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Imagine you have a magic compass that will draw any equation (whereas an ordinary compass draws the equation $x^2 + y^2 = r^2$). What equation(s) would you have this magic compass draw in order that you could do third roots of lengths using it, a straight-edge, and an ordinary compass.

diligar
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    How about $x^3=r$? – Gregory Grant Aug 05 '16 at 23:10
  • They did do some estimates with cube roots, although this may not have entered into geometry as such. Evidently there is this business of twisting ropes, in the same way we twist rubber bands to propel a toy airplane. There is a cubic function in there somewhere; some weaponry was involved. – Will Jagy Aug 05 '16 at 23:22
  • The ancient greeks didn't just have a word for it (neusis) but they did it: see https://en.wikipedia.org/wiki/Doubling_the_cube#Using_a_marked_ruler and https://en.wikipedia.org/wiki/Neusis_construction – Rob Arthan Aug 05 '16 at 23:57
  • Catapults and cube roots: http://www.koslow.co.uk/JohnMcCoy/Catapults%20and%20Cube%20Roots.pdf – John Wayland Bales Aug 06 '16 at 00:58
  • @JohnWaylandBales thank you. I knew it was weapons, I was never sure about the rope aspect. – Will Jagy Aug 06 '16 at 01:10
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    @WillJagy I remember reading an article about Roman catapults and cube roots decades ago in Scientific American. The romans used a fixed ruler in a box with a right angle. The ruler began in a vertical position inside the box and the bottom end was moved horizontally like the classical freshman problem of the ladder sliding down a wall. The envelope of straight lines traced out a curve which, if I recall, has equation $x^{1/3}+y^{1/3}=1$. They used the curve to extract cube roots which told them how thickly to wind the ropes about the two "springs" to catapult a bolus the desired distance. – John Wayland Bales Aug 06 '16 at 01:16
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    @WillJagy "Ancient Catapults" Scientific American March 1979 (paywalled unfortunately) http://www.scientificamerican.com/article/ancient-catapults/ – John Wayland Bales Aug 06 '16 at 02:22
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    @JohnWaylandBales https://www.amazon.com/Catapult-Harry-Build-Siege-Weapon/dp/0156005565 – Will Jagy Aug 06 '16 at 02:35

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