The primary instrument I’ve used for my observing is an astronomical quadrant. That instrument is designed primarily to measure the angular distance of an object above the horizon1, otherwise known as its altitude. However, this isn’t the only instrument good for this sort of thing. Brahe’s Astronomiae Instauratae Mechanica is filled with instruments that essentially fill this same purpose, but in different ways.
One design, he describes as a “parallactic instrument” but it was also known as a triquetrum in period. This design dates back to Ptolemy and is described in Chapter $12$. Here’s a drawing of it from Toomer:
Let’s break down how this instrument works, starting by breaking down the parts. First, we have the upright pillar ($1$). This pillar has a base ($4$) and a plumb (d) to help level it. It’s also inscribed with a line (e) that is parallel to the face of the pillar which marks the plane the instrument needs to be placed in which is the plane of the meridian.
The face of this pillar has a scale divided into $60$ parts and each part is subdivided as much as possible. The overall scale of this is described to be “no less than $4$ cubits, which is about $6 \frac{1}{2}$ft, so I’d imagine it could be subdivided quite readily. The scale starts from the bottom at $0$ and increases as you go up until you hit $60$. At both the $0$ and $60$ mark, a peg in inserted about which another arm rotates. To help keep them separate, I’ll call the top one the arm, and the bottom one the rod.
The top arm rotates on one of the pegs (c) and has a set of sights on it with the one nearest to the observer having a small hole, and the further one having a larger one, sufficiently large that the full diameter of the moon can be viewed through it when looking through the smaller hole. Down the center of this arm a line is drawn. While this arm should be able to rotate with some pressure from the user, it should be tight enough that it holds its position when not being held. The rod also rotates on a peg (f) and also has a center line drawn on it.
To use the instrument, the observer would sight down the arm. Once the moon is sighted, the user then takes the rod and rotates it so its center line meets the center line on the edge of the arm (as shown in the lower image). When that is done, we have essentially laid out a triangle in the plane of the meridian2.
If we inscribe a circle around it, with the upper vertex at the center, that circle effectively represents the meridian and that upper angle is a vertical angle, and thus equal to the angle of the object being sighted from the zenith.
So that’s one difference from the quadrant: It’s not measuring angle above the horizon, but angle from the zenith. Of course, the zenith is $90º$ from the horizon, so you can quickly get from one to the other3.
However, we’re not directly measuring the angle. Rather, this instrument determines the length of the rod between the pillar and the arm, shown above as $x$. But you’ll notice Ptolemy didn’t say there’s any scale on this rod. Instead, it’s on the pillar. While later period instruments certainly did have the scale on that rod, Ptolemy instead marks the rod, once aligned, and then rotates it until it’s upright on the pillar so he can take the reading there.
Immediately, there’s an issue that comes to my mind that isn’t discussed which is that, while Ptolemy describes the length of the rod as “long enough to reach the end of the line on the [arm] when [the arm] was rotated to its maximum distance [from the base].”
To me, this suggests that the arm could be rotated up to $90º$ from the base4. If doing so, this would form a right triangle with the rod as the hypotenuse which would be the longest side. If so, then there’s no way the rod could be rotated to the scale as the peg (c) would be in the way. I can imagine a way around this by carving out a slot in the rod to be able to fit around the peg, but in reality, this instrument was designed to be used for the moon which, at Ptolemy’s latitude, would never be more than $60º$ from the zenith while on the meridian. Thus, the rod could be shortened to the same length as the arm and would be entirely usable.
But how to get from this straight line distance to the actual angle? Quite easily. There’s a reason that Ptolemy chose $60$ parts as the scale here as it’s the scale for his generic circle. The length measured of the rod is the chord length from which the angle can be determined readily from our table of chords.
That covers how the instrument is constructed and used. In the next post, we’ll explore Ptolemy’s lunar observations using the instrument and his rationale on when they were taken.
- I say primarily because I did add an azimuthal scale. However, it can be argued that this just lets me measure the altitude of objects not on the meridian.
- This is why the instrument was called the triquetrum which literally means “three cornered”.
- There’s been a few times I’ve sighted down the wrong side of the quadrant which does the same thing. Yseult and I got several stars in during one observing session before I realized a star that was certainly higher than the previous one had a lower reading and discovered the error. But it wasn’t a problem since I could easily subtract that number from $90º$ to get the correct value.
- Which would be observing an object on the horizon.