270 



NATURE 



[January 20, i8c8 



the number in the ist Column, i.e. when you subtract ,('- times 

 this number from the 2nd period with its prefix, it will some- 

 times happen that the subtrahend exceeds the minuend. In 

 this case the subtraction will end with a minus digit, which may 

 be indicated by an asterisk. Now find what number of Divisors 

 must be added to the 2nd Column to cancel this viinus digit, 

 and enter that number, marked with an asterisk, in the ist 

 Column, and that multiple of the Divisor in the 2nd ; and then 

 draw a line below the 2nd Column, and add in this new item. 



As an example, let us take a new Dividend, but retain the 

 previous Divisor, changing the sign of k, so that it will become 

 7003 {i.e. 7. i^^ + 3). The sum, as set for working, would stand 

 thus : — 



7003 II 6504 I 318 I 972 I 526 



Quot. I II 



I I "T 



After working, it would stand thus :- 



Rem. 



7003 



the Mental Process being as follows :— 



Divide the 6504 by 7, and enter the Quotient, 929, in the 

 1st Column, and the Remainder, i, above the 2nd period. 

 Then subtract, from the 1318, 3-times the 929, entering the 

 result in the 2nd Column, thus. "27 from 8 I ca'n't, but 27 

 from 28, I." Enter the i, and carry the borrowed 2. "8 

 from 1 I ca'n't, but 8 from 11, 3." Enter the 3, and carry 

 the borrowed i. "28 from 3 I ca'n't, but 28 from 33, 5." 

 Enter the 5, and carry the borrowed 3. "3 from i, viinus 

 2." Enter it, with an asterisk. Observing that, to cancel 

 this mimis 2, it will suffice to add ouce the Divisor, enter a 

 ( - i) in the ist Column, and 7003 in the 2nd ; and then draw 

 a line below the 2nd Column, and add in this new item : the 

 result is 5534. Then add up the ist Column, and enter the 

 result, 928, in the Quotient. Now take the 5534 as a new ist 

 period, and the 3rd period, 972, as a new 2nd period, and 

 proceed as before. 



The Rules for case (2) may be derived, from the above, by 

 making k=\ ; and those for case (3) by making /^=I. I will 

 give worked examples of these ; but it will not be necessary 

 to give the Mental Processes. 



By making i=i, we get Divisors of the form (^./« + i) : 

 let us take (11/* - i) and (6/* + i). 



109999 II 107523 I 8168 



10 4 



9662 I 0985 



Quot. 9774 I 9813 I " 0861 I 41846 Rem. 



9774 I 107942 



6000c I I 

 Quot. 



7239 



3 

 51798 



I 19474 

 I 



_947S 



861 



2 6004 



3 

 13825 



In this last example, there is no need to enter the Quotient, 

 produced by dividing the 7239 by 7, in the ist Column : we 

 easily foresee that the number at the top of the 2nd Column wii/ 

 be less than the Divisor, so that there will be no new item in the 

 1st: hence we at once enter the 1206 in the Quotient. 



NO. 1473. VOL. 57] 



By making /; = i, we get Divisors of the form {t" ± A-) : let 

 us take {t* - 7) and (^+12). 



9993 II 867 I 8159 I 2485 I 7031 I 



Quot. 868 I 4238 I 2153 I 2104 



^76 

 0004 



4092 

 4120 Rem. 



867 I 14228 

 I I 7 



4235 

 3 



100012 II 7185 I 6 2039 I 10327 I 531 18 



The first of these two sums is the one I gave to illustrate Mr. 

 Collingwood's Method of working with Divisors of the form 



It may interest the Reader to see the 3 Methods of working 

 the above example — ordinary Division, Mr. Collingwood's 

 Method, and my version of it — compared as to the amount of 

 labour which each entails in the working : — 



I am assuming that any one, working this example by ordinary 

 Division, would begin by making a Table of Multiples of 9993, 

 for reference : so that he would have no Multiplications to do. 

 Still, the great number of digits he would have to write, and of 

 Additions and Subtractions he would have to do, involving a far 

 greater risk of error than either of the other Methods, would 

 quite outweigh this advantage. 



By whatever process a Question in Long Division has been 

 worked, it is very desirable to be able to test, easily and quickly, 

 the correctness of the Answer. The ordinary test is to multiply 

 together the Divisor and Quotient, add the Remainder, and 

 observe whether these together make up the given Number, as 

 they ought to do. 



Thus, if JV be the given number, Z> the given Divisor, Q the 

 Quotient, and Ji the Remainder, we ought to have 



JV=:D.Q + /i. 



This test is specially easy to apply, when D = {k.t" + /:) ; 

 for then we ought to have 



N=z{Lf'±/^). Q + /?; 

 = (kQ.t" + Ji)±kQ. 



Now JiQ.t" may be found by multiplying Q by h, and tacking 

 on n ciphers. Hence {hQ.t" + R) may be found by making Ji 

 occupy the place of the n ciphers. If A' contains less than n 

 digits it must have ciphers prefixed ; if more, the overplus must 

 be carried on into the next period, and added to hQ. 



Having found our " Test," viz. (hQ.t" + R), we can write it 

 on a separate slip of paper, and place it below the working of 

 the example, so as to come vertically below N, which is at the 

 top. When the sign in D is ' - ,' we must add kQ to N, and 

 see if the result = T; when it is ' + ,' we must add kQ to T, 

 and see if the result = N. 



Now it has been already pointed out that when, in the new 

 Method, the ist and 2nd Columns have been worked, the 1st 

 period of the Quotient and the number at the foot of the 2nd 

 Column are the Quotient and Remainder that would result if the 

 Dividend ended with its second period. Hence the Test can be 

 at once applied, before dealing with the 3rd Column. This 

 constitutes a very important new feature in my version of Mr. 

 Collingwood's Method. Every two adjacent columns contain a 

 separate Division-sum, which can be tested by itself. Hence, in 

 working my Method, as soon as I have entered the ist period 

 of the Quotient, I can test it, and, if I have made any mistake, 

 I can correct it. But the hapless computator, who has spent, 

 say, an hour, in working some gigantic sum in Long Division — 



