Oct. 5, 1883,] 



- KNOWLEDGE 



221 



technical education. Among these, the large number who want 

 mathematics at all require geometry much more than algebra and 

 analysis, and geometry as applied to drawing and mensuration. 



The teaching of geometry especially, as judged by the text-books 

 which have come before me, is somewhat deplorable. And this is 

 so, principally, because the spirit of Euclid and the methods of the 

 ancient Egyptians and Greeks, rather than the fundamentally 

 differc-nt ideas and methods of modern geoinetry, still rule supreme ; 

 though the latter have had their origin partly in technical wants. 

 In wiiat is called geometrical drawing, or practical geometry, for 

 instance, there are first given a number of elementary con- 

 structions — such as drawing parallels and perpendiculars, or 

 bisecting the distance between two given points. They are 

 solved by aid of those instruments onlj' which Euclid 

 knew — viz., the pair of compasses for drawing circles, and the 

 straight-edge for drawing straight lines. Bnt there is no draughts- 

 man who would not, as a matter of course, use set squares for 

 tlie former problem, and solve the latter by trial rather than by 

 construction. Then again, there come constructions like the 

 division of the circumference of the circle into seven parts, which 

 cannot be solved accurately, but which is very easily solved bj- 

 trial. Instead of that, a construction is given which takes much 

 more time, and is by no means more accurate. For, after all, our 

 lines drawn on the paper are not without thickness, so that, for 

 this reason alone, every part of the construction is affected by 

 .some small error ; and it is absurd to employ a construction, 

 though theoretically true for ideal figures as conceived in our 

 mind, in preference to a much simpler one which, within our prac- 

 tical limits, is equally', or perhaps more, correct. 



This is very much like the manner in which I found problems on 

 decimal fractions treated by the candidates ior the matriculation 

 oxamin.ation at the London University, and which reflected little 

 credit on the manner in which the important subject of decimals is 

 handled at our schools. It is so characteristic that I may be ex- 

 cused for giving it here. The problem, for instance, being to give 

 the product of two decimal fractions, exact to, say, four decimals, 

 each nf the factors having the same number of places, this was 

 almost regularly performed as follows : — First, the decimals are 

 converted into vulgar fractions, then these are duly multiplied, 

 nnmerator by numerator, and denominator by denominator, and 

 then tliC resulting fraction is again converted to a decimal, with as 

 many places as it may yield ; and, lastly, of these the first four 

 -are taken and put down, duly marked " Answer ! " Or a candidate 

 (standing, however, on a far higher level) multiplies both decimals 

 out in the proper fashion, but to eight places, and cuts off four 

 places at the end. No wonder that the public at large will hear 

 nothing of the decimal system of weights and measures if the very 

 essence of the decimal system of numbers is so little understood by 

 the men who have to train the minds of the young generation. 



It is by the neglect of pure geometry and of its applications to 

 geometrical drawing that Cambridge has lost, or rather has never 

 had, contact with the practical needs of the nation. All the 

 marvels of modern engineering have sprung into existence without 

 its help. The great engineers have had to depend to a degree now 

 unheard of upon costly experiments, until they themselves gra- 

 dually discovered mathematical methods adapted to tlieir purposes. 

 Only the electrical engineer found ready to his hands a complete 

 theoiT of which the mathematical part has been to a very great 

 extent developed at Cambridge, or by men who liave had their 

 mathematical training there. This theory is, however, in its very 

 nature less geometrical. 



The engineer will always prefer geometrical methods to analysis, 

 and has invented for himself a great variety of them. Originally 

 these are disjointed, being invented for special purposes. It is the 

 business of the mathematician afterwards to connect, simplify, and 

 extend them, as has been done to a great extent by Culmann in 

 Zurich, and by Cremona at the Polytechnic School at Rome. Of 

 these methods a few may bo mentioned. First of all the geographi- 

 cal determination of stresses in certain girders invented both by 

 mathematicians and by engineers. Its application i.s so simple that 

 jio engineer will ever use any other method if once he knows this 

 one. It is so well adapted to its purpose, that I venture to say that 

 a simpler method is impossible, being fully aware how dangerous 

 euch a statement is. Nay, if I were asked to give the formula" to 

 obtain the stresses by calculation, I should write these down from a 

 sketch of the diagram, this being the simplest way of obtaining 

 them. Another problem which occurs again and again is the deter- 

 mination of the area of a figure representing perhaps a plot of land, 

 <ir the section of a beam. Here also the advantage is altogether on 

 the side of the graphical method. 



It is unnecessary to multiply these examples. But to make full 

 tiso of graphical methods, the draughtsman ought to have a 

 llioronghly geometrical education. 



That the old-established mode of teaching the elements of 



geometry based on Euclid requires a thorough and fundamental 

 change has been often acknowledged, among others, at Exeter and 

 Bradford, by two of the most eminent mathematicians who have occu- 

 pied the chair, and besides by the many teachers who constitute the 

 Association for the Improvement of Geometrical Teaching. I 

 have hesitated on entering on this somewhat delicate qnesion, 

 because I fear that I have little to offer but criticism, 

 which might seem hostile to the association just named. But I 

 hope that the many earnest workers, who have devoted much time 

 and thought to the drawing up of syllabuses on different parts of our 

 subject will excuse the remarks of one who has himself tried his hand 

 at the same work, and who therefore may be supposed somewhat to 

 know the difficulties that have to be overcome. When the syllabus 

 on the elements of plane geometry appeared, I resolved to give it 

 a thorough trial, and took the best means in my power to form an 

 opinion on its merits, by introducing it into one of my classes. The 

 fact that it did not quite satisfy me, and that I gave up its use 

 again, does not, of course, prove that it fails also for use in schools, 

 for which it was originally intended. 



The more I have become acquainted with the difEculty of the 

 whole subject, the greater has become my admiration for Euclid's 

 book, while my comdction of its unfitness as a school book has 

 equally gained in strength. In considering the merits of Euclid as 

 a text-book, it is desirable to distinguish clearly between the general 

 educational value of its teaching and the gain of geometrical 

 knowledge. It is with the latter chiefly that I am concerned, while 

 it is, of course, through the former that Euclid has got so firm a 

 hold at all schools; and to the great majority of boys this is un- 

 doubtedly of most importance, and no reform would have the 

 slightest chance of becoming generally introduced which neglects 

 this. But improvement in both directions may well go together, 

 and the logical reasoning employed in Euclid would gain to many 

 boys much, both in clearness and interest, if the subject-matter 

 reasoned about became in itself better understood. 



If acquaintance with geometrical objects, particularly through 

 the medium of geometrical drawing and the many methods used in 

 the Kinder-Gartens, were secured, then a systematic course of 

 geometry would become both easier and more useful. Much, 

 indeed, may be done by introducing simple geometrical teaching 

 into the nursery, and into the earliest instruction of children, fol- 

 lowing the example of the Kinder- Garten, and it is pleasing to see 

 that the latter are rapidly gaining ground in England. It is true 

 that these schools may still be improved. In geometry they seem 

 to, and perhaps at present are bound to, work mostly towards 

 Euclid. But many able men and women are actively engaged in 

 perfecting them, and it is of interest to know that Clifford had it 

 in his mind to write a geometry for the nursery and the Kinder- 

 Garten. 



In a curious contrast to the mode of teaching geometry stands 

 that of teaching algebra. In the first, everything is sacrificed to 

 logic. Axioms and definitions without end are given, though to 

 the beginner a more rapid dive into the subject would be much more 

 suitable. In algebra, on the other hand, the boy is at once plunged 

 into the midst of it. No axiom is mentioned. A number of rules 

 are stated, and the schoolboy is made to practise them mechanically 

 until he can perform, and that often with considerable skill, a 

 number of most complicated calculations — but calculations which 

 are often of very little use for actual applications. Simplifications 

 of equations follow in senseless monotony, until the poor fellow 

 really thinks that solving a simple equation does not mean the 

 finding of a certain number which satisfies the equation, but the 

 going mechanically through a certain rcgulai- process which at the 

 end yields some number. The connection of that number with the 

 original equation remains to his mind somewhat doubtful. Then 

 there are processes, like the finding of the G.C.M., which most of 

 the boys never have any opportunity of using, excepting, perhaps, 

 in the examination room. A more rational treatment of the sub- 

 ject, introducing from the beginning reasoning rather than calcula- 

 tion, and applying the results obtained to various problems taken 

 from all jiarts of science as well as from everyday life, would be 

 more interesting to the student, give him really useful knowledge, 

 and would be at the same time of true educational value. 



The chief jirogress in geometrical teaching has to bo sought in 

 the introduction of modern ideas and methods into the very ele- 

 ments, and modern teaching ought to take full account of this. In 

 favour of this view I might bring forward the opinions of many 

 teachers and mathematicians from England, as well as from abroad, 

 but I will confine myself to ono quotation. Professor Sylvester 

 gives his opinion thus : — " I should rejoice to see mathematics 

 taught with that life and animation which the presence and ex- 

 ample of her young and buoyant sister (viz., natural and experi- 

 mental science) could not fail to impart, short roads preferred to 

 long ones, Euclid honourably shelved or buried ' deeper than over 

 plunimet sounds ' out of the schoolboy's reach, morphology intro- 



