Utilizing Compact Logarithmic Tables. 227 



in the preceding section, the first eight figures of the required 

 number. Determine the factors. 



As a matter of convenience write out the logarithms of 

 the factors beneath the given logarithm, but separated from 

 it by a score, when the product is less than the correspond- 

 ing figures of the number to eight figures. If the product 

 be greater, write them above; so that whichever be the less, 

 the given logarithm or the sum of the logarithms of the fac- 

 tors, the less shall always be below, but separated from the 

 other by a score*. Here the factors selected are 4645 and 

 4571, and their product 21232295 ; their logarithms are there- 

 fore placed above. Subtracting then the given logarithm with 



-1346 456 . .. a 

 4645.. 3-66698 571 83 297 

 4571 . . 3-66001 12212 893 



7-32699 69396 190 



7-32699 68049 73387 

 1 

 •36221 57 

 3-12919 22 



0-81840 48 log 



. . sum 



. . given logarithm 



• • i a 



. . constant. 



. . log a 



21232295 



-6-582710 

 21232288 427290 



No. 



its index replaced by 7 from the sum of these, we have a, nega- 

 tive since the given logarithm is the less, and one half of it 

 gives to the nearest unit the correction 1 in the 7th place, which 

 is written beneath the given logarithm, and under these the con- 

 stant and logarithm of a ; the sum 0*8184048 is the logarithm of 

 6-582710, the number to be subtracted, since a is negative, 

 from the product (21232295) of the factors, to have the number 

 answering to the given logarithm, and it is to be pointed off 

 into whole number and decimal in accordance with its own 

 proper index. Had the factors 4924 and 4312 been adopted, 

 the number would have been found to be 212 32884272897. 



Lastly, to find a logarithm to twenty places of figures. Make 

 the preceding processes subsidiary, in the accompanying ex- 

 ample using the logarithm and numbers above found. The 



M 



value of log — will be found by subtracting the logarithm of the 



first factor from 9-63778 43113 00536 78912. 



* This step is simply to make the correction \a always additive. 



Q2 



