Placed at either R1 or R8 the Rook occupies the central point of a great valley, 15 points in length, which winds around the slope of an immense and inaccessible mountain range. This latter, in extent, includes the remainder of the Topographical Zone. With the Rook placed at R2 or R7, this great mountain wall becomes pierced by a long valley running at right angles to the first, but the area open to the movement of the Rook is not increased. Placed at Kt2, B3, K4, or Q4, the Rook becomes enclosed amid impassable natural barriers. But although in such cases it always occupies the point of union of four easily traversed although unequal valleys, its area of movement is neither increased nor diminished, remaining always at fourteen points open to occupation. Unless impeded by the presence of kindred or adverse corps on its logistic radii, the Rook always may move either in two, three, or four directions, and it may reach any desired point on the Chess-board in two moves. The weakness peculiar to the Topographical Horizon of the Rook lies in the fact that it never commands the origins of diagonals or obliques. Hence it is open to unopposed attack along the first from adverse Queen, King, Bishops and Pawns, and along the second from adverse Knights. TOPOGRAPHY OF THE BISHOP From the view-point of the _Bishop_, the surface of the Chess-board takes on a topographical aspect most forbidding. To this Chess-piece at least one-half of the Topographical Zone is inaccessible, and under any circumstances his movements are limited to the thirty-two squares of his own color. Thus, the Topographical Horizon of the Bishop takes the form of a broken country, dotted with high hills, deep lakes, impenetrable swamps, and thick woodlands. But between these obstacles thus set about by Nature, run level valleys, varied in extent and easy of access. This fact so modifies this harshest of all sensible horizons as to make the Bishop next in activity to the Rook. Within its limited sphere of action, the Bishop may move in either one or four directions with a minimum of nine and a maximum of fourteen points open to his occupation. Unimpeded by other corps blocking his route of march, the Bishop may reach any desired point of his own color on the chess board in two moves. The weakness peculiar to the topographical horizon of the Bishop is its liability to unopposed attack along verticals and horizontals by the hostile King, Queen and Rooks; and along obliques by the hostile Knights. TOPOGRAPHY OF THE KNIGHT From the view-point of the _Knight_, the surface of the Chess-board takes on the aspect of a densely wooded and entirely undeveloped country; made up of a profusion of ponds, rivulets, swamps, etc., none of which are impassable although sufficient to impede progress.