ON METHODS OF INSTRUCTION IN ELEMENTARY GEOMETRY. 9 



claims of the teacher to greater freedom with the necessity of one fixed and 

 definite standard for examination purposes. They also expressed their con- 

 viction that " no text-hook that has yet heen produced is fit to succeed Euclid 

 in the position of authority ; " and that in the absence of such a text-book, 

 whether the existence of a standard authority in the future such as Euclid 

 has been in the past be regarded as desirable or not, it is important to secure 

 " the requisite degree of uniformity and no more by the pubhcation of an 

 authorized Syllabus " of propositions in a definite sequence, which should be 

 regarded as a standard sequence for examination purposes, and subject to 

 which alone any amount of variety in demonstration and general treatment 

 of the subject should be admissible. 



As it was understood that the Association for the Improvement of Geo- 

 metrical Teaching was engaged in the task of drawing up such a Sj'Uabus, no 

 further action was taken by the Committee until the present year, when, the 

 Syllabus having been completed and published, they have proceeded to con- 

 sider the same in accordance with the instructions contained in the resolution 

 reappointing the Committee. 



The Committee have not considered it to bo their duty to examine the 

 Syllabus in minirte detail, but rather to report on its general character and 

 its fitness as a basis for an authorized standard sequence of propositions. 



The Committee have no hesitation in stating at the outset, as the result of 

 their consideration of the Syllabus as a whole, that it appears to have been 

 drawn up with such care, and with such regard to the essential conditions of 

 the problem, as to render it highly desirable that it should be considered in 

 detail by authorized representatives of the Universities and the other great 

 examining bodies, of the United Kingdom with a view to its adoption, subject 

 to any modifications which such detailed consideration may show to be 

 necessary, as the standard for examinations in Elementary Geometry. 



It may be well to observe that the adoption of this or some such standard 

 SyUabus would not necessitate the abandonment of the 'Elements of Euclid' 

 as a text-book by such teachers as still preferred it to any other, as it would 

 at the utmost involve only such supplementary teaching as is contained in 

 the notes appended to many of the editions of Euclid now in use ; while it 

 would greatly relieve that large and increasing body of teachers, who demand 

 greater freedom in the treatment of geometry than under existing conditions 

 they can venture to adopt. 



Having thiis expressed their opinion of the general merits of the Syllabus 

 as a whole, the Committee have only further to add a few remarks on its 

 more important features, which may serve to call attention to those points in 

 which it differs from Euclid, and which give it a claim on the consideration 

 of aU who are interested in the improvement of instruction in Geometry. 



1. Geometrical. Constructions. 



It has been found, in the experience of many who have taught Geometry to 

 young beginners, that the attainment of a firm grasp of its fundamental con- 

 ceptions and methods is much facilitated by a series of exercises in con- 

 structions made with the ruler and compasses, such exercises being given 

 either as preliminary to, or simultaneously with, the study of the earlier parts 

 of Theoretical Geometry. A judicious selection of such exercises is prefixed 

 to the Syllabus ; and the Committee remark with approval that here, as well 

 as in the Postulates of Book I., the use of the compasses for direct transference 

 of distances is formally admitted. 



