THE FUTURE OF MATHEMATICS—POINCARE. Loy 
instrument to us. This esthetic satisfaction is therefore connected 
with the economy of thought. Thus the caryatides of the Erech- 
theum engender in us the same feeling of elegance, for example, 
because they carry their heavy load with such grace, or we might 
-say so cheerfully, that they produce in us a feeling of economy of 
effort. 
It is for the same reason that when a somewhat long calculation 
has led us to a simple and striking result we are not fully satisfied 
until we have shown that we could have foreseen, if not the whole 
result, at least its most characteristic details. Why? What is it 
that prevents our satisfaction with this accomplished calculation 
giving all which we seemed to desire? It is because our long calcu- 
lation would not again serve in another analogous case and because 
we have not used that mode of reasoning, often half intuitive, which 
would have allowed us to foresee our result. When our process is 
short we may see at a glance all its steps, so that we may easily 
change and adapt it to whatever problem of the same nature may 
occur, and then, since it allows us to foresee whether the solution 
of the problem will be simple, we can tell at least whether the prob- 
lem is worth undertaking. 
What we have just said suffices to show how vain would be any 
attempt whatever to replace by any mechanical process the free initia- 
tive of the mathematician. To obtain a result of real worth it will 
not suffice to grind it out or to have a machine for putting our facts 
in order. It is not alone order but the unexpected order which is 
of real worth. The machine may grind upon the mere fact, but the 
soul of the fact will always escape it. 
Since the middle of the last century mathematicians have been 
more and more anxious for the attainment of absolute rigor in their 
processes; they are right, and that tendency will increase more and 
more. In mathematics rigor is not everything, but without it there 
would be nothing; a demonstration which is not rigorous is void. 
I believe no one will contest this truth. But to take this too literally 
would bring the conclusion, for example, that before 1820 there 
Was no mathematics. That is surely going too far; then the geome- 
tricians assumed willingly what we explain by a prolix discussion. 
This does not mean that they did not realize their omission, but they 
passed it over too rapidly, and for greater surety they would have 
had to go through the trouble of giving this discussion. 
But is it necessary to repeat every time this discussion? Those who, 
first in the field, had to be preoccupied with all this rigor have given 
us demonstrations which we could try to imitate; but if the demon- 
strations of the future must be built upon this model our mathe- 
matical treatises would become too long, and if I fear this length 
it is not only because I dread the incumbrance of our libraries, but 
