636 
SIR FREDERICK POLLOCK ON THE MYSTERIES OF NUMBERS 
Now the above observation on a number of the form An-\- 2 shows that Lagrange’s 
method may be used to prove that every number is composed of four trigonal numbers 
as well as of four squares, and thus is brought into immediate connexion with The Square, 
and enables it to divide the first term (whatever odd number it may be) into 4 squares 
the algebraic sum of whose roots shall be 1, and the consequence of that is, that the 
I st AT 1 — - will be a number composed of 3 trigonal numbers or less ; it may be as well 
here to insert the proof of this. 
If the sum of two of the roots differ from the sum of the other two by 1, the 2 sums 
of the roots must be of the form 2«+l and 2 a, and the four roots will be of the form 
a-\-p- 1-1, a—p, a + q, a—q, and the sum of the roots squared will be 
4a 3 +2p 2 +2£ 2 +2«-|-2p+l ; 
deducting 1, and dividing by 2, the number will be 2a 2 -\- a-\-p 2 -\-p q 2 \ 2a 2 -\-a is a tri- 
gonal number, and p 2 -\-p-^-q 2 is an expression for 2 trigonal numbers. Now if 1 be the 
first number in The Square , every number in it will be of the form \-\-2a 2 -\-2a-\-2b 2 , 
which is 4 squares, and expressed by their roots is a-\- 1, a,b,b\ and the sum of any two 
terms added together will be an even number ; and as every possible value of a and b is 
to be found in The Square, every even number may be obtained by adding together some 
two of the numbers in The Square. 
Another result is that, as every number is of the form a 2 -\-a-\-b 2 -\-c 2 -\-c-\-d 2 , every 
even number may be composed of 4 of the terms of the series 0, 2, 4, 8, 12, 18, 24, &c., 
the terms of which are alternately 2 b 2 and 2<f-\-2 a. 
The law of the series is obvious enough ; beginning with 0, the differences are 
2, 2, 4, 4, 6, 6, . . . 2n, 2 n. It is a convenient mode of using this series to place the 
terms in two columns, putting all the 2b 2 in one column and all the 2a 2 -\-2a in another, 
as below. 
2 4 
8 
12 
18 
24 
32 
40 
50 
60 
72 
84 
&c. 
&c. 
Every even number may be made by some 2 terms of each column, and any 2 squares, 
if equal, may be increased by 2, 8, 18, 32, &c., that is, by 2b 2 , by changing the roots — 
Increase. 
n, n into n — 1, n -\- 1 2 
n— 2, n-\- 2 8 
n— 3, n-\- 3 18 
&c. &c. 2b 2 
