( 502 ) 
where we obtain exactly the sums, found on page 329 by adding 
2 or 3 numbers at the same time. 
So the operation gives a good insight into the reason of the 
addition of the numbers in groups of 2 or 3; it admits however 
still of a simplification if we pay attention to the last two figures of 
the sums. 
The terminal figure 4 or 9 of b? must be preceded hy an even 
figure, so the terminal figure 2 or 7 of the product must likewise 
be preceded by an even figure. If on the place of the tens an odd 
figure appears, we need not continue. 
We directly see that this is the case for 2 X 1436, and 4 X 1438, 
but then also for 12 X 1446, 14 x 1448, 22 « 1456, 24 « 1458, in 
general for (2 + 10) (1436 + 10x) and (4 + 10x) (1438 + 10x). 
So there remains: 
7X 1441 + 422 = 10087 + 422 = 10509 
9 x 1443 12987 13409 
17 Xx 1451 24667 25089 
19 x 1453 27607 28029 
etc. f 
Only the first two multiplications need be executed; after that 
additions suffice. 
For (7 +4 10n) X (1441 4 10n)=7 X 1441 + 14480 n + 100 n2, 
so that we can arrive at 17x 1451 by adding 14480 + 100 or 
14580 to 10087; in the same way we find 27 x 1452 by adding 
to the obtained result 
14480 + 300 or 14780, 
etc.; each following number to be added is 200 more than the 
preceding. 
This holds good for the products with factors 9, 19, 29 etc. and 
the operation becomes 
7X 1441 + 422 = 10509 9 s< 1443 + 422 = 13409 
(7 + 1441) X 104.100 = 14580 | (941443) 10+100= 14620 
25089 28029 
14780 14820 
39869 42849 —=2072, 
