117 



COMPUTATION. 



COMPUTATION. 



113 



7296 54', 86', 26', 6'5'; 30', 48', 14', 3'6'; 48', 76,' 23', 5'8 ; 

 859 4'; 6'; 8, 16, 22'; 8, 12, 17'; 4, 10, 16'; 9, 12'; '6. 



65664 

 36480 

 58368 



6267264 



The usual process of caitiny out the nines, as it is called, though no 

 an absolute verification, is a useful one. Without entering upon its 

 demonstration, we describe it. 



Casting out the nines, means adding together the digits of a number 

 and throwing out 9 as fast as it arises. Thus in 26647895 we have 2 

 and 6 are 8 and 6 are 14, throw out 9 and 5 remains ; take in the next 

 figure 4, giving 9, throw out 9 and remains ; then 7 and 8 are 15 

 throw out 9 and 6 remains ; pass over the 9 and take in 5 which gives 

 11, on which two remains. All that is necessary to be repeated is 2, 

 8, 14, 5, 9, 7, 15, 6, 11, 2. Cast the nines out of both factors, multiply 

 the results, and cast out nines ; the answer should give the same figure 

 as the reputed product gives when the nines are cast out. Thus 7296 

 and 859 give 6 and 4, the product of which is 24, giving 6 : and the 

 reputed product 6267264 also gives 6. This is a high presumption 

 that the result is correct : the error, if any, lies in this, that the figures 

 of 6267264 which are too great are exactly compensated by others 

 which are too small. Now it is very unlikely that this exact com- 

 pensation of errors should exist ; and in this unlikelihood consists the 

 strong presumption of verification which the rule of casting out the 

 nines affords. 



It may be observed that this rule applies to all operations, thus : 

 Do with the result of casting out the nines the same as was done with 

 the numbers from which the nines were cast out ; the answer, with its 

 nines cast out, should give the same result as arises from casting out 

 the ninos of the reputed answer. Thus in the previous example of 

 addition, the nines cast out from the several numbers to be added give 

 1, 6, 2, 2, 0, 5, the sum of which, with nines cast out, is 7, the same as 

 from the reputed answer 95515. In subtraction, it may be necessary 

 to take in a nine before subtracting. 



IHrifion. This rule is considerably shortened, and (in our opinion, 

 though we think many would differ from us) increased in safety, by 

 performing each subtraction without setting down the multiplication. 

 As a preliminary, suppose it required to subtract 7 times 29398 from 

 410843. Beginning with 7x8 instead of putting it down as 56, look 

 at 3 and pass from 56 to 63, writing down the 7 and carrying the 6 

 tens just used. Take these 6 tens and 7 times 9, giving 69, and make 

 them up to 74, putting down the 5, and carrying 7. All that need ba 

 repeated is as follows, dwelling on the tens carried : 



From 410843 56 and 7' make 63 ; 69 and 6' make 74 ; 



Take 29398 x 7 



205057 



28 and 0' make 28 ; 65 and 5' make 

 21 and 0' make 21 ; 2 and 2' make 4. 



70; 



Suppose 



usual plan, write the divisor on the left, or the quotient on the right : 

 t any rate, the divisor and the quotient should be near one another. 



Divisor 47698)3293614829 

 Quotient 69051 431734 



245282 

 87929 

 20231 remainder. 



The work is as follows : The first quotient figure being 6, subtract 

 6 times 47698 from 329361, thus : 48 and 3' are 51, 59 and 7' are 66, 

 42 and 1' are 43, 46 and 3' are 49, 28 and 4' are 32. The 4 being 

 brought down, and 9 ascertained to be the next quotient figure, we 

 have 72 and 2' are 74, 88 and 5' are 93, 63 and 4' are 67, 69 and 2' are 

 71, 43 and (unnecessary) are 43. Then 8 and 2 are brought down, 

 and 5 written as quotient figures, and 40 and 2' are 42, 49 and 9' are 

 56, 35 and 7' are 42, 39 and 6' are 45, 24 and are 24. The last 



?uotient figure is 1, and 8 and 1 are 9, 9 and 3' are 12 ; 7 and 2' are 9, 

 and 0' are 7, 4 and 2' are 6. 



Casting out the nines, the products obtained from the divisor and 

 quotient, increased by the result of the remainder, ought, after casting 

 out nines, to agree with the dividend. Here the divisor gives 7, the 

 quotient 3, and the remainder 8 : 7 times 3 and 8 is 29, giving 2 ; the 

 dividend ;il.-o gives 2. 



We have put down every step of the work in all the rules. The 

 very basis of this method is the acquisition of such expertness in the 

 previous exercises as will enable the computer to dispense with a large 

 quantity of the ordinary halting-places. We need not enter upon the 

 rules for the extraction of the square and cube root [SQUARE], or of 

 the solution of equations [INVOLUTION AND EVOLUTION], at length ; 

 but we shall annex an example of each, leaving the reader to apply the 

 seventh exercise in its proper place. In the contracted part of Horner'a 

 method, we add to the method of the last article cited, a provision for 

 making the first figure on the right of the vertical h'ne of contraction 

 as correct as the process will allow it to be. Tl.is last process (Homer's 

 metLod) is the bmt exercise of computation. It involves all rules in a 

 form ofwonriderable detail and dispersion. 



First, we put down the process for the extraction of the square root 

 of 157. 



157)12-52996408614166778849537 

 22 57 

 245 1300 

 2502 7500 

 25049 249600 

 250589 2415900 

 2505986 16059900 

 25059924 102398400 

 2505992808 21587040000 

 25059928166 153909758600 

 250599281721 355018460400 

 2505992817224 10441917867900 

 2505992817228 417946599004 



167347317281 



16987748248 



1951791845 



197596378 



22176876 



2128934 



124140 



23900 



1347 



94 



19 



Now let it be required to find the positive root of X s + X> 

 12 = 0. 



1 

 4 



7 



100 

 104 

 108 

 1120 

 1126 

 1132 

 11380 

 11384 

 11388 

 113920 

 113921 

 113922 

 11392300 

 11392301 

 11392302 

 11392303 



-12 

 



2100 

 2516 

 294800 

 301556 

 30834800 

 30880336 

 3092588800 

 3092702721 

 30928166430000 

 30928177822301 

 30928189214603 

 30928196049985 

 30928202885367 

 3092820299929 

 3092820311321 

 S092S2-;:31702 

 309282032272 



12(3-4641016151377545871 

 12000 

 1936000 

 126664000 

 3142656000 



49953279000000 



19025101177699 



468183547708 



158901517715 



4260501864 



1167681541 



239835444 



22338022 



1688280 



141870 



18157 



2693 



219 



3 







In both these examples the result is carried to about three times as 

 many figures as are usually wanted. But this is what should be dona 

 in computing for exercise. No one does his very highest with ease or 

 with certainty ; and a person who can safely and rapidly knock off the 

 seven or eight figures which are generally requisite must be one to 

 whom a much larger amount of correctness is, or has been, familiar. 



It is necessary to insist particularly upon acquiring the habit of rapid 

 computation by attempts at rapidity. Taking any exercises which 

 verify one another, as a multiplication and a division, or a raising of a 

 square and extraction of the square root, the learner who has acquired 

 a little familiarity with the rules should try them at the top of his 

 speed. If the verification be attained ; if the division, for instance, 

 reproduce the multiplier for a quotient, without remainder, it is 

 ihousands to one that the whole process is correct. But if there 

 should be a failure, as may, and almost certainly will at first, happen 

 several times running, there is no occasion to examine closely into the 

 reason of the failure. The whole question should be thrown aside, 

 and another should be taken. In this course it will soon appear that 

 .he attempts become more and more nearly correct, until at last failure 

 s the exception, and not the rule. There must be no fear of error, 

 whether there be cause for it or not. The young calculator should 

 >roceed as boldly as if he were infallible ; for he may depend upon it, 

 ,hat it is not inaccuracy which is to be avoided by precaution, but 

 accuracy which is to be obtained by habit. 



With regard to minor points, every one must decide for himself 

 whether actual vocal repetition of the words and numerals necessary 

 o the process does or docs not tend to accurate working. Some 

 icrsons cannot compute without repeating to themselves; but it is 

 worth while to give silence a fair trial. It would also be of assistance 

 o some persons to invent a habit of signifying the multiplier, which 

 s in actual use, by some arbitrary position of the left hand. Thus, 1, 

 *, 3, and 4 might be signified by placing the corresponding number of 

 ngers of ihe left hand on the paper, 5 by placing the thumb only, 

 nd 6, 7, 8, and 9 by the thumb, with a corresponding number of 

 hngers. 



In all those parts of computation which relate peculiarly to fractions, 



