How did John Napier invent his sliderule?

[markkrebs]

First, I'll note this interesting thread: #4158384.
However I unfortunately don't have privileges to post there so I'll start this new one.

I'm trying to make my own sliderule. How did they do it? How did Napier invent his "bones?" A more particular question would be: If I make a table of powers of 2, arraying them along a numberline, and a similar table of powers of 3 (9,27,81, etc) then, could I combine these? I know the answer is "yes" since that's what a sliderule DOES, but I don't know how to reconcile the two. Where, on the table of x2 logarithms, do I etch in the 3?

 

[jedishrfu]

The thread has no reference, I'm assuming its a PF thread. It would be best if you posted its actual URL.

Making a slide rule is simply using the log values as "feet" starting with 1 at 0 and 2 at 0.301 feet and 3 at 0.477 feet...

Here's an article about making one from a paper template:

http://www.instructables.com/id/Making-Your-Own-Slide-Rule/

 

[markkrebs]

How would you do it from first principles? Assume you can only add and multiply: I can definitely type log(2) into my venerable HP15c simulator and the the answer but what if I was trying to INVENT logs, instead of using them to define themselves? That seems like cheating: circular.

 

[markkrebs]

I don't think the original post too relevant but it was this.

 

[jedishrfu]

Then you would use the series for logarithms input 1, 2, 3... and use the output scaled to measure lengths on the rule.

However since we have this advanced technology called a calculator we can get the values that way. However if you sent back in time and had to construct one from memory then you'd need to use the Taylor series for the natural log.

 

[markkrebs]

Napier invented that series?

 

[jedishrfu]

Napier knew of how to generate logarithms but I don't know of the series he used.

http://en.wikipedia.org/wiki/John_Napier

and this one on Napier's log

http://en.wikipedia.org/wiki/Napierian_logarithm

 

[markkrebs]

Hey thanks! So I came up with this:
1) Make a slide rule based on base 2 as follows. .25, .5, 1,2,4,8,16...
2) Question is where does the 3 go? Well, we know other powers of 2. For instance sqrt(2) is just half the distance between the marks on the base 2 rule.
3) So find some sum of distances whose powers, 2^(1 + a/2 + b/4 + c/8) multiply to get 3. Not all the a,b,c have to be 1: some are zero.

 

[SteamKing]

Napier knew of how to generate logarithms but I don't know of the series he used.

http://en.wikipedia.org/wiki/John_Napier

and this one on Napier's log

http://en.wikipedia.org/wiki/Napierian_logarithm

This article:

http://en.wikipedia.org/wiki/Logarithm#History

explains how Napier calculated his logarithms. Given the calculation tools of Napier's time, parchment and quill pens, this was a lengthy process, which reportedly took Napier some 20 years to complete.

It was Henry Briggs who seized on Napier's ideas and produced the first table of logarithms. He also made logs more convenient to use, by switching from Napier's natural logarithms to the so-called common logs based on the number 10:

http://en.wikipedia.org/wiki/Henry_Briggs_(mathematician)

Napier did all of his natural log calculations without knowledge of the base of the natural logs, e ( ≈ 2.71828 ...), which is now called Euler's constant, but it was in fact discovered by Jacob Bernoulli many years before Euler was born:

http://en.wikipedia.org/wiki/E_(mathematical_constant)

Taylor series, which were, of course, discovered before Taylor came along, was published by him in 1715, not long before Taylor himself had expired at the age of 46 in 1731:

http://en.wikipedia.org/wiki/Brook_Taylor

http://en.wikipedia.org/wiki/Taylor_series

It was an Anglican minister, William Oughtred, who devised the first rudimentary "slip stick" with which we are (or were) familiar:

http://en.wikipedia.org/wiki/William_Oughtred

In his spare time, Oughtred invented the multiplication sign (×) and the double colons to indicate proportion, e.g. (1 :: 2).