5i 



KNOWLEDGE, 



[Maboh 1, 1900. 



position, and in the o 

 row. 



Bj- pushing out the 

 rows and columns, so 

 circles or octagons we 

 tical, and obtain a fi 

 Natural Chain, eqiiall 

 Circle or Octagon, and 

 the transposition of tw 

 6 in one case, 2 and 8 



ther in a middle position, of a 



middle positions of the outside 

 as to convert the squares into 

 may take the two figures iden- 



gure which may be called the 



y convertible into the Natural 

 tlie Magic Circle or Octagon by 



o opposite even numbers, 4 and 



in the other. 



Fig 7, 



The law of polarity between the Natural Circle, or 

 Octagon, and the Magic Circle, or Octagon, is of course 

 pi-ecisely the same as that of polarity between th.5 

 Natui-al and Magic Squares, but it admits of a differ- 

 ent exjsression. 



For in the Circle, or Octagon, the external rows and 

 columns of the squares have become arcs, or triangles, 

 while the middle rows and columns, as well as the 

 middle diagonals have become diameters. The polarity 

 of the two figures may therefore be stated as follows: — 



In the Natural Circle the Natural Rows and Columus 

 have become four arcs each equal to a quadrant and 

 two diameters : the Magic Rows and Columns have 

 become four arcs each equal to three quadrants and two 

 diameters. In the Magic Square the statement must 

 be reversed. 



In the Natural Octagon, the Natural Rows and 

 Columns have become four obtuse angled triangles, 

 each on a side of a square, and two diameters; the 

 Magic Rows and Columns have become four acute 

 angled triangles, each on a side of the square, and two 

 diameters. In the Magic Octagon the statement must 

 be reversed. 



In the Natural Ti-iads in either figure an even num- 

 ber forms the vertex of a triangle, in the Magic Triads 

 an odd number forms the vertex. 



This connection of a Natural Figure with its com- 

 plementary Magic Figure through the intermediary 

 Natural Chain leads to the enquiry whether there is 

 any other Natural Figure connected thus closely, or 

 more so, with its complementary figure, and to the dis- 

 covery that 12 3 4 5 6 7 can be arranged as a natural 

 chain in a figure which may be described as perfectly 

 natural, and at the same time perfectly magic. 



If a hexagon be formed of six equilateral triangles 

 placed with their vertices at a point, the numbers 

 12 3 4 5 6 7 may be placed as in figure 10, where 



Fig 10 



The numbers form a continuous chain, all the links 

 of which are of equal length. 



Each of the five natural triads, or triads of numbers 

 in natural order, lies on three successive positions, 

 1—2—3, 2—3—4, 3—4—5, 4—5—6, 5—6—7. 



Each of the five magic triads or triads of numbers 

 having equal Summation (12) lies on three successive 

 positions, 1--1— 7, 1—5—6, 2—4—6, 2—3—7, 3—4—5. 



It is somewhat surprising that this figure was not 

 adopted by the Ancient Mystics and Astrologers as a 

 jjerfect presentment of the sacred and jjlanetary number 

 seven combined with the equally sacred and funda- 

 mental number twelve. 



V. Paths. — We have already had occasion to regard 

 the Series 123456789 as a recurring Series, and 

 on the same principle we may regard a number-square 

 as capable of extension, without alteration in its 

 character, by repeating its rows and columns m 

 their original order. Let the Natural and Magic 

 Squares of 3 then be extended, first by repeating two 

 rows above and below the square, and then by re- 

 peating two columns of the resulting rectangle of 

 numbers to the right and left. 



We shall then have the squares shown in the follow- 

 ing figures 11, 12, 13, 14, 15, 16. 



Fig. II. 



Fig. JZ. 



Fig 13. 



Fig. 14. 



Fig 15 



Fig 16. 



In Figs. 11, 12, we see that in the extended Natural 

 Square the Magic Square is latent, with its rows and 

 columns in Bishop's paths, while in the extended Magic 



