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Napier and logarithms
From: J Cora
Date: 2009 May 28, 06:06 -0700

```I finished reading "The Story of Numbers" by John McLeish
chapter 12 is about Napier.  Unfortunately there is no mention
of Briggs.

On page 171 there is a description of how Napier calculated the logarithm
of 2.  He raised 2 to the power of 10,000 which comes out to a
number of 3011 places. He then subtract 1 from 3011 and gets 3010
which is the logarithm of two good to 4 decimal places.  log10(2) = 0.3010
I really don't understand why this works and hope somebody can
explain it?

In any case Napier wanted logarithm values to seven decimal places
and he worked out the following process.

Starting with the logarithm of 100 and 1000 which are 2 and 3
the next logarithm value is the arithmetic mean of the two original
logarithms so the next log value nxlog = (2+3) / 2 = 2.5

and the number represented by nxlog is  nxnum = 10**2.5

The amazing insight is that nxnum is the geometric mean of the
product of the orignal two numbers.

nxnum = squareroot( 100 x 1000) = squareroot( 10**2 + 10**3)

nxlog = 2.5 then nxnum = 316.22776601683796    (from pythons sqrt func)
using the nxlog value you can continue the process with either 2 or 3

I used some software called pari/gp   to get the exact value of 2**10000.
I hesitate to attach the file as the number is a single long string so it needs
some formatting.

I am going to try and work the square root of 100000 by hand to torture myself
and to see  how far I get. My calculator shows the square root of 100000
as 316.227766  with 6 decimal places.

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