# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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. --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---