TRANSACTIONS OF SECTION A. 399 
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 geo- 
metrical 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 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. 
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 used in the Kinder- 
gartens, were more 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, following the example of the 
Kindergarten, 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 Kindergarten. 
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 per- 
form, and that often with considerable skill, a number of most complicated caleu- 
lations—but calculations which are often of very little use for actual application. 
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 
regular 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 subject, introducing from the beginning reasoning 
rather than calculation, and applying the results obtained to yarious problems 
taken from all parts 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 progress in geometrical teaching has to be sought in the introduction 
of modern ideas and methods into the very elements, and modern teaching ought 
to take full account of this. sea * 
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 one quotation. Professor Sylvester gives his opinion thus: ‘I should 
rejoice to see mathematics taught with that life and animation which the presence 
and example of her young and buoyant sister (viz., natural and experimental 
science) could not fail to impart, short roads preferred to long ones, Euclid 
honourably shelved or buried “deeper than did ever plummet sound” out of the 
‘schoolboy’s reach, morphology introduced into the elements of algebra—projec- 
tion, correlation, motion accepted as aids to geometry—the mind of the student 
