32 



KNOWLEDGE. 



[Febeuaey 1. 1900. 



numbers so aiTauged that the numbers in each of its 

 rows, columns, and diagonals, amount to the same sum, 

 as in Fig. 1, where the numbers 123456789 are 



Fig I. 



Fig. 2. 



so arranged in the form of a square that the rows 

 6—1—8, 7—5—3, 2—9—4, the columns 6—7—2, 

 1—5—9, 8—3—4, and the diagonals 6—5 — 4, 8—5—2, 

 all amount to 15. 



This definition calls for some comment. In the first 

 place it presupposes a square, apart from the numbers, 

 in which a certain construction has been made, a geo- 

 metrical square which has been divided by lines parallel 

 to its sides into a number of equal rows, and the same 

 number of equal columns, of small squares, or positions, 

 as they will be called. 



Furthermore the definition involves a classification of 

 the parts into which the whole figure is divided, as 

 (1) rows of positions, (2) columns of positions, (3) 

 diagonal lines of positions. A moment's consideration 

 shows that this classification is incomplete. The word 

 diagonal is not of the same extension as the words row 

 and column. 



The rows comprise all the positions of the Square, 

 taken three at a time ; so do the columns ; but not so 

 the diagonals, which in one direction comprise three 

 positions, and in another direction three also, one of 

 which is common to both diagonals. The classification 

 is therefore not exhaustive. It may be made so how- 

 ever by extending the meaning of the word diagonal so 

 as to include parallel to a diagonal. For with this 

 extension the diagonals will comprise all the positions 

 of the square, taken three at a time, in +wo oblique 

 directions, related to one another in precisely the same 

 way as the rows and columns are related. Let the 

 positions of the square Fig. 2, be numbered in the usual 

 or natural order. 



We may then arrange the positions in four classes, 

 according to their direction. 



(1) 3 rows of 3 iiositions eiich, 1—2—3, 4—5—6, 7—8—9 



(2) 3 columns of 3 positions eaot, 1—4—7,2—5-8,3—6—9 

 (3J 3 diagonals of 3 positions each, 



descending to the right, 1—5-9, 2—6—7, 3— 1—8 



(4) 3 diagonals of 3 positions each, 



descending to the left, 1-6-8,2-4—9,3—5—7 



If we wish to distinguish (3) and (4) we may call 

 (3) positive diagonals or -)- diagonals, and (4) negative 

 diagonals or — diagonals. 



We may also distinguish the diagonals in the usual 

 sense from the diagonals in the extended sense by call- 

 ing the former the middle diagonals. And we may 

 class together the rows and columns on the one hand, 

 and the two kinds of diagonals on the other, as laterals 

 (for they are measured by the sides of the square), and 

 diagonals. 



We shall now be prepared for an analysis of the magic 

 square of 3, and for a comparison of the magic square 

 with the complement which by the universal law of 

 things must somewhere exist. The square which stands 

 in this relation of polatity to the magic square is shown 



in Fig. 3, and is called the Natural Square, and the 

 object now in view is to establish and illustrate the 

 completeness of the polarity existing between these 

 two squares. 



The law might be called in general terms the law of 

 polarity in direction, but, as might be expected, it 

 shows itself under various aspects, ■Which will have to 

 be considered separately. 



I. Summation. — Equal summation of all rows and 

 columns is the special note of the magic square ; for in 

 the equal summation of its mean diagonals and mean 

 laterals it is undistinguishable from the natural square. 



Now if the square in Fig. 1 be compared with tha'. 

 in Fig. 2 it will be seen that the -I- diagonals of the 

 first are the columns of the second, and its — diagonals 

 the rows. 



The diagonals therefore of the Natural Square, and 

 the laterals of the Magic Square have equal summation, 

 and polarity of direction as regards summation exists 

 between the two. 



II. Difference. — Let the series 123456789 be 

 regarded as a recurring series, that is to say a series in 

 which we may begin at any point, read in either 

 direction to either end, revert to the other end, and 

 read in the same direction to the starting point as 

 4 5 6 7 8 9 1 2 3 or 5 4 3 2 1 9 8 7 6. In all these 

 readings of the series the difference is said to be 1, for 

 successive terms are taken at intervals of one position. 

 Now let the series be varied by taking successive terms 

 at intervals of 2, 3 and 4 positions respectively ; it will 

 be unnecessary to go further since by so doing we shall 

 only obtain the same variations inverted. 



The possible variations for these differences will be 

 found to be — for 

 or 



123|456|789 

 1 I 357 I 924 |681 

 147|258|869 

 159|483|726 



When the difference is 3 or 6, it is impossible to com- 

 plete the series without beginning at three different 

 starting points since the third position after 4 is 7, the 

 third after 7 is again 1. 



Now if these valuations of the series be divided each 

 into three triads, beginning with 1 in all cases except 

 where the difference is 2 or 7, when a triad must begin 

 with a multiple of 3, the triads will be found to be 

 identical with the lines of the Natural and Magic 

 squares, and the distinction between the squares to lio 

 in the direction of the differences. 



The subjoined table shows the directions of the differ- 

 ences in each square : — 



Thus polarity of direction as regards differences exists 

 between the two squares. 



III. Odd and Even Cross. — If the Natural and magic 

 squares be compared as regards the position of odd and 

 even numbers, it will be observed : 



That odd and even numbers are alternate in the out- 

 side rows and columns and either in the middle laterals 

 or middle diagonals of each. 



