Colloquium publications.

32

TIHE BOSTON COLLOQUIUM.

which he is free to make as he pleases, provided that they are
self-consistent. The test of the validity of the hypotheses lies in
their results. We make at first hypotheses which follow the ideas
of Riemann's famous Hlablitationsschrift.*
It is admitted that questions may be raised which lie back of
these hypotheses, as, for example, the possibility of reducing them
to simpler axioms, but the discussion of such questions lies outside
our present province. The Riemann method has for us the double
advantage of allowing the immediate use of analytic methods and
of restricting the discussion at the outset to a small region of space.
A geometry having thus been developed in a restricted portion
of space is extended to all space by means of new hypotheses,
which are essentially those used by Killing in his Grunldlagen
der Geometrie. In the further development the ideas of the last
named treatise have been largely followed.
1. THE FIRST Two HYPOTHESES.
As already said, we adopt in our investigations the method of
Riemann by which our objective space is assumed to be an example
of an extent (M3annigfaltikeit) of three dimensions in which an
element may be determined by means of coordinates. We assert
this explicitly in the following words:
FIRST HYPOTHESIS. Space is a continuum of thi're timenlsions
in which a point may be determiied by three independent real coSrdinates (zl, z, z3)' If a )prope)rly restricted por'tion of space is considered, the correspondence between point and crodinate is o0le-to-one
and continuous.
Within our space, we may pick out at pleasure one-dimensional
extents or lines. We shall restrict ourselves to lines which may
be expressed by the equations
Z1 =f(t), z = f2(t), = f(t),
where t is an arbitrary parameter and f,.f, and f3 are continuous
*Riemann, B., " Ueber die HIypothesen, welche der Geometrie zu Grunde
liegen," Gesammelte WTorke, 1st ed. p. 2354; 2d ed. p. 272.