TWEXTY-SIXTH ANNUAL MEETING. 



67 



of 30,000 squares, no two alike, adding 34 in 10 or more different ways, that 

 may be built up from that one series. 



Now, under these considerations (and they are capable of demonstration), 

 it will be evident that in case of squares adding 38, which may be done by 

 three different series, there may be three times as many, or 90,000 made; as 

 to squares adding 42 or 44, there may be seven times as many, or over 200,000 

 made; and when it comes to squares adding 100, which can be done by 895 

 different series, the number of squares that may be constructed, all different, 

 is 895 times 30,000, or say 27,000,000, and of squares adding 102 no less than 

 33,000,000, no two alike, all magic, and at least one-fourth of them, or say 

 8,000,000, perfectly harmonic. Such numbers are beyond easy comprehension, 

 and almost beyond belief. By way of better conceiving the import of such 

 immense numbers, it may be stated that if a person were to construct at the 

 rate of say 25 16-space squares an hour (and that would be found reason- 

 ably rapid), and work four hours a day for 270 days in the year, and not get 

 sick or tired, and live long enough, it would take him just 1,000 years to make 

 the 27,000,000 different squares, "each adding 100 in 10 or more different ways." 

 Can human penetration farther go? Ah, yes, immeasurably farther. 



Square of Six. 



The square of six is produced by multiplication, in various ways, of the square 

 Df three and the square of two. For instance: the thirty-six places may be divided 

 into four quarters, and each quarter arranged according to the square of three; or 

 the square may be divided into nine sections, and each section filled according to 

 the square of two. 



I here produce several as samples: 



No. 1. 



No. 2. 



No. 3. 



23 3 29 



28 17 12 



7 35 14 



22 4 31 



26 19 10 

 5 33 15 



21 2 30 



25 18 11 



8 34 13 



21 1 32 



27 20 9 

 6 36 16 



These squares are all laid like the scheme in fig. 2, Square of Three. 



So far as the arrangement of the four numbers in each section, or in the 

 respective places in the four quarters, is concerned, it is necessarily various; 

 but since there are twenty-four methods of laying these numbers, as shown 

 in Square of Two, and only nine of them can possibly be used, it allows a 

 very wide latitude in the selection of methods. 



The first step necessary in producing such a square is to arrange the 

 numbers 1, 2, 3 and 4 into sections in such a way that they will add equally 

 in each line, column, diagonal, and such other directions as may be chosen, 

 thus: 



