July 11, 1902.] 



SCIENCE. 



73 



The number of auxiliary digits is 27 in the 

 last as against 17 in the first. We have 

 further a tedious addition to perform. It is 

 moreover clear that if only four decimals are 

 sought we have written down a mass of fig- 

 ures in the ordinary method on,ly to throw 

 them away in the end. 



In the preceding examples the two factors 

 have contained each the same number of 

 digits. If this is not the case we may imagine 

 the vacancies filled with zeros and proceed as 

 before. For example, 



187-23.5 

 213 



2252914 



17351915 



39881055 



The successive operations are: 



1-2=2 



l-l + 8-2=17 

 l-3-|-8-l + 7-2=25 

 8-3 + 7-l + 2-2 = 35 

 7-3 + 2-l + 3-2=29 

 2-3 + 3-l + 5-2=;19 

 3-3 + 5-l = 14 

 3-5 = 15 



If the digits are somewhat large it may hap- 

 pen that the product sum contains three 

 digits. Three rows of auxiliary figures are 

 then necessary. Thus: 



396 



994 



2T~ 



10890 



14724 

 393624 

 Or in detail: 



3-9 = 27 



3-9 + 9-9 = 108 



3-4 + 9-9Hr6-9 = 147 



9-4 + 6-9 = 90 



6-4=24 



The same rule must always be observed in 

 arranging the product sums. 



When there are two or more equal digits in 

 the multiplier the ordinary method would 

 seem to be preferable, since the corresponding 

 partial products are equal. This advantage 

 is more than balanced in the new method by 



the resulting simplification in the product 

 sums. Thus in the last example the opera- 

 tions may be written, 



3-9 = 27 

 (3 + 9)9 = 108 

 3-4+ (9+6)9 = 147 

 (6 + 4)9 = 90 

 6-4 = 24 



It is seen that the simplification occurs not 

 only when there happens to be a pair of equal 

 digits in the multiplier, but also when there 

 is a pair in the multiplicand or even when 

 one is in the multiplier and one in the multi- 

 plicand. A little practice enables one to catcli 

 sight of these pairs and the labor is materially 

 decreased in this way. This feature makes 

 the method particularly advantageous in 

 squaring a number. Thus : 



3.1415 

 3.1415 



9.6141810 

 .25484125 



9.86902225 



The operations are, 



3-3 = 9 

 (3 + 3) -1=6 

 (3 + 3)4 + 1-1=25 

 etc. 



A formal proof of the above method is 

 hardly necessary. The method itself was dis- 

 covered by inspecting the coefficients in the 

 product of two polynomials, 



a-ix' -\-a,x -\-a3 

 l)ia;- + 6.a; + 63 

 The product is 



a^bix' + (0^)2 + a^b^)!!)' 



+ {aj),-}- a.,li2-\- o-fii) X' 



+ ( a,6., + 036, ) a; + Oa^s 



Since we may write any number as 375 in the 

 form 



3-10= + 7-10+5 



the reason for the method is obvious. 



It is not difiicult also to work out a similar 

 short method of division which seems to pos- 

 sess advantages over the ordinary method. 



The method of multiplication described 

 above is to be carefully distinguished from the 

 familiar ' cross-multiplication ' (multiplicato 



