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\origpage{133}
\section*{Plan of the Investigation}

It is well known that geometry presupposes both the concept of space and the first
fundamental concepts for constructions in space as something given. It gives of them
only nominal definitions, while the essential determinations appear in the form of
axioms.

The relationship of these presuppositions thereby remains obscure; one sees neither
whether, and to what extent, their connection is necessary, nor \emph{a~priori}
whether it is possible.

\origpage{134}
This obscurity was, from Euclid down to Legendre --- to name the most celebrated modern
reviser of geometry --- removed neither by the mathematicians nor by the philosophers
who occupied themselves with it. The reason for this lay presumably in the fact that
the general concept of multiply extended magnitudes, under which spatial magnitudes are
contained, remained wholly unelaborated.

I have therefore first set myself the task of constructing the concept of a multiply
extended magnitude out of general notions of magnitude.

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