Copernicus and the Tusi Couple – Or was it Proclus?

Posted: February 9, 2015 by tallbloke in Celestial Mechanics
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Some scholars say that Copernicus was influenced in his astronomical theory by 13th century Persian mathematician Nasir al-din al-Tusi, who discovered (or re-discovered) a principle for converting angular motion to reciprocating linear motion. Watch the neat video below to see the basis of this motion conversion

TusiCoupleThe ‘Tusi couple’ is covered by this wiki page, and has this simpler animation to show the principle. But in relation to Copernicus, an his supposed ‘borrowing’ of this device to explain the apparent equant motion of planets as viewed from Earth there is also reference to a paper by I. N. Veselovsky, “Copernicus and Nasir al-Din al-Tusi”, Journal for the History of Astronomy, 4 (1973): 128-30

In that paper, Veselovsky notes that Proclus, the 5th century Neoplatonist philosopher, wrote a trestise on Euclid’s geometry which also contains the circular to linear motion principle, and that a passage from Proclus is repeated in Copernicus’ work. It’s possible Proclus included this aspect of Euclid because it was highlighted by a slightly earlier Greek scholar, Eudoxus.

Those interested in the shift from the classical and Arabic geocentric world to the heliocentric model of Copernican astronomy can find a useful passage in this google books link to Toby Huff’s work ‘Intellectual curiosity and the scientific revolution: A global perspective’.

With reference to the post on resonance rings in galaxies immediately preceding this one, consider the path taken by the point traveliing along the line in the wiki animation if the outer circle and the line were also revolving, at half the rate of the inner circle. It’s a bit hard to visualise, but I think it might make a figure 8 as it completes an ‘orbit’. Anyone got the time to model it?

  1. Ian Wilson says:


    Mesmerizing! I think there is more to this than meets the eye!

  2. PeterF says:

    That linear motion turning into a circular motion makes me wonder, whether this could become the principle for a car engine, like some super-wankel engine?

  3. oldbrew says:

    Time to re-think that old 2D model of planetary movements perhaps.

  4. nzrobin says:

    Reminds me of a synchronous generator. Where a single rotating magnetic field makes three AC voltages in three sets of coils displaced around the stator.

  5. tallbloke says:

    Ian: I owe you a hat tip for the youtube animation. I searched my inbox for it but couldn’t remember the source.
    The ‘simple harmonic motion’ of each of the balls is what you would see if they were following circular orbits at a fixed angular velocity with your view ‘edge on’ to those orbits i.e. like a small moon orbiting a big planet’s equator. That’s part of the fascination here, a really interesting puzzle.

  6. oldbrew says:

    TB: see the helical motion video above.

  7. Graeme No.3 says:


    At least one motor tried in the 1930’s but ran into problems with lubrication. Involved a single cylinder and (from memory) driving an epicyclic gear wheel inside a larger gear wheel exactly double the size. Remembered from a book which I got rid of a few years ago.

  8. PeterF says:

    Isn’t the radial engine of early aircraft such an engine?

    As they aren’t in use any more, they may not have performed good enough?

  9. linneamogren says:

    @Oldbrew That’s an awesome video from NASA you found! I shared it in class.

  10. graphicconception says:

    I have an Excel 2013 model like the Wiki one that you can plug parameters into. If you would like a copy I could email it.

  11. dynam01 says:

    Reblogged this on I Didn't Ask To Be a Blog and commented:
    Cool illusion!

  12. Sphene says:

    I’ve been following your website for a little while, especially your work on planetary resonances. Very intriguing.

    I was not familiar with the Tusi couple. Great animation. With one full circuit of the larger circular track by the inner wheel, the inner wheel edge points have all travelled 8r along straight lines. It’s interesting to compare that to the curve generated by rolling a wheel once along a straight line track, which is the cycloid of length 8r.