ae 
DECEMBER 6, 1918] 
our motto as teachers and our own practise 
should convince the world of our sincerity. 
We have thus far considered only the ex- 
isting means for the scientific improvements 
of mathematics teachers. It may be desirable 
to consider also possible new means, for our 
science is one of infinite progress and hence 
we naturally look for new things. Possibly 
the new means for scientific development 
which I shall outline briefly will appear to 
you as too idealistic, but high ideals are es- 
sential for great progress. Hence I venture 
to propose that high school teachers should be 
required to give evidence at the end of every 
seventh year, until they are forty years old, 
of having made during the preceding seven 
years scientific progress equivalent to at least 
one year of university work. 
In fact, this might commonly be in the form 
of a sabbatical year spent in study at some 
university. In special cases it might be in 
the form of attendance at summer sessions, or 
the publication of scientific work. In all 
eases is should be understood that the proper 
authorities would go over the records care- 
fully every seventh year and would insist on 
such progress as a necessary condition for re- 
appointment. If the young teacher does not 
grow scientifically at least at the rate of one 
seventh of the normal growth of the university 
student he does not possess the type of mind 
that inspires his pupils properly. 
While our young university instructors are 
not formally subjected to such a rule they are 
practically subject to a more severe scientific 
test in our better universities by means of a 
considerable series of grades, such as instruc- 
tor, associate, assistant professor, associate pro- 
fessor, professor. In the better institutions 
each higher grade normally implies scientific 
attainments which are superior to those re- 
quired for the next lower grade. It is, of 
course, dificult to enforce high standards in 
these times of scarcity of teachers, but with 
the return of peace we may naturally look for 
greater competition and higher standards. 
To meet these higher standards it is not 
sufficient that we learn more mathematical 
SCIENCE 559 
facts. Mathematical growth is not based so 
much on the number of facts as on the kind 
of facts. The facts must be general and far 
reaching. A formula involving a parameter 
is more general than a large logarithmic table 
because the former contains potentially an 
infinite number of special values while the 
latter represents only a finite number of such 
values. It is, however, necessary to exercise 
care in regard to the use of the word general 
in mathematics, for, what is often called gen- 
eral is really very special. 
Tf one established theorem includes another 
it is evidently proper to speak of the former 
as the more general, but if one undeveloped 
theory embraces another it is not so clear that 
the former should always be regarded the more 
general. It may be that the generality of the 
principles underlying this theory is too great 
to permit of much progress. A theory ought 
to be regarded as general in proportion to its 
possible development and not in proportion to 
the generality of the definitions underlying it. 
It is evident that such a use of the word 
general is attended by great difficulties, but it 
is hard to see how this word can maintain its 
position of respect in the mathematical litera- 
ture unless we do make an effort to restrict its 
use to the potentially larger things. My 
thought may become clearer if I note the 
fact that the most general definition of the 
term group is too broad to serve as the basis 
of a theory. The most general group theory 
is therefore of zero extent and will probably al- 
ways be of this extent. There is a type of 
definitions which give rise to the most gen- 
eral theory, but it is practically impossible to 
fix the limitations imposed by such definitions. 
As an instance of a tendency to generalize 
unduly for the pedagogical purposes in ele- 
mentary mathematics we may refer to one of 
the oldest among the somewhat complicated 
mathematical formulas, viz., the Heron for- 
mula expressing the area of a triangle in 
terms of its three sides. This formula is found 
in the majority of our text-books on trigonom- 
etry but it is questionable whether it can be 
regarded as a useful formula for the ordinary 
