ON teachinct elementary mathematics. 475 



Teaching of Prctctical Geometry. 



The former Committee recommended (and the present Committee 

 desire to emphasise the recommendation) that the teaching of demonstra- 

 tive geometry should be preceded by the teaching of practical and 

 experimental geometry, together with a considerable amount of accurate 

 drawing and measurement. This practice should be adopted, whether 

 Euclid be retained or be replaced by some authorised text-book or sylla- 

 bus, or if no authority for demonstrative geometry be retained. 



Simple instruments and experimental methods should be employed 

 exclusively in the earliest stages, until the learner has become familiarised 

 with some of the notions of geometry and some of the properties of 

 geometrical figures, plane and solid. Easy deductive reasoning should 

 be introduced as soon as possible ; and thereafter the two processes 

 should be employed side by side, because practical geometry can be made 

 an illuminating and interesting supplement to the reasoned results 

 obtained in demonstrative geometry. It is desirable that the range of the 

 practical course and the experimental methods adopted should be left in 

 large measure to the judgment of the teacher ; and two schedules of 

 suggestions, intended for different classes of students, have been submitted 

 to the Committee by Mr. Eggar and Professor Perry respectively, and are 

 added as an Appendix to this Report. 



Should there be a Single Authority in Geometry 1 



In the opinion of the Committee it is not necessary that one (and only 

 one) text-book should be placed in the position of authority in demonstra- 

 tive geometry ; nor is it necessary that there should be only a single 

 syllabus in control of all examinations. Each large examining body might 

 propound its own syllabus, in the construction of which regard would be 

 paid to the average requirements of the examinees. 



Thus an examining body might retain Euclid to the extent of requiring 

 his logical order. But when the retention of that order is enforced, it is 

 undesirable that Euclid's method of treatment should always be adopted ; 

 thus the use of hypothetical constructions should be permitted. It is 

 equally undesirable to insist upon Euclid's order in the subject-matter ; 

 thus a large part of the contents of Books III. and IV. could be studied 

 before the student comes to the consideration of the greater part of 

 Book II. 



In every case the details of any syllabus should not be made too 

 precise. It is preferable to leave as much freedom as possible, consistently 

 with the range to be covered ; for in that way the individuality of the 

 teacher can have its most useful scope. It is the competent teacher, 

 not the examining body, who best can find out what sequence is most 

 suited educationally to the particular class that has to be taught. 



A suggestion has been made that some Central Board might be insti- 

 tuted to exercise control over the modifications made from time to time in 

 every syllabus issued by an examining body. It is not inconceivable that 

 such a Board might prove useful in helping to avoid the logical chaos 

 occasionally characteristic of the subject known as Geometrical Conies. 

 But there is reason to doubt whether the authority of any such Central 

 Board would be generally recognised. 



Opinions differ as to whether arithmetical notions should be introduced 

 into demonstrative geometry, and whether algebraic methods should be 



