THE FUTURE OF MATHEMATICS—POINCARE. 133 
reduced and also that the form F transformed is reduced. It then 
follows that if the form F can be transformed to itself it can have 
many reductions; but this inconvenience is essential and can be 
avoided by no subterfuge. On the other hand these reductions do 
not prevent a classification of the forms. It is clear that this idea 
which has hitherto been applied only to limited classes of forms and 
transformations can be extended to groups of nonlinear transforma- 
tions and we may yet hope to have a harvest greater than has ever 
been reaped from it. 
An arithmetical domain where unity seems absolutely absent is 
found in the theory of prime numbers; the laws of asymptotes have 
been found and we must not hope for others; but these laws are 
isolated and are reached only by different paths which seem to have 
no intercommunication. I believe that I have a glimpse of the 
wished for unity, but I see it only vaguely; all leads back without 
doubt to the study of a family of transcendental functions which, 
through the study of singular points and the application of the 
method of M. Darboux, will permit the calculation asymptotically of 
certain functions of very great numbers. 
Il. ALGEBRA. 
The theory of algebraic equations will still hold for a long while 
the attention of geometricians; the sides from which it may be ap- 
proached are numerous and diverse; the most important is that of the 
theory of groups, to which we will return. But there is also the ques- 
tion of the calculation of the numerical value of roots and the discus- 
sion of the number of real roots. Laguerre has shown that not all 
was said upon this point by Sturm. Then there is the study of the 
system of invariants which do not change sign when the number 
of real roots remains the same. We may also form series of powers 
representing functions which may have for singular points the 
various roots of an algebraic equation (for instance, rational func- 
tions of which the denominator is the first member of this equation) ; 
the coefficients of the terms of high order will furnish one of the 
roots with an approximation more or less close; there is here the 
germ of a process of numerical calculation to which a systematic 
study could be given. 
During a period of forty years the study of invariants of algebraic 
forms seems to have absorbed all algebra; they are to-day laid aside, 
although the subject has not been exhausted; but we must no longer 
limit the study to the invariants of linear transformations; it is to be 
extended to those referring to any group whatever. The theorems 
acquired in the past have suggested others more general which are 
grouping about them much as a crystal grows from a solution. And 
as to the theorem of Gordan that the number of distinct invariants 
