138 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1909. 
to simplify the demonstrations and to coordinate and classify the 
results. 
The study of the groups of Galois is much less advanced, and for 
a very simple reason, that same reason which makes arithmetic be- 
hindhand to analytical geometry, that lack of continuity which is 
of such great use for our advances. But happily there is a manifest 
parallelism between the two theories and we must try to put this 
more and more in evidence. This analogy is exactly parallel to 
that which we have noted between arithmetic and algebra and we 
should derive from it similar aid. 
VIII. GEOMETRY. 
It seems at first sight as if geometry could contain nothing which 
is not already presented to us in algebra and analytical geometry; 
for the facts of geometry are nought else than the facts of algebra 
_ and analytical geometry expressed in another language. One might 
think then, after the review which we have just made, that there 
would remain nothing further to say specially about geometry. 
But we would then be unmindful of a well-built language, mode of 
argument, of something which adds to the things themselves a mode 
of expressing them and consequently of grouping them. 
And, moreover, geometrical considerations lead us to propose new 
problems; they are, indeed, if you so choose to call them, analytical 
problems, but they would never have been proposed through ana- 
lytical geometry alone. Meanwhile analytical geometry profits from 
these just as it has profited from the problems it has been called upon 
to solve for physics. 
Common geometry has a great advantage in that the senses may 
come to the help of our reason and aid it in finding what path to 
follow, and many minds prefer to put their problems of analytical 
geometry in the ordinary geometrical form. Unfortunately our senses 
can not lead us so very far, and they fail us when we try to escape 
from the classical three dimensions. Must we say that, departing 
from the limited domain where our senses seem to wish to confine us, 
we must no longer count upon pure analysis and that all geometry of 
more than three dimensions is vain and useless? In the generation 
which preceded us the greatest masters would have replied “ yes.” 
We have nowadays become so familiar with this notion of more than 
three dimensional space that we may speak of it even in the university 
without arousing astonishment. 
But what purpose can geometry serve? It gives us, close at hand, 
a most convenient language which can express very concisely what 
the language of analytical geometry can express only in very prolix 
phraseology. Moreover, its language gives the same name where 
