402 



NATURE 



[Mar. 2 1, 1872 



Euclid, as the one best fi'ited to establish in the youthful 

 mind the habit of thoroughly rigorous reasoning ; above 

 all, he is not to impair the purity of the ancient geometry 

 by transforming geometrical theorems into algebraic 

 formula;, that is to say, by substituting in place of concrete 

 magnitude — such as lines, angles, superfices, volumes — 

 their respective measures ; on the contrary he is to ac- 

 custom his pupils to reason always on the magnitudes 

 themselves even when their ratios are under contempla- 

 tion. It is only after the propositions of Euclid and of 

 Archimedes, mentioned in the programme, have been 

 mastered that formula; are to be deduced for practically 

 determining the areas of rectihneal figures, the area of 

 the circle, the length of its circumference, and the magni- 

 tudes of the surfaces and volumes of prisms, pyramids, 

 cylinders, cones, and spheres." 



The measures taken by the Italian Government in 

 1867 have, I am informed, fully answered the expectations 

 of the mathematicians who recommended them. A taste 

 for rigorous and purely geometrical methods has been 

 revived, and the ground has been cleared for further ad- 

 vances. That such advances were contemplated from the 

 first is obvious from the following passages, with which 

 the Professors Betti and Brioschi — two of the most dis- 

 tinguished mathematicians of Italy — concluded their 

 preface to the nciv edition (based on that of Viviani) of 

 the Elements of Euclid, with which classical schools were 

 supplied in 1867 ■ " Profoundly convinced that it is only 

 through the eminent qu\lities of precision and clearness 

 which distinguished Euclid's Geometry that we can hope, 

 in seeking to promote the intellectual development of our 

 youth, to secure those results at which all civilised nations 

 aim when they give to geometrical instruction so im- 

 portant a place in public instruction, we have undertaken 

 the publication of an edition of the elements with the 

 fixed intention of improving it whenever new foreign 

 publications and the experience gained m our own schools 

 shall have shown that improvements are desirable. We 

 trust that professors in L/cei will help us in this work. 

 We shall gratefully accept their observations and sug- 

 gestions." 



Experience, however, has gone further than was here 

 aniicipated ; already there appears to be a demand for 

 something beyond a revision of Viviani's edition of Euclid's 

 Eltments. In the Gazzctta Uffichxlc of the kingdom of 

 Italy, published at Rome, 1 find that on the 2nd of Decem- 

 ber last an announcement was made by the authority of 

 the Minister of Public Instruction, to the effect that in 

 1873 a prize of 2,500 lire (about 100/) would be given to 

 the author of the best " Treatise on Elementary Geometry 

 which shall adhere rigorously to the method of Euclid, 

 and contain, besides the subject-matter in the programme 

 of 1867, those portions of the science, developed since 

 Euclid's time, which are now to be found in all elements 

 of geometry adopted as text-books in the classical schools 

 of the most cultured nations." 1 forbear to attempt to 

 determine what would be the rank of England amongst 

 cultured nations if she were judged by this standard of 

 the introduction of post Euclidean matter into school text- 

 books. I prefer to see in the announcement merely an 

 encouragement to proceed with our self-imposed task of 

 endeavouring to bring up the teaching of geometry and 

 the text-book we employ to the level of the science of our 

 day. In Italy this can be done more promptly than in 

 England. Our Government cannot, with a stroke of the 

 pen, alter the entire character of the instruction given in 

 English schools. With us improvements are of slower 

 growth, and it is by operations less surgical in their cha- 

 racter that obstructions to their growth have first to be 

 removed. It is, in fact, the function of associa'ions like 

 our own to endeavour to remove unreasonable prejudices 

 ag.iinst changes in the English habit of teaching geometry 

 by bringing prominently forward the defects which we 

 find to exist, and the improvements which we desire to 



see introduced. Let me now turn, therefore, to the work 

 done by this association during the past year. You will 

 recollect that members were invited to prepare pro- 

 grammes and syllabuses of text-books on geometry in ac- 

 cordance with their own views. The primary object in 

 making this request was to ascertain what amount of 

 unanimity at present prevails amongst teachers. The in- 

 vitation was accepted by many, and the syllabuses received 

 were referred to two committees, one meeting at London 

 and the other at Rugby. Although the occupations of 

 many of us, and our distances asunder, rendered it very 

 difficult to secure concerted action, a report has at length 

 been prepared, and will be this day submitted to ycai. 

 With respect to the resolutions and recommendations 

 embodied in this report, I will for the present confine my- 

 self to the statement that the main object they are in- 

 tended to further is a practically useful degree of confor- 

 mity amongst teachers during the present transitional 

 state of matters. No attempt has been made to prepare 

 any detailed scheme or programme of elementary geome- 

 trical study. This last difficult task, however, although 

 postponed, is not, as you will hereafter see, abandoned. 



Although the assertion may partake of the character of 

 a truism, it cannot be too often insisted upon, that how- 

 ever necessary it may be to have good text-books, it is far 

 more necessary to have good teacliers ; that, in fact, good 

 text-books are useful principally by the aid they render in 

 forming good teachers and in furnishing students with an 

 accurate record of what they have been taught. In teach- 

 ing, one might say, there is vis vi'iui — actual energy ; 

 whereas in a text-book, however good it may be, the dis- 

 ciplinal energy is at most potential. The text-book, 

 indeed, to be properly used, should always be subordinated 

 to the teaching ; but to do this it is absolutely essential 

 that the teacher should, by his own study, have risen not 

 merely up to, but above, the level of the text-book he em- 

 ploys. Until he has so mastered the subject that it has 

 become plastic in his hands, his teaching must necessarily 

 remain defective; for geometrical truth, it must be le- 

 membeied, has, like all other truth, many sides, and no 

 text-bool; can present all, or necessarily the one which, to 

 individual pupils, is the most accessible. Alternative 

 methods of demonstration, inquiries into the interdepen- 

 dence of propositions, judicious variation of data, and 

 just discrimination between the contingent and necessary 

 properties of figures ; these and numerous other matte: s, 

 all essential to geometrical culture, can only be properly 

 supplied by the teacher ; no te.xt-book could be weighted 

 with them. Above all, it is to him that we must mainly 

 look for the cultivation of that scientific method of inquiiy 

 under whose guidance solely problem-solving can be 

 raised in character above what has been termed " exalted 

 conundrum guessing," and acquire its full educational 

 value. 



The interdependence of geometrical propositions above 

 alluded to, as one of the subjects to which teachers should 

 habitually direct the attention of their pupils, is mainly 

 logical in character, but nevertheless most essential to 

 geometrical culture. Every one will admit the primary 

 importance of habituating the student to extract its full 

 logical significance from every proposition he establishes, 

 to recognise eajh proposition readily under different, 

 although logically equivalent forms of enunciation, and 

 thus to discriminate accurately between the cases where 

 mere logical deduction from antecedent propositions is 

 requisite, from those which require the introduction of 

 iurihex geoiiiet/ical considerations. Obvious as this may 

 be, it is rarely sufficiently attended to by teachers, and 

 even in approved text-books, ancient as well as modern, 

 we not unfrequently find remarkable instances of the 

 absence of the discrimination to which I refer. The ninth 

 proposition of the third book of Euclid is now a well- 

 known case of the kind. Geometrical apparatus is there 

 employed to demonstrate, indirectly, what had virtually ; 



