eer ee 
(507) 
a (4v—1) 69, so a? (=0? + G) a (4 v) 00, so that a? can terminate 
only in .400 or .600, and by this J? only in .769 or .969. So wê 
could have added the numbers 2316 and 2324 at once, the numbers 
2356 and 2364 likewise, etc. Of every 4 additions one is dropped. 
After adding 44 numbers, so after about 36 additions, we find 
GF 10915969: = 3313" 
so that 
G = 6660? — 3313? =.9973 x 3347. 
The second and third columns were not continued for a reason to 
be mentioned in the following §. 
According to the common method of § I 6669 — 5778 or 882 
numbers would have to be added, which can be taken in groups, 
but which would require a considerably larger number than 36 
additions. 
Also the method of § VII would require a greater number of 
operations. 
A table showing the 6 or 8 last figures which may appear in 
a square, would undoubtedly lead to further abbreviations. 
§ XIV. Property of a and b?. 
2 u 2 
If for shortness’ sake we call (5) and = c and d?, 
then G = c?—d*, If moreover G = a?—b*, we have a®—b? = c?—d?, 
Now b? and d? can never have the two terminal figures alike; 
neither can a? and c?. 
To show this we consider the table, giving the four last figures, 
which can appear in a square; immediately the following theorem 
strikes us: dn a column under an even number and in a column 
under an odd number, there is always one of the two XX, but 
never more than one at the same height. 
Or: In the columns under an even number the two XX are 
placed immediately under each other, in the other columns there is 
every time a space of a line. We must moreover fancy that under 
the fourth line of the table the two first lines have been repeated. 
The consequence is evident from an example: 
On pag. 504 a number a° of the form. G ) 04 is given, from 
v 
4y—1 
which 7803, a (4v + 2) 03 is subtracted. 
