TRANSACTIONS OF THE SECTIONS. 5 



Having now stated, from the point of view of a mathematiciau, the reasons 

 which appear to me to justify the existence of so composite an institution as Sec- 

 tion A, and the advantages which that very compositeness sometimes brings to 

 those who attend its meetings, I wish to refer very briefly to certain definite ser- 

 vices which this Section has rendered and may yet render to Science. Tlie im- 

 provement and extension of scientific education is to many of us one of the most 

 urgent questions of the daj' ; and the British Association has already exerted 

 itself more than once to press the question on the public attention. Perhaps 

 the time has arrived when some further eftbrts of the same kind may be desirable. 

 Without a rightly organized scientific education we cannot hope to maintain our 

 supply of scientific men, since the increasing- complexity and difficult}' of science 

 renders it more and more difficult for untaught men, by mere power of genius, to 

 force their way to the front. Every improvement, therefore, which tends to ren- 

 der scientific knowledge more accessible to the learner, is a real step towards the 

 advancement of science, because it tends to increase the number of well quali- 

 fied workers in science. 



For some years past this Section has appointed a committee to aid the improve- 

 ment of geometrical teaching in this country. The Report of this committee wiU 

 be laid before the Section in due course ; and without anticipating any discussion 

 that may arise on that Report, I think I may say that it will show that we have 

 advanced at least one step in the direction of an important and long-needed reform. 

 The action of this Section led to the formation of an Association for the improve- 

 ment of geometrical teaching ; and the members of that Association have now 

 completed the first part of their work. They seem to me, and to other judges 

 much more competent than mj'self, to have been guided by a sound judgment in 

 the execution of their difficult task, and to have held, not unsuccessfully, a middle 

 course between the views of the innovators who would uphold the absolute 

 monarchy of Euclid, or, more properly, of Euclid as edited by Simson, and the 

 radicals who would dethrone him altogether. One thing at least they have not 

 forgotten, that geometry is nothing if it be not rigorous, and that the whole edu- 

 cational value of the study is lost if strictness of demonstration be trifled with. 

 The methods of Euclid are, by almost universal consent, unexceptional in point of 

 rigour. Of this perfect rigorousness his doctrine of parallels, and his doctrine of 

 proportion, are perhaps the most striking examples. That Euclid's treatment of 

 the doctrine of parallels is an example of perfect rigorousness, is an assertion 

 which sounds almost paradoxical, but which I nevertheless believe to be true. 

 Euclid has based his theory on an axiom (in the Greek text it is one of the postu- 

 lates ; but the difference for our purpose is immaterial) which, it may be safely said, 

 no unprejudiced mind has ever accepted as self-evident. And this uuaxiomatic 

 axiom Euclid has chosen to state, without wrapping it up or disguising it, not, for 

 example, in the plausible form in which it has been stated by Playfair, but in its 

 crudest shape, as if to warn his reader that a great assumption was being made. 

 This perfect honesty of logic, this refusal to varnish over a weak point, has had its 

 reward ; for it is one of the triumphs of modern geometry to have shown that the 

 eleventh axiom is so far from being an axiom, in the sense which we usually attach 

 to the word, that we cannot at this moment be sure whether it is absolutelv and 

 rigorously true, or whether it is a very close approximation to the truth. Two of 

 those whose labours have thrown much light on this difficult theory are at present at 

 this Meeting — Prof. Cayley, and a distinguished German mathematician, Dr. Felix 

 Klein ; and I am sure of their adherence when I say that the sagacity and insio-ht of 

 the old geometer are only put in a clearer light by the success which has attended the 

 attempt to construct a system of geometry, consistent witli itself, and not contradicted 

 by experience, upon the assumption of the falsehood of Euclid's eleventh axiom. 



Again, the doctrine of proportion, as laid down in the fifth book of Euclid, is 

 probably still unsurpassed as a masterpiece of exact reasoning, although the cum- 

 brousness of the forms of expression which were adopted in the old geometry has 

 led to the total exclusion of tliis part of the elements from the ordinary course of 

 geometrical education, A zealous defender of Euclid might add with truth that 

 the gap thus created in the elementary teaching of mathematics has never been 

 adequately supplied. 



