THE FUTURE OF MATHEMATICS—POINCARE. Neil 
whether the solution will be useful to the engineer of the twenty- 
second century; we feel otherwise, and are sometimes as happy if 
we have saved for our grandson as for our contemporaries. 
Sometimes, trying this way and that, empirically, we might say, 
we happen upon a formula sufficiently convergent. “ What more do 
you want?” we ask the engineer; and yet, despite that, we are not 
satisfied ourselves. Why? Could we have foreseen it the first time, 
we might a second. We have reached a solution; that is a small 
matter to us if we have no sure hope of getting it a second time. 
As a science grows it becomes more and more difficult to know it 
all. Then we cut it up into bits and each one contents himself with 
a bit; in a word, we specialize. If this process continues it will 
become a vexatious obstacle to the progress of our science. We have 
said that it is the unexpected bringing together of diverse parts of 
our science which brings progress. Too much specialization prevents 
this. Let us hope that a congress like this, bringing us into closer 
relationships with each other and spreading before the eyes of each 
his neighbor’s fields, obliging us to compare these fields, so that we 
set forth for awhile from our own little villages, will annul this 
danger to which I have just called attention. 
But I have stopped too long over generalities. Let us pass in re- 
view the diverse parts which form the whole science of mathematics, 
let us see what each branch has done, whither each tends and what 
we may hope from each. If the views we have just expressed are 
right, the great advances of the past will be found where two of 
these branches have approached each other, where the similarity of 
their forms despite the dissimilarity of material has become evident, 
where one has been modeled upon the other in such manner that each 
takes profit from the other. At the same time we should foresee the 
progress of the future in interlockings of the same nature. 
I. ARITHMETIC. 
The progress of arithmetic has been slower than that of algebra 
or analytical geometry, and the reason is very evident. Arithmetic 
does not present to us that feeling of continuity which is such a 
precious guide; each whole number is separate from the next of its 
kind and has in a sense individuality; each in a manner is an excep- 
tion and that is why general theorems are rare in the theory of num- 
bers; and that is why those theorems which may exist are more hid- 
den and longer escape those who are searching for them. 
But if arithmetic is less developed than algebra and analytical 
geometry it may well model itself upon those branches and take profit 
by their advances. The arithmetician must take for his guide the 
analogies with algebra. These analogies are many, and if often they 
