On the right is all the process of forming the second line, which completes the multiplication by 76, as the third line completes that by 876, and the fourth line that by 9876.

Division. Make each multiplication and the following subtraction in one step, by help of the process in the 9th exercise, as follows:

The number of words by which 26577 is obtained from 165402 (the multiplier being 5) is as follows: 15 and 7′ are 2″2; 47 and 7′ are 5″4; 35 and 5′ are 4″0; 39 and 6′ are 4″5; 14 and 2′ are 16.

The processes for extracting the square root, and for the solution of equations ([Appendix XI].), should be abbreviated in the same manner as the division.[57]

APPENDIX II.
ON VERIFICATION BY CASTING OUT
NINES AND ELEVENS.

The process of casting out the nines, as it is called, is one which the young computer should learn and practise, as a check upon his computations. It is not a complete check, since if one figure were made too small, and another as much too great, it would not detect this double error; but as it is very unlikely that such a double error should take place, the check furnishes a strong presumption of accuracy.

The proposition upon which this method depends is the following: If a, b, c, d be four numbers, such that

a = bc + d,

and if m be any other number whatsoever, and if a, b, c, d, severally divided by m, give the remainders p, q, r, s, then