A Multiplication Nomograph

This is a puzzle.

If you look at the picture below:

you see numbers on the left, in red, and numbers on the right, in red.  If you connect the number 2 on the left with the number 4 on the right, the connecting (blue) line hits the vertical line at 8.  If you connect the number 3 on the left with the number 2 on the right, the connecting line hits the vertical line at 6.  This picture seems to do multiplication!

Does it work if you connect 3 on the left with 3 on the right?   Can you extend this to work for larger numbers?  Decimal numbers?

The puzzle?  Figuring out how this works.

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2 Responses to A Multiplication Nomograph

1. rr says:

y = x^2 a parabolic curve iirc ferdinand mobius multiplication table

2. Bert Speelpenning says:

rr,
I like your approach of leaving the merest hint that you think you know how it works, in a way that doesn’t spoil the fun for anybody else.

I had never seen this nomograph attributed to Möbius. I’ll follow up on that. I’ve seen it attributed to many other people.
In any case, I didn’t discover this on my own. But I did reconstruct how it works in a number of ways. (1) assuming that it works, you can find the curve and the distribution of grid points on the curve that are required to make it work. (2) you can verify that it works, algebraically. (3) you can verify that it works, geometrically. (4) you can look for variations of the scheme, or different applications of the scheme.

Any or all of these give interesting results in their own right.