878 
economies, dynamical geology, dynamical 
meteorology and the statistical parts of 
various biological sciences. Visitors usu- 
ally expect the best that can be provided 
for them, and the efforts to please them 
frequently lead to a more careful study of 
available resources than those which are 
put forth in providing for the regular 
household. 
We have thus far spoken only of what 
might be called the materialistic incentives 
for mathematical development. While 
these have always been very significant, it 
is doubtful whether they have been the 
most powerful. Symmetry, harmony and 
elegance of form have always appealed 
powerfully to dame mathematics; and a 
keen curiosity, fanned into an intense flame 
by little bits of apparently incoherent in- 
formation, has inspired some of the most 
arduous and prolonged researches. Incen- 
tives of this kind have led to the mathe- 
matics of the invisible, relating to refine- 
ments which are essentially foreign to 
counting and measuring. The first impor- 
tant refinement of this type relates to the 
concept of the irrational, introduced by the 
ancient Greeks. As an instance of a com- 
paratively recent development along this 
line we may mention the work based upon 
Dedekind’s definition of an infinite aggre- 
gate as one in which a part is similar or 
equivalent to the whole.’ 
Mathematics is commonly divided into 
two parts called pure and applied, respect- 
ively. It should be observed that there 
are various degrees of purity and it is very 
difficult to say where mathematics becomes 
sufficiently impure to be called applied. 
The engineer or the physicist may reduce 
his problem to a differential equation, the 
student of differential equations may re- 
duce his troubles to a question of function 
2¢<Fneyclopédie des sciences mathematiques,’’ 
Vol. I., part 1, 1904, p. 2. 
SCIENCE 
\ 
[N.8. Vou. XXXV. No. 910 
theory or geometry, and the workers in the 
latter fields find that many of their difficul- 
ties reduce themselves to questions in num- 
ber theory® or in higher algebra. Just as 
the student of appled mathematics can not 
have too thorough a training in the pure 
mathematics upon which the applications 
are based so the student of some parts of 
the so-called pure mathematics can not get 
too thorough a training in the basic subjects 
of this field. 
As mathematics is such an old science 
and as there is such a close relation between 
various fields, it might be supposed that 
fields of research would lie in remote and 
almost inaccessible parts of this subject. 
It must be confessed that this view is not 
without some foundation, but these are 
days of rapid transportation and the stu- 
dent starts early on his mathematical jour- 
ney. The question as regards the extent of 
explored country which should be studied 
before entering unexplored regions is a 
very perplexing one. A lifetime would 
not suffice to become acquainted with all 
the known fields, and there are those who 
are so much attracted by the explored 
regions that they do not find time or cour- 
age to enter into the unknown. 
In 1840 C. G. J. Jacobi used an illustra- 
tion, in a letter* to his brother, which may 
serve to emphasize an important point. He 
states that at various times he had tried to 
persuade a young man to begin research in 
mathematics, but this young man always 
excused. himself on the ground that he did 
not yet know enough. In answer to this 
statement Jacobi asked this man the fol- 
lowing question: Suppose your family 
would wish you to marry, would you then 
5“¢Der Urquell aller Mathematik sind die ganzen 
Zahlen, Minkowski, Diophantische Approxima- 
tion,’’ 1907, preface. 
4«<Briefwechsel zwischen C. G. J. Jacobi und 
M. H. Jacobi,’’ 1907, p. 64. 
