Tusi couple

Tusi's diagram of the Tusi couple, 13th century[7]

Tusi's two mutual inversions of the couple. Note that, contrary to the description, the circles in the lower animation must rotate in the same direction.

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.[8]

The description is not coherent and appears to arbitrarily combine features of several both possible and impossible inversions of the geometric model.

Algebraically, the model can be expressed with complex numbers as

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

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.

Copernicus' perception of the Tusi couple. "The two circles move ... in opposite direction."

The term "Tusi couple" is a modern one, coined by Edward Stewart Kennedy in 1966.[9] 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,[10] The 16th century scientist and traveler Guillaume Postel has been suggested as one possible facilitator.[11][12]

Since the Tusi-couple was used by Copernicus in his reformulation of mathematical astronomy, there is a growing consensus that he became aware of this idea in some way. It has been suggested[13][14] that the idea of the Tusi couple may have arrived in Europe leaving few manuscript traces, since it could have occurred without the translation of any Arabic text into Latin. One possible route of transmission may have been through Byzantine science; Gregory Chioniades translated some of al-Tusi's works from Arabic into Byzantine Greek. Several Byzantine Greek manuscripts containing the Tusi-couple are still extant in Italy.[15]

There are other sources for this mathematical model for converting circular motions to reciprocating linear motion. It is found in Proclus's Commentary on the First Book of Euclid[16] and the concept was known in Paris by the middle of the 14th Century. In his questiones on the Sphere (written before 1362), Nicole Oresme described how to combine circular motions to produce a reciprocating linear motion of a planet along the radius of its epicycle. Oresme's description is unclear and it is not certain whether this represents an independent invention or an attempt to come to grips with a poorly understood Arabic text.[17]

Although the Tusi couple was developed within an astronomical context, later mathematicians and engineers developed similar versions of what came to be called hypocycloid straight-line mechanisms. The mathematician Gerolamo Cardano designed a system known as Cardan's movement (also known as a Cardan gear).[18] Nineteenth-century engineers James White,[19] Matthew Murray,[20] as well as later designers, developed practical applications of the hypocycloid straight-line mechanism.

A practical and mechanically simple version of the Tusi couple, which avoids the use of an external rim gear, was developed in 2021 by John Goodman in order to provide linear motion.[21] It uses 3 standard spur gears. A rotating (blue) arm is mounted on a central shaft, to which a fixed (yellow) gear is mounted. A (red) idler gear on the arm meshes with the fixed gear. A third (green) gear meshes with the idler. The third gear has half the number of teeth of the fixed gear. An (orange) arm is fixed to the third gear. If the length of the arm equals the distance between the fixed and outer gears = d, the arm will describe a straight line of throw = 2d. An advantage of this design is that, if standard modulus gears that do not provide the required throw, the idler gear does not have to be colinear with the other two gears

The ellipses (green, cyan, red) are hypotrochoids of the Tusi couple.

A property of the Tusi couple is that points on the inner circle that are not on the circumference trace ellipses. These ellipses, and the straight line traced by the classic Tusi couple, are special cases of hypotrochoids.[22]

• Murray's Hypocycloidal Engine, utilising a Tusi couple as a substitute for crosshead guides or parallel motion
• Epicyclic gearing
• Straight line mechanism
• Spirograph
• Geometric lathe
• Guilloché
• Deltoid curve

• Di Bono, Mario (1995). "Copernicus, Amico, Fracastoro and Tusi's Device: Observations on the Use and Transmission of a Model". Journal for the History of Astronomy. 26: 133–154. Bibcode:1995JHA....26..133D. doi:10.1177/002182869502600203. S2CID 118330488.
• Kennedy, E. S. (1966). "Late Medieval Planetary Theory". Isis. 57 (3): 365–378. doi:10.1086/350144. S2CID 143569912.
• Kren, Claudia (1971). "The Rolling Device of Naṣir al-Dīn al-Ṭūsī in the De spera of Nicole Oresme". Isis. 62 (4): 490–498. doi:10.1086/350791. S2CID 144526697.
• Ragep, F. J. "The Two Versions of the Tusi Couple," in From Deferent to Equant: A Volume of Studies in the History of Science in Ancient and Medieval Near East in Honor of E. S. Kennedy, ed. David King and George Saliba, Annals of the New York Academy of Sciences, 500. New York Academy of Sciences, 1987. ISBN 0-89766-396-9 (pbk.)
• Ragep, F. J. Nasir al-Din al-Tusi's "Memoir on Astronomy," Sources in the History of Mathematics and Physical Sciences,12. 2 vols. Berlin/New York: Springer, 1993. ISBN 3-540-94051-0 / ISBN 0-387-94051-0.

• Dennis W. Duke, Ancient Planetary Model Animations includes two links of interest:
  • An interactive Tusi couple
  • Arabic models for replacing the equant
• George Saliba, "Whose Science is Arabic Science in Renaissance Europe?" Discusses the model of Nasir al-Din al-Tusi and the interactions of Arabic, Greek, and Latin astronomers.