THE FUTURE OF MATHEMATICS—POINCARE. 139 
there are resemblances and affirms analogies so that we do not forget 
them. And even more, it guides us into that space which is too vast 
for us and which we may not see; it does this by ever bringing to mind 
the relationship of the latter space to our ordinary, visible space, 
which without doubt is only a very imperfect image, but which never- 
theless is an image. Here further, as in all the preceding instances, 
this analogy with what is simple allows us to comprehend that which 
is complex. 
This geometry of more than three dimensions is not a simple 
analytical geometry; it is not purely quantitative; it is also qualita- 
tive, and it is in the latter sense that it becomes especially interesting. 
The importance of the Analysis Situs is very great; I can not insist 
too much on that; the advance which it has taken from Riemann, 
one of its chief creators, is enough to indicate this. It is essential 
that it should be constructed completely in hyperspace. We would 
be then furnished with a new sense, one capable of seeing really into 
hyperspace. 
The problems of the Analysis Situs would perhaps not have been 
thought of had there been only the language of analytical geometry ; 
or rather, I am wrong, they would certainly have been set, since their 
solution is necessary for many of the questions of analytical geom- 
etry; but they would have been set one after another with no indi- 
cation of a common bond between them. 
It is the introduction of the ideas of transformations and groups 
which has contributed especially to the recent progress in geometry. 
We owe to these that geometry is no longer an assemblage of more or 
less curious theorems which follow each other with no resemblances; 
they have now acquired a unity; and, furthermore, we must not forget 
in our history of science that it was for the sake of geometry that 
a systematic study was started of continuous transformations, so that 
pure geometry has contributed its part to the development of the idea 
of the group so useful in the other branches of mathematics. 
The study of groups of points upon an algebraic curve, according 
to the method of Brill and Noether, has given us also fruitful 
results either directly or as serving as models for analogous theories. 
We have thus seen develop a whole chapter of geometry where the 
curves traced upon a surface play a role similar to that of a group 
of points upon a curve. And from this very day on, we may hope 
to see in this way light thrown on the last mysteries which exist in 
the study of surfaces and which have been so difficult to solve. 
The geometricians have thus a vast field from which to reap a 
harvest. I must not forget enumerative geometry, and especially 
infinitesimal geometry, cultivated with such brilliancy by M. Dar- 
boux, and to which M. Bianchi has added such useful contributions. 
