• Registered: 2022-01-21
  • Posts: 382

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

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

https://i.postimg.cc/xJ86Km9J/image-2022-04-07-135315694.png https://i.postimg.cc/WtQknZ2R/image-2022-04-07-134436876.png

  • Alfred
  • Alfred
  • Administrator
  • Offline
  • Registered: 2022-01-21
  • Posts: 1,333

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

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

"Has it ever struck you that life is all memory, except for the one present moment that goes by you so quick you hardly catch it going?"
― Tennessee Williams
  • Registered: 2022-01-21
  • Posts: 382

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

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.

  • Registered: 2022-01-21
  • Posts: 1,588

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

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

5 Reply by GB

  • GB
  • GB
  • Member
  • Offline
  • Registered: 2022-01-22
  • Posts: 351

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

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.

  • Registered: 2022-01-21
  • Posts: 1,588

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

Thank you.

William