226 Mr. R. Templeton on a Method of 



Seek for the first remainder and successive sums in a Table 

 of squares, and take out such squares as correspond most closely 

 with each, writing them alongside the particular sums to which 

 they approximate most closely. Select such as have closest 

 agreement, rejecting the others. In the example before us, 152 2 

 differs by a single unit from the sum 23105 ; add then 23105 

 to the e;iven eight figures, and we have 15382084, a complete 

 square, yiz. 3922* ; and 3922 ±152, or 4074 and 3770 are the 

 factors required, their product (3922' 2 -152 2 ) being 15358980*. 



Now to find the logarithm of 1535897932384626. 



4074x3770: 



15358980 

 15358979*32384626 





-•67615374 



n... log w 9-83004 55 



4074 . , 



, . 361002 10246 64145 



... log- 602776 33 



X 



3770 ., 



, . 3-57634 13502 05793 



. . . A.C. 6-42365 86 





7-18636 23748 69938 



2-28146 74 

 Correction + 





-191 19100 



No 2-28146 74 



7-18636 23557 50838 



Mark off the first eight figures, omitting for the time the num- 

 ber's proper decimal point if it have one, and regard these eight 

 figures as a whole number, the remainder as decimal. Next take 

 out the logarithms of the factors, prefixing to each the index 3, the 

 proper index to the four digits of the factor ; take out also the 



M 



logarithm of — for the first of the factors, writing it under log n ; 

 oo 



also place beneath it the arithmetical complement of the other 



factor ; add these three logarithms together, correct the sum f, 



and we have the logarithm 2'2814674, which is the logarithm 



of the quantity to be applied to the sum of the logarithms of 



the factors ; it is in this instance subtractive, since n is negative. 



The proper index is now to be supplied. 



Next let it be required to find the number answering to a given 



logarithm 



2-32699 68049 73387. 



Reject the index of the given logarithm and replace it with 

 the index 7. Find from the common Tables, or by the method 



* There is no reason why the square should he limited to eight places 

 other than that it would require an additional factor if they were extended 

 to ten or twelve, and that primes beyond the limits of the present Tables 

 inconveniently present themselves. 



t Vide preceding Section. 



