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

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

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Date Dec 30, 2004

As second step we have only to apply the well-known identity
which is a special case of y = e"...
l
both positive; the case when they are both negative can
readily be treated by changing the signs throughout...
Among the different classes belonging to the same value of
D, there is one class of particular importance, which I call the
principal class...
It is not impossible that Gauss himself made use of similar
considerations in deducing his law, which, taken apart from this
geometrical illustration, bears such an abstruse character...
All equations whose solutions
cannot be expressed by radicals were classed simply as insoluble,
although it is well known that the Galois groups belonging to
such equations may be very different in character...
This reduction is possible because
the Galois group of our quintic equation (the square root of the
discriminant having been adjoined) is isomorphic with the group
72 LECTURE IX...
To begin with the older method, we have the fundamental
elliptic functions in the Jacobian form
±M, cosam[»,C\ Дат(да,^-
fz',
\
), Дат (да, —
as depending on two arguments...
Beginning with the discontinuous group mentioned before
и' = и + mvmY + м2щ,
our first task is to construct all its sub-groups...
It is well known that Gb'pel and Rosenhain established that
theory in 1847 in a manner closely corresponding to the Jaco-
bian theory of elliptic functions, the integrals
и - С dx и -С
V/e(*)
taking the place of the single elliptic integral u...
The equation is of the 40th degree;
and Burkhardt has given the general law for the formation
of the coefficients...
As mentioned before, from the point of view of projective
geometry, von Staudt's system should be adopted as the basis...
Now this space can of course be continued, and
the question is to see what kind of connection of space may
result from this continuation...
that infinity of space is a consequence of zero curvature),
so that we are forced to the opinion that our geometrical
demonstrations have no absolute objective truth, but are true
only for the present state of our knowledge...
Astronomy is
also recommended as showing an important application of
mathematics; and I believe that the technical branches, such
as applied mechanics, resistance of materials, etc., would form
a valuable aid in showing the practical bearing of mathematical
science...
A student having
nothing but an elementary knowledge of the differential and
integral calculus, usually coupled with hardly a moderate famil-
familiarity with the German language, makes a decided mistake in
attempting to attend my advanced lectures...
My lectures may then serve to form
the wider background on which these special studies are pro-
projected...
And yet I prefer to rank Gauss with
the great investigators of the eighteenth century, with Euler,
Lagrange, etc...
Besides these men the most promi-
prominent figure is that of Steiner (connected with the university
from 1835 t0 1864), the founder of the German synthetic
geometry...
But with
this the significance of Clebsch for the development of our
science is not completely characterized...
For the devel-
developments that now arise are not yet finished; the persons whom
we should have to name are still in the midst of their creative
activity...
In conclusion a few words should here be said concerning the
modern development of university instruction...