126 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1909. 
eralization, then it is not merely a new answer which I have acquired ; 
it is a new force. 
An example which comes at once to mind is the algebraic formula 
which gives us the solution of a class of numerical problems when 
its letters are replaced by numbers. Thanks to the formula, a sin- 
gle algebraic demonstration spares us the pains of going over the 
same ground time after time for each new calculation. But this 
gives us only a very rough illustration. Everyone knows that there 
are analogies, some most valuable, which can not be expressed by a 
formula, 
If a new result has value it is when, by binding together long- 
known elements, until now scattered and appearing unrelated to 
each other, it suddenly brings order where there reigned apparent 
disorder. It then allows us to see at a glance the place which each 
one of these elements occupies in the ensemble. This new fact is 
not alone important in itself, but it brings value to all the older facts 
which it now binds together. The brain is as weak as the senses, 
and it would be lost in the complexities of the world were there 
not harmony in that complexity. After the manner of the short- 
sighted, we would see only detail after detail, losing sight of each 
detail before the examination of another, unable to bind them 
together. Those facts alone are worthy of our attention which bring 
order into this complexity and so render it comprehensible. 
Mathematicians attach great importance to the elegance of their 
methods and results; nor is this pure dilettanteism. Indeed, what 
brings to us this feeling of elegance in a solution or demonstration ? 
It is the harmony among the various parts, their happy balancing, 
their symmetry; it is, in short, all that puts order among them, 
all that brings unity to them and which consequently gives us a 
certain command over them, a comprehension at the same time both 
of the whole and of the parts. But as truly it is that which brings 
with it a further harvest, for, in fact, the more clearly we compre- 
hend this assemblage, and at a glance, the better we will realize its 
relationships with neighboring groups, the greater consequently will 
be our chances of divining further possible generalizations. Ele- 
gance may arise from the feeling of surprise in the unexpected asso- 
ciation of objects which we had not been accustomed to group to- 
gether; it occurs frequently from the contrast between the simplicity 
of the means employed and the complexity of the given problem; we 
consequently reflect as to the reason of this contrast and almost with- 
out fail we find the cause not in pure hazard, but in some unexpected 
law. In a word, the sentiment of mathematical elegance is naught 
else than the satisfaction due to some, I know not just what, adapta- 
tion between the solution just found and the needs of our mind, and 
it is because of this adaptation itself that the solution becomes an 
