Our modern graphing calculators can draw any curve given its algebraic equation. Everyone is familiar with the compass, a tool invented to draw perfect circles. The ancients needed to invent similar types of instruments to draw other curves.
To illustrate, below are some ancient ellipse drawers.
| String Construction | Archimedes' Trammel | Proclus' Ellipse Drawer | Van Schooten's Ellipse Drawer |
Description of the Four Ellipse Drawers
- In the string construction, two ends of a string are attached to two fixed pins $F$ and $F'$. Then a pen holds the string taut as it rotates around the pins. The first person known to have written about the string construction of the ellipse was Anthemius of Tralles in the fifth century.
- The next ellipse drawer, attributed to Archimedes, is the trammel construction. The point $B$ on segment $AP$ is attached to the horizontal axis in such a way that it can slide along the axis. Similarly, the point $A$ is attached to the vertical axis and allowed to slide along it.
- The next ellipse drawer would have been known to Proclus (418-485), who knew that an ellipse could be generated by tracing a point $P$ inside a circle that rolls without slipping inside and tangent to another circle whose radius is twice as long. Such a construction brings to mind the popular spirograph game that children use to generate cycloidal curves.
- Our last ellipse drawer is due to the Dutch mathematician Frans Van Schooten (1615-1660). Van Schooten was very interested in conic section drawers and wrote a treatise on them. He devised several instruments for drawing conic curves, including the ellipse drawer shown.
Exercises
Consider the string construction of an ellipse. Assume a coordinate system with $F$ and $F'$ on the $x$ axis and the origin half way between $F$ and $F'$. Then the coordinates of $F$ and $F'$ are $(\pm c,0)$ for some $c$. If $P = (x,y)$ is a point on the ellipse, then we must have $|PF|+ |PF'| = d$ where $d$ is a constant (the length of the string in the string construction).
1. Use the distance formula to translate $|PF| + |PF'| = d$ into an algebraic equation involving $x$, $y$, $c$, and $d$. Solve for $y$ to obtain two equations describing the top and bottom halves of an ellipse. Find the values of $c$ and $d$ such that the ellipse has $x$ intercepts at $(\pm 5,0)$ and $y$ intercepts at $(0,\pm 3)$. Substitute these values into your equations and simplify. Use the sliders in the applet below to set $c$ and $d$ and verify that the resulting ellipse has the correct intercepts. Enter your equations into the input field to obtain graphs and verify that you get the ellipse shown.
2. The applet below shows the ellipse drawer devised by the Dutch mathematician Franz Van Schooten. Click the Play button at the lower left hand corner to generate the curve. What should be the lengths of $a=CD$ and $b=DP$ so that the ellipse generated will have $x$ intercepts at $(\pm 5,0)$ and $y$ intercepts at $(0,\pm3)$. Use the sliders to set $a$ and $b$ to these values and verify that the resulting curve has the correct intercepts. Enter the equations that you found in Exercise 1 into the input field to verify that Van Schooten's ellipse drawer does generate an ellipse.
3. The applet below shows the ellipse drawer devised by Proclus. Click the Play button at the lower left hand corner to generate the ellipse. What should be the lengths of $a=OA$ and $b=PC$ so that the curve generated will have $x$ intercepts at $(\pm 5,0)$ and $y$ intercepts at $(0,\pm3)$. Use the sliders to set $a$ and $b$ to these values and verify that the resulting curve has the correct intercepts. Enter the equations that you found in Exercise 1 into the input field to verify that Proclus's ellipse drawer does generate an ellipse.
4. Finally, the applet below shows the ellipse drawer devised by Archimedes. Click the Play button at the lower left hand corner to generate the curve. What should be the lengths of $a=BP$ and $b=AB$ so that the curve generated will have $x$ intercepts at $(\pm 5,0)$ and $y$ intercepts at $(0,\pm3)$. Use the sliders to set $a$ and $b$ to these values and verify that the resulting curve has the correct intercepts. Enter the equations that you found in Exercise 1 into the input field to verify that Archimedes's ellipse drawer does generate an ellipse.
