634 
SIR FREDERICK POLLOCK ON THE MYSTERIES OF NUMBERS 
Roots. 
25 = 3, 0, 0, 4 
27 = 3, 1, 1, 4 
29 = 3, 0, 2, 4 
31 = 2, 1, 1, 5 
33 = 2, 0, 2, 5 
But 33 is in the other series, and is represented by 
4, 0, 1, 4 = 33 
3, 0, 1, 5 = 35 
4, 1, 2, 4 = 37 
3, 1, 2, 5 = 39 
4, 0, 3, 4 = 41 
The first five terms have eaeh two roots whose difference is 7 ; the second five terms 
have each two roots whose difference is 8 ; and this process may be continued throughout 
the two series. 51 is the middle term between 41 and 61; 73 is the middle term 
between 61 and 85 ; and each of these admits of the same treatment as 25, 33, and 41. 
If a Supplemental Square be appended to The Square , and be constructed on the 
reverse principle of diminishing towards the left as the other increases towards the right, 
beginning with precisely the same odd number (see Diagram No. 1), it is manifest that 
a series of numbers may be obtained less than the original number, and which will 
finally terminate, giving all the numbers from which the odd number in the beginning 
of The Square may proceed, if each be placed in succession in the first position of The 
Square ; for it is clear that if the given odd number be lessened by 4, 8, 12, &c., and 
each of these numbers be again lessened by 2, 6, 10, &c., till the operation can be carried 
no further, the process which takes place in the formation of The Square itself will be 
reversed, and therefore if any number so obtained in The Supplemental Square be placed 
in the first position in The Square, and 4, 8, 12, &c. be added, and then to each of these 
2, 6, 10, &c., at length the given odd number will be reached; and in this way it will 
be seen that the odd number will be found in any position whatever that its magnitude 
(as a number) would enable it to fill and properly occupy. And now, if all the numbers 
less than the given number have the properties, which it is alleged belong to all odd 
numbers, viz. of differences of two roots and sum of all the roots, then the given odd 
number will be accompanied by numbers less than itself, having such properties that 
they properly (that is, according to the laws of The Square) occupy their place in The 
Square ; and it must follow that their composition will indicate what roots the given odd 
number ought to have in order to comply with the exigencies of the place in The Square 
that it occupies ; for, in respect to any one of the series that compose The Square , if one 
