Tusi-couple
Encyclopedia
The Tusi-couple 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.
It was developed by the 13th-century Persian
astronomer
and mathematician
Nasir al-Din al-Tusi
as part of a planetary model which avoided the equant
introduced over a thousand years earlier in Ptolemy
's Almagest
.
Some modern commentators also call the Tusi couple a "rolling device" and describe it as a small circle rolling
inside a large fixed circle. However, Tusi himself described it differently:
, including his Mercury
model and his theory of trepidation
. Historians suspect that Copernicus or another European author had access to an Islamic astronomical text, but an exact chain of transmission has not yet been identified, although the 16th century scientist and traveler Guillaume Postel
has been suggested.
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
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. 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.
Since the Tusi-couple was used by Nicolaus 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 both by a historian of Medieval European astronomy and by a historian of Arabic astronomy 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
, which 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.
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
rotates inside a larger circle twice the diameter
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
of the smaller circle. Rotations of the circles cause a point on the circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
of the smaller circle to oscillate
Oscillation (mathematics)
In mathematics, oscillation is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; that is, oscillation is the failure to have a limit, and is also a quantitative measure for that.Oscillation is defined as the...
back and forth in linear motion
Linear motion
Linear motion is motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with constant velocity or zero acceleration; non uniform linear motion, with variable velocity or non-zero...
along a diameter of the larger circle.
It was developed by the 13th-century Persian
Persian people
The Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient Indo-Iranians and themselves part of...
astronomer
Islamic astronomy
Islamic astronomy or Arabic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age , and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and...
and mathematician
Islamic mathematics
In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600...
Nasir al-Din al-Tusi
Nasir al-Din al-Tusi
Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī , better known as Naṣīr al-Dīn al-Ṭūsī , was a Persian polymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed...
as part of a planetary model which avoided the equant
Equant
Equant is a mathematical concept developed by Claudius Ptolemy in the 2nd century AD to account for the observed motion of heavenly bodies....
introduced over a thousand years earlier in Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
's Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
.
Original description
Roulette (curve)
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
inside a large fixed circle. However, Tusi himself described it differently:
- 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.
Other sources
The term "Tusi couple" is a modern one, coined by Edward Kennedy in 1966. It is one of several late Islamic astronomical devices bearing a striking similarity to models in De revolutionibus orbium coelestiumDe revolutionibus orbium coelestium
De revolutionibus orbium coelestium is the seminal work on the heliocentric theory of the Renaissance astronomer Nicolaus Copernicus...
, including his Mercury
Mercury (planet)
Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
model and his theory of trepidation
Trepidation
According to a medieval theory of astronomy, trepidation is oscillation in the precession of the equinoxes. The theory was popular from the 9th to the 16th centuries....
. Historians suspect that Copernicus or another European author had access to an Islamic astronomical text, but an exact chain of transmission has not yet been identified, although the 16th century scientist and traveler Guillaume Postel
Guillaume Postel
Guillaume Postel was a French linguist, astronomer, Cabbalist, diplomat, professor, and religious universalist.Born in the village of Barenton in Basse-Normandie, Postel made his way to Paris to further his education...
has been suggested.
There are other sources for this mathematical model for converting circular motions to reciprocating linear motion. It is found in Proclus's
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
Commentary on the First Book of Euclid
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
and the concept was known in Paris by the middle of the 14th Century. In his questiones on the Sphere
De sphaera mundi
De sphaera mundi is a medieval introduction to the basic elements of astronomy written by Johannes de Sacrobosco c. 1230...
(written before 1362), Nicole Oresme described how to combine circular motions to produce a reciprocating linear motion. 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.
Since the Tusi-couple was used by Nicolaus Copernicus
Nicolaus Copernicus
Nicolaus Copernicus was a Renaissance astronomer and the first person to formulate a comprehensive heliocentric cosmology which displaced the Earth from the center of the universe....
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 both by a historian of Medieval European astronomy and by a historian of Arabic astronomy 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
Byzantine science
Byzantine science played an important role in the transmission of classical knowledge to the Islamic world and to Renaissance Italy, and also in the transmission of medieval Arabic science to Renaissance Italy...
, which translated some of al-Tusi's works from Arabic into Byzantine Greek
Medieval Greek
Medieval Greek, also known as Byzantine Greek, is the stage of the Greek language between the beginning of the Middle Ages around 600 and the Ottoman conquest of the city of Constantinople in 1453. The latter date marked the end of the Middle Ages in Southeast Europe...
. Several Byzantine Greek manuscripts containing the Tusi-couple are still extant in Italy.
External links
- Dennis W. Duke, Ancient Planetary Model Animations includes two links of interest:
- 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.