He groans with disappointment in the video that the longimeter and the map measurer don't give the same result, but we don't find out which was right. When he measures the curve he drew, the longimeter gives 28.9 mm but the map measurer (a.k.a. opisometer, a hand-held instrument for measuring curves on paper) says 34.5 mm; that's a 16% difference, so pretty big.

To do a real test, you could print out a curve with a mathematically calculated length, say a sine curve, then measure with both methods (and other methods like placing a string on top) to find out what works best in practice.

Hm. That 16% difference reminds me of the 60% point Miles Mathis makes in his infamous 'pi'='4' [single quotes INTENTIONAL.] paper (which ISN'T what people think. Numerous people tried to debunk it and I would absolutely LOVE to see a proper debunking of it, but each 'debunker' actually only skimmed his writing, or only read one of his papers on it instead of all of them, and constructed bad counter arguments as a consequence.)

http://milesmathis.com/pi.html http://milesmathis.com/pi2.html http://milesmathis.com/pi3.html https://sagacityssentinel.wordpress.com/ (the debunk reply there is from 2011 and doesn't factor in the fact that Mathis indeed didn't realized the rather obvious link to the Hilbert/Taxicab metric until 2012, see the pi2.html page there.)

The short version has a simple but fundamental error in the use of limits, and so as long as the author claims that this contains the essence of his argument, there seems little point in wading through the full work, especially as it is written in an extremely discursive manner, to put it charitably.

The interesting part is the explanation on how it works and on the accuracy you can expect:

https://books.google.it/books?id=sFzCAgAAQBAJ&pg=PA121#v=one...

This is a great youtube channel as well! I liked this review on a planar planimeter and how it ties in to Green's Theorem.

https://www.youtube.com/watch?v=jMvEOmpy8Kw

Wow it really is great. Plan to share with everyone I know who would find this interesting. Already watched a few of them and they are all just as enjoyable. Hope he gets a bunch more subs and keeps making videos.

Presumably this is the length of some discretisation of the curve, perhaps one of millimeter-long line segments. (The exact length is uncomputable!) Really cool though!

Yeah, one could draw a tight spiral in any of the open spaces of unbounded length. But "discretization" is probably a good angle to interpret it, since such a counterexample would be discretized to a point.

It also has the corner cases of curves that exactly align with the overlay and hence never cross any lines.

For increased accuracy, one can displace the overlay slightly and measure again, and take the average of multiple measurements.

> uncomputable

[citation needed]

There exist plenty of mathematical curves whose lengths are undefined. The term to search is "non-rectifiable curve".

This tool is for measuring curves drawn on a piece of paper, not abstract curves.

Oh, that doesn't make it any easier: https://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Brita...

Well, it doesn't in theory, but it does in practice. Natural phenomena are only fractal over a certain range of scales. Outside that range they follow normal rules.

That doesn't fit on a piece of paper, either.

A cartographer might disagree.

See https://en.wikipedia.org/wiki/Crofton_formula

Also see page 41 of Differential Geometry of Curves and Surfaces by Manfredo Do Carmo

Also interesting is https://en.wikipedia.org/wiki/Opisometer

That one is easy to understand. hettps://en.wikipedia.org/wiki/Planimeter is more interesting: trace the perimeter to measure an area.

I got you: http://www.youtube.com/watch?v=ifKrwF47uP8

It's like a small version of a Rolatape (used for measuring race courses etc)

http://www.rolatape.com/us/en/home/measuring-wheels.html

How is this better than using a string? (Actual piece of string, not text).

This is brilliant.