PRINCIPLE
To make the first of a series of movements, each of which shall increase the mobility of the kindred pieces and correspondingly decrease the mobility of the adverse pieces.
As the effect of such policy, the power for resistance appertaining to Black, ultimately must become so insufficient that he no longer will be able adequately to defend:
- 1. His base of operations.
- 2. The communications of his army with its base.
- 3. The communications of his corps d’armee with each other, or,
- 4. To prevent the White hypothetical force penetrating to its Logistic Horizon.
To produce this fatal weakness in the Black position by the advantage of the first move is much easier for White than commonly is supposed.
The process consists in making only those movements by means of which the kindred corps d’armee, progressively occupying specified objectives, are advanced, viz.:
- I. To the Strategetic Objective, when acting against the communications of the adverse Determinate Force and its Base of Operations.
- II. To the Logistic Horizon, when acting against the communications between the adverse Determinate and the adverse Hypothetical Forces.
- III. To the Strategic Vertices, when acting against the communications of the hostile corps d’armee with each other.
To bring about either of these results against an opponent equally equipped and capable, of course is a much more difficult task than to checkmate an enemy incapable of movement.
Yet such achievement is possible to White and with exact play it seemingly is a certainty that he succeeds in one or the other, owing to his inestimable privilege of first move.
For the normal advantage that attaches to the first move in a game of Chess is vastly enhanced by a peculiarity in the mathematical make-up of the surface of the Chess-board, whereby, he who makes the first move may secure to himself the advantage in mobility, and conversely may inflict upon the second player a corresponding disadvantage in mobility.
This peculiar property emanates from this fact:
The sixty-four points, i.e., the sixty-four centres of the squares into which the surface of the Chess-board is divided, constitute, when taken collectively, the quadrant of a circle, whose radius is eight points in length.
Hence, in Chessic mathematics, the sides of the Chessboard do not form a square, but the segment of a circumference.
To prove the truth of this, one has but to count the points contained in the verticals and horizontals and in the hypothenuse of each corresponding angle, and in every instance it will be found that the number of points contained in the base, perpendicular, and hypothenuse, is the same.
For example:
Let the eight points of the King’s Rook’s file form the perpendicular of a right angle triangle, of which the kindred first horizontal forms the base; then, the hypothenuse of the given angle, will be that diagonal which extends from QR1 to KR8. Now, merely by the processes of simple arithmetic, it may be shown that there are,
- 1. Eight points in the base.
- 2. Eight points in the perpendicular.
- 3. Eight points in the hypothenuse.
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: