304 H. F. TALBOT ON A GENERAL SOLUTION OF 
Here reduct of sum is 4, and sum of reducts is 6+ 7 =13=4 (if the sign = 
denote equivalence). 
Add together 8746795 . (reduct 1) 
and 4893781 . (reduct 4) 
13640576 . (reduct 5) . 
Add together or, reduct 1 
911 . reduct 2 
7(2. reduct 7 
_ 487. reduct 1 
Sum 2549 . reduct 2. 
Here the reduct of the sum is 2, and the sum of the reducts is 1 +2+7+1=11, 
whose reduct is 2. This theorem is of great utility, for suppose a very long 
column of figures has to be added up, and the calculator does not feel certain 
that he has done it correctly, then take the reduct of each number and add them 
together, neglecting the nines. If the result is not equal to the reduct of the 
sum, the calculator is thereby warned that he has made a mistake, and must 
recommence the operation. But this is only a negative test; for if the results 
agree, that is no proof that the calculation is right, but only that there has been 
a kind of compensation of errors. 
Examples of Theorem I1.—Multiply 89 by 17, the product is 1513, whose 
reduct is 1. But the reduct of 89 is 8, and that of 17 is 8, therefore multiply 
8 by 8, and the product 64 has for its reduct 1, which is correct. But instead 
of multiplying 8 by 8, it is simpler to multiply —1 by —1, which gives 1 for 
the reduct. And, in general, we may always substitute for a number its 
complement to 9. 
Multiply together 87 . 11. 13, the product is 12441, whose reduct is 3. But 
the reduct of 87 is 6, that of 11 is 2, and that of 13 is 4. Multiply therefore 
6.2.4 and we get 48, whose reduct is 3, which is correct. 
Theorem IT. is as useful as Theorem I., for if the calculator has many long 
numbers, which are all to be multiplied together, he can try if the veduct of his 
product equals the product of all the partial reducts, and if not, he has com- 
mitted an error of calculation. 
But all the factors may be equal, and this leads us to the consideration of 
the powers of numbers. 
Example.—The cube of 23 is 12167, whose reduct is 8, and since that of 23 
is 5, that of 23° will be 5° or 125=8. 
The cube of 43 is 79507 = 1, and the reduct of 43 being 7, that of 43° is 7° 
or 343 = 1. 
The construction of a few tables is very useful, of which the following may 
be taken as a specimen :— 
és 
