Nov. 4, 1881.] 



♦ KNOWLEDGE • 



19 



#ur i¥latl)fmatiral Column. 



MATHEMATICAL columns in magazines arc too apt to de- 

 generate into puzzledom corners, collections of problems of 

 considerable difficulty, but having no real interest, and not valuable 

 as illustrating principles. We wish ours to be of real use to the 

 student of mathematics, but still more to those who, when studying 

 other branches of science, find that rightly to understand their 

 subject they require to be familiar with certain special departments 

 of mathematics. It must be confessed that most of our treatises on 

 mathematical subjects pay Uttlc attention to the requirements of 

 students of this kind. They are excellently adapted to prepare 

 students for examination ; because at an examination a student of 

 algebra must not be surprised if he be asked questions relating to 

 infinite series, a student of trigonometry if he be asked to establish 

 Dr. Moivre's Theorem, and so forth. But they do not meet the 

 wants of one who requires to know the methods and principles of 

 algebraical, trigonometrical, or other forms of calculation. There is 

 bttle in such works to show the use and value of the processes con- 

 sidered in them. Many a ready student has passed a fair examina- 

 tion in the differential calculus, who had not the remotest idea of 

 the practical value of its methods, or that in hundreds of simple 

 subjects of inquiry the calculus can be employed easily and advan- 

 tageously. There is no occasion for mystery respecting the use of 

 mathematical methods; but it would really seem, to read some 

 mathematical treatises, as though the last idea in the world the 

 student should associate with any mathematical subject was the 

 idea that it may actually be useful. Even the problems which are 

 given for solution are, for the most part, either useless or absurd. 



This is not the way to render mathematical subjects inviting, or 

 to encourage the student to master the difficulties which are 

 inherent in them. 



Of course, we cannot here give mathematical essays which can 

 render the student independent of systematic treatises. Such 

 treatises ho must have, and must carefully study. But we hope 

 to show that a number of departments of mathematical research, 

 supposed to be either too profound, or of too little practical value 

 to be worth taking up, can be readily and usefully studied. In 

 every case we wish to come as quickly as possible to the practical 

 application of the various methods we shall successively deal with. 



We shall probably begin with a simple paper on the use of loga- 

 rithms, seeking to show that, instead of being regarded as a mere 

 mysterious collection of numerals, a book of logarithms should be 

 considered the great simjjlifier of all forms of calculation. Scarcely 

 anyone who has much to deal with figures, otherwise than in mere 

 processes of addition or stibtraction, should be without such a book ; 

 and a very brief study of the subject will suffice to enable anyone 

 to make ready and intelligent use of the tables which a book of 

 logarithms contains. 



We may then discuss the Laws of Chance, the simple applications 

 of Trigonometry, and other such matters, avoiding, as far as possible, 

 those parts of a sr.bject which a student can readily study in set 

 treatises. 



We shall be glad to receive problems of interest, either for solu- 

 tion or discussion, preferring, he never, those which illnstrato 

 general principles to those which are merely, as it were, casual. 



In many cases where problems are sent to us for solution, wc 

 shaU only give hints, not complete solutions, believing that, so dealt 

 with, they are likely to be of more use to the student than if a 

 complete solution were at once given. 



We need hardly say that this is not a suitable place for the dis- 

 cussion of very difficult mathematical problems, albeit those of our 

 readers who do not take interest in mathematics must not be angry 

 with ns if from time to time we devote a column, or even a page 

 or two, to matters of no interest, or even absolutely unintelligible 

 to them. They must remember that each of our lighter essays here 

 may be equally interesting to proficients in the subject dealt 

 with ; that, in fact, no one can expect all the contents of snch a 

 journal as the present to be interesting to himself individually. Wo 

 shall endeavour, however, to keep within close limits all matter 

 likely to be " caviare to the general." 



Sunlight a-.-o Heat.— The intensity of sunlight at the sun's 

 surface is calculated to be 190,000 times that of a candle-flame ; 

 5,,30O times that of the glowing metal in a Bessemer converter ; 

 146 times that of the limelight ; 3'4 times that of the electric arc 

 at the brightest yet obtained. The heat emitted by the sun in a 

 single second would suffice to melt a shell of ice covering the entire 

 surface of the earth to a depth of 1 mile 1,457 yards, or to burn a 

 layer of anthracite coal 17'7 yards thick over the earth's entire 

 surface. This would be equivalent to a consumption of about 

 16,436 millions of millions of tons of such coal per second. 



(Bur Cftrss Column. 



MANY weekly and monthly periodicals in this country and in 

 others now contain a chess column, but it may be a question 

 how far any general knowledge of the game is encouraged in this 

 way. Usually these chess contributions are limited to problems, 

 games between first-class proficients, and replies to queries, with 

 occasional items of chess news. The problems are such as only 

 good players can readily solve, so that the learner, unable to 

 master them without an expenditure of time which he can ill spare, 

 is rather deterred from the study of chess than otherwise by these 

 masterpieces of chess strategy. The games, again, even when they 

 are sufficiently annotated, are generally too profound to have much 

 interest for the learner. He cannot see the purpose of moves whose 

 real significance depends generally on results five or six moves at 

 least in advance, and almost always on moves which are not played 

 on either side. The chess expert recognises the beautiful positions, 

 the brilliant attacks, the complex defences, and so forth, which 

 would result if a particular move were not met in that way in 

 which, in the game itself, it actually is met. A volume would be 

 required to show all such results of each move played on either 

 side by two first-class players ; and the beauty of a game between 

 two such players can only be properly appreciated by those who, 

 as each move is played, can follow oat the various consequences, 

 according to the way in which the move may be met. 



We wish to cater for a wider circle of chess players — for those 

 who enjoy the game and can appreciate its beauties, while they 

 have no ambition to be able to meet a Steinitz, a Blackbume, or a 

 Zukertort without receiving odds. 



It must be admitted that nine out of ten who call themselves 

 chess-players know very little about the game. They move their 

 pieces without any adequate idea of the value of position, or of the 

 manner in which the powers of the various pieces can best be 

 brought out. Of the importance of time, again, in chess strategy, 

 they seem to have scarcely any conception. A player of this class 

 will move his Queen out to some square where she can be attacked 

 by a piece which in attacking her will take up a strong position, 

 then to some other square where she can be similarly attacked by 

 another piece, and, perhaps, after four or five such moves, be fortu- 

 nate in being able to return her to her own square. He will then 

 complacently remark that he must try some other way of opening 

 his attack, utterly overlooking the fact that his opponent has 

 gained four or five moves, and that even a single move early in the 

 game often makes all the difference between a strong attack and 

 an unsatisfactory defence. If such a player moves his Knight 

 from King's Bishop's third to King's Knight's fifth, and on 

 the opponent moving Fawn to King's Rook's third (attacking the 

 Knight), finds no better move than to return him to his former 

 position, he would be incredulous if told that he had thus lost all 

 chance of winning against correct plaj'. Yet there is scarcely an 

 imaginable case, in the early stage of a game, where, if the game 

 had been equal before this had happened, it would not afterwards 

 have been seriously compromised. The opponent, be it observed, 

 has made a move of great utility (though often too slow, except as 

 in this case when a move is given away), while it is now his turn 

 to play instead of the first player's, who stands just where he did 

 before he rashly moved forth his Knight. (The opening, of wliich 

 the accompanying game is an illustration, shows that, even when 

 by venturing forth the Knight to the square in question a Pawn is 

 gained, the counter attack, after the Knight has been driven back, 

 compensates the second player fully for the loss. It also presents 

 at move 33, a case in which loss of time is equivalent to loss of a 

 game which might probably have been drawni.) 



It seems to us that we may do something to encourage sound 

 chess play by giving our readers chess material of a different kind 

 from that which has usually been presented in chess columns. We 

 propose to explain in a scries of short papers the principal open- 

 ings, discussing their advantages for attack and defence, and show- 

 ing how the opening moves illustrate the general princijiles on 

 which sound play depends. We shall illustrate these openings by 

 games played by good players, but not played with quite so much 

 study (and, therefore, not needing .such skUl for their interpretation) 

 as the match games commonly selected for these columns. For this 

 purpose we have made arrangements with the proprietor of the 

 mechanical chess-player, Mephisto, to have games played with 

 Mephisto specially for our chess columns. By an extension of Mr. 

 Giimpel's kindness, the guiding spirit of that mysterious player has 

 been persuaded to make his own comments on the game. It need 

 hardly be said that only such games will be selected as have real 

 chess interest. The multitudinous contests in which tyros have 

 succumbed to Mephisto (and would have succeeded at the odds of 

 a Queen) would be quite as much out of place here as in the 



