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.: