i 9 2 SCIENCE PROGRESS 



table of logarithms. Third, there is the profound mathematical 

 insight which enabled Napier not only to grasp the significance 

 of a logarithmic function, but also to indicate with precision the 

 great superiority of what is now known as the common 

 logarithm. 



In regard to the first of these, it must be remembered that 

 considerable progress had been made before Napier's day in the 

 recognition of the laws of exponents or indices. This is indeed 

 the subject on which Prof. Eugene Smith discourses with 

 erudition and lucidity. And we know that Buergi, following up 

 the line of some of these conceptions, constructed a few years 

 after Napier a practical table of what we would now call anti- 

 logarithms, by which certain arithmetical short-cuts could be 

 taken. 



The fundamental idea was to establish a correspondence 

 between the successive terms of two sets of increasing or 

 diminishing numbers. We may take the example given long 

 before Napier's day, and familiar enough in our elementary text- 

 book. 



The two sets or series of numbers 



are constructed on an obvious principle. The numbers in the 

 upper row increase by successive multiplications by two, and 

 form what is known as a geometrical progression. On the other 

 hand the numbers in the lower row form an arithmetical pro- 

 gression, and increase by the successive additions of unity. The 

 lower numbers serve to mark the positions in the upper row of 

 the successive powers of two. They are, in fact, the numbering 

 of the ratios, the meaning which lurks etymologically in the 

 word Logarithm. To multiply together any two numbers in 

 the first row we have simply to add their corresponding numbers 

 in the second row, and above the sum we find the desired 

 product. In like manner, to divide one number in the upper 

 row by another we subtract the corresponding numbers in the 

 lower row, and above the difference we find the quotient. 



Such a table is obviously of no practical value, for the 

 numbers it contains are too far apart and the interval between 

 each contiguous pair grows rapidly in magnitude as we pass to 

 higher terms. Nevertheless the idea of the logarithmic table is 



