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., The Static surface of the Chess-board being a square, its least division is into two great right angled triangles having a common hypothenuse. The Dynamic surface being the quadrant of a circle, its least division also is into two great sections, one of which is a right angled triangle and the other a semi-circle. Comparing the two surfaces of the Chess-board thus divided, it will be seen that these three great right angled triangles are equal, each containing thirty-six points; and having for their common vertices, the points KR1, QR1 and R8. Furthermore, it will be seen that the hypothenuse common to these triangles, also is the chord of that semi-circle which appertains to the Dynamic surface. Again, it will be perceived that this semi-circle, like the three right angled triangles, is composed of thirty-six points, and consequently that all of the four sub-divisions of the Static and Dynamic surfaces of the Chess-board are equal. Thus it obviously follows, that: 1. The great central diagonal, always is one side of each of the four chief geometric figures into which the Chess board is divided; that: