128 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1909. 
also because I fear that in this lengthening of our demonstrations 
they will lose that appearance of harmony of which I have just 
shown the so serviceable role. 
We should always aim toward the economy of thought. It is not 
enough to give models for imitation. It must be possible to pass 
beyond these models and, in place of repeating their reasoning at 
length each time, to sum this in a few words. And this has now and 
then been already accomplished; for instance, there was a whole 
type of demonstrations which were perfectly similar and repeatedly 
occurring; they were perfectly rigorous, but tedious; one day some 
one thought of applying the word convergence and that word has 
taken their place. There is now no need of repeating these proc- 
esses, for they are understood. Those who have cut our difficulties 
in quarter have rendered us double service—first, they have taught 
us to do as they have done when there is need, but above all to 
avoid this process as often as we can without the loss of this rigor. 
We have just seen, through an example, the importance of words 
in mathematics, but I could cite many more cases. It is scarcely 
credible, as Mach said, how much a well-chosen word can economize 
thought. I do not know whether or not I have said somewhere that 
mathematics is the art of giving the same name to different things. 
We must so understand it. It is meet that things different in sub- 
stance but like in form should be run in the same mold, so to speak. 
When our language is well chosen it is astonishing to see how all 
the demonstrations made upon some known fact immediately become 
applicable to many new facts. Nothing has to be changed, not even 
the words, since the names are the same in the new cases. 
There is an example which comes at once to my mind; it is 
quaternions, upon which, however, I will not dwell. A word well 
chosen very often causes the disappearance of exceptions to rules as 
announced in their former forms; it was for this purpose that the 
terms negative quantities, imaginary quantities, infinite points, have 
been invented. And let us not forget that these exceptions are per- 
nicious, for they conceal laws. 
Very well then, one of those marks by sane we recognize the 
pregnancy of a peat is In that it permits a happy innovation in our 
language. The mere fact is oftentimes without interest; it has been 
noted many times, but has rendered no service to science; it becomes 
of value only on that day when some happily advised thinker per- 
celves a relationship which he indicates and symbolizes by a word. 
The physicists also do just the same way. They invented the 
term energy, a word of very great fertility, because through the 
elimination of exceptions it established a law; because it gave the 
same name to things differing in material but similar in form. 
