THE FUTURE OF MATHEMATICS—POINCARE. 125 
discerning those which conceal something and recognizing that which 
is concealed ; minds which under the bare fact see the soul of the fact. 
That is exactly what we do in mathematics; out of the various 
elements at our disposal we could evolve millions of different com- 
binations, but one of these combinations by itself alone is absolutely 
void of value. Oftentimes we take much trouble in its construction, 
but that serves absolutely for naught, unless possibly to give a task 
for further consideration. But it will be wholly different on the 
day that that combination takes its place in a class of like results 
and we have noted this analogy. We are no longer in the presence 
of a bare fact but of a law. And the true inventor is not the work- 
man who has patiently built some few of these combinations, but 
he who has shown their relationships, their parentage. The former 
saw only the mere fact, the other alone felt the soul of the fact. 
Oftentimes for the indication of this parentage it has served the 
inventor’s purpose to invent a new name and this name becomes 
creative; the history of science will supply us with innumerable such 
instances. 
The celebrated Viennese philosopher, Mach, states the réle of 
science to be the production of economy of thought just as a machine 
produces economy of labor. And that is very just. The savage 
counts with his fingers or with his assemblage of pebbles. By teach- 
ing the children the multiplication table we spare them later in- 
numerable countings of pebbles. Someone, sometime, has discovered 
with his pebbles, or otherwise, that 6 times 7 makes 42; it occurred 
to him to note the fact and he thus spared us the necessity of doing 
it over again. He did not waste his time even though his calcula- 
tion was only for his own pleasure; his operation cost him but two 
minutes; it would have cost two thousands of millions of minutes 
had a thousand of million of men to recompute it after he had. 
The importance of a fact is known by its fruits, that is to say, 
by the amount of thought which it enables us to economize. 
In physics, the facts of great fruitage are those which combine 
into some very general law, because they then allow us to predict 
a great number of other facts, and it is just the same with mathe- 
matics. I have devoted myself to a complicated calculation and 
have come laboriously to a result; but I will not feel repaid for my 
pains if I am not now able to foresee the results of other analogous 
calculations and to pursue such calculations with sure steps, avoiding 
the hesitations, the gropings of the first time. I shall not have 
wasted my time, on the contrary, if these gropings have ended in 
revealing to me in the problem which I have just treated some 
hidden relationship with a far more extended class of problems. If 
at the same time they have shown me resemblances and differences; 
if, in short, they have made me forsee the possibility of a gen- 
