82 



KANSAS ACADEMY OF SCIENCE. 



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In figure 73 the square series is inserted with the same result. Diameters 

 equal 703. 



These series, trigonal and square, are placed in the form of Greek crosses, 

 in which the columns and lines all add equally. (Figs. 74 and 75.) 



Trigonal Series. 



Square Series. 



Fig. 74. 



Fig. 75. 



Besides columns and lines adding equally in the above crosses, the fol- 

 lowing pairs of equal sums are obtainable: Opposite quarters are equal; 

 curves, having for their centers the centers of the crosses, or the middle points 

 of the termini of the crosses, and consisting of the means of one column or 

 line and the extremes of the adjacent one, are equal to a parellel curve in 

 the opposite half of the square. For instance: 78 + 120 + 36 + 6 = 28 + 10 + 66 

 -\- 136. The corner numbers, outside the cross, are to be considered in calcu- 

 lating quarters and curves. 



In figure 76 is presented a parallelogram transformed into a circle, in 

 which all diameters, circles, and prolate cycloidal lines (indicated by dotted 

 lines) add equally, as they would do if the numbers were arranged in a rec- 

 tangle. 



