132 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1909. 
have not so far proved very useful yet they have at least been known 
for some time; the language itself of the two branches shows this; 
for instance when we speak of transcendental numbers and when we 
take into account that the future classification of these numbers 
images that of transcendental functions; still it is difficult to see 
how we can pass from one classification to the other; however, the 
step has already been taken, so it is no longer the task of the future. 
The first example which comes to mind is the theory of congruents 
where we find a perfect parallelism with that of algebraic equations. 
And we will certainly complete this parallelism which must exist 
between the theory of algebraic curves and that of congruents of two 
variables, for instance. And when the problems relative to con- 
gruents of several variables are solved we shall have taken the first 
step toward the solution of many of the questions of indeterminate 
analysis. 
Another example where the analogy has not always been seen at 
first sight is given to us by the theory of corpora and ideals. For a 
counterpart let us consider the curves traced upon a surface; to the 
existing numbers correspond the complete intersections, to the ideals 
the incomplete intersections, and to the prime ideals the indecompos- 
able curves; the various classes of ideals thus have their analogs. 
There can be no doubt that this analogy can throw light upon the 
theory of ideals, or upon that of surfaces, or perhaps on both at the 
same time. 
The theory of forms, and in particular that of quadratic forms, is 
intimately bound with that of ideals. Among the theories of arith- 
metic this was one of the first to take shape and it came when the 
arithmeticians introduced unity through the considerations of groups 
of linear transformations. 
These transformations permitted classification and consequently the 
introduction of order. Perhaps we have obtained all the fruit which 
could be hoped for; but if these linear transformations are the parents 
of geometrical perspectives, analytical geometry may furnish many 
other transformations (as, for example, the birational transforma- 
tions of an algebraic curve) for which it may be well worth our while 
to look for arithmetical analogs. Doubtless these will form discon- 
tinuous groups of which we must first study the fundamental parts 
as the key to the whole. I have no doubt that in this study we will 
make use of Minkowski’s Geometrie der Zahlen (Geometry of Num- 
bers). 
An idea from which we have not yet taken all that is possible is 
the introduction by Hermite of continuous variables in the theory of 
numbers. Let us start with two forms F and F’, the second quadratic 
determinate, and apply to both the same transformation; if the form 
F’ transformed is reduced, we will say that the transformation is 
