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:
PRINCIPLE
The King is the SOURCE from whence the Chess-pieces derive all power of movement; and from his ability to move, emanates ALL power for attack and for defence possessed by a Chessic army.
This faculty of mobility, derived from the existence of the kindred King, is the all essential element in Chess-play, and to increase the mobility of the kindred pieces and to reduce that of the adverse pieces is the simple, sure and only scientific road to victory; and by comparison of the Static with the Dynamic surface of the Chess-board, the desired principle readily is discovered, viz.,