(334) 
So the factors of a number prove to be the roots of the squares 
that can be reached by proceeding from this number in the two 
directions inclined under 45°. 
The prime numbers present themselves only once (in the hypo- 
thenusa), the composed odd numbers present themselves still one = 
or several times more. 
As it is not very convenient to make out if a given number is 
contained in the table, if this table is continued much farther than 
here, the classification of Table IL recommends itself more. Here 
equal numbers are placed in the same row, the head of each column 
bearing the dominating square. The prime numbers appear only in 
the inclined line at the right side; their rows are denoted by 
horizontal lines. 
Beneath any square a? are arranged the numbers a? — 1, a? — 4, 
a? — 9, etc. with the differences 1,3,5,7, ete. The last number 
of each column is always zero, the last but one a? — (a—1)? = 2a — 1, 
and this number appears always in the inclined line at the right side. 
If we assume any number of this line, e.g. 19, then we find above 
it: 194-1796, 19417-+15==51, 194171541864, 6441175, 
751-984, 84+-7=—91, 91415—96, 964399, 991+1—100. 
Now 19 can be called the base of the numbers 36, 51, 64 etc. 
Above the base 2a—1 we then find: 
(2a—1) +(2a—3)=4(a—1), (2a—1)-++-(2a—3) + (2a—5) = 8(2a—3), 
3(2a—3)-+(2a—7) =8(a—2), 8(a—2)-+(2a—9) =5(2a—5), 
5(2a—5)-+-(2a—11)=12(a—3), 12(a—3)4+(2a—13)=7(2a—7), ete. 
Therefore the 3-folds, 5-folds, ete. of the odd numbers are situated 
on oblique lines passing through the numbers 3, 5, 7, etc., whilst 
the 4-folds, 8-folds, ete. are situated on intermediate oblique lines 
commencing at 4, 8, 12, etc. 
Twofolds of prime numbers do not present themselves. 
For immediate application this table has the inconvenience, that it 
cannot be continued far enough without becoming unmanageable. 
However it leads to an important abbreviation of the method given 
in § I by means of the simple remark that between the oblique lines 
no numbers can present themselves. For illustration a small number 
is chosen; the application to a large number will be evident. 
Example V = 83 = 10? — 17. 
To 17 we must successively add 2 X 10 4+ 1= 21, 23, 25, etc. 
