1902.] on Magic Squares and other Problems on a Chess-Board. 57 



between arithmetic and algebra ; the former may be regarded as an 

 algebra iu which the numerical magnitudes undsr consideration are 

 restricted to be integers ; the two Iranches contemplate discontinuous 

 and continuous magnitude respectively. Similarly, in geometry we 

 have the continuous theory, which contemplates figures generated by 

 points moving from one place to another and in doing so passing over 

 an infinite succession of points, tracing a line in a plane or in space, 

 and also a discontinuous theory, in which the position of a point 

 varies suddenly, j9er saltum, and we are not concerned with any con- 

 tinuously varying motion or position. The present problems are 

 concerned sometimes with this discontinuous geometry and sometimes 

 with an additional discontinuity in regard to numerical magnitude, 

 and the object is to count and not to measure. Far removed as these 

 questions are, apparently, from the subject-matter of a calculus of 

 infinitely small quantities and the variation of quantities by infini- 

 tesimal increments, my purpose is to show that they are intimately 

 connected with them and that success 

 is a necessary consequence of the re- 

 lationship. I must first take you to a 

 much simpler problem than that of the 

 Latin square, to one which in a variety 

 of ways is very easy of solution, but 

 which happens to be perhaps the sim- 

 plest illustration of the method. In the 

 game of chess a castle can move either 

 horizontally or vertically, and it is easy 

 to place 8 castles on the board so that 

 no piece can be taken by any other 

 piece. One such arrangement is shown 

 in Fig. 10. The condition is simply Fig. 10. 



that one castle must be in each row and 



also in each column. Every such arrangement is a diagrammatic 

 representation of a certain mathematical process performed upon a 

 certain algebraical function. For consider the process of differen- 

 tiating x^ ; it may be performed as follows : Write down x^ as the 

 product of 8 x's, 



JL JL Jb *,L- J^ «{/ >// iA^« 



and now substitute unity for x in all possible ways and add the results ; 

 the suhstitution can take place in eight difi'erent ways, and the addi- 

 tion results in 8ic^, which will be recognised as the differential co- 

 efficient. Observe that the process of differentiation is thus broken 

 up into eight minor processes, each of which may be diagrammatically 

 represented on the first row of the chess-board by a unit placed in the 

 compartment corresponding to the particular x for which unity has been 

 substituted. If we now perform differentiation a second time, we may 

 take the results of the above minor processes and in each of them again 

 substitute unity for x in all possible ways ; since in each the sub- 



