208 



[June 19, 



As an example, 51=4 2 + 4 2 +19 



=3 2 + 5 2 + (19-2)17 



The series 2-w + l, 

 1 3 



l8)l 2 . O 2 . 

 , 2ra + 19, &c., has similar proper- 



5 7 9 11 



10 



ties, not necessary to be stated in detail in order to understand the 

 effect of placing the series in a square. 



If an odd number, asl 1 (see diagram, No.l), be placed in the upper 

 line on the left-hand square of a larger square divided into small 

 squares, and be made the first term of a series, increasing by 4, 8, 1 2, 

 &c., and then each term of that series be made the first term of a 

 series increasing by 2, 6, 10, 14, &c., the square will be completed, 

 and every odd number in the small squares will be a term in both 

 series ; but if a diagonal be drawn from 11 to 121, and parallel from 



