Consequently the _three_ sides of this given right angled triangle are
_equal_ to each other, which is a geometric _impossibility_.
Therefore, it is self-evident that there exists a mathematical
incongruity in the surface of the Chess-board.
That is, what to the eye _seems_ a right angled triangle, is in its
relations to the _movements_ of the Chess-pieces, an equilateral
triangle. Hence, the Chess-board, in its relations to the pieces when
the latter are at _rest_, properly may be regarded as a great _square_
sub-divided into sixty-four smaller squares; but on the contrary,
in those calculations relating to the Chess-pieces in _motion_, the
Chess-board must be regarded as the _quadrant_ of a circle of eight
points radius. The demonstration follows, viz.:
Connect by a straight line the points KR8 and QR8. Connect by another
straight line the points QR8 and QR1. Connect each of the fifteen points
through which these lines pass with the point KR1, by means of lines
passing through the least number of points intervening.
Then the line KR8 and QR8 will represent the segment of a circle of
which latter the point KR1 is the center. The lines KR1-KR8 and KR1-QR1
will represent the sides of a quadrant contained in the given circle and
bounded by the given segment, and the lines drawn from KR1 to the fifteen
points contained in the given segment of the given circumference, will be
found to be fifteen equal radii each eight points in length.
* * * * *
Having noted the form of the Static or positional surface of the
Chess-board and its relations to the pieces at rest, and having
established the configuration of the Dynamic surface upon which the
pieces move, it is next in sequence to deduce that fundamental fact and
to give it that geometric expression which shall mathematically harmonize
these conflicting geometric figures in their relations to Chess-play.
As the basic fact of applied Chessic forces, it is to be noted, that: