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

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Logarithms by Hand
From: Noell Wilson
Date: 2014 Jun 3, 09:29 -0700

I’m an unlikely advocate of math history but the CelNav solution by logs has made me interested in how we got to this point and how logs were first calculated.
The first answers I got on the source of logs to the base 10 were on solving by log to the base e or log to the base 2 using binary routines. I wanted the pencil and paper method and found the information below.

I used:
http://lamarotte2.blogspot.com/2011/04/calculating-log-of-number-by-hand
as my first guide for this.
John Napier ( 1550 – 1617) did some initial logs but to a base 10,000,000. We won’t go there.

Leonard Euler ( 1707 – 1783) showed that, as opposed to calculating logs, you can FIND them by starting with, relatively easy, logs that bracket the number and then closing in on the number desired by using math rules on a number above and below and using log rules to do the same on the logs of the numbers above and below. This will give a number closer to that desired and allow another calculation.

The example in the guide referenced above is the log of 2.
We know the logs above and below:
Number - -- - - -- - - - log (I’m using the dashes to try to maintain column formatting)
1 - ----------------------- 0 This says that 10^0 power = 1.
10 --------------------- 1 This says that 10^1 power = 10.

My non-math version is that we square the numbers on the left and right and take the square roots.
We’re actually going to ‘average’ by picking and multiplying two numbers on the left and take the square root of them. We do the same on the right using rules for logs.
If we do this using a number above 2 and a number below two, we’ll get a number closer to 2 and its log. Then we’ll repeat.

Using math on the numbers, SQRT ( 1 x 10) = 3.16238 (SQRT is Excel for Square Root)
Using log rules to do the same to the logs ( 1+0)/2 = 0.5 which is the log of 3.16238. (Add logs to multiply and divide logs for roots.)

List these as above:
3.16238------------------0.5
Picking 1 and 3.16238 as bracketing the 2, SQRT (3.16238 x 1) = 1.778828, (0+0.5)/2 = 0.25 the log of 1.778828. Listing these:
1.778828------------------0.25
Continue picking the closest above and below 2, calculating, and listing the new values.
2.37137---------------------0.375
2.05353---------------------0.3125
1.91095---------------------0.28125
11 more lines and:
2.00001---------------------0.301033
2.00000---------------------0.301029 This is NOT a converging series calculation. You manually pick the closest numbers above and below 2 to use each time.

You could expand from the logs of 1 and 10, and those numbers between them.
Log 100--------------------2
Log 1000-------------------3
Log 1,000,000-------------6

All of these are greater than 1. In CelNav the trig functions are often less than 1 or fractional.

The above gives us the basis to calculate the logs of numbers as is usually shown near the beginning of a book of logs and trig functions.
Sin of 30 degrees is 0.5 so the number by itself is 5, 50, 500 - the same no matter how many digits we wish to use.
Looking up the log of 5/50/ 500, we get 69897.
The characteristic of 0.5 is -1. Written in engineering, in the olden days, as 1(with a minus sign over the 1).69897.
Combining the -1 and the .69897 we get -.30103 which is the same as you get from a calculator if you put in 0.5 and press “log”. I was mistaken in my 29 May post attachment when I said that this
( -.30103) was only available from a calculator.
The log and trig book writes this value as 9.69897 -10 which is also -.30103. The -10 is not listed in the book but is understood.
If we look up the log of Sin 30 in a trig book, the answer listed is 9.69897. The -10 is still there.

Looking back up to the start of this, the missing part is how to take square roots by hand. That’s possible and only about twice as difficult as long division. Not for now.
Just keep in mind how much work went into this. The log of 2 to six decimal places took 19 lines of calculated data including 17 manual square roots.

There is more on the history in Google Books under “The Calculation of Logarithms” by James K. Whittlemore including references to Briggs’ calculation from 1614 to 1617 manually calculating the logs of 1 to 1000 by hand to 14 places.
A salute to all of those guys.
Regards, Noell

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