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Most students meet trigonometry as a tool for heights and distances, and simple harmonic motion as a chapter of springs and pendulums. Nasir al-Din al-Tusi connects the two. He built one of the most ambitious observatories of the medieval world, wrote trigonometry as a subject of its own and found a neat way to make straight-line motion out of circles. This post covers his life, the Maragha observatory and a worked version of the Tusi couple you can check with Class 11 maths.
Who was Nasir al-Din al-Tusi?
F. Jamil Ragep's entry in the Biographical Encyclopedia of Astronomers gives his dates as 17 February 1201 (born in Tus, north-east Iran) to 25 June 1274 (died in Baghdad). He wrote over 150 works in Arabic and Persian, on mathematics, astronomy, philosophy and religious sciences.
His life ran through some of the most violent decades of the 13th century. After his education he found patrons at the Ismaili courts of eastern Iran and eventually moved to their fortress capital, Alamut. The MacTutor biography records that he was in Alamut in 1256 when it was attacked by the Mongol leader Hulegu, and that the Mongols entered Baghdad in February 1258. Al-Tusi then joined Hulegu's service as an adviser.
What was the Maragha observatory?
Al-Tusi presented Hulegu with plans for an observatory at Maragha, in north-western Iran. MacTutor states that construction began in 1259 and that it became operational in 1262. It had a 4-metre copper wall quadrant, an azimuth quadrant invented by al-Tusi himself, and a large library. Ragep describes it as a library and school as well as an observatory, one that "may be considered the first full‐scale observatory", which drew scholars from across the Islamic world and even from China.
Its main product was the Zij-i Ilkhani, a set of astronomical tables. MacTutor says al-Tusi published it after 12 years of observation, with tables for planetary positions and a star catalogue. Ragep is frank that the tables themselves were not especially original. The originality lay elsewhere.
What is the Tusi couple?
Ptolemy's planetary models broke a rule that ancient astronomers cared about: every motion in the heavens should be built from uniform circular motion. Al-Tusi wanted models that kept the rule and still matched observations. Ragep describes his device: a small circle rolling inside a circle twice its size, turning twice as fast in the opposite direction. Al-Tusi proved that a point on the small circle moves back and forth along a straight line.
Worked example: why does the point move in a straight line?
Let the big circle have radius R and centre at the origin. The small circle has radius R/2 and touches the big circle from inside.
As the small circle rolls, its centre moves around a circle of radius R/2. After it has turned through angle θ, the centre is at (R/2) cos θ, (R/2) sin θ.
The marked point is R/2 from that centre. Because the small circle turns twice as fast in the opposite direction relative to the arm, the point's direction from the centre is at angle −θ. Its offset is (R/2) cos θ, −(R/2) sin θ.
Add the two: x = (R/2) cos θ + (R/2) cos θ = R cos θ, and y = (R/2) sin θ − (R/2) sin θ = 0.
So the point stays on the x-axis and moves as x = R cos θ. If θ increases steadily with time, θ = ωt, then x = R cos ωt, which is exactly simple harmonic motion with amplitude R. Two uniform circular motions add up to a straight-line oscillation. Our post on simple harmonic motion problems uses this same "SHM is the shadow of circular motion" picture.
A common mistake is to make both circles turn the same way. Try step 2 with the point at angle +θ and you get x = R cos θ, y = R sin θ, a circle of radius R, not a line. The opposite rotation is the whole trick.
Did al-Tusi influence Copernicus?
Ragep writes that al-Tusi's models influenced later Islamic astronomers such as Qutb al-Din al-Shirazi and Ibn al-Shatir, and early-modern European astronomers including Copernicus. MacTutor adds that the Tusi couple appears in Copernicus's work about 250 years later, but that not all historians agree he took it from al-Tusi; one view is that it came from Proclus's commentary on Euclid. This is a live question in the history of science, and it is fine to say so. For the background to the models al-Tusi was correcting, see our post on Ptolemy's Almagest.
Why is al-Tusi important for trigonometry?
MacTutor calls his Treatise on the quadrilateral the first work in history to treat trigonometry as an independent branch of mathematics, giving the whole system of plane and spherical trigonometry, including the sine rule. In modern form, for any triangle, a / sin A = b / sin B = c / sin C.
A quick check: if A = 30°, B = 45° and a = 10 cm, then b = a sin B / sin A = 10 × 0.7071 / 0.5 ≈ 14.14 cm. Students often divide the wrong way round. A quick sense check helps: the larger angle must face the longer side.
What can a student learn from al-Tusi?
Al-Tusi kept a strict rule (only uniform circles) and searched for a construction that met it, instead of dropping the rule when it became inconvenient. That is good practice in problem solving too: state your constraints, then look for a construction that satisfies all of them.
If seeing two circles add up to a line is easier than reading about it, the Fermi Simulation Lab has interactive simulations for ideas easier to see than read.
Try it now: sketch the Tusi couple at θ = 0°, 45°, 90° and 180°, and mark where the point is each time.
