The first volvelle in Astronomicum Caesareum is the constellation volvelle. It depicts a view of the northern constellations to a declination of approximately -20º. The depiction is based on a 1515 star map by Albrecht Durer.
Although this volvelle (as well as all of the others) are illustrated to be reminiscent of an astrolabe, this one is not capable of depicting the sky at a specific time as an astrolabe can1. Rather, the purpose of this volvelle is related to the precession of the equinoxes – a slow change of the position of the vernal equinox in the zodiac over a 26,000 year cycle. Determining this is important because it carries the line of apsides and the nodes of the planets along with it.
The understanding that the equinox precessed is generally attributed to Hipparchus around the 2nd Century BCE. However, most of Hipparchus’ writings have been lost and what we know of them generally comes to us through Ptolemy’s writings in the 2nd Century CE. In the Almagest, Ptolemy tells us that Hipparchus believed the equinox to move 1º per century. Ptolemy checks this number and eventually concludes the same2. The value was revised over subsequent centuries, but was eventually complimented by the concept of trepidation sometime in the 9th century. This hypothesis proposes that the position of the equinox has a back and forth motion in addition to the linear precession3 of Ptolemy. While incorrect, this hypothesis was widely accepted as part of medieval cosmology although there were disagreements in how large the back and forth motion was and its period. Thus, the use of this instrument first requires two parts: Adjusting the volvelle for the linear precession and an adjustment for the trepidation.
The first of these is done by first finding a hidden scale beneath the rotating disc along the right side (shown at right). The horizontal line which divides the years before Christ (ANNI ANTE NATIVITATEM CHRISTI) from the years after (ANNI POST NOTIVITATEM CHRISTI) is marked with a cross (✠). A corresponding tab on the rim of the rotating disc is also marked with the same symbol and should be turned to this position, which sets the dial for the year 1 CE. Then, the number of years before or after this is determined and the tab on the disc rotated to the corresponding year4. Obviously, the scale is far too small to turn to the exact year, as each division represents a full century.
Next, the disc must be adjusted to account for the trepidation. This is done by locating a small, oval shaped scale immediately next to the tab with the cross (shown right). Again, this oval is marked with the years before and after Christ. To make use of it, a thread from the center of the volvelle is pulled to the correct year, held in place, and then the disc rotated so that the flat side of the tab marked “Aux Y” is lined up with the thread. Once this is done, the disc is set properly.
Apian tells us that we can learn several things once the disc is aligned. The first is the ecliptic coordinates of the stars depicted. The ecliptic longitude can be determined by pulling the thread through the star in question and finding where it falls along the outer zodiac ring on the base. For the latitude, Apian instructs the user to again draw the thread through the star, and then use the feet of a compass to measure the distance between the star and the ecliptic (which is depicted as a solid black line). This distance can then be compared to a scale near the tail of Sagittarius.
But, more importantly, the dial can now indicate the positions of apogee for each of the five naked eye planets. This is done via a series of tabs placed around the border. This information will be important in later calculations for the planets.
Issues & Restoration
Gingerich identifies two issues with this volvelle. The first is that the central thread is missing. Second, some of the tabs around the circumference of the disc were improperly trimmed off. In my copy, I also found that the disc was not properly centered. Lastly, there may have been a small seed pearl on the missing thread that could be slid back and forth to be used as an indicator as Apian’s instructions appear to mention a pearl (margaritam).
Reviewing the digital copies listed above, there does not appear to be a standard color for the threads used. Red and green are the most common colors. None show any evidence of a pearl.
To restore the volvelle, I first removed the disc to investigate how it was constructed. Removing the disc resulted in some tearing to the disc itself (seen in the image at right), although the paper was thick enough that it did not pull all the way through to the top surface. The image at right shows the top of the volvelle removed making the misalignment obvious. In the image you can see the true center (blue), which is clearly offset from the center of rotation (red). Although slight, this difference could create an error of 1-2º in the results from using this volvelle. Given that the positions of the apogees of the planets are used later, it is important that this volvelle be as accurate as possible lest the error propagate. As such, I decided to remove this assemblage and recreate it more carefully.
Removing the remaining layers that allowed the disc to rotate was quite difficult as the glue holding down the larger circle with the hole was still in quite good condition5 resulting in the paper, both of the assemblage and the base page tearing as can be seen (below) where the page tore towards the top and the circle tore near the bottom. Fortunately, the paper on the page was sufficiently thick that this did not tear through to the reverse side of the page.
Examining the pieces, I found this volvelle was attached to a circular shaft of thick card stock affixed to another larger circle beneath it, also on card stock. This was placed on the base of the page, but not affixed.
To hold it in place while it rotated, another larger circle of similarly heavy card stock with a hole cut out for the central shaft was placed over it with another ring of thinner card stock around the edge. In this way, both the central shaft and its base were constrained. A cross section of the layers is shown below.
I then took measurements of all of the pieces
- Central shaft: 7.5cm (radius = 3.75cm)
- Shaft Base: 14cm (radius = 7cm)
- Covering Disc: 20.5cm (radius 10.25cm)
In removing all of the pieces, I also discovered a central registration mark. To make sure the covering disc was properly centered this time, I first cut it without cutting out the central hole. This was done using a rotary cutting compass on 6
5lb cardstock. Despite the thickness of the cardstock, I found that this tool still tended to pull the paper and thus found it necessary to tape the corners down to my cutting mat. Once it was cut, I sketched quartering marks onto the circle.
Since the fixed center of the compass left a small hole in the cardstock, I could insert a fine sewing pin through this onto the registration mark to ensure it was centered. I then traced the circle onto the base of the page and extended the quartering marks, labeling them A-D to ensure that I could align it properly again once the central hole was cut.
I then cut a circle of the same size out of 32lb paper and then removed the center from this with the diameter of the shaft base. This was then glued to the covering disc using generic school glue. However, this glue makes the paper somewhat flexible and there were some wrinkles in this which I didn’t like, but not so much that I felt it necessary to redo.
I then cut the shaft base from the heavier cardstock and then removed the central shaft from the upper disc. These were attached together using school glue with a pin through their centers to make sure they were concentric. This meant that the covering disc and the central assembly were complete. Both of these layers were placed under heavy books and allowed to sit for some time to hopefully avoid warping.
Once sufficiently flat and dry, I placed the covering plate face down on my table and inserted the central shaft assembly (also face down), rotating it to ensure that its edges did not protrude onto the ring around the covering plate. A few edges were trimmed slightly and the central shaft was taped in place temporarily.
I then rubbed glue on the ring around the bottom of the covering plate and aligned it on the calibration marks I had created. Next, I removed the tape to ensure that the central shaft could still rotate freely. Although it was hard to rotate by just pushing on the top of it, I knew this was the case from when I had removed the constellation disc originally, and was satisfied that, once I re-attached the disc, it would be fine. However, I continued to rotate it as much as possible as the glue dried to ensure that, if any glue had gotten on it, that it wouldn’t become permanently stuck in place. Fortunately, that didn’t seem to be the case.
I then turned my attention back to the constellation disc itself. I first needed to insert the missing thread. For this, I selected some lightly-waxed leather stitching thread I had on hand as it appeared to be an appropriate thickness. I poked a small hole in the center of this disc using the sewing pin and was able to slip the thread through without the need of a needle. The tail was then lightly glued to the reverse of the disk with a drop of super-glue.
At this point, I was ready to reattach the disc to the layers beneath, but I was concerned that the tearing that had reduced the thickness of the paper when I removed it (seen at right at center) may cause it to tear further when the disc was rotated. As such, I decided to reinforce this disc by affixing another circle of card stock to its reverse. I again used the 65lb cardstock for this. However, because I was concerned about how the school glue allowed the paper to warp previously, I instead used a gel-super glue. While this did provide an excellent bond without warping, I feel that it did end up showing through the constellation disk slightly (see the slight discoloration just below Ursa Major’s head on the image on the next page). As such, I’m not sure I would use it again on layers that are to be presented, or on the base page.
With that done, I again used the sewing pin to align the centers and attached the constellation disc to the central shaft, again using the gel-super glue. I closed the book to apply pressure and let it dry for a few minutes.
Once done, I opened it again to test functionality. The disc seemed to rotate smoothly and the problem with it being off center seemed mostly resolved.
However, one edge (near the constellation of Argo, upper right in the image at right6. Although it looks like the left side spills over the scale as well, this is caused by the disc hovering over the page as it tucks into the binding), still seemed to cover the scale more than it should. However, its opposite edge was not inside the circle created by the scale as it had previously been, so overall the fit was a significant improvement and this issue is likely caused by some non-circularity to the disc itself.
The final remaining step was to attach the missing tabs. A sheet of replacement parts was included in Gingerich’s restoration kit. These were again printed on heavy card stock and were cut out in preparation for attaching.
However, finding the correct position for them was more difficult than previous steps as there is nothing to indicate where they should have been. Instead, Gingerich states that,
If stub X beside the trepidation oval on the planisphere is set at 0s0º, the sub ‘Y Aux Comunis’ must be completed at 0s18º30’, and the sub ‘Aux ♂’ must be added at 4s14º (just below the spar for the mast of Argo Navis); the stub for ‘Aux ☿’ must be completed at 6s29º30’ (just below the hindmost foot of Centarus).
The notation here is somewhat confusing and in need of explaining. The first value (followed by a superscript “s”) indicates which sign is being referred to on the outer scale. Thus, Gingerich is instructing the reader to place tab X at the beginning of Aries which is on the right of the dial. Then, the position for the other tabs can be determined using the scale. They again denote the sign and then how far into that sign it should be placed in degrees and minutes (where there are 60 minutes in a degree).
Thus, there are three tabs that need to be replaced. The first of them, (“Y Aux Comunis”) was not fully trimmed off by those that assembled the book. However, it was cut such that it only included the word “Comunis” as shown at right with the replacement tab behind it but not attached yet. That this was the correct position for the tab was verified by comparing to the digital copies and the replacement tab placed behind it to make as much use of the original tab as possible.
The second tab (“Aux ♂”) was fully missing by the base of Argo. This was replaced in whole.
The final tab (“Aux ☿”) also was not fully trimmed. However, this one was reduced to just the letter A, next to Centaurus’ hoof (shown at right). Again, the replacement tab was affixed behind this making use of the existing “A”.
These changes completed the restoration of this work.
Testing the Results
In order to test that the changes had the desired effects, I followed the examples that Apian gives as part of the text (see the Appendix for my working translation). There are two examples given: One for Charles V (Holy Roman Emperor) and a second for his brother, Ferdinand I. In both cases, the dial needs to be set to the correct year (as closely as possible).
To do so, I look under the dial for the scale giving the precession and find where it is 1500 after (post) Christ, the year of Charles’ birth7 and close enough for Ferdinand. I could not pull the string directly to this since I had to lift the disc to see it, but it is immediately adjacent to the zodiac scale, so I noted where on that scale it falls and placed the thread across that once I placed the disc down again. I then pulled the tab marked X to that and then found where on the trepidation oval 1500 is as well, pulled the string taut, and moved tab Y to that position.
This example is to find the ecliptic longitude of the “lion’s heart” (Regulus) which is somewhat obscured by the ecliptic line. The thread was pulled through this and the longitude read off the outer dial. I found this slightly challenging as the bottom of Argo slightly obscured the scale on that side, although the opposite side of the disc (with the water poured from Aquarius) does not pull inside the scale, indicating that the disc is simply not quite circular. However, doing this, I came up with a longitude of 22.5º into Leo. The value Apian gives is 22º8’ (22.13º). Apian’s value was most likely calculated as this is too precise for this instrument. As such, our values are in good agreement.
For the example problem for Ferdinand, Apian simply reads off the positions of all the tabs around the edges. In all cases, I found agreement within 1º, which is quite reasonable.
Conclusion
The volvelles of in Astronomicum Caesareum are elaborately constructed. Disassembling this and putting it back together took far more time than I had initially estimated. Some of the complication was getting the circles cut sufficiently well that I was willing to use them. While taping the paper to my cutting mat helped, any lean caused by switching hands using the circular cutting tool resulted in it not lining up correctly when it came around. Thus, I had a large number of rejected circles cut.
Although I’m generally pleased with the results, I think for future volvelles, I will find a different adhesive. As I neared completion, it was recommended to use a pH neutral PVA adhesive which is commonly used for archival purposes. Before continuing with this project, I plan on testing that as I am told it is more reversible should future errors need to be corrected.
- To do so would require another layer to represent the local horizon.
- It should be noted that the math Ptolemy does to arrive at this conclusion is among the worst in the Almagest as steps along the way are repeatedly rounded in odd ways. My opinion is that Ptolemy was intentionally doing so to arrive at the same value as Hipparchus.
- There was an earlier version of the hypothesis attributed to Theon of Alexandria (4th Century CE) who cites unknown even more ancient but unnamed astronomers, in which there was no linear component and the position of the equinox only varied cyclically.
- The rate of precession that is used is 1º per 136 years. This is extremely far off of the correct value of 1º per 72 years, but the effect of the presumed trepidation had been calibrated such that the inclusion of this effect gave a good approximation to the correct value during the medieval period.
- For some, later volvelles, the glue had lost its hold and one of the volvelles was already partially disassembled when I received the book, with no apparent damage, indicating that the glue on this volvelle had simply deteriorated.
- This appears to be the case in some of the reference images as well. I suspect that the woodcut was not truly circular.
- The text states that Charles was born “Augustus 23, Februarii natus est, 15. hora. 44 m pomeridianis, post Christi annum sesquimillesimum”. The number of years (sesquimillesimum) doesn’t translate cleanly. As such, I am basing the year off the known birth year instead of the text for this example.