TWENTY-SIXTH ANNUAL MEETING. 



75 



ments will afford several hundred thousand ways of footing up 14,828. 



"Divide the entire square into four equal parts. Each quarter is har- 

 monic by itself. Each line and each column in each quarter sums up 7,414. 

 Each transverse or direct semi-diagonal of two places, added to an opposite 

 transverse or direct semi-diagonal of the same quarter, adds 7,414. 



"Divide the square into sixteen equal parts. The sum of the four numbers 

 of each such part is 7,414. Any pair of numbers, added to a similar pair 

 diametrically opposite, equals 7,414. Two numbers taken anywhere, added 

 to the two numbers diametrically opposite, equal 7,414. 



"Finally, any number, added to the number diametrically opposite, is equal 

 to 3,707. This number is composed of the prime factors 11 and 337. 



"It is believed that the footings indicated, together with many others not 

 mentioned, can be obtained in more than a million different ways, or say as 

 many as there were men in the federal army." 



SECTION II. OTHER PLANE FORMS. 

 HEXAGONS. 



I present a harmonic hexagon (fig. 51), which will, no doubt, win admira- 

 tion for its beautiful symmetry and perfect harmony. 



c?y. S^f. 



In this hexagon any full line of three places adds 30, whether along one 

 side or through the center; any line of four places equals 40; of five places 50. 

 The inner circle of six adds 60; the outer circle of 12 adds 120. The numbers 

 in the middle of each of the sides, being at the apices of two large triangles 

 (as 11, 12, 7), add 30 for each three. The same six numbers, each taken 

 separately in connection with the two adjacent numbers of the inner circle, 

 thus forming six small triangles, adds each 30, as 11, 14, 5. Any two opposite 

 numbers add 20; any four, forming a parallelogram (rectangle, rhomb, or 

 rhomboid) having its center at center of hexagon, add 40; the extremes of 

 any outer line and the means of the opposite interior line, as 1, 18, 6, 15, 

 add 40; any pair at one end of an outer line, added to the adjacent pair of an 

 alternating outer line, as 1, 11, 16, 12, add 40. The three numbers on any side, 

 added to either of six other possible pairs, add 50. Six numbers in any 

 order, three on one side and three diametrically opposite, add 60; the three 



