MULTIPLICATION. The following, put into words, is all that need be repeated in the multiplying part; the addition is then done as usual. The unaccented figures are carried. 670383 9876 ------- 4022298 18′, 49′, 22′, 2′, 42′, 4′0′, 4692681 21′, 58′, 26′, 2′, 49′, 4′6′, 5363064 24′, 66′, 30′, 3′, 56′, 5′3′, 6033447 27′, 74′, 34′, 3′, 63′, 6′0′. ---------- 6620702508 Verify each line of the multiplication and the final result by casting out the nines. (_Appendix_ II. p. 166.) It would be almost as easy, for a person who has well practised the 8th exercise, to add each line to the one before in the process, thus: 670383 9876 ------- 4022298 50949108 587255508 6620702508 8; 21 and 9 are 30′; 59 and 2 are 61′; 27 and 2 are 29; 2 and 2 are 4′; 49 and 0 are 49′; 46 and 4 are 5′0′. 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: 27693)441972809662(15959730 165042 265778 165410 269459 202226 83756 6772 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] [57] The teacher will find further remarks on this subject in the _Companion to the Almanac_ for 1844, and in the _Supplement to the Penny Cyclopædia_, article _Computation_. 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 _p_ and _qr_ + _s_ give the same remainder when divided by _m_ (and perhaps are themselves equal). For instance, 334 = 17 × 19 + 11; divide these four numbers by 7, the remainders are 5, 3, 5, and 4. And 5 and 5 × 3 + 4, or 5 and 19, both leave the remainder 5 when divided by 7.