A MATHEMATICAL ERROR IN THE CENTURY 

 DICTIONARY. 



BY FLORIAN CAJORI. 



The Century Dictionary, under the head of "Logarithm," 

 gives a table in which are found the two left hand columns of 

 numbers given below. The second column is wrong, all the 

 numbers being too small by 61180956. The relation between 

 Napier's and natural logarithms is expressed by the well- 

 known formula, 



logjv « = 10' ^^Se — ' 



Calculating by this formula the logarithms of .1, 1, etc., we get 

 the numbers in the columns on the right, which are the true 

 Napier's logarithms. Napier's original publication, the Descrip- 

 iio canonis of 1614, contained only logarithms of sines. The 

 logarithms of the particular numbers given below can therefore 

 not be found in the table. But by our formula we are able to 

 reproduce the logarithms in Napier's tables, and we are, there- 

 fore, sure that our calculated logarithmic figures in the column 

 on the right are correct. For example, the sine of 44° 19' is 

 given in Napier's table as 6986235 and its logarithm, 3586432. 

 Quite the same logarithm is obtained by calculation by our 

 formula. Again, the Century Dictionary gives a minus logar- 

 ithm for 10^ while Napier's logarithms of all numbers below 

 10' are positive. 



Natural Napier's logarithms. The true Napier's 

 Numbers as given in C. D. Logarithms. 



0.1 123025851 184206807 



1 100000000 16118095& 



10 76974149 138155105 



100 53948298 115129254 



1000 30922447 92103403 



10000 7896596 69077552 



100000 —15129255 46051701 



