Alfred's Serif Users' Forums — Reinventing the magic log wheel: How was this missed for 400 years?

Re: Reinventing the magic log wheel: How was this missed for 400 years?

null@example.com (William) — Fri, 08 Apr 2022 19:09:45 +0000

Thank you.

William

Re: Reinventing the magic log wheel: How was this missed for 400 years?

null@example.com (GB) — Fri, 08 Apr 2022 18:17:58 +0000

What am I missing please?

The parts of the rubber band no longer stretch after they contact with the wheel. Imagine the wheel is coated with superglue.

Re: Reinventing the magic log wheel: How was this missed for 400 years?

null@example.com (William) — Fri, 08 Apr 2022 17:19:52 +0000

If the rubber band has stretched along its length, why are the numbers not spaced evenly around the wheel?

What am I missing please?

William

Re: Reinventing the magic log wheel: How was this missed for 400 years?

null@example.com (jackneve) — Fri, 08 Apr 2022 14:27:57 +0000

I have been trying to rationalise why the scale is natural log...

My mind is sadly too feeble at this moment to originate a proof, but I can post a comment from the Mathologer piece:

Proof of the length of the wheel being ln(10) :
When the number x reaches the wheel, the rubber band has stretched by a factor 1/x (we consider that the rubber band is numbered from 0 to 1 like in the video). If you now wind the band just a bit more so that you reach x+dx (with dx infinitesimal), then the length added on the wheel is dl=a(x)dx. So the total length of the wheel is the integral of dl for x=0.1 to 1 that is int(dx/x, x=0.1..1), which is equal to ln(10).

I was getting there myself, but couldn't put it together. Obviously my cogs don't always mesh.

Re: Reinventing the magic log wheel: How was this missed for 400 years?

null@example.com (Alfred) — Thu, 07 Apr 2022 16:22:10 +0000

Interesting and informative (as Mathologer so often is). Thanks for the link, Jack!

Reinventing the magic log wheel: How was this missed for 400 years?

null@example.com (jackneve) — Thu, 07 Apr 2022 12:55:02 +0000

A Mathologer video.
A novel way of demonstrating the circular slide rule - stretch a rubber band round the disk.

Today is about reinventing a really cool mathematical wheel and its many different slide rule incarnations, just using a rubber band.

https://www.youtube.com/watch?v=ZIQQvxSXLhI