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COMPUTATION. 



COMPUTATION. 



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cloth, imitate bahiou*, but be can follow direction*, and he U the nun 

 who actually makes the coat Nor do we mean that the person who 

 0*0 merely perform the four nils ia a complete arithmetician : ha 

 can nut know how to apply thoae rule*, he ha* not learned how to 

 think of their use*. But, under direction*, he can really manage the 

 laat and executive proem* of any mathematical inquiry. Until be 

 i* *""H*Mng batter, he U but a tool in the hand* of other*, but 

 no one think* the lea* of tool* beoauae they cannot work of them- 



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I* thi* then, we ahall be asked, all that you propose, namely, to 

 deecribe addition, subtraction, multiplication, and diu- . v. , 

 answer. ye, and on theae aimple ground*, that nothing more ia wanted, 

 and that in what we have to aay upon them, we ahall Unich the reaaou 

 why *o many person* are incapable of and disgusted with arithmetioal 

 ni HUM There are but few who find much difficulty in comprehend- 

 ing what i* to be done in a question of arithmetic : but there are 

 many who find that, however clearly they may nee the way, there i* a 

 heavy and broken ascent between them and the aniwer. The figures 

 will not come right, there ia nothing but mistake after mistake : and 

 though everything elae i* gained, the result ia not Fow of thoae who 

 are thiu perplexed are aware that nothing U wanting but attention to 

 the elementary processes with a better method than a beginner invent* 

 for himself : every one know* that book* and teacher* give no method 

 at all 



The following system of exercises constitute* the whole of what in 

 necessary. The degree of attention with which each U to be practised 

 must depend upon the circumstances of the learner's case. Koch one 

 contains a difficulty which most persons will suppose they could avoid, 

 some in one manner, some in another. Their several suppositions are 

 true ; and it is just as true that the trouble and risk with which an 

 infant learns to walk upon its feet could be avoided ; some could con- 

 tinue to crawl, others could contrive to walk on their knees ; all would 

 get on in a certain manner. Any person who i* determined to succeed, 

 and who ha* reason to know that his method has not answered a* yet, 

 should try our plan with the faith of a learner. But we can promise 

 him no success unless he will make up his mind, from the outset, 

 entirely and at once to abandon every habit which we condemn, and to 

 adopt t very mode which we prescribe : and this, though it should seem 

 to him that what he has to take up is more difficult than what he is to 

 leave oft He may take any time, and must not be discouraged by 

 finding that the new operations are at first longer than the old ones. 

 He must continually attempt more and more rapidity, remembering 

 that quickness of operation will never come of itself by practising with 

 deliberate caution. Slow and sure is better than quick and wrong, no 

 doubt, if one or the other must be the end of it; but ijvirk and sure 

 must be the motto of an arithmetician. All very correct computers 

 that we have seen have been rather rapid workers : we believe the 

 reason to bo that those who cannot acquire rapidity give it up in 

 disgust. 



1. Presuming that the learner can count one, two, three, IK., a* fast 

 a* he van speak the words, he must then try if he can do the same 

 backward*, as fifty, forty-nine, forty-eight, forty-seven, tc. He must 

 then practise counting forward by two at a time, as in 2, 4, 6, tc., 

 1,8, 5, Ac. by three at a time, as in 1, 4, 7, Ac., 2, 5, 8, Ac., 8, 6, 9, 

 Ac. by four at a time, as in 1, 5, 9, Ac., 2, 6, 10, Ac. and so on up to 

 11 or 12 at a time. The same should be done backwards, as in the 

 following by sevens, 90, 88, 76, 69, Ac. In doing this he must not use 

 any description, either vocal or mental : it must not be 22 and 6 are 

 28, 28 and 6 are 34, Ac. ; but 22, 28, 34, Ac. These various processes 

 should be carried to or from 100 at least. 



2. Proceed to form with rapidity the number which must be added 

 to a given number to make up the next number which ends with a 

 given digit Thus, one of the questions asked, at its fullest length, is 

 " Given 38 and 4, how much must be added to 38 to give the next 

 number that ends with 4. " But all that must be repeated, orally or 

 mentally, U "88 and six are forty-four." Write down a row of 

 numbers, as in 



729632841 



and practise- thus : 72 and 7 are 79, 29 and 7 are 36, 96 and 7 are 103, 

 63 and 9 are 72, Ac. ; taking 72, 29, 96, Ac., for the successive lesser 

 numbers, and 9, 6, 3, Ac., for the successive unit figures of the greater 

 numbers. 



8. Endeavour occasionally, in the exercise Immediately preceding, 

 and in those which follow, to fix the thought* particularly upon the 

 Ifni of the result. It will generally happen that the units are imme- 

 diately to be written down and discarded, while the tens are to be 

 retained in memory. Practise repeating a number, so a* while repeat- 

 ing it to write down the unit* and think of the tens : thus, in 76, at 

 the moment of writing down 6, think of 7. 



4. Learn the multiplication table up to 12 tunes 12, but not with 

 the usual practice of wording all the result*, as in " 7 times 8 make 

 66," " 4 times 6 are 20," Ac. The table nrust be so learnt that the 

 two factors suggest the product instantaneously. Thus, 8 and 9 must 

 give 72 the instant they come together in the mind ; and so on. Take 

 a row of numbers a* before, and looking at the successive pain, repeat 

 the product*. Thu*, 2 9 8 7 4 8 5, Ac., is to suggest 18, 72, 56, 28, 12, 

 IS. Ac., - fact n the word* can be Rpoken. Those who have been 



accustomed to learn the multiplication table from the half of it, alwayt 

 putting either the greater factor firnt, or else the leaser, mu*t now 

 accustom themselves to the whole : 5 and 9, or 9 and 6, mutt niggesl 

 45 with equal ease. 



5. Augment the last exercise a* follow* : Having three digit* learn 

 to pus in thought immediately to the product of the finrt two aug- 

 mented by the third; thus 7, 9,and 8 must lead to 7 times 9 increased 

 by 8, or 71. Take a row of figure*, a* before, aay, 2497163. Ac . 

 must be made the mean* of suggesting immediately 17, 43. 64, 13, 9, 

 Ac. The usual repetition, " twice 4 are 8 ad 9 are 17," Ac., mu*t not 

 be tolerated for one moment 



6. Acquire the power of combining the fifth and second exercise* a* 

 follows : Having four digits, learn to add the third to the product of 

 the first and second, and to pass on to the next number which hai the 

 fourth in its unit'* place. Thus, with 7, 8, 5, and 0, think of 61(7 

 time* 8 and 5), a* in the last exercise, and as in the second, get " 61 

 and 9 are 70." Repeat only a* much a* i* in the hut phruae. dealing 

 with the lirst three numbers by the habit acquired in the last exercise. 

 Thus, with the row of numbers 19728663, Ac., should be rapidly sug- 

 gested 16 and 6 are 22, 66 and 3 are 68, 22 and 4 are 26, 54 and 9 are 

 63, Ao. 



7. Having four numbers, deal with the first three a* in the fifth 

 exercise, and then repeating the result, add the fourth. Thus, from 

 2, 7, 5, 8, get 19 and 8 are 27. Thus, the row of figure* 7984361 , Ac., 

 must give 71 and 4 ore 75, 76 and 3 are 79, 35 and 6 are 41, 18 and 

 1 are I 1 . 1 . 



8. Having a digit and a number of two place*, learn to arrive 

 speedily, and with few word* at the number of times which the 

 second contains the first (when not more than 9), and at the remainder. 

 Thus, " 7 in 53, 7 times and 4," " 8 in 29, 3 time* and 5," Ac. 



A person who really desires to become a good computer matt arrive 

 at the power of performing these exercises easily, quickly, and accu- 

 rately. It is possible to dispense with them, and it is possible to dis- 

 pense with rules and numerals altogether and to use pebbles. But we 

 ore very confident that when these exercises are once made very rapid 

 and safe, the computer has gone through nine parts out of ten of his 

 training : he can walk, and will soon learn to find his way. The 

 common error lies in imagining that learning to find the way is Kirn- 

 ing to walk. 



We now take the four rules, insisting on the details of the mode 

 of performing each of them, and presuming the usual process to be 

 known. 



Numeration. Learn to distinguish tens, hundreds, Ac., not by the 

 places in which their digits come, but by the numbers of place* which 

 oouie after those places. Instead of connecting thousands in the mind 

 with the fourth, place, connect them with three fluctt cut off. Thus 

 176493 has 1764 hundreds, 176 thousands, Ao. 



A ddition. Add as in exercise 1 ; thus, if the figures were to be 7 2 

 3 4, Ac., it should not be 7 and 2 are 9 and 3 are 12 and 4 are 16, Ac., 

 but 7, 9, 12, 16, Ac. Dwell upon the tens to be carried, in writing 

 down the units. In the following question, every word that should be 

 repeated, orally or mentally, is written down, and each figure that is 

 to be written down is marked with an accent ; each ten that is to be 

 dwelt on for a moment is in italics : 



8, 17, 22, 80, 86, forty-five' 

 10, 19, 20, 27, dirty-one' 

 11, 18,24,27,29, M/rty-five' 



8, 6, 9, 18, <wi<y-five' 



3, 7, nine' 



9276 

 MM 



44615 

 MM 



95515 I 



Let the method of verification be simply taking the columns in the 

 opposite order, or downwards, a* 9, 15, 23, 28, 37,/orty-five', Ac. 



Xtilitrai-lioM. Let the several subtractions be made by mental re- 

 covery of additions as in exercise 2 : not " 7 from 16 leaves 9," but " 7 

 and 9," not repeating 16, which is unnecessary, inasmuch as win n :> U 

 found, 16 is done with, and the carriage of one is all that is to be 

 remembered, The following detail has every word of the process : 



274681807 8 and* 9' 

 95318948 3 and 3' 



5 and 5' ; 10 and 8' ; 9 and 2' ; 2 and 1' 

 5 and 9'; 10 and 7' ; 1 ami 1'. 



179312859 



According to even recent books the computer should my, " 8 from 

 7 I cannot, but 8 from 17 and there remains 9, set down y and carry 

 one," Ac. It would be shorter than this to say, " 8 from the next 

 higher number ili.it ends with 7 leaves 9." Our plan is the abbrevia- 

 tion of " 8 and 9 make up the next number that ends with 7 " repeating 

 only the first three words. 



tfi(!ii/>liraiinn. Apply the exercises 4 and 5 to the usual process. 

 Here, as before, the detail given has every w<ml which need be 

 repeated. The figure* to be written down are accented ; all others are 

 cirri ed. 



