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68 H. S. CARSLAW. 
In his third method Napier explains how a close approx- 
imation to the logarithm of any given number can be 
obtained by finding the number of figures in the result 
obtained by raising the given number to a power equal to 
the assumed logarithm of 10. 
As an example, he takes the case of the logarithm of 2, 
in the system where the logarithm of unity is 0 and the 
logarithm of 10 is 10,000,000,000. 
‘* Suppose it is asked,’’ he says, ‘“ what number is the 
logarithm of 2? I reply, the number of places in the result 
obtained by multiplying together 10,000,000,000 of the 
number 2. 
But, you will say, the number obtained by multiplying 
together 10,000,000,000 of the number 2 is innumerable. 
I reply, still the number of places in it, which I seek, is 
numerable. 
Therefore with 2 as the given root, and 10,000,000,000. 
as the index, seek for the number of places in the multiple, 
and not for the multiple itself; and by our rule you will 
find 301029995 etc. to be the number of places sought, 
and the logarithm of the number 2.”’ 
This should be 3010299957, and the logarithm of 2, on 
this system, lies between 3010299956 and 3010299957. 
When Napier says the logarithm is the number of places 
in the result obtained by multiplying together 10,000,000,000 
of the number 2, he means that the logarithm is nearly 
this. He understands that it lies between this number 
and the one next below it. 
Having agreed upon a reconstruction of the logarithm 
tables, the calculation was to be left altogether in the 
hands of Briggs. And on Napier’s death in 1617, the whole 
1 Constructio English Trans., p. 61 (Briggs’ note). 
