630 
SIR FREDERICK POLLOCK OK THE MYSTERIES OF NUMBERS 
increases which may belong to the sum of the roots, the number of odd numbers in 
succession which may be formed increases faster than the interval between the terms 
of the gradation series. This will appear by an examination of the numbers themselves ; 
but the interval may be filled up more readily in the manner above mentioned ; and if 
any two terms in the gradation series be taken, the odd numbers from the one to the 
other, both inclusive, may be made up of the squares of roots whose sum shall be the 
number indicated by the sum of the roots of the first of the two terms. 
Thus 3, 3, 3, 4 are the roots whose squares make the number 43 ; and from 43 to 57, 
which is composed of 3, 4, 4, 4, every odd number can be found, the sum of whose roots 
shall equal 13. Thus 
From 43 to 57 is but 7 steps ; 
but the series of odd numbers 
consisting of squares whose roots 
equal 1 3 goes on 6 steps further. 
' 0 0 1 
3, o, 3, 4 
— 
43 
1 1 0 
2, 3, 4, 4 
— 
45 
1 0 2 
2, 3, 3, 5 
= 
47 
0 2 1 
2, 2, 4, 5 
= 
49 
< 
2 1 1 
1, 3, 4, 5 
_ 
51 
0 1 3 
2, 2, 3, 6 
— 
53 
2 0 3 
1, 3, 3, 6 
= 
55 
1 2 2 
i 1, 2, 4, 6 
— 
67(= 
3 2 0 
0, 3, 5, 5 
= 
59 
3 1 2 
0, 3, 4, 6 
= 
61 
1 1 4 
1, 2, 3, 7 
— 
63 
2 3 1 
0, 2, 5, 6 
zz: 
65 
0 3 3 
1, 1, 4, 7 
— 
67 
2 2 3 
0, 2, 4, 7 

69 
The differences between the roots are marked in order to compare them with those in 
the next number 
