Explanatory Bemarks on Diagram No. 2. 
The number in the first square is 35, a number of the form 4n+3 ; the nearest division in whole 
numbers of 35 is 17, 18, and the nearest division of these is 8, 9, 9, 9, which are the roots of the 
(»+l)th term down the diagonal (in this case the 9th) ; these roots, when 8 and 9 are negative, give 
a difference of 1, and therefore the roots traverse from the place they occupy to the top comer, becoming 
there 0, 1, 17, 17, and also to the opposite corner, becoming there 1, 1, 16, 17. If the roots ascend 
in the diagonal towards 35, diminishing each root every step by one, they will require a numeral to be 
added, viz. 4, 6, 12, 16, &c., to make them equal to the number that should be found in the square they 
occupy. This may be easily done by the assistance of the 2 columns of 2b 2 and 2a 2 -\-2a ; a few of them 
are corrected and marked thus, = 0, 2, 5, 6 = by way of example. The term in the gradation series next 
below 8, 9, 9, 9 has the roots 8, 8, 8, 9, but it requires 2 to be added, and when corrected is 7, 9, 8, 9. 
If the roots 8, 8, 8, 9 (diminished by 1 each) at every step be taken up the 2nd diagonal, they will 
require a numeral to be added, viz. 2, 6, 10, 14, &c., to make them equal to the required number ; this 
also may be done in the same way as the other, whenever the correction produces 1 as a root ; the 
roots of the 1st number may be obtained ; thus in the 4th square of the diagonal the corrected roots are 
3 4 0 
1, 6, 2, 6, the sum of which is 15 ; but by making 1 negative, —1, 2, 6, 6 = 13, and lessening the roots 
by 3 each, they become —4—133, the sum of whose squares is 35, and the algebraic sum of the roots 1 ; 
35 q 
the number — - — , that is 17, is composed of 3 trigonal numbers, viz. 10, 6, 1. All the squares in 
the line have a similar property, 12, 13, 5, 5 ; the 4th below to the left from 8, 9, 9, 9 may be 
carried up, diminishing the roots by 1 each every step, but adding as a numeral 4, 8, 12, &c., the roots 
will always (one or more) have 1 among them ; but it may not always be easy to find out which ; the 
following mode cannot fail : 8, 8, 8, 9 is 34 less than 8, 9, 9, 9, each term in the Gradation Series is 
less than the next above it by the sum of the roots of the next, minus 1 ; 34 will be added by adding 
18, 12, and 4 ; 18 will be added by changing 8, 8 into 5, 11 ; 12 may be added by changing 8, 9 into 
6, 11 ; the roots will then be 5, 6, 11, 11, and to these 4 will be added by charging 5, 6 into 4, 7 ; 4, 
7, 11, 11, when squared (each of them), =8, 9, 9, 9 when squared and added together; but the sum 
of the roots of 4, 7, 11, 11 is 33, the index of the square in which 8, 9, 9, 9 is ; it will therefore 
traverse upwards to the left and diminish each root by 8; the roots will be — 4 — 133 as before. 
Note . — Every square that is crossed by a red line, whether perpendicular, horizontal, or transverse, 
will have one or more forms of roots, the sum of the squares of which will be the number belonging 
to that square. 
