SCIENCE. 



[N. S. Vol. XVI. No. 393. 



appears to offer marked advantages over other 

 methods vphere logarithms are inadequate. A 

 vital defect in the methods commonly used 

 lies in the fact that the result is obtained from 

 the right; that is, the digits of lower order in 

 the product are obtained iirst. The following 

 method is free from this defect and has the 

 further advantage that the approximation may 

 be carried to any degree of accuracy. Those 

 methods which require the writing of the 

 digits of the multiplier in inverse order are 

 objectionable in that such a process invites 

 error. The summing of a number of partial 

 products is not only objectionable in itself, 

 but renders uncertain the magnitude of the 

 error arising from the dropping of final digits. 

 The continued attention required in obtaining 

 a long partial product is again a fruitful 

 source of error. It will be seen that none of 

 these objectionable features appear in this new 

 method. 



The method is easiest explained by a few 

 examples. Let it be required to multiply 324 

 by 516. The process is shown thus : 



324 

 516 



154024 



1316 

 167184 

 The work in detail, which of course is all 

 done mentally, is as follows : Obtain the follow- 

 ing products, and sums of products: 



3-5 = 15 



3 •1 + 2-5 = 13 



3-6 + 2-1+4-5 = 40 



2-6 + 4-1 = 16 



4-6 = 24 



Set these results down in order, placing the 

 units figures of each result one place to the 

 right of the units figure of the preceding result. 

 Then add. The operation might be written: 



15 

 13 

 40 

 16 

 24 



167184 



The rule is entirely similar for numbers 

 of four or more digits. Thus the product 

 1543 ■ 2789 may be exhibited as follows : 



1543 

 2789 



Arrange as before and add. The product of. 

 two numbers containing five digits each is 

 obtained as follows : 



3.1415 

 2.7183 



6.18382515 

 2.3557143 



8.53953945 

 or in detail : 



3^2 = 6 



3^7 + 1^2 = 23 



3-l + 1^7 + 4-2 = lS 



3-8 + l-l+4-7 + l-2 = 55 



3^3 + 1 •8 + 4^1 + 1-7 + 5^2 = 38 



1^3 + 4-8 + l^l + 5^7 = 71 



4^3 + l-8 + 5-l = 25 



l-3 + 5^8 = 43 



5^3 = 15 



Arrange as before and add. If the result 

 were desired to four decimals only, the work 

 would be : 



3.1415 



2.7183 



6.1838 



2.3557 



8.5395 



It is interesting to make this last multiplica- 

 tion by the ordinary method and compare. 



3.1415 

 2.7183 



but the arrangement shown above is clearly 

 neater. 



94245 

 251320 

 31415 

 219905 

 62830 

 8.53953945 



