Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab.  When I say they are of equal length, I mean that one can be laid on any other without the ends overlapping.  We next lay four of these little rods on the marble slab so that they constitute a quadrilateral figure (a square), the diagonals of which are equally long.  To ensure the equality of the diagonals, we make use of a little testing-rod.  To this square we add similar ones, each of which has one rod in common with the first.  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.  I only need state how many rods I must pass over when, starting from the origin, I proceed towards the " right " and then " upwards,” in order to arrive at the corner of the square under consideration.  These two numbers are then the " Cartesian co-ordinates " of this corner with reference to the " Cartesian co-ordinate system” which is determined by the arrangement of little rods.

By making use of the following modification of this abstract experiment, we recognise that there must also be cases in which the experiment would be unsuccessful.  We shall suppose that the rods " expand " by in amount proportional to the increase of temperature.  We heat the central part of the marble slab, but not the periphery, in which case two of our little rods can still be brought into coincidence at every position on the table.  But our construction of squares must necessarily come into disorder during the heating, because the little rods on the central region of the table expand, whereas those on the outer part do not.