( 431 } 
X. Abbreviation of the method of § I in special cases. 
In the series 1 + 3 + 5 ete. each term is a 4-fold + 1 or a 4-fold — 1, 
so that the sum of two successive terms is always a 4-fold. If we 
leave out the first term, then the sum of 2 successive terms 
3-+5,7-+ 9, etc. is always two times their mean, i.e. an 8-fold. 
This leads to an abbreviation. 
Example: G = 953 = 31° — 8, 
In § I is given that to 8 the following had to be added: 
63-+65 + 67-69-7173 + 754-774-794 81+ 88485 + ......... + ete.; 
—— ~— — —— ~~ —— ——— 
2 terms 4. terms 6 terms 4 terms 
or 16 X8 435 X 8 + 608, ete. 
The groups of 4 terms give: 35 X 8, 45 & 8, 55 X 8, etc. 
The groups of 6 terms give: 60 x 8, 75 x 8, 90 x 8, etc. 
The entire sum must be a square and so necessarily divisible by 
16 and as the first addition would give 8 + 16 x 8 or 17 *8, the 
numbers 17, 35, 60, 45, 75, 55, 90, ete. must be taken in such 
wise together, that they form even numbers. 
It is evident that to 8 we can first add 2 + 4 terms, then 6 terms, 
then 10, 14, 10, 6, 10, 14, 10, 6, 10, 14 etc. terms at once. This 
gives rise to a considerable decrease in the number of additions ; 
however, the method can be applied only when the (negative) 
remainder of the extraction of the root is an 8-fold. 
XI. The added factor. 
If a number can be written down in more than one manner as 
a product of two factors, the method of § I always gives the factors 
closest to the root, thus those differing the least. 
Boris oe fon la we und. 1321, for) 1155 3 Ko xix 
likewise 33 X 35. 
This observation immediately furnishes us with an abbreviation as to 
the decomposition of one of the factors of a number, when the other 
factor has becn decomposed into its factors. 
Example: In § I is found G = 513667 — 539 x 953, and 
bedr a ay meen 
Now, 953 cannot possibly contain a factor, which multiplied by 
one or more of the factors 7, 7 or 11 of 539, furnishes a number 
closer to YG, namely 716, than 539. For, if that were the case 
