January 20, 1S98] 



NATURE 



269 



LETTERS TO THE EDITOR 



[ The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can lie undertake 

 to return, or to correspond with the writers of rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications. "X 



Abridged Long Division. 



A Brief Method of dividing a given Number by a Divisor of 

 the Form {h. ID" ± k), where at least one of the two 

 numbers, h and k, is greater than i . 



My former paper on this subject, which appeared in Nature 

 for October 14, 1897, dealt only with the case where h = 1 and 

 /• — I. It elicited, from other correspondents of Nature, 

 several interesting letters, which the editor kindly allowed me 

 to see. One, from Mr. Alfred Sang, quotes Mons. L. Richard's 

 " Stenarithmie," as containing my Rule for dividing by 11. 

 Mons. Richard's book, which I had not previously met with, does 

 certainly contain the rule, but the author has failed to see that 

 the test, which this Method furnishes, for the correctness of the 

 working, is absolutely definite. He says " La derniere 

 difference, ou cette difference augmontee de i, egalera le chiffre 

 de gauche du nombre propose." So ambiguous a test as this 

 would of course be useless. But the "difference" he is speak- 

 ing of is really the last but one : the very last will always (as I 

 stated in my former paper) be equal to zero. Another corre- 

 spondent, >Ir. Otto Sonne, says that my Rules, both for 9 and 

 for II, are to be found in a school-book, by a Mr. Adolph Steen, 

 which was published at Copenhagen in 1847. So I fear I must 

 reduce my claim, from that of being the first to discover them, to 

 that of being the first to publish them in English. 



The Method, now to be described, is applicable to three 

 distinct cases : — 



(i) where h > i, k = i ; 



(2) where -^ = i, ,^ > i ; 



(3) where // > i, k > i. 



With certain limitations of the values of k, k, and n, this 

 Method will be found to be a shorter and safer process than that 

 of ordinary Long Division. These limitations are that neither 

 // nor k should exceed 12, and that, when k > i.n should not be 

 less than 3 : outside these limits, it involves difficulties which 

 make the ordinary process preferable. 



In this Method, two distinct processes are required — one, for 

 dealing with cases where h > i, the other, for cases where k> 1. 

 The former of these processes was, I believe, first discovered by 

 myself, the latter by my nephew, Mr. Bertram J. Collingwood, 

 who communicated to me his Method of dealing with Divisors of 

 the form (10" - k). 



In what follows, I shall represent 10 by t. 



Mr. Collingwood's Method, for Divisors of the form (/" - k), 

 may be enunciated as follows : — 



"To divide a given Number by (/" - k), mark off from it a 

 period of n digits, at the units-end, and under it write /(-times 

 what would be left of it if its last period were erased. If this 

 number contains more than ;/ digits, treat it in the same way ; 

 and so on, till a number is reached which does not contain more 

 than n digits. Then add up. If the last period of the result, 

 plus /C'-times whatever was carried out of it, in the adding-up, be 

 less than the Divisor, it is the required Remainder ; and the rest 

 of the result is the required Quotient. If it be not less, find 

 what number of times it contains the Divisor, and add that 

 number to the Quotient, and subtract that multiple of the 

 Divisor from the Remainder." 



For example, to divide 86781592485703152764092 by 9993 

 \i.e. by i^ — 7), he would proceed thus : — 



9993 II 867 8159 2485 7031 5276 I 4092 



6074 7114 7399 9220 6932 



4 2522 9803 1799 4540 



29 7660 8622 2593 



208 3626 0354 



1458 5382 



I 0206 



7 



Quot. 868 4238 2153 2104 0004 II 4106 -f 14 = 4120 Rem. 



The new Method will be best explained by beginning with 

 NO. 1473 VOL. ^j] 



case (3) : it will l)e easily seen what changes have to be made in 

 it when dealing with cases (i) and (2). 



The Rule for case (3), when the sign is " - ," may be 

 enunciated thus : — 



Mark off the Dividend, beginning at its units-end, in periods 

 of « digits. If there be an overplus, at the left-hand end, less 

 than h, do not mark it off, but reckon it and the next n digits as 

 one period. 



To set the sum, write the Divisor, followed by a double 

 vertical : then the Dividend, divided into its periods by single 

 verticals, with width allowed in each space for (« + 2) digits. 

 Below the Dividend draw a single line, and, further down, a 

 double one, leaving a space between, in which to enter the 

 Quotient, having its units-digit below that of the last period but 

 one of the Dividend, and also the Remainder, having its units- 

 digit below that of the last period of the Dividend. In this 

 space, and in the space below the double line, draw verticals, 

 corresponding to those in the Dividend ; and make the last in 

 the upper space double, to separate the Quotient from the 

 Remainder. 



For example, if we had to divide 5984407103826 by 6997 (i.e. 

 7- fi - 3), the sum, as set for working, would stand thus : — 



6997 II 598 4 I 407 I 103 I 8^ 

 Quot. I I II Rem. 



I \ \ 



To work the sum, divide the ist period by// : enter its quotient 

 in the ist Column below the double line, and place its Remainder 

 above the 2nd period, where it is to be regarded as prefixed to 

 that period. To the 2nd period, with its prefix, add .('-times the 

 number in the 1st Column, and enter the result at the top of the 

 2nd Column. If this number is not less than the Divisor, find 

 what number of times it contains the Divisor, and enter that 

 number in the ist Column, and /C'-times it in the 2nd ; and then 

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

 deducting from the result /"-times the number just entered in the 

 1st Column ; and then add up the ist Column, entering the 

 result in the Quotient. If the number at the top of the 2nd 

 Column is less than the Divisor, the number in the ist Column 

 may be at once entered in the Quotient. The number entered 

 in the Quotient, and the number at the foot of the 2nd Column, 

 are the Quotient and Remainder that would result if the Divi- 

 dend ended with its 2nd period. Now take the number at the 

 foot of the 2nd Column as a new 1st period, and the 3rd period 

 as a new 2nd period, and proceed as before. 



The above example, worked according to this Rule, would 

 stand thus : — 



6 5 3 



6997 II 5984^1 407 I 103 1^^ 

 ■ Quot. 8.S1; I '2817 849 II 6373 Rem. 



1972 

 281 



5946 

 849 



the Mental Process being as follows : — 



Divide the 5984 by 7, entering its Quotient, 854, in the ist 

 Column, and placing its Remainder, 6, above the 2nd period. 

 Then add, to the 6407, 3-times the 854, entering the result in 

 the 2nd Column, thus. " 7 and 12, 19." Enter the 9, and 

 carry the i. "i and 15. 16." Enter the 6, and carry the i. 

 " 5 and 24, 29." Enter the 9, and carry the 2, which, added 

 to the prefix 6, makes 8, which also you enter. Observing that 

 this 8969 is not less than the Divisor, and that it contains the 

 Divisor once, enter i in the 1st Column, and 3-times i in the 

 2nd, and then draw a line below, and add in this ijew item, re- 

 membering to deduct from the result 7-times t'^, i.e. 7000 : the 

 result is 1972. Then add up the 1st column, as far as the 

 double line, and enter the result, 855, in the Quotient. Now 

 take the 1972 as a new ist period, and the 3rd period, 103, as a 

 new 2nd period, and proceed as before. 



The Rule for case (3), when the sign is "-f-," may be 

 deduced from the above Rule by simply changing the sign of k. 

 This will, however, introduce a new phenomenon, which must 

 be provided for by the following additional clause : — 



When you add, to the 2nd period with its prefix, ( - /(-j-times 



