464 



NATURE 



{March 14, 1889 



EXAMINATIONS IN ELEMENTARY 

 GEOMETRY. 



IN the spring of 1870 a letter appeared in the columns 

 of Nature suggesting the formation of an Association 

 which should have for its object the improvement of 

 geometrical teaching. As to the great desirability of 

 such improvement there could be no doubt. It was felt 

 that one of the greatest obstacles in the way of effecting it 

 was the nature of the examinations, and that no change 

 in that nature was likely to be brought about except by 

 the combined action of those who desired it The idea 

 took root, and a circular was drawn up in which the 

 objects of the proposed Association were stated to be — 



(i) To collect and distribute information as to the pre- 

 vaihng methods of instruction in geometry practised in 

 this and other countries, and to ascertain whether the 

 desire for change was general. 



(2) To use its influence to induce examining bodies to 

 frame their questions in geometry without reference to any 

 particular text-book. 



(3) To stamp with its approval some text-book already 

 published, or to bring out a new one under its own 

 auspices. 



The movement led to a conference at University 

 College in 1871, which resulted in the formation of the 

 Association for the Improvement of Geometrical Teach- 

 ing. Many difficulties beset the way of the desired 

 improvement. In addition to all the arguments which may 

 be used against radical change in any department of 

 affairs, the uncomprising supporters of Euclid (or rather 

 of Simson's edition of his works) had the very cogent one 

 that a Commission appointed by the Italian Government 

 to inquire into the state of geometrical teaching in Italy 

 found it " so unsatisfactory, and the number of bad text- 

 books so great and so much on the increase, as to compel 

 them to recommend the adoption of Euclid pure and 

 simple,"^ and were able to quote the authority of dis- 

 tinguished French mathematicians as to the great merits 

 of Euclid as compared with modern French writers. We 

 need not wonder, then, and need perhaps scarcely regret, 

 that the outward and direct progress of the Association 

 towards the realization of the aims set forth in its 

 programme was slow. 



One of its first steps was to attempt to draw up a Syllabus 

 of Geometry in place of Euclid's scheme of propositions, 

 which might form the basis of future text-books. Members 

 were requested to send in, and the Committee of Man- 

 agement to report upon, programmes of the subjects 

 which a text-book ought to include. Eleven programmes 

 were sent in, which agreed upon many fundamental points; 

 and after much work by a sub-committee, and much dis- 

 cussion from time to time at the general meetings of the 

 Association, a Syllabus of Geometry was at last published 

 as the work of the Association. It is worthy of remark 

 that a Committee of the British Association, containing 

 such eminent mathematicians as Profs. Cayley, Clifford, 

 H. J. S. Smith, and Sylvester, appointed for the purpose 

 of considering the possibility of improving the methods of 

 instruction in elementary geometry, and reappointed to 

 consider this Syllabus, reported 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 other 

 great examining bodies of the United Kingdom, with a 

 view to its adoption, subject to any modification which 

 such detailed consideration may show to be necessary, as 

 the standard for examinations." 



The secretaries then applied to many of the principal 

 examining "bodies, submitting the Syllabus for their con- 

 sideration, with a view to its adoption as the basis of 

 examination. 



' See First Annual Repo.-t of the A.I.G.T. 



The direct result of this application was small. It was 

 felt that the non-existence of a text-book based on the 

 Syllabus was a bar to its adoption, and a sub-committee 

 was appointed to prepare proofs of the propositions. A 

 text-book embodying these, and entitled " The Elements 

 of Plane Geometry," was published by Messrs. Sonnen- 

 schein and Co. The appearance of Part II. of this work 

 in 1886 seemed to afford a fitting opportunity for me- 

 morializing the Universities as to the advisability of 

 relaxing their regulations. The following petition was 

 drawn up, and received about 180 signatures, among 

 which may be noticed those of Sir R. S. Ball, Prof. 

 Chrystal, Prof. Henrici, Dr. Hirst, and Prof. Tait :— 



" We, the undersigned members of the Association for 

 the Improvement of Geometrical Teaching, and others 

 interested in the teaching of elementary geometry, believ- 

 ing that greater freedom in the teaching of that subject 

 than is consistent with a rigid adherence to the letter of 

 Euclid's ' Elements ' is highly desirable, would welcome 

 such a change in the examinations in elementary geo- 

 metry conducted by the Universities and other examining 

 bodies as would admit of the subject being studied from 

 text-books other than editions of Euclid, without the 

 student being thereby placed at a disadvantage in those 

 examinations." 



The Council forwarded the petition to the Universities, 

 praying them to take the subject of the petition into their 

 favourable consideration, with a view to adapting the 

 regulations for the examinations in elementary geometry 

 conducted by the Universities so as to meet the desire 

 for greater freedom felt by a large number of teachers, 

 and supported by the judgment of many eminent 

 mathematicians. 



The Hebdomadal Council of the University of Oxford 

 having referred the application of the Association to the 

 Board of the Faculty of Natural Science, the Board 

 reported as follows : — 



(i) " That a rigid adherence to the ordinary text-books 

 of Euclid should no longer be insisted on, but that a 

 greater freedom of demonstration should be allowed, 

 both in geometrical teaching and in examination. 



(2) "That, nevertheless, Euclid's method should be 

 required in all pass examinations in geometry in so far 

 as that no axioms other than those of Euclid shall be 

 admitted, and that no proof of a proposition be allowed 

 which assumes the truth of any proposition which does 

 not precede it according to Euclid's order. 



(3) " That the University should not prescribe the use 

 of any particular text-book or text-books." 



This Report was adopted by the Hebdomadal Council 

 on June 20, 1887. 



The Senate of the University of Cambridge having 

 referred to the Special Board for Mathematics, the Board 

 reported as follows : — 



" The majority of the Board are of opinion that the 

 rigid adherence to Euclid's text is prejudicial to the 

 interests of education, and that greater freedom in the 

 method of teaching geometry is desirable. As it appears 

 that this greater freedom cannot be attained while a 

 knowledge of Euclid's text is insisted upon in the 

 examinations of the University, they consider that such 

 alterations should be made in the regulations of the 

 examinations as to admit other proofs besides those of 

 Euclid, while following, however, his general sequence of 

 propositions, so that no proof of any proposition occurring 

 in Euclid should be accepted in which a subsequent 

 proposition in Euclid's order is assumed. 



"To give effect to this view, the Board recommend 

 that for Regulation 10 (2) of the previous' examination 

 {Ordlnationes, p. 8) the following be substituted : — 



" Elementary geometry, viz. the substance of the first 

 three books, the definitions i-io of Book V., and the 

 substance of the first nineteen propositions of the Sixth 

 Book of Euclid's ' Elements.' Euclid's definitions will be 



