573 



a 2 + a -|_6 2 . For if 2 numbers be both odd or both even, they may 

 always be represented by + 6 and 06; if one be odd and the 

 other- even, they may always be represented by a + 6+1 and a 6, 

 or by a + b and a 6 + 1 ; and if the two numbers be made the bases 

 of trigonal numbers, the sum of the two trigonal numbers will 

 always be of the form a 2 + a + 6 2 , or 2 + b + b 2 : now when any 

 number in the natural series of numbers is composed of two trian- 

 gular numbers, it may be represented by 2 + a + 6 2 , and 4n + 1 will 

 then equal 40 2 + 40 + 1 +46 2 , obviously the sum of an odd and an 

 even square, whose roots are 26 and 20+ 1 ; and 2w+ 1, the corre- 

 sponding odd number, will equal 20 2 + 20 + 1 + 26 2 , obviously com- 

 posed of 4 square numbers, whose roots are 6, 6, 0, + 1 ; and if they 



be arranged thus, 



26, (0 + 6)20+1 

 6, 6, 0, 0+1, 



so that the sum of their roots may equal 1, the exterior differences 

 of the roots will be 26 and 20+1, the roots of the two squares into 

 which 4w+ 1 is divisible ; and the middle difference will be (a + 6), 

 the smaller half of the sum of the roots (26 + 20+1) with a nega- 

 tive sign ; if the exterior differences be reversed and the middle dif- 

 ference be increased by 1, the differences will be 20 + 1, (a + 6 1), 

 26, and the roots whose sum will equal 1 will be, with their differ- 

 ences above them, 



20 + 1, -(0+6-1), 26 

 -(0+1), -(6-l), 6 + 1, 



and the sum of the squares of the roots will be 2 more ; from these 

 two sets of roots all the rest may be obtained, by adding one to each 

 of two roots and subtracting 1 from each of the other two roots ; the 

 exterior differences of the roots will therefore always be the same, 

 and the middle difference will increase by 1 at each step ; the sum 

 of the squares of the roots will increase by 

 2, 4, 6, 8, &c. 



As the sum of any two square numbers of which one is odd and the 

 other even (40 2 + 40+l +46 2 ) must be of the form 4ra+l, every 

 possible case of an odd square combined with an even square must 

 occur somewhere in the series 



1, 5, 9, 13, &c., 

 and the Table (if extended) must contain every possible case of odd 



