NAPIER COMMEMORATIVE LECTURE. 67 
ealculation of the logarithms fell, did not employ this 
method. 
The second method Napier explains as follows :— 
*‘Any common number being formed from other common 
numbers by multiplication, division, (raising to a power) 
or extraction (of a root); its logarithm is correspondingly 
formed from their logarithms by addition, subtraction, 
multiplication, by 2, 3, etc. (or division by 2, 3, etc.): 
whence the only difficulty is in finding the logarithms of 
the prime numbers; and these may be found by the 
following general method. 
For finding all logarithms, it is necessary as the basis of 
the work that the logarithms of some two common numbers 
be given or at least assumed; thus in the foregoing first 
method of construction, 0 or a cypher was assumed as the 
logarithm of the common number one, and 10,000,000,000 
as the logarithm of one-tenth or of ten. These therefore 
being given, the logarithm of the number 5 (which is a 
prime number) may be sought by the following method. 
Find the mean proportional between 10 and 1, namely 
316227766017 | 
100000000000” also the arithmetical mean between 
10,000,000,000 and 0, namely, 5,000,000,000; then find the 
316227766017 
geometrical mean between 10,000,000,000 and 100000000000 
562341325191 
namely 100000000000" also the arithmetical mean between 
10,000,000,000, and 5,000,000,000, namely 7,500,000,0003...”’ 
Expressed in a few words, this method consists in 
inserting geometrical means between the numbers, and 
arithmetical means between the logarithms. It is the 
method which Briggs employed in calculating his tables. 
it is also the method which most teachers will now use in 
explaining to their classes, with the index notation, the 
principle on which the theory of logarithms is based, and a 
way in which they might be calculated. 
