ON INSTEUCTION IN ELEMENTARY GEOMETllY. 459 



Report of the Committee appointed to consider the possibility of Improv- 

 ing the Methods of Instruction in Elementary Geometry, the 

 Committee consisting of Professor Sylvester, Professor Cayley, 

 Professor Hirst, Rev. Professor Bartholomew Price, Professor 

 H. J. S. Smith, Dr. Spottiswoode, Mr. R. B. Hayward, Dr. 

 Salmon, Rev. R. Town send. Professor Fuller, Professor Kelland, 

 Mr. J. M.Wilson, awf? Professor Clifford [Secretary). 



Until recently the instruction 'm elementary geometry given in this country 

 was exclusively based upon Simson's modification of the text of Euclid. Of 

 late years, however, attempts have been made to introduce other text-books, 

 agreeing with the ancient elements in general plan, but differing from it in 

 some important details of treatment. And, in particular, the Association for 

 the Improvement of Geometrical Teaching having considered the whole 

 question with great labour and deliberation, is engaged in the construction of 

 a syllabus, part of which is already completed. The Committee had thus to 

 consider, ^rs<, the question of the plurality of text-books; secondly, certain 

 general principles on which deviation from the ancient standard has been 

 recommended; and, thirdly, the SyUabus of the Geometrical Association. 



1. On the Plurality of Text-BooTcs. 



It has already been found that the practical difficulty of examination stands 

 in the way of allowing to the geometrical teacher complete freedom in the 

 methods of demonstration and in the order of the'propositions. The difficulty 

 of demonstrating a proposition depends upon the number of assumptions which 

 it is allowable to start from ; and this depends upon the order in which the 

 subject has been presented. When different text-books have been used, it 

 thus becomes virtually impossible to set the same papers to aU the candidates ; 

 and in this country at present teaching is guided so largely by the require- 

 ments of examinations, that this circumstance opposes a serious barrier to in- 

 dividual attempts at improvement. On the other hand, the Committee think 

 that no single text-book which has yet been produced is fit to succeed Euclid 

 in the position of authority ; and it does not seem probable that a good book 

 could be written by the joint action of selected individuals. It therefore 

 seems advisable that the requisite uniformity and no more shoidd be obtained 

 by the publication of an authorized Syllabus, indicating the order of the pro- 

 positions, and in some cases the general character of the demonstrations, but 

 leaving the choice of the text-book perfectly free to the teacher ; and the 

 Committee believe that the authorization of such a syllabus might properly 

 come from the British Association. 



2. On some Principles of Improvement. 



The Committee recommend that the teaching of Practical Geometry shoxild 

 precede that of Theoretical Geometry, in order that the mind of the learner 

 may first be familiarized with the facts of the science, and afterwards led to 

 see their connexion. With this end the construction in practical geometry 

 should be directed as much to the verification of theorems as to the solution 

 of problems. 



It has been proposed to introduce what are called redundant axioms — that 

 is to say, assumptions whose truth is apparently obvious, but which are not 



