451 
tes, 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 unaxiomatic axiom Euclid has chosen to 
| state, without wrapping it up or disguising it,—not, for example, 
i ‘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 absolutely and rigorously true, or 
whether it is only a very close approximation tothe truth. Two 
_ of those whose labours have thrown much light on this difficult 
theory are at present at this meeting—Prof. Cayley, and a distin- 
guished German mathematician, Dr. Felix Klein; and Lam sure of 
their adherence when I say that the sagacity and insight 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, con- 
sistent with 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 cumbrousness of the forms of 
_ expression which were adopted in the old geometry has led to 
the total exclusion of this part of the elements from the ordi- 
nary 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. op 
id But after all has been said that can be said in praise of Euclid, 
_ the fact remains that the form in which the work is composed 
renders it unsuitable for the earlier stages of education, Euclid 
wrote for men ; whereas his work has been used for children, 
and it is surely no disparagement to the great geometer to sup- 
| pose that after more than 2,000 years the experience of generations 
of teachers can suggest changes which may make his Elements, 
_ I will not say more perfect as a piece of geometry, but more 
_ easy for very young minds to follow. The difficulty of a book 
or subject is indeed not in itself a fatal objection to its use in 
education, for to learn how to overcome difficulties is one great 
part of education : Geometry is hard, just as Greek is hard, and 
-one reason why Geometry and Greek are such excellent educa- 
tional subjects is precisely that they are hard. But in a world 
in which there is so much to learn, we must learn everything in 
the easiest way in which it can be learnt ; and after we have 
smoothed the way to the utmost of our power, there is sure to be 
enough of difficulty left. I regard the question of some reform 
in the teaching of elementary geometry as so completely settled 
| bya great concurrence of opinion on the part of the most compe- 
tent judges, that I should hardly have thought it necessary to 
direct the attention of the section to it, if it were not for the 
following reasons :— 
First, that the old system of geometrical instruction still re- 
mains (with but few exceptions) paramount in our schools, 
colleges, and universities, and must remain so until a very great 
-consensus of opinion is obtained in favour of some one definite 
text-book. It appears to me, therefore, that the duty will 
eventually devolve upon this section of the British Association, 
of reporting on the attempts that have been made to frame an 
improved system of geometrical education ; and if it should be 
found that these attempts have been at last successful, I think 
that the British Association should lend the whole weight of its 
authority to the proposed change. I am far from suggesting that 
any such decision should be made immediately. The work 
undertaken by the Association for the improvement of geometrical 
teaching is still far from complete ; and even when it is complete 
it must be left to hold its own against the criticism of all comers 
before it can acquire such an amount of public confidence as 
would justify us in recommending its adoption by the great 
teaching and examining bodies of the country. 
Secondly, I have thought it right to remind the section of the 
part it has taken with reference to the reform of geometrical 
teaching, because it appears to me that a task, at once of less 
difficulty and of more immediate importance, might now be 
undertaken by it with great advantage. There is at the present 
moment a very general agreement that a certain amount of 
natural science ought to be introduced into school education ; 
and many schools of the country have already made most laud- 
_ able efforts in this,direction. As far as I can judge, there is 
further a general agreement that a good school course of natural 
science ought to include some part or parts of physics, of chemis- 
try, and of biology ; but I think it will be found that while the 
courses of chemistry given at our best schools are in the main iden- 
tical, there is great diversity of opinion as to the parts of physics 
and of biology which should be selected as suitable for a school 
education, and a still greater diversity of opinion as to the methods 
which should be pursued in teaching them. Under these circum- _ 
stances it is not surprising to find that the masters of those schools 
into which natural science has hardly yet found its way (and some 
of the largest and most important schools in the country are in 
this class), are doubtful as to the course which they should take ; 
and from not knowing precisely what they should do, have not 
as yet made up their minds to do anything of importance. There 
can be no doubt that the masters of such schools would be glad 
on these points to be guided by the opinion of scientific men ; 
and I canriot help thinking that this opinion would be more 
unanimous than is commonly supposed, and further, that no 
public body would be so likely to elicit an expression of it, as a 
Committee appointed by the British Association. I believe 
that if such an expression of the opinion of scientific men were 
once obtained, it would not only tend to give a right direction 
to the study of natural science in schools, but might also have 
the effect of inducing the public generally to take a higher and 
more truthful view of the objecis which it is sought to attain by 
introducing natural science as an essential element into all courses 
of education, All knowledge of natural science that is imparted 
to a boy, is, or may be, useful to him in the business of his after 
life ; but the claim of natural science to a place in education 
cannot be rested upon its practical usefulness only. The great 
object of education is to expand and to train the mental faculties, 
and it is because we believe that the study of natural science 
is. eminently fitted to further these two objects, that we urge its 
introduction into school studies. Science expands the minds of 
the young, because it puts before them great and ennobling 
objects of contemplation ; many of its truths are such as a child 
can understand, and yet such that, while in a measure he under- 
stands them, he is made to feel something of the greatness, some- 
thing of the sublime regularity, and of the impenetrable mystery, 
of the world in which he is placed. But science also trains the 
growing faculties, for science proposes to itself truth as its only 
object, and it presents the most varied, and at the same time the 
most splendid examples, of the different mental processes which 
lead to the attainment of truth, and which make up what we call 
reasoning. In science, error is always possible, often close at 
hand ; and the constant necessity for being on our guard against 
it is one important part of the education which science supplies. 
But in science, sophistry is impossible ; science knows no love ot 
paradox ; science has no skill to make the worse appear the better 
reason ; science visits with a not long deferred exposure all our 
fondness for preconceived opinions, all our partiality for views 
that we have ourselves maintained, and thus teaches the two 
best lessons that can well be taught—on the one hand the love 
of truth, and on the other, sobriety and watchfulness in the use 
of the understanding. 
In accordance with these views I am disposed to insist very 
strongly on the importance of assigning to physics, that is to say 
to those subjects which we discuss in this section, a very 
prominent place in education. From the great sciences of 
observation, such as botany, or zoology, or geology, the young 
student learns to observe, or more simply, to use his eyes ; he 
gets that education of the senses which is after all so impor- 
tant, and which a purely grammatical and literary education so 
wholly fails to give. From chemistry he learns, above all other 
things, the art of experimenting, and of experimenting for him- 
self, But from physics, better as it seems to me than from any other 
part of science, he may learn to reason with consecutiveness and 
‘precision, from the data supplied by the immediate observation 
of natural phenomena. I hope we shall see the time when each 
successive portion of mathematical knowledge acquired by the 
pupil will be made immediately available for his instruction in 
physics ; and when everything that he learns in the physical labora- 
tory will be made the subject of mathematical reasoning and cal- 
culation. In some few schools I believe that this 1s already the 
case, and I think we may hope well for the future, both of 
mathematics and physics in this country, when the practice be- 
comes universal. In one respect the time is favourable for such 
a revolution in the mode of teaching physical science. During 
the past few years a number of text-books have been made avail- 
able to the learner, which far surpass anything that was at the 
