View source for Tusi couple

{{Short description|Mathematical device}}
[[Image:TusiCouple.gif|right|thumb|250 px|An animated model of a Tusi couple]]
The '''Tusi couple''' (also known as '''Tusi's mechanism'''<ref>{{Cite book |url=http://archive.org/details/RoshdiRasheded.EncyclopediaOfTheHistoryOfArabicScienceVol.3Routledge1996 |editor=Roshdi Rashed |title=Encyclopedia Of The History Of Arabic Science |language=en}}</ref><ref>{{Cite journal |last=Saliba |first=George |date=2002-07-01 |title=Greek astronomy and the medieval Arabic tradition: the medieval Islamic astronomers were not merely translators. They may also have played a key role in the Copernican revolution |url=https://go.gale.com/ps/i.do?p=AONE&sw=w&issn=00030996&v=2.1&it=r&id=GALE%7CA88098478&sid=googleScholar&linkaccess=abs |journal=American Scientist |language=English |volume=90 |issue=4 |pages=360–368|doi=10.1511/2002.27.360 }}</ref><ref>{{Cite journal |last=Nosonovsky |first=Michael |date=2018-08-14 |title=Abner of Burgos: The Missing Link between Nasir al-Din al-Tusi and Nicolaus Copernicus? |url=https://brill.com/view/journals/zuto/15/1/article-p25_3.xml |journal=Zutot |volume=15 |issue=1 |pages=25–30 |doi=10.1163/18750214-12151070 |s2cid=135358186 |issn=1571-7283}}</ref>) 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 [[Oscillation (mathematics)|oscillate]] back and forth in [[linear motion]] along a diameter of the larger circle. The Tusi couple is a 2-cusped [[hypocycloid]].

The couple was first proposed by the 13th-century [[Persian people|Persian]] [[Islamic astronomy|astronomer]] and [[Islamic mathematics|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<ref>George Saliba (1995), "[https://books.google.com/books?id=mOquCzBX3xcC&pg=PA152 A History of Arabic Astronomy: Planetary Theories During the Golden Age of Islam]", pp.&nbsp;152–155.</ref> and later used extensively as a substitute for the [[equant]] introduced over a thousand years earlier in [[Ptolemy]]'s ''[[Almagest]]''.<ref>"Late Medieval Planetary Theory", E. S. Kennedy, ''Isis'' '''57''', #3 (Autumn 1966), 365–378, {{JSTOR|228366}}.</ref><ref>Craig G. Fraser, "[https://books.google.com/books?id=3tJr_vl6rYsC&pg=PA39 The cosmos: a historical perspective]", Greenwood Publishing Group, 2006 p.&nbsp;39.</ref>

== Original description ==
[[Image:Tusi couple.jpg|right|thumb|250 px|Tusi's diagram of the Tusi couple, 13th century<ref>Vatican Library, [http://wps.prenhall.com/esm_chaisson_astronomytoday_5/0,9185,1383966-content,00.html Vat. ar. 319 fol. 28 verso math19 NS.15] {{Webarchive|url=https://web.archive.org/web/20141224122410/http://wps.prenhall.com/esm_chaisson_astronomytoday_5/0,9185,1383966-content,00.html |date=2014-12-24 }}, fourteenth-century copy of a manuscript from Tusi</ref>]]

[[File:2 Tusi inversions 90X20.gif|thumb|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.<ref>Translated in F. J. Ragep, ''Memoir on Astronomy'' II.11 [2], pp. 194, 196.</ref>

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 
:<math>  \left( 1-  \frac{1}{2} \right) e^{i \theta} - \frac{1}{2} e^{-i \theta} = i \, \sin \theta.</math>
Other commentators have observed that the Tusi couple can  be interpreted as a [[Roulette (curve)|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 ==
[[File:Tusi by Copernicus 90x20.gif|thumb|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.<ref>E. S. Kennedy, "Late Medieval Planetary Theory," p. 370.</ref> It is one of several late Islamic astronomical devices bearing a striking similarity to models in [[Nicolaus Copernicus]]'s ''[[De revolutionibus orbium coelestium|De revolutionibus]]'', including his [[Mercury (planet)|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,<ref>E. S. Kennedy, "Late Medieval Planetary Theory," p. 377.</ref> The 16th century scientist and traveler [[Guillaume Postel]] has been suggested as one possible facilitator.<ref>{{Citation | last = Saliba | first = George | author-link = George Saliba | title = Writing the History of Arabic Astronomy: Problems and Differing Perspectives | journal = Journal of the American Oriental Society | volume = 116 | issue = 4 | pages = 709–18 | date = 1996 | jstor = 605441 | doi=10.2307/605441}}, pp. 716-17.</ref><ref>[http://www.columbia.edu/~gas1/project/visions/case1/sci.4.html Whose Science is Arabic Science in Renaissance Europe?] by [[George Saliba]], Columbia University</ref>

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<ref>Claudia Kren, "The Rolling Device," p. 497.</ref><ref>[[George Saliba]], "Whose Science is Arabic Science in Renaissance Europe?" [http://www.columbia.edu/~gas1/project/visions/case1/sci.1.html]</ref> 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 [[Medieval Greek|Byzantine Greek]]. Several Byzantine Greek manuscripts containing the Tusi-couple are still extant in Italy.<ref>{{cite web|author=George Saliba|author-link=George Saliba|title=Islamic Science and the Making of Renaissance Europe|website=[[Library of Congress]] |date=April 27, 2006|url=https://www.loc.gov/today/cyberlc/feature_wdesc.php?rec=3883|access-date=2008-03-01}}</ref>  Another possibility is that he encountered the manuscript of the "Straightening of the Curves" (Sefer Meyasher 'Aqov) while studying in Italy.<ref>Ruth Glasner and Avinoam Baraness, "Alfonso's Rectifying the Curved: A Fourteenth-Century Hebrew Geometrical-Philosophical Treatise." Sources and Studies in the History of Mathematics and Physical Sciences Berlin/New York: Springer, 2021.</ref>

There are other sources for this mathematical model for converting circular motions to reciprocating linear motion.  It is found in [[Proclus|Proclus's]] ''Commentary on the First Book of [[Euclid's Elements|Euclid]]''<ref>{{cite journal | last1 = Veselovsky | first1 = I. N. | year = 1973 | title = Copernicus and Nasir al-Din al-Tusi | url = http://articles.adsabs.harvard.edu/full/seri/JHA../0004//0000128.000.html | journal = Journal for the History of Astronomy | volume = 4 | issue = 2 | pages = 128–30 | bibcode = 1973JHA.....4..128V | doi = 10.1177/002182867300400205 | s2cid = 118453340 }}</ref> and the concept was known in Paris by the middle of the 14th Century.  In his ''questiones'' on the [[De sphaera mundi|''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.<ref>Claudia Kren, "The Rolling Device," pp. 490-2.</ref>

== Later examples ==

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'').<ref>{{Cite journal |bibcode = 1973JHA.....4..128V|title = Copernicus and Nasir al-Din al-Tusi|journal = Journal for the History of Astronomy|volume = 4|pages = 128–130|last1 = Veselovsky|first1 = I. N.|year = 1973| issue=2 |doi = 10.1177/002182867300400205|s2cid = 118453340}}</ref>  Nineteenth-century engineers James White,<ref>{{Cite web | url=https://books.google.com/books?id=JlxRAAAAYAAJ&q=James+White+engine+1801&pg=PA490 |title = Appleton's dictionary of machines, mechanics, engine work, and engineering|year = 1857}}</ref> [[Matthew Murray]],<ref>{{Cite web | url=http://www.pollymodelengineering.co.uk/sections/stationary-engines/anthony-mount-models/murrays-Hypocycloidal-Engine.asp | title=Polly Model Engineering: Stationary Engine Kits - Anthony Mount Models | access-date=2016-12-30 | archive-date=2019-03-27 | archive-url=https://web.archive.org/web/20190327085725/https://www.pollymodelengineering.co.uk/sections/stationary-engines/anthony-mount-models/murrays-Hypocycloidal-Engine.asp | url-status=dead }}</ref> as well as later designers, developed practical applications of the hypocycloid straight-line mechanism.
[[File:Goodman Tusi Couple.webm|thumb|Goodman modification of the Tusi couple, using 3 spur gears]]

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.<ref>{{Cite web |title=The Equation of Time - Introduction |url=https://equation-of-time.info/introduction-1 |access-date=2022-09-04 |website=equation-of-time.info |language=en-US}}</ref>{{Failed verification|reason=source does not say he invented the circa 1550AD Cardan straight line mechanism, only that he made a video of one|date=February 2023}}{{dubious|date=February 2023}}  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

== Hypotrochoid ==

{{Main|Hypotrochoid}}

[[File:Tusi couple ellipses.gif|right|400px|thumb|The [[ellipse]]s (green, cyan, red) are [[hypotrochoid]]s of the Tusi couple.]]
A property of the Tusi couple is that points on the inner circle that are not on the circumference trace [[ellipse]]s. These ellipses, and the straight line traced by the classic Tusi couple, are special cases of [[hypotrochoid]]s.<ref>{{Citation
| last       = Brande
| first      = W.T.
| year       = 1875
| title      = A Dictionary of Science, Literature, & Art
| publisher  = Longmans, Green, and Company
| page       = 181
| url        = https://books.google.com/books?id=Q1pFAQAAMAAJ&q=Hypotrochoid&pg=PA181
| access-date = 2017-04-10
}}</ref>

==See also==
* [[Murray's Hypocycloidal Engine]], utilising a Tusi couple as a substitute for [[crosshead guide]]s or [[parallel motion]]
* [[Epicyclic gearing]]
* [[Straight-line mechanism]]
* [[Spirograph]] 
* [[Geometric lathe]]
* [[Guilloché]]
* [[Deltoid curve]]

== Notes ==
{{commons category|Tusi-couple}}
{{reflist|30em}}

== References ==

* {{cite journal | last1 = Di Bono | first1 = Mario | year = 1995 | title = Copernicus, Amico, Fracastoro and Tusi's Device: Observations on the Use and Transmission of a Model | journal = Journal for the History of Astronomy | volume = 26 | issue = 2 | pages = 133–154 | doi=10.1177/002182869502600203| bibcode = 1995JHA....26..133D| s2cid = 118330488 }}
* {{cite journal | last1 = Kennedy | first1 = E. S. | year = 1966 | title = Late Medieval Planetary Theory | journal = Isis | volume = 57 | issue = 3| pages = 365–378 | doi=10.1086/350144| s2cid = 143569912 }}
* {{cite journal | last1 = Kren | first1 = Claudia | year = 1971 | title = The Rolling Device of Naṣir al-Dīn al-Ṭūsī in the ''De spera'' of Nicole Oresme | journal = Isis | volume = 62 | issue = 4| pages = 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}}.

== External links ==

* Dennis W. Duke, [http://www.csit.fsu.edu/~dduke/models Ancient Planetary Model Animations] includes two links of interest:
** [http://people.scs.fsu.edu/~dduke/tusi.html An interactive Tusi couple]
** [http://people.scs.fsu.edu/~dduke/arabmars.html Arabic models for replacing the equant]
* George Saliba, [http://www.columbia.edu/~gas1/project/visions/case1/sci.2.html "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.

{{Islamic astronomy}}

[[Category:Astronomy in the medieval Islamic world]]
[[Category:Linear motion]]
[[Category:Roulettes (curve)]]
[[Category:13th-century inventions]]