76 KANSAS ACADEMY OF SCIENCE. 



of any outer line and the three of an alternating outer line, as 1, 11, 18, 2, 12, 

 16, add 60. Any eight, four on either side, complementary to four opposite, 

 add 80. Alternate outer lines sum up 90; adjacent pairs of alternate outer 

 lines, -with the transverse central row, as 1, 11, 16, 12, 2, 15, 10, 5, 18, add 90; 

 anj' small triangle and two adjacent sides equal 90; same with two alternate 

 sides equal 90; same with opposite and one adjacent side equal 90; two alternate 

 small triangles and either of four sides equal 90. Any ten, half complementary 

 to other half, equal 100; any twelve, ditto, equal 120; any fourteen, the same, 

 140; any sixteen 160. 



Thus, in the above hexagon, we may find nine sums of 20 each; 23 sums of 

 30 (14 Independent and the nine sums of 20 added to the central number); 

 57 sums of 40; 93 sums of 50; 147 sums of 60; the same of 70 (the same sums with 

 central number added); 126 sums of 80; 82 sums of 90; 70 sums of 100; the 

 same of 110; 60 sums of 120; the same of 130; 28 sums of 140; 42 sums of 150; 23 

 sums of 160; 9 sums of 170; one sum of 180 and one of 190. In all cases, what- 

 ever numbers are added, if they are added as indicated, will add 10 for each 

 number taken, the same as if the numbers were all lO's, 



TRANSPOSITIONS. 



By inverting, reversing, etc., as in the squares, twelve apparently different 

 harmonic hexagons may be obtained from this one; but which are in reality 

 only different views of tl},e same hexagon. 



<^i^ sz.. 



Again, by transposing the inner circle for the outer corners, number for 

 number, an apparently new hexagon (fig. 52) is formed from this; but which, 

 a careful study will show is only one form of transposition; and which may 

 also, by reversion, inversion, etc., be transformed into twelve other hexagons, 

 making twenty-four in all. All these hexagons have the same harmonic 

 properties as the first. Whatever is true of one is true of all. 



Four of those hexagons, showing apparent differences, are here presented: 



2 12 16 6 7 17 3 12 15 18 11 1 



9 15 3 13 9 19 4 8 1^5 1(3 2 9 8 5 14 13 



19 6 10 14 1 15 2 10 18 5 14 1 10 19 6 4 17 10 3 16 



7 17 5 11 12 16 1 11 11 18 4 7 7 6 15 12 



4 8 18 3 13 14 5 8 17 19 9 2 



