 a reduction of the degree of analytical skill now re- 

 quired, an extension of the range of mathematical 

 studies, closer connection with other subjects, more 

 regard for after-careers, encouragement of original 

 research. Recent reforms in school mathematics 

 sometimes meet with approval, sometimes with dis- 

 approval. Oxford and Cambridge are working, in 

 their examination regulations, towards a greater range 

 and less analytical skill; Cambridge also towards 

 meeting the needs of students of physics and engineer- 

 ing. 



Prof. Bryan deplores the indifference of the prac- 

 tical man to the value of mathematics. Of this in- 

 difference there is no doubt, or of the fact that the 

 practical man frequently meets a problem in which 

 the mathematician could help him. .The engineer has 

 an outfit of mathematical tools sufficient for his 



