This process of casting out the nines does not detect any errors in which the remainder to 9 happens to be correct. If a process be tedious, and some additional check be desirable, the method of casting out _elevens_ may be followed after that of casting out the nines. Observe that 10 + 1, 100-1, 1000 + 1, 10000-1, &c. are all divisible by eleven. From this the following rule for the remainder of division by 11 may be deduced, and readily used by those who know the algebraical process of subtraction. For those who have not got so far, it may be doubted whether the rule can be made easier than the actual division by 11. Subtract the first figure from the second, the result from the third, the result from the fourth, and so on. The final result, or the rest of 11 if the figure be negative, is the remainder required. Thus, to divide 1642915 by 11, and find the remainder, we have 1 from 6, 5; 5 from 4,-1;-1 from 2, 3; 3 from 9, 6; 6 from 1,-5;-5 from 5, 10; and 10 is the remainder. But 164 gives-1, and 10 is the remainder; 164291 gives-5, and 6 is the remainder. With very little practice these remainders may be read as rapidly as the number itself. Thus, for 127619833424 need only be repeated, 1, 6, 0, 1, 8, 0, 3, 0, 4,-2, 6, and 6 is the remainder. When a question has been tried both by nines and elevens, there can be no error unless it be one which makes the result wrong by a number of times 99 exactly. APPENDIX III. ON SCALES OF NOTATION. We are so well accustomed to 10, 100, &c., as standing for ten, ten tens, &c., that we are not apt to remember that there is no reason why 10 might not stand for five, 100 for five fives, &c., or for twelve, twelve twelves, &c. Because we invent different columns of numbers, and let units in the different columns stand for collections of the units in the preceding columns, we are not therefore bound to allow of no collections except in tens. If 10 stood for 2, that is, if every column had its unit double of the unit in the column on the right, what we now represent by 1, 2, 3, 4, 5, 6, &c., would be represented by 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, &c. This is the _binary_ scale. If we take the _ternary_ scale, in which 10 stands for 3, we have 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, &c. In the _quinary_ scale, in which 10 is five, 234 stands for 2 twenty-fives, 3 fives, and 4, or sixty-nine. If we take the _duodenary_ scale, in which 10 is twelve, we must invent new symbols for ten and eleven, because 10 and 11 now stand for twelve and thirteen; use the letters _t_ and _e_. Then 176 means 1 twelve-twelves, 7 twelves, and 6, or two hundred and thirty-four; and 1_te_ means two hundred and seventy-five.