As Ptolemy begins building his model of the celestial sphere, there are a few fundamental values that will need to be known. One of them is the angle between the celestial equator and the ecliptic (denoted in the below diagram as $\angle{\theta}$.
To do this, Ptolemy points to a design for a simple instrument1.
First, note that the ring is aligned along the North-South line. In other words, the circle is aligned to the meridian. The outer ring is marked with the degrees of a full circle (360º with as many subdivisions as possible. The inner ring is allowed to rotate freely within the outer ring. The two pegs on the inner ring overlap slightly on the outer ring and have narrow pins used to point at the scale.
The tool is used at local noon, by aligning the two pegs such that the shadow of the upper one falls on the lower one. When that happens, it is pointing directly at the sun allowing the observer to take a measurement of the angle of the sun’s altitude (above the horizon).
So how does that translate to the angle between the celestial equator and ecliptic?
Ptolemy doesn’t get in to it, but observing the altitude of the sun on the solstice and subtracting its altitude on the equinox gives the correct answer. Since Ptolemy didn’t, I won’t go into the geometry either. However, I will note that it’s not perfect and only provides a lower bound.
The reason is that, while we think of the solstices and equinoxes as full days, they’re actually instants in time as the sun is always changing position. Thus, the chance that the moment that the equinox or solstice occurs is your local noon (since that’s the only time you can use this instrument) is rather small. However, the difference is going to be rather small. I plugged the days into Stellarium and took the difference between the altitude at noon of the summer solstice for 2018 and the autumnal equinox and got 23;18,33º. The true value is currently 23;26,12.8º (23.4369º). That’s a difference of 0;7,39.8º (0.128º) which is well below the instrumental error of such a device. It wasn’t until Tycho Brahe’s time that such slight differences could be measured.
However, assuming better precision, the method could be improved upon by observing the sun’s altitude at the summer and winter solstices and dividing the difference in half. Using that method I got a result of 23;26,9.15º in Stellarium. That’s a difference of 0;0,3.65º (0.001º).
Ptolemy’s value came out slightly high at 23;51,20º (23.855º). In addition, he uses an approximation for the summer to winter value (so twice the angle between the ecliptic and equator) which is $\frac{11}{83}$ of 360º.
Ptolemy also devised another, more straightforward instrument2:
This one is simply a quadrant on a block of stone or wood (with 0º being upper right in this diagram and 90º being lower left), again aligned along the meridian. A round peg is placed at the center and another at the bottom edge of the quadrant. A plumb line is hung from the top one to the bottom such that it would hang just over the lower peg when it is level. When the sun is in the right position, the shadow of the peg would point to the angular height.
While simpler, I find this design less useful because there is no way to affirm the sun was directly on the meridian. With the previous one, the aligning of the shadows would indicate that it was the appropriate time to take the measurement. With this, it would be very hard to determine with accuracy.
In both instances, Ptolemy concerns himself greatly with ensuring that the tools are level. In fairness, being level is not a paramount concern since the difference in altitudes is being take which means any offset from level would cancel out. What is important is that it be well centered on the meridian. Oddly, Ptolemy did not describe how to do this3.
- Image taken from Almagest translation by Toomer
- Image again from the Toomer translation
- However, methods to do so were known to the ancient Egyptians.