328 TRANSACTIONS OF THE CANADIAN INSTITUTE. [Vou. VIII. 
extensive and intensive magnitudes,is untenable. All magnitude is in- 
tensive and yet it presupposes as a conditio sine qua non the separation 
in extension, consequently one can not simply say with Grassmann* that 
the intensive magnitude rises through the production of the equal, the 
extensive through the production of the different, because the production 
of the equal presupposes space, as we have seen above. For Grassmann 
extension is still in first line a magnitude or ‘ liquid number”’ through the 
separation of the elements. This separation of the elements is what we 
call extension. A pure theory of magnitude independent of extension 
(i. e. of space) such as so many analytical mathematicians claim to possess 
is consequently impossible. No matter whether we employ number 
chiefly as a representative of the quantitative principle (cardinal number) 
or as a representative of the principle of order (ordinal number), no matter 
‘whether the series is continuous or consists of discrete parts, pure number 
always contains extension and intensity. 
It is a different question whether we can have a pure theory of exten- 
sion, a geometry without any reference to magnitude. It is true, mag- 
nitude presupposes extension but not vice versa; for the characteristic 
property of extension (i.e. space) is not quantity but quality, and indeed 
a quality more different from all other qualities than they are from one 
another. This quality cannot be expressed otherwise than, ‘‘this place 
in space is not that place.””’ This makes an affirmative answer unques- 
tionable. A theory of extension or space without metrical elements is 
very well possible. It must abandon every use of intensive considerations. 
In such a non-metrical geometry we may deal with points, lines, planes, 
with loci, directions, and systems of directions, but not with distances, 
angular or other magnitudes. In such a geometry there is no place for 
size, (magnitude, extension) nor for similarity of figures. For of what we 
call shape or form in ordinary geometry—shape and form rest partially on 
relations of magnitude—there is nothing left but ‘‘collination.”” The 
so called geometry of position, projective geometry, comes close to such 
a non-metrical geometry, though its promoters are not always consis- 
tent in preserving its purely non-metrical character. 
We have seen above that space can be considered from two different 
standpoints, from the qualitative and quantitative point of view. The 
qualitative and that is the extensive, is the very nature of space, the 
quantitative, that is measurement by virtue of intensive considerations, 
is carried into it. For all qualitative continuous manifoldnesses admit 
of quantitative considerations. 
*Grassmann, Ausdehnungslehre 1 (Engel Ausgabe) Intro. P. 26. 
