Tusi couple

The Tusi couple (also known as Tusi's mechanism) is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and forth in linear motion along a diameter of the larger circle. The Tusi couple is a two-cusped hypocycloid.

The couple was first proposed by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in his 1247 Tahrir al-Majisti (Commentary on the Almagest) as a solution for the latitudinal motion of the inferior planets and later used extensively as a substitute for the equant introduced over a thousand years earlier in Ptolemy's Almagest.

Original description

The translation of the copy of Tusi's original description of his geometrical model alludes to at least one inversion of the model to be seen in the diagrams:

If two coplanar circles, the diameter of one of which is equal to half the diameter of the other, are taken to be internally tangent at a point, and if a point is taken on the smaller circle—and let it be at the point of tangency—and if the two circles move with simple motions in opposite direction in such a way that the motion of the smaller [circle] is twice that of the larger so the smaller completes two rotations for each rotation of the larger, then that point will be seen to move on the diameter of the larger circle that initially passes through the point of tangency, oscillating between the endpoints.

Algebraically, the model can be expressed with complex numbers as

(

1

−

1

2

)

e

i

θ

−

1

2

e

−

i

θ

=

i

sin

⁡

θ

.

{\displaystyle \left(1-{\frac {1}{2}}\right)e^{i\theta }-{\frac {1}{2}}e^{-i\theta }=i\,\sin \theta .}

Other commentators have observed that the Tusi couple can be interpreted as a rolling curve where the rotation of the inner circle satisfies a no-slip condition as its tangent point moves along the fixed outer circle.

Other sources

The term "Tusi couple" is a modern one, coined by Edward Stewart Kennedy in 1966. It is one of several late Islamic astronomical devices bearing a striking similarity to models in Nicolaus Copernicus's De revolutionibus, including his Mercury model and his theory of trepidation. Historians suspect that Copernicus or another European author had access to an Arabic astronomical text, but an exact chain of transmission has not yet been identified.