Page:Notes and Queries - Series 2 - Volume 6.djvu/441

2"* S. VI. 152., Nov. 27. '58.] NOTES AND QUERIES. 435 Old Romney and Brookland. I have in my possession three small volumes of Sermons in MS., preached in the above two places between the years 1691 and 1694. Can any of your readers tell me the author's name ? SAMPSON. [Perhaps our correspondent may obtain a clue to the author if AVC state that the Rev. John Defray was Rector of Old Romney from 1690 to 1738 ; and that the Rev. Thomas Johnson was Vicar of Brookland from 1677 to 1727.] CHESS CALCULUS. (2 nd S. vi. 347.) The question asked is whether it be " practica- ble to construct a Chess Calculus, so that every position in a game may be expressed by a func- tion of the positions and powers of the pieces, by operating on which the best move for the next player might be evolved." The following pre- sumptions in favour of the practicability are raised : First, that chess is evolved from axioms and definitions ; secondly, that the power of a piece may be expressed by coordinates. To say that such a calculus must be impossible, would be to speak beyond knowledge ; and more- over would not be conclusive : for impossible things are done from time to time. A very sim- ple game might be proposed of which the calculus is not impossible : and if a simple game admit of such treatment, in what should a more compli- cated game differ from it except in complication ? Take the common game which in my school days used to be called by some noughts and crosses, and by others tit-tat-toe, which were the formular words of victory, just as check-mate are those of chess. There are nine squares in rank and file, in one of which the first player enters a nought, the second player enters a cross in another, and so on ; the game being won when either player can point out his marks three in a row, whether horizontal, ' vertical, or diagonal. Now the number of pos- sible games must very considerably fall short of 362880, the product of the first nine numbers, the total number of orders in which the squares can be filled up. The number of rationally played games probably does not^exceed a few hundreds. A calculus is conceivable : but it would be of very intricate expression. Given the state of things at the nth move, it is possible that a formula might, by inserting the value of n, give out all the ways in which a player might afterwards win, distinguish- ing the few in which the new move reduces his winning to a certainty." But the chess calculus is beyond human ima- gination." In the first place chess is not entirely evolved from definitions and postulates. A geo- meter who plays with these things as he finds them in Euclid, must play every proposition of every book : but the chess player is dictated to by an adversary. Suppose all possible rational games to be, one with another, of 30 moves on each side, 60 moves in all, which is rather low. Suppose that at each of 50 moves the player in action has two good choices, which is not much, considering how many choices he frequently has. This supposes more than eleven hundred mil- lions of millions of games, and a calculus supposes a formula containing in its structure an implicit ac- count of the progress of every one of these games. For a formulary contains not merely what shall emerge in any case ; but all that by possibility might emerge. That the use of such a formula should involve the solutions of equations of the ten-thousandth degree is probably very much be- low the mark. Again, how are we to express the powers of the several pieces? I remember seeing an attempt which was based on the number of squares com- manded : but the proposer acknowledged himself incapable of representing the additional power derived by a knight from his not being stopped by other pieces. This, however, would be far from enough, even if it could be satisfactorily done. The power of a piece depends upon the neighbours it may have, and the opponents who check it. A protected pawn immediately before a castle limits its power and value, except in those rare cases in which it will be worth while to sacri- fice the castle for the pawn. Whether or no the sacrifice would be worth while depends upon the prospects of the game. Hence the power of the pieces, in any given position, will depend upon the whole structure of the game ; while the formula for the game will depend upon the mode of ex- pressing the power of the pieces. Such compli- cations of the ignotum per ignotum it is the daily business of mathematical analysis to unravel : but I confess that I should expect, in the expression of the chess problem, a complexity far exceeding that of any problem which was ever successfully dealt with up to this time. A. DE MOKGAN. MARSTON S WORKS. (2 nd S. vi. 368.) I have just seen in " N. & Q." some rather severe strictures on Mr. Halliwell's late edition of this poet. I do not think they are merited ; for Mr. Halliwell's object was, as he says, to give these pieces " as nearly as possible in their ori- ginal state," and thus to give people who, like myself, cannot or will not lay out large sums in the purchase of old and scarce books, or spend days in the Museum, an opportunity of seeing how books came out of the hands of the old prin- ters, even when, as was evidently the case with Marston, the proofs were read by the author,