Klein F. Lectures on mathematics (Macmillan, 1894)(T)(117s)_M

Klein F. Lectures on mathematics (Macmillan, 1894)(T)(117s)_M_.djvu

Size 0.7Mb
Date Dec 30, 2004

Then, if the exponential curve have any alge-
algebraic point (x, y), (besides x=O, y=i), the equation
О(л-О(л-<1) ¦¦¦-]
^Нл-^Нл-^) ¦¦¦ I o
J
must evidently be fulfilled...
We now select any one of the intersections, or vertices, as
origin O, and denote every other vertex by the symbol (x, y),
x being the number of sides л/а, у that of sides л/с, which
must be traversed in passing from О to (x, y)...
This is the year in which
Hermite and Kronecker, together with Brioschi, found the
solution of the equation of the fifth degree, at least in its
fundamental ideas...
The results are
well known :
A) All the n roots can be expressed as linear functions
of any one of them, 17:
e being a primitive nth root of unity...
Quite recently the French mathematicians have turned their
attention to the general question of the representation of sur-
surfaces by means of hyperelliptic functions, and a long memoir by
Humbert on this subject will be found in the last volume of the
Journal de Mathfanatiques.^
I turn now to the abstract theory of hyperelliptic functions...
The lectures I gave on this subject in 1887-88
have been developed very fully by Burkhardt in the Math...
It is, of course, possible by a similar
process to obtain the geometry with a positive measure of
curvature, first suggested by Riemann; it is only necessary
to formulate the axioms so as to make the length of a line
finite, whereby the existence of parallels is made impossible...
Indeed, all the researches of Riemann, Helmholtz,
Lie, consider only a portion of space surrounding the origin;
they establish the existence of analytic laws in the vicinity of
that point...
one geodesic line that is of finite length, and while in the plane
two geodesic lines always intersect in one point, if at all, on
the cylinder there may be 00 points of intersection...
As regards the former class of students, it is my opinion that
in Germany (here in America, I presume, the conditions are
very different) the abstractly theoretical instruction given to
94
THE STUDY OF MATHEMATICS AT GOTTINGEN...
This general tendency you will, I trust, also find
expressed in this colloquium, in which I have tried to present,
within certain limits, a general programme of my individual
work...
Our professors, find-
finding themselves limited to a small number of students who, as
future teachers and investigators, would naturally take great
interest in matters of pure theory, were able to follow the bent
of their individual predilections with much greater freedom
than would have been possible otherwise...
Jacobi came originally from Berlin and returned there for
the closing years of his life (died 1851)...
For henceforth,
besides the special achievements of individual workers, the
principle of co-operation, with its dependence on local condi-
conditions, comes to have more and more influence on the advance-
advancement of our science...
Let it above all be emphasized that even within the limits
here chosen, we have by no means exhausted the subject...
If we survey the entire field of development described above,
this, at any rate, appears as the obvious conclusion, in Germany
as elsewhere, that the number of those who have an earnest
interest in mathematics has increased very rapidly and that, as a
consequence, the amount of mathematical production has grown
to enormous proportions...