THE GENESIS OF LOGARITHMS 163 



proposition is employed, that, given two sines s t and s 2 , the 

 difference between the logarithms of the two sines lies between 



the limits (s x — s a ) — and (s x — $$) — , which difference, if the 



two sines differ but slightly, may be taken with sufficient accuracy 

 to be equal to the arithmetic mean of the above limits. 1 This 

 proposition, which Napier proves in much the same manner as 

 the proposition quoted previously, may be verified readily by 

 putting 



y t* s 



D = log jv s 2 - log a- s t = r log, — - r log, -j~ r lo °' "7* 



and putting D in the form 



D = r log, (1 + S - -—') = r\o ge (i - ±=^) 



and expanding, the truth of the proposition is easily shown. 



Now the last term of the First Table being q,999,90o - 0O0495o 

 and the second term of the Second Table 9,999,900, when these 

 numbers are substituted in the expression for the arithmetic 

 mean of the limits given above, it is found that the difference of 

 the logarithms of these two terms is, to the approximation 

 considered, '0004950 ; and, adding this number to the logarithm 

 of the last term of the First Table gives 100*0005000 at the 

 logarithm of the second term of the Second Table. Since the 

 logarithm of the first term of the Second Table is zero, this 

 number gives us also the common difference for all the terms in 

 the Second Table. 



By a precisely similar process we can pass from the Second 

 Table to the first column of the Third Table and fill in the 

 logarithms of the twenty-one terms of this column. Then, using 

 the theorem again to pass from the twenty-first term of the first 

 column to the first term of the second column, we find that the 

 logarithm of this latter term is ioo503"3 ; this number, it must 

 be noticed, is the common difference of the logarithms of the 

 first terms of the first, second . . . sixty-ninth columns, of the 

 second terms of the various columns, and so on; so that, knowing 

 the logarithms of all the terms of the first column and the 

 common difference between all terms on the same line in the 

 various columns, we can fill in the logarithms of all the terms 

 of the Third Table. The Third Table, with its logarithms so 



1 Construction Prop. XXXIX.-XL. 



