368 



NATURE 



[May 26, 1910 



were throughout their greater part of the usual fair weather 

 type, the potential being neither specially high, specially 

 low, nor specially variable. There were, however, 

 two intervals, between 8.40 and 9.20 p.m. on May 19, 

 and between 1.30 and 3 a.m. on May 20, when there were 

 rapid oscillations and negative potentials, which were not 

 accompanied — as is usually the case — by a rainfall visible 

 in the rain-gauge curves. Thunderstorms were, however, 

 in active progress at the time at no great distance, a good 

 many peals of thunder being audible in Richmond ; there 

 was thus nothing in the electrical phenomena that is not 

 adequately accounted for by the observed meteorological 

 conditions. C. Chree. 



May 21. 



The Magic Square of Sixteen Cells. A New and 

 Completely General Formula. 



The ancient problem : To construct a Magic Square with 

 sixteen consecutive integers, may be regarded as a special 

 case of the general problem : To construct a Magic Square 

 with any sixteen positive integers, no two of which shall 

 be identical. The solution of the problem thus generally 

 enunciated throws much new light upon the ancient special 

 one, and will, in fact, enable us to classify and tabulate 

 its 880 known solutions (8x880, if we admit reversals and 

 reflections of the same square to be " different ") much 

 more scientifically than has hitherto been done. 



The following is the completely general formula for the 

 Magic of Sixteen Cells : — 



For (i) this formula obviously represents a Magic 

 Square, since every row, every column, and both the 

 central diagonals sum to A+B+C+D. 



Also (2) it is a function of eight independent variables. 



Let S be the sum of our sixteen unknown quantities ; 

 then the constant total of the square will =8/4. If 

 three of the rows sum to S/4, the fourth row must do 

 the same ; similarly with the columns. 



Hence only eight of the ten given conditions are 

 independent ; we have to solve eight simultaneous linear 

 equations involving sixteen unknown quantities. The 

 solution, if general, must thus involve eight arbitrary 

 constants. Therefore the above solution, which does 

 involve eight arbitrary constants, is a perfectly general 

 one. 



I proceed to a numerical example. If A=io, B = i2, 

 C = 8, D = 5, a = 8, 6=— 9, c= — 10, d = 2, our formula 

 gives us a Magic summing in every direction to 35 :— 



It will be noticed that the number 19 is used, and the 

 number 15 is not. 



We have here an example of a Magic in its simplest 

 form, with none of the superfluous (accidental) relations 

 such as appear among the components when those numbers 

 happen to be consecutive ; and we see that the " comple- 

 mentary pairs " (each summing to half the constant total) 

 upon which previous writers have laid such stress are a 

 purely adventitious feature, and have no real connection 

 with the laws of construction of the square. 



In the fourth volume of the " Recreations Math^- 

 matiques " of Edouard Lucas (Paris, 1894) ^^^ set out 

 three theorems and three corollaries, enunciating various 

 equalities which must exist between the component 

 numbers of every Magic of Sixteen Cells. The proof of 

 these takes up four pages and a half, and requires twelve 

 illustrative diagrams. My formula proves them all by 

 simple inspection. 



If, in the formula, a=h, the square assumes the type 

 which Freiiicle designated by the letter 5.' If a=—h, it 

 assumes the type which Fr^nicle, in his table, left un- 

 marked. Of the latter type, there/ 1 are exactly i2p in 

 consecutive numbers. I append an example of each 

 type :— 



NO. 2 II 7, VOL. ^2>\ 



(A=7; B=9; C=4; 0=14; 

 a=6; 6=6; c=2 ; d=^.) 



(A=4; 8 = 15; C = io; D = 5; 

 a- 3; 6= -3; c= -4; (i=r) 



It must be borne in mind, however, that a complete 

 numerical solution of the 5 type necessarily includes the 

 squares which Frenicle marked a and /3, because both of 

 these are, algebraically, particular cases of the 5^ 

 form. 



My formula readily supplies an infinity of solutions of 

 the problem. To construct a Magic Square with sixteen 

 different prime numbers. The following example (first 

 published by me in the Pall Mall Gazette of February 26 

 last) omits two only out of the first eighteen odd primes,, 

 and sums to a far smaller constant than any other investi- 

 gator has been able to obtain : — 



(A = 7; 8 = 17; C = 3i;D = 59; 

 a=6; 6=6; c=io; ^=4.) 



It is obvious that every 4^ Magic formed by the addition 

 of two Latin squares is divided into equal quarters. No 

 proof, however, has up to now been given of the " con- 

 verse " of this proposition. I will deduce the theorem from 

 my general formula. 



Theorem. — Every 4^ Magic in equal quarters can be 

 expressed as the sum of two Latin squares. 



That the form of the result may be more convenient, I 



1 "Ouvragesde Math^matique." Par M. Frcnicle. (La Haye, 1731.) 



