( 333 ) 
In the following table are indicated under each sauare the num- 
bers dominated by it: 
BEB UE 
ipo ores coe. OF. AS IN TOF). PIF TREE 
fa a Oo te dae oO. 49 64-8100 121. 144 
a os whom eaten. 48-65 80 99> 120 143 
Paster aes 40-560). 0s 290 TEA 2140 
Heme) OD! fe OI 112 135 
SAU, de 40 Ou | 84> BOD.) “128 
Piet DO fame. ois LEO 
For 26, 45104 85" 108 
LS 325 os 720505 
Lip vO OTS 
19 40, 2 63 
21 44 
23 
The classification of the numbers in this table is very remarkable: 
1°. In the columns as well as in the rows the successive 
differences are 1, 3, 5, 7, etc. 
2°. Parallel to the odd numbers in the’ hypothenusa of the triangle 
we find the fourfolds 4, 8 12, 16 etc. 
3°. Ascending from one of the odd numbers of the hypothenusa 
in a direction perpendicular to it one finds 3, 5, 7, etc. times this 
number, e.g. starting from 9 one finds 27, 45, 63, 81. 
Now descending from 81 in the direction of the hypothenusa we 
find the continuation 11, 13, 15 etc. times 9 or 99, 117, 135, ete. 
So all the odd multiples of 9 are to be found in two lines passing 
through 81 and inclined under 45°. 
4°. Ascending from one of the fourfolds, e.g. 16, in a direction 
perpendicular to the hypothenusa we find 2 & 16, 3 x 16, 4 « 16 = 8? 
and from this point parallel to the hypothenusa 5 x 16, 6 16, etc. 
The proof of all these properties is easily given. 
From the remark sub 3° ensues that the number 7 x 11 will be 
found in the line passing through 7? parallel to the hypothenusa 
and also in the line passing through 11° perpendicular to it. 
