953 = sles — 8 
2x 31-+1= 63 
65 
a? — 332 136 
Aere 0 = coe 
ot 416 
Gr xX 77180 = 480 
43? 896 
bey OO = : 360 
47? 1256 
GX 100 = 600 
532 1856 
etc. 
The importance of this last abbreviation is self-evident. 
The numbers obtained by addition must be looked for in a table 
of squares. For shortness’ sake one can make use of the following 
table (see next page) representing all the possible groups of four 
figures, in which a square can end). 
For if this table shows that the four terminal figures of a number 
cannot occur in a square, it is unnecessary to use the table of squares. 
II. Classification of the row of natural numbers 
according to their divisibility. 
By diminishing any square, e.g. 13° by the squares 1%, 27, 32, 
ete. we obtain the composed numbers 168, 165, 160, 153, etc. or 
if we pay attention to the odd numbers only: 165, 153, 133, 105 
69, 25. 
It is immediately evident that in the factorisation of any of 
these numbers 13° may, but not that 13° necessarily must present 
itself as a’. 
For shortness’ sake we will say that 13° dominates-these numbers. 
So a number admitting of more than two factors is dominated by 
more than one square, e.g. 273 by 1377, 412, 237, 17%. 
In following the method developed in § I one always finds the 
least dominating square. 
) 
1) In his “Théorie des Nombres” Lucas states that Prester has published 
a table for the same purpose in his “Nouveaux Eléments de Mathématiques, 1689. It 
has been impossible tor us to make out if this table was constructed in an analogous 
manner. 
