222 INTERNATIONAL CONGRESS OF EDUCATION. 
school in which these two subjects, algebra and geometry, are to be taught. If the aim 
of the school is to prepare for the college, then it seems to me that the college may, to 
2 large extent, assume the responsibility of securing the organic union between these 
two subjects with the student, and of giving him full and practical control for life 
conduct of both these subjects. But if the aim of the secondary school, as is usually 
the case in our public high-schools, is to use the studies of algebra and geometry for what 
they are worth in securing a rounded education ; and when the secondary school does 
itself assume the responsibility, as it does in these high-schools, of dismissing the 
student with this organic union among the subjects well established, and with fairly 
good control of the connections with the subjects of general life conduct, then the ques- 
tion becomes a different one. 1 have placed myself rather upon the latter of these 
standpoints; and it has seemed to me of greater importance that we should examine 
this question with reference to the aim which affects the public at large. I think it is 
time that in this subject, as well as in regard to many others, the secondary public 
schools should abandon the practice of giving subjects in isolation to children, simply 
for their own sake. They should strive to give more of its subjects in organic connec- 
sion, one having necessarily a bearing upon the other. 
Now I would say the elementary school has to furnish, first of all, the conditions of 
apperceptional development at the hands of well-regulated experience. In the second 
place, it must enable the child to make orderly arrangement of all this experience in an 
approximately conventional and approximately scientific grouping. His mind must 
be set in the direction of order. The business of the secondary school, I should say, is 
to give him the experiences and sciences of the race ; to place him squarely and firmly, 
conscientiously and intelligently upon the basis of the past, and upon the basis of 
achievements already attained. Giving to the pupil that which the race possessed in 
conventional, scientific attainment, and with reference to life conduct, seems to me has 
to be the predominant task, almost specific task, of the secondary public high-school. 
The tertiary phase of education should then afford to the student an opportunity, on 
‘he basis of this historic and scientific grasp of knowledge, for independent research. 
With such a scheme of education no phase of it would be fragmentary. Wherever 
he scholar leaves the school, and leaves the scheme, he leaves it well rounded with 
seference to his stage of development. Does he leave the school at the elementary 
stage, he goes out with his mind and his habits clearly set toward an appreciation and 
ove of the historical stage. Does he leave it at the historical stage, he leaves with his 
mind fully set toward the point of independent research, and in such a way that he 
can be helped to self-culture. -Yet, again, if he leaves the elementary school, and has 
no opportunity to do schoolwork in the secondary school, the attitude of his mind is in 
that direction, and he will seek of his own accord to supply that which he has not 
obtained. 
What interests us to-day is to find what would be the natural place of algebra and 
geometry in such a scheme. The highest phases of life conduct are certainly self- 
expression, self-expression in benevolence, I say everything that is done must grow 
out of self-expression in benevolence. That which stimulates self-expression in the 
historic law of the race comes from literature and art, in its highest sense—technical 
arts as well as representative arts. History is an account of the development of the 
feelings of kinship among races. 
The knowledge of physics and chemistry is a closed book until it is opened by mathe- 
matical studies: until we can view it in the light of mathematics, of quantity and 
number. The study of numbers is the route, and it furnishes the key for all else. The 
study of geometry and the study of form will become clear to us only in the study 
»f the rules of quantity. Of course algebra is a high development of the quantitative 
mathematics. Algebra is not a study of principles. It is a study of the laws and the 
anetions of the laws. It is through algebra, or through the study of the functions of 
number, that we are enabled to appreciate the marvels of mathematics. And these facts 
can be appreciated only at the hand of algebraic equation. Kvery form, every line, 
avery surface has its equation, and quantity reveals it. It is through equation that we 
coms to the soul of the form. If we have superficial geometry, which fills that external 
-elation and only the external phases, then possibly we might not need this algebraic 
form. But to get at the soul of it, and that which can never be changed by any con- 
sideration whatever, it must bereached through algebraic formula. In every stage, form 
study must rest upon quantity study ; and, in a general way, geometry must work up 
on some corresponding phase of algebra, using this term in its very highest sense. 
We sometimes see the mechanical way of drawing a circle by making a six-sided 
form, and then an octagon, and a thirty-two-sided form ; and then we have some 
resemblance to a circle : but the life of a circle is not there. The circle grows out from 
the c¢ 
stick 
in the 
tion « 
30-cal 
arith: 
const 
laws - 
exact 
may 1 
secon 
eom: 
geom 
Let 
whicl 
purpc 
It is « 
to for 
parti 
child: 
the a 
of th 
IIL 
have 
probl 
AN 
ion, 
so Tay 
wher 
vers 
that 
matic 
seem: 
AZO, 
seach 
sion 
ough 
crest 
the bh 
Wha 
But t 
ret 1 
Pr 
deati 
pleas 
vani: 
simp 
deal 
they 
com 
inver 
subj 
other 
I h 
entire 
sake 
wait] 
cover 
shall 
eran. 
that 
ttle