Sept. 20, 1883] 



NATURE 



499 



the writer, forgetting the subject of his letter, burst out into an 

 enthusiastic praise of a German text-book, the "Geometrie der 

 Lage," by Keye, through which Maxwell, evidently for the first 

 time, got any idea of this subject. 



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 pur- 

 poses. 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, or by Cremona at the Polytechnic 

 School at Rome. 



Of these methods a few may be mentioned. First of all the 

 graphical determination of stresses in certain girders invented 

 both by mathematicians and by engineers. Its application is so 

 simple that no 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 such a statement is. Nay, if I were asked 

 to give the formulae 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 recurs again and again is the deter- 

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

 land or 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 use of graphical methods the draughtsman ought to have a 

 thoroughly geometrical education. For instance, the real nature 

 of the reciprocal diagrams already mentioned is only understood 

 by aid of a peculiar reciprocal relation between points and planes 

 in space closely connected with the theory of the linear complex, 

 as has been shown by Cremona. 



I have mentioned already the "Analytical Mechanics" of 

 Lagrange, which is without any trace of geometry, although 

 there is scarcely a branch of applied mathematics which is in its 

 very nature more geometrical. In fact one part of it, now sepa- 

 rated as kinematics, treats solely of changes in position and shape 

 of geometrical quantities, and differs from pure geometry only in 

 this, that the changes are considered as referring not to space 

 alone, but also to time. 



What mechanics gains by introducing geometry to the full 

 will be apparent to all who have become acquainted with modern 

 Continental text-books on the subject. 



Let us compare the analy.ical with the geometrical reduction 

 of a system of forces acting on a rigid body, or, to use Clifford's 

 nomenclature, the reduction of a system of rotors, which may 

 represent either forces or rotations, or any other quantities which 

 have certain fundamental properties in common with those, so 

 that they may lie represented by rotors. In the analytical process 

 the system is reduced to a rotor and a vector, that is a resultant 

 force and a couple. In the geometrical treatment we see that 

 this is only one way of reducing the rotors to two, viz. the one 

 which is best fitted to be treated by analysis. But there is a 

 multitude of other reductions. These all appear as of equal 

 importance in the geometrical method. Furthermore, this 

 method shows us in the simplest way possible how all the line 

 pairs which may be the lines of action of two resultant rotors, 

 although there are infinities of infinities of such pairs, are 

 arranged in space, so that one gets a clear picture of all these 

 reductions in one's mind. 



Again, compare Mobius's geometrical investigation of the rays 

 of light passing through a system of lenses with that of Gauss, 

 whose very name suggests simplicity and elegance. The cele- 

 brated " cardinal points " appear in Gauss's original paper as 

 the result of a somewhat long though certainly elegant analysis, 

 whilst by Mbbius they are the natural outcome of his geometry, 

 so that any student once started on this method is bound to 

 come across these points, or rather across pairs of points, 

 of which the cardinal points of Gauss are only one special 

 case. The whole is, in fact, contained in the following easily 

 proved proposition : The rays of light starting from a point in 

 the axis of the system before entering the first lens, and after 

 leaving the last, form two homographic pencils in perspective 

 position. 



This is only one small part of the advantage which optics can 

 derive from geometry. 



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 occupied this chair, and besides by the many teachers who 



constitute the Association for the Improvement of Geometrical 

 Teaching, which itself grew out of the action of our Section. 1 

 know, therefore, of no opportunity better suited to review the 

 progress made in this direction than the present one, as the sub- 

 ject has on several occasions occupied the attention of our Sec- 

 tion. Nevertheless I have hesitated on entering on this some- 

 what delicate question, 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 ap- 

 peared, I resolved to give it a thorough trial, and took the best 

 means in my power to form an opinion on its merits by intro- 

 ducing it into one of my cla ses. 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 origin- 

 ally intended. 



Let me add that the more I have become acquainted with the 

 difficulty of the whole subject the greater has become my ad- 

 miration for Euclid's book, whilst my conviction of its unfitness 

 as a school book has equally gained in strength. 



In considering the merits of Euclid as a text-book it is desir- 

 able 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, whilst 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 undoubtedly 

 of most importance, and no reform would have the slightest 

 chance of becoming generally introduced which neglects this. 

 But improvement in both directions mi y well go together, and 

 the logical reasoning employed in Euclid would gam to many 

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

 reasoned about became in itself better understood. 



Probably a great deal could be done by introducing some of 

 the elements of logic into the teaching of language. I have 

 been assured by an eminent scholar that the laws of forming a 

 sentence — the fact that a sentence in its simplest form consists of 

 subject, object, and copula — was not explained in English schools. 

 If this grammatical part of logic were properly treated of in 

 connection with language, and if at the same time acquaintance 

 with geometrical objects, particularly through the medium of 

 geometrical drawing and the many methods u ed in the Kinder- 

 Gartens, were more secured, then a systematic course of geo- 

 metry 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, following the example of the Kinder-Gartens, 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 iinstly 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 m )st complicated calculations— 

 but calculations which are often of very little use for actual appli- 

 cations. Simplifications of equations follow in senseless mono- 

 tony, 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 regular process which at the end yields some number. 

 The connection of that number with the original equation re- 

 mains 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 subject, introducing 

 from the beginning reasoning rather than calculation, and apply- 

 ing the results obtained to various problems taken from all parts 

 of science as well as from everyday life, would be more interestin 



