History of Astronomy in a Nutshell
- ~350 BC, Aristotle philosophizes that the celestial bodies must move in constant circular motion. He is fully aware of retrograde motion and challenges his students to explain the apparent paradox. This belief essentially still stands, however Aristotle's idea of the Earth at the center of the universe has been dismissed.
- ~150 AD Claudius Ptolemy writes the Amagast. The equant is used to explain the observed speed change in planetary orbit during different stages of the orbit. This device allows Ptolemy to remain true to Aristotle despite massive mathematical contortions.
- 1247 Nasir al-Din Tusi writes Tahrir al-Majisti (Commentary on the Almagest) which offers an alternative, more elegant technique to account for the apparent motion of the planets.
- 1543 AD Copernicus publishes (on his deathbed) Dē revolutionibus orbium coelestium placing the Earth in the center of the universe and utilizing Tusi couples to make it work.
Simulation
The simulation below keeps the outer circle stationary as the inner circle rotates: This shows how a perfectly straight line can be obtained using two circles (as idea that will latter be exploited in the invention of planing machines).
However, Tusi himself envisioned both circles rotating to produce retrograde motion (see sidebar).
Tusi's Explanation
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.
The Tusi Couple is a special case (where \(R = 2r\)) of what are now called hypercycloids described by the following equations for circles of radius \(R\) and \(r\): $$x(\theta) = (R-r)\cos{\theta} + r \cos \left( \frac{R-r}{r}\theta \right)$$ $$y(\theta) = (R-r)\sin{\theta} - r \sin \left( \frac{R-r}{r}\theta \right)$$
Explore
The ratio \(\frac{R}{r}\) is extremely interesting
See if you can figure out what happens when this raiio is a whole number vs when it has a fractional part.
di·dac·tic
/dīˈdaktik/
adjective intended to teach, particularly in having moral instruction as an ulterior motive.
