We proceed in like manner with each of
these squares until finally the whole marble slab is laid out with
squares. The arrangement is such, that each side of a square belongs
to two squares and each corner to four squares.
It is a veritable wonder that we can carry out this business without
getting into the greatest difficulties. We only need to think of the
following. If at any moment three squares meet at a corner, then two
sides of the fourth square are already laid, and, as a consequence,
the arrangement of the remaining two sides of the square is already
completely determined. But I am now no longer able to adjust the
quadrilateral so that its diagonals may be equal. If they are equal of
their own accord, then this is an especial favour of the marble slab
and of the little rods, about which I can only be thankfully
surprised. We must experience many such surprises if the construction
is to be successful.
If everything has really gone smoothly, then I say that the points of
the marble slab constitute a Euclidean continuum with respect to the
little rod, which has been used as a " distance " (line-interval). By
choosing one corner of a square as " origin" I can characterise every
other corner of a square with reference to this origin by means of two
numbers.
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