ON CHESS
XIX - On The Power of Pieces and Pawns (Concluded)
Another variation of value arises from the following circumstance: Suppose a bishop to be at one end of a diagonal line of squares cleared of pieces: a queen or rook could not be placed on any square of that diagonal without being en prise, a circumstance which, from their superior value, would be avoided, whether they were supported or not. Also a bishop, knight, or pawn could not be placed on that diagonal, without capture, unless supported. A similar power is possessed by the other pieces, and may be thus expressed: if a piece commands a certain range of squares, the opposite party cannot place on any square of that range a superior piece, or an unsupported equal or inferior piece, without immediate loss. It will be observed, that this is not the power of moving along a line of squares, but of preventing the antagonist from occupying any square of that line without loss. Supposing the board to be about half-cleared of mett, the power of the relative pieces in thus preventing the opponent from occupying any square in a particular line, has been calculated to be
Pawn ... = 2
Knight ... = 5 1/4
Bishop ... = 7 1/4
Rook ... = 10 1/2
Queen ... = 17
But if wo now omit all hostile proceedings, and consider simply how many squares a piece may command, without taking any opposing piece, we arrive at different results, principally because the pawn moves straight forward when merely making a move, but diagonally when capturing; The proportionate number of optional squares within the reach of the piece at one move, supposing the board, as before, to be about half-cleared of combatants, have been calculated at
Pawn ... = 1
Knight ... = 8
Bishop ... = 7
Rook ... = 10
Queen ... = 16 1/2
Suppose we wish to attack a particular piece with one of our own. If ours happen to be a pawn, we can do so by moving it to one square only; but if it be a bishop, the diagonals may be so far clear as to allow of our doing it in either of the directions. Place the black king on his own square, and the antagonist white bishop on its queen's bishop's 2nd.: the bishop can give cheek at two different squares. With the king in the same position, and the antagonist rook on its own square: the rook can check at two different squares. With the black king in the same position, place the white queen on her bishop's second: she can check at six different squares. Place the white knight on his king's fourth; he can check the king on two squares. In all these cases, we suppose the attacking piece to be free from any obstruction, either from an ally or an antagonist. From this enumeration of powers it is seen, that when a particular piece is to be employed to make an attack on a particular antagonist piece, it may often be done on more than one square. But as the intervention of other pieces would in some degree prevent this from being done, and as the presence of other pieces blocks out some more than others, according to their different modes of movement, we have hence a new scale of powers. The comparative power of the different pieces, in choosing what point to select as a position of attack, has been estimated at
Pawn ... = 2
Knight ... = 6
Bishop ... = 6 1/2
Rook ... = 11
Queen ... = 24
Let us assume that a piece is actually attacked. In order to save it, one of three things must be done: 1 st., to capture the attacking piece: 2nd., to interpose another piece: 3rd., to remove. Now different pieces have these several powers in different degrees; and to compare them it will be convenient to suppose that the attacking piece cannot be captured without loss: there will then remain two modes of releasing the piece. If the attack be made by a pawn, nothing can be interposed, since the belligerent pieces are close together: the assailed party has, therefore, nothing to do but to remove to a more distant square. If the attacking piece be a knight, no interposition will ward off the attack; on account of the peculiar privilege of this piece in leaping over other pieces. If a bishop attack a rook, interposition will not save it, because thte bishop may take the interposed piece, without being re-captured by the rook; this arises from the circumstance that the rook has not the diagonal power of the bishop: removal is the only way of saving the rook. For a somewhat similar reason, if a rook attack a bishop, no interposition will save it, because the bishop and the rook move in different ways : interposition is, therefore, of no avail. In all these examples it is assumed that the attacked and the interposed pieces, are not supported or defended by others. From a minute calculation of the various kinds and degrees of this power, it is found, that the dislodging faculty, or the power of an assailant to compel the removal of an assailed piece, is greater in the pawn and the knight than in the other pieces in comparison with their generally inferior power, being in the ratio of
Pawn ... = 0.8
Knight ... = 2.8
Bishop ... = 1.0
Rook ... = 2.9
Queen ... = 4.7
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From the circumstance of a pawn being capable of promotion to the rank of a piece, its value is greater than it Would be If that privilege were denied; and as it alone is capable of this sudden increase of power, the ratio of its value when compared with the pieces, is higher. The pawn has likewise an increase of comparative power resulting from its use as a support for a superior piece. If a piece make an attack on another of inferior value, a pawn may be as effectual a support for the latter as a superior piece would be: this circumstance also increases the ratio of the pawn's power. On the contrary, if two pawns become, by capture or exchange, placed one behind the other, or "doubled," the power of the hindmost one is much decreased, particularly if on the rook's file; so much, indeed, that the two together arc deemed not to be worth more than one pawn and a quarter under usual circumstances. Another circumstance which modifies the power of a pawn is the contiguity or not of another pawn on the adjoining file; if a pawn is isolated, that is, if neither of the adjacent files is occupied by a pawn, the pawn's value is below the average hitherto expressed; but if it be supported by pawns on both the contiguous files, its value is greatly enhanced. These details show how much the value of a pawn depends on position.
Lastly, there is a difference of power in different pieces in giving checkmate to the adverse king. When the king has no pieces or pawns left for his defence, the attacking pieces show degrees of power very different from those which they possess in the usual course of the game. A rook is of almost infinite value compared with a bishop or a knight; for while the former, acting in conjunction with the king, may give checkmate, and must, do so if proper care be taken, a knight or a bishop cannot. Under such circumstances a rook is nearly as valuable as the queen, for the latter has now a surplus amount of power which cannot be brought into use; and checkmate is given nearly in the same way by the rook as by the queen, only rather more slowly.
The reader will now be in a condition to understand, from this brief and necessarily imperfect sketch, how many circumstances must be taken into account before we can correctly estimate the relative value and power of the combatants in the chess battle-field. In order to elicit something like a practical rule which may be valuable in play, all the several lists which we have given, and a few more besides, are added together, and the total balance of each power compared with that of the others. The values of any particular piece, in moving over the open board, in moving over a board about half cleared by play, in keeping off an antagonist from a particular set of squares, in making an attack on two or more different squares, in dislodging an antagonist from a particular square, in giving mate without the aid of other pieces, &c., are added together: this is done for each piece; and finally, the whole are reduced to smaller numbers by making a pawn = 1. The final relative values then are as follows:
Pawn ... = 1.00
Knight ... = 3.05
Bishop ... = 3.50
Rook ... = 5.48
Queen ... = 9.94
As, from the nature of the game, the king is invaluable, since he is never exchanged or captured, he is excluded from the computation. It will be seen, from this list, that a knight is worth about three pawns; and that a rook is worth a bishop and two pawns, or five pawns and a half. There appears to be nearly half a pawn difference of value between the knight and the bishop; but the most experienced players are generally willing in an indifferent part of the game, to exchange one for the other, thereby implying that the two are valued equally. This would appear to show that the computed values are not quite correct; but the discrepancy has been explained in a remarkable manner. Suppose a bishop and a knight to be on the board, but not immediately attacking each other. Take the average state of the board, and the bishop could attack the knight in a smaller number of moves than the knight could attack the bishop, arising principally from the knight being unable to act at a distance. This smaller number of moves is often sufficient to give "the move," the advantage of which in an average state of the game is reckoned to be equal to half a pawn: this value, added to that of the knight, would account for the superior value of the bishop.
The result arrived at in this manner is found to be sufficiently near to that which experience points out to the player, to merit attention; still the mode in which it is arrived at is too uncertain and conjectural to give it a scientific character. The time has not yet arrived for applying the rigour of mathematics to the game of chess, so as to demonstrate the excellence of one move over others, in the precise ratio of the powers possessed by the pieces. The great dependence of the player's success on position, independent of the number of his pieces, and the striking effect which the single move will often produce, have hitherto prevented any attempt to include the whole game in a system of mathematical laws. Until this can be done, we doubt whether chess ought to be termed a "science ;" since we are accustomed to apply this term to those subjects only which fall under the influence of general laws or principles which are universally admitted.
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Chess Archeology 
