218 PROFESSOR THOMSON'S INVESTIGATION OF A NEW SERIES FOR 



The last two series, besides the simplicity and elegance of their form, are re- 

 markably convergent, when x is large, compared with nor 1. The latter of them 

 gives, with great facility, the logarithm of a whole number from the logarithms 

 of the two numbers immediately preceding and following it, when the number is 

 considerable : and this, as we shall presently see, is a case of continual occurrence 

 in the computation of logarithmic tables. 



To exemplify the use of formula (7), suppose that the common logarithm of 

 2 has been computed by any of the known methods : * then, by doubhng and 

 trebling it, the logarithms of 4 and 8 are obtained ; while that of 5 is found by 

 subtracting it from 1, the logarithm of 10. From the logarithms of 8 and 10, the 

 logarithm of 9 is obtained by means of series (7), as, by taking ^ = 9, that for- 

 mula gives 



,„ no + /8 . no — /8 . ,, / 1 1.2 1 „ \ 



In this the convergence is so rapid, that to find the logarithm true for seven deci- 

 mals, it is not necessary to proceed beyond the first term in the vinculum ; and 

 by employing additional terms, any assigned degree of accuracy is easily obtained. 

 By halving the logarithm of 9, we get that of 3 ; from which, and from the loga- 

 rithm of 2, that of 6 is found. Then, by series (7), 



,„ /8 + /6 , /8 — /6 , .. / 1 1,2 I ,, \ 



— a series of rapid convergence. 



Now, by adding the logarithm of 2 to the logarithms of 6, 7, 8, 9, and 10, we 

 get those of the even numbers 12, 14, 16, 18, and 20 ; and the logarithm of 15 is 

 the sum of the logarithms of 3 and 5. We should then find with great ease, by 

 means of (7), the logarithms of the prune numbers 11, 13, 17, and 19. By add- 

 ing the logarithm of 2 to the logarithms of 11, 12, 13, 20, we should have 



those of the even numbers from 20 up to 40 ; and those of the primes between 

 the same limits would be computed by means of (7). In a similar manner, we 

 should first obtain the logarithms of the even numbers from 40 up to 80, and then 

 those of the intermediate primes ; and thus we might proceed as far as we please, 

 the computations for the primes becoming easier and easier, as the numbers be- 

 come larger. The logarithm of any whole number, indeed, from 40 upwards, 

 would be obtained by (7), true for seven or more places of decimals, merely by 

 means of the logarithms of the two numbers immediately preceding and foUow- 



* If the modulus of the common logarithms be supposed to be known, the common logarithm of 2 

 may be computed with great ease by finding, by Mercator's series, the logarithm of 1 -J- 0-024 ; by 

 adding to the result 3, the logarithm of 1000, and thus finding the logarithm of 1024 ; and, lastly, by 

 dividing by 10, because 1024 is the tenth power of 2. 



