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Friday 24 March 2023

Astronomical Adventures With An Astrolabe

It's time for another post about my weird hobbies. Hooray !

Last year I want on a redecorating spree to turn my home office space from a soulless IKEA nightmare* to somewhere I would voluntarily choose to be. Naturally this included a lot of astronomical elements, one of which now deserves its own post.

* This is not to be confused with a John Lewis nightmare, which is only possible for very rich and stupid people.

As a present to myself for finishing a paper I bought an astrolabe. Why ? Mainly because when I was searching for suitable furnishings I stumbled on similar items, but these in particular caught my eye. I just think they look lovely :

I was also curious to learn how they worked, how astronomers in days of yore actually did things. Now I'd absolutely love a proper shiny brass version, but those tend to be offensive prices, or of a design style I just don't like, or worst of all... non–functional. Urrgh ! Which is a bit silly in this age of mass manufacturing, but I guess the bottom fell out of the astrolabe market even before Jacob Rees–Mogg was a thing.

Still, one day...

There being not enough hours in the day to get everything done, I figured out only the absolute basics before I got distracted by other projects. But now it's time to return to this medieval marvel and figure out what it's all about.

The version that I have is from Dreipunkt and is definitely nice to look at, wooden materials notwithstanding – though why their deluxe versions are so outrageously expensive is beyond me. Unfortunately the instructions that come with it are limited to a 4–page guide, including the assembly instructions (oh, and and a tiny quickstart card for some reason). So, this required quite a lot of Google searching on my part, because the astrolabe community is apparently an oxymoron. But I won't bore you with the narrative. Instead, let's get right down to business by describing how to actually use the bloody thing.

In Theory

0) Convert calendar to Zodiacal date

All of the astrolabe's celestial calculations use a special measure of the date which corresponds to the Sun's angular position along the ecliptic, the path it traces across the sky. Since this is just a circle it spans 360 degrees, but of course the actual calendar year is either 365 or 366 days long. This means we need to do a conversion between these two dating systems. For this, we start on the back.

The information density here is quite high, but what we need to do is actually really easy. There are three circular scales towards the outer edge of the back piece. The outermost is for measuring the angle of a target above the horizon and we don't need this right now. We just need the other two.

All we do is align the alidade (the rotatable pointer) on the back with the innermost of the three scales shown, the calendar date scale – each of the segments is divided into the correct number of days for each month. From this we simply read off where the alidade is pointing on the zodiacal date scale, taking care not to confuse it with the altitude scale on the very outer edge. This will give a value in degrees within a constellation, e.g. 11th March is 21° in Pisces.

Here shifting things a wee bit just so the scales can be seen.

And that's all we have to do for step zero. But you're probably curious, I hope, about what this actually means, so I shall tell you. Yes, even if you don't want me to.

Supposedly it corresponds to the position of the Sun in the zodiac. With each constellation defined to span 30 degrees, the Sun should on 11th March be about 2/3rds of its journey through Pisces. Now obviously the constellations of the zodiac don't really span exactly 30 degrees each, but even so, there's quite a discrepancy, because on 11th March the Sun is  actually in Aquarius. At least it's still a watery constellation, I suppose. 

Pisces is only just above it but clearly the Sun is nothing like "2/3rds" of the way through it !

Why is this ? This is be due to the slow precession of the Earth's orbit, meaning that the Sun's position is never quite in the same place relative to the constellations each year. On human timescales (i.e. the time over which anyone would ever use an astrolabe) this is negligible, moving by about 1 degree every 72 years. But the conventions for setting where the constellations are defined for this purpose were devised about 2,000 years ago, long enough that there has been a considerable shift.

Finally a note on terminology. I've called it "zodiacal date" following this link, but there doesn't seem to be an agreed-upon convention. "Astronomical" or "astrological" date would also work, maybe even sidereal date. The guide included calls it "star sign date" which is for obvious reasons very hard to Google. Another document calls it the ecliptic longitude of the Sun, which is probably the correct term.

1) Calculating sunrise and sunset times

Here the official guide is really quite a muddle. It's a very small section, which is odd because sunrise and sunset times are probably the easiest values to look up to verify the results. It doesn't help that the images in the guide say they're using an astrolabe calibrated to a latitude of 50°N but actually aren't, because I couldn't reproduce their values despite setting everything exactly as shown in the images. So I pieced this together from the other parts of the guide and from Google searching.

But again it turns out to be easy. Having completed step zero, we turn to the front of the astrolabe. There's quite a bit of information to process here, but it's all very small steps. Here's the bits we'll need to use :

The "rete" is also known as the spider and can be turned. The circular ring in the rete, which has the zodiacal date marked on its edge, corresponds to the ecliptic. To find the sunrise time, we turn the rete so that the zodiacal date (using the value we calculated in part 0) on the ecliptic ring intersects the horizon line on the left side of the astrolabe. This can be quite fiddly, and some times of year are harder than others because of the higher density of markings on the date scale. But it's possible to be fairly accurate without that much effort.

Using the example of 11th March = 21° Pisces as before.

If we wanted to find sunset time, we'd turn the rete to intersect the horizon line on the right hand side of the astrolabe instead.

Next we turn the pointer to align with this point of intersection. We then use it to read off the time from the outermost scale on the astrolabe.

I guess maybe it's a German convention (?) but we have to take care with this particular model because it uses "IIII" instead of the standard Roman numeral "IV". Other than that it's simple enough, with each small tick marking off 5 minute intervals. So in the example of 11th March, we find that the time is somewhere between 6:15 and 6:20 am. The actual sunrise time for this date is 6:25 am, which is not too bad at all.

But this is misleading. Now I'm doing this write-up some time after my first experiments, but when I did them I was getting values which were rather less impressive. And this is where the guide has an important omission. Being curious as to what was going on, the only way to proceed was to redo the calculation for a whole bunch of different dates throughout the year. Lo and behold, it's usually wrong : considerably worse than the five minute precision should allow. 

As you can probably guess, the major ticks mark the end of each month with the minor ticks marking each week. There's not really any obvious relation of the offset to the solstices or equinoxes.
Also, the fact that I was able to calculate so many points is testament not so much as to my obsessive zeal as to how easy the instrument is to use with a bit of practise.

It's decent enough for about half the year, within about 5 minutes accuracy. Pretty good ! But the rest of the time it's nigh-on miserable, being up to 20 minutes out – and systematically so, not just because of random errors.

This led me on a merry dance to figure out why this should be. The graph shows quite a distinct trend in the offsets, something like a sinusoid but not exactly (I tried to make it fit but failed). Now coordinate systems are an area of geekdom I am resolutely uninterested in; this level of precision orbital mechanics gets a bit tediously maths-heavy for me. But this article led me on the right path : 
Now this is solar time, of course, and by May, US Daylight Savings Time will be in effect, so I add one hour. Then there is the time zone issue: I have to compensate for the difference between Houston’s longitude, 95 degrees west, and that of the longitude to which its time zone is pegged, 90 degrees west. For every degree I need to add four minutes: a total of 20. Finally, I have to find the “equation of time chart” that compensates for arcane astronomical eccentricities, and it says that I have to add three more minutes for May 6. 
The hour compensation for daylight savings is trivial and already included in the above graph. The correction for longitude is likely unimportant, though the guide doesn't say what this particular astrolabe is set to. I'd guess Berlin, which is at a very similar longitude to Prague so this makes not much difference. Playing around with online sunrise calculators, I couldn't find any longitude that would give be better values on one date without making them worse on another.

But the article also mentions a third correction for the "equation of time", which in their example happens to be a small difference. However this is just happenstance. This difference is the correction for the Earth's orbit not being a perfect circle, which means the Sun doesn't move across the sky at the same rate each day. And lo, the form of this correction from mean solar time (which effectively pretends that the Sun does move at a fixed rate across the sky) to actual solar time looks like this :

Very much like the offset from the astrolabe ! Exact values for any date can be found here. These do vary annually, but only slightly : the value any year in my lifetime is going to be good enough to use for every other year I'm likely to be alive, barring significant medical advancements.

In the end, all we have to do is to read off the time, add an hour if we're in daylight savings, subtract 4 minutes for longitude, and subtract the offset from the equation of time. And this gets a much better result. Now our results are typically accurate to within 5 minutes, below the precision of the scale ! For our 11th March example, the offset is -10 minutes, so our reading of ~6:17 am becomes 6:27 am (the offset is negative and we subtract it, i.e. add ten minutes), only two minutes away from the actual value of 6:25 am.

For the longitude correction I guessed the astrolabe was calibrated to Berlin. The difference from Prague is 1.0328° => 4.1312 minutes offset. Gaps are where the struts from the ecliptic ring to the rest of the rete make it impossible to get a reading.

It's interesting that there still appears to be a residual pseudo-sinusoid. Perhaps this is an error in the longitude correction, a problem with the accuracy in reading the scale, or with the accuracy of the device itself (e.g. with the rete not being exactly centred). However since we're already below the precision of the scale, there doesn't seem any point in worrying about this too much.

In summary, the steps are :
  1. Convert the calendar to zodiacal date.
  2. Align the zodiacal date on the rete to the horizon line.
  3. Align the pointer with the zodiacal date and horizon line on the rete, read off the clock time.
  4. If necessary, add one hour to account for daylight savings time.
  5. Subtract the equation of time offset for the current date.
  6. Correct for longitude. For every degree east of the astrolabe's calibration, subtract four minutes (for every degree west, add four minutes). 

2) Finding the time from the Sun

Okay, if we know the date we can predict the sunrise or sunset time, wait for these and then we'll know the time. This is hardly any better than the proverbial stopped clock, but fortunately we can also convert the Sun's position at any time of day.

Calculating the sunrise time is equivalent to predicting the time the Sun will have an altitude of 0°. We can generalise this to finding the time the Sun will have any given altitude, but since the Sun being at 37° is of no particular significance to anyone, it's more useful to reverse the process : when the Sun is at 37°, what's the time ?

Well, let's make this easier on ourselves. First, we can measure the altitude of the Sun using the alidade, which has a hole on each end we can sight through. But... let's not do that. Instead, let's use the online tools to find some nice values, so we can verify this all works before getting our hands dirty. 

The solar position calculator says that in Prague on 5th March 2023, at 1:50 pm the Sun is at an altitude of almost exactly 30°. The zodiacal date for 5th March  is 15° Pisces. So what we do is align the rete's zodiacal date of 15° Pisces not with the horizon line, but with the 30° altitude line.  As with sunrise and sunset, there's a degeneracy here since there are two positions at which this is possible, one in the morning and one in the afternoon. So we need to take it as known that it's the afternoon value we want. This is easy enough because we could just take two altitude readings of the Sun a few minutes apart; if the altitude increases it's morning, if it decreases it's afternoon. Simples.

Here choosing the rete to intersect the line to the right of the XII marker on the top, since we want the afternoon time.

And once again we align the pointer and read off the time from the outer scale :

Which is a value of about 1:37 pm. Not great, but we need to apply the corrections : add 11.5 minutes because of the equation of time, then an additional 4.13 minutes because of longitude => 1:52 pm. Two minutes out ! Not too shabby at all. Well within range of 99% of everyday practical uses of knowing the time. Jeez, I could use this thing to set the clock on my VCR...

To summarise, the steps for this procedure are :
  1. Convert the calendar to zodiacal date.
  2. Use the alidade to measure altitude of the Sun, if necessary twice to see if it's before or after noon.
  3. Align the zodiacal date on the rete to the corresponding altitude line.
  4. Align the pointer with the zodiacal date and altitude line on the rete, read off the clock time.
  5. If necessary, add one hour to account for daylight savings time.
  6. Subtract the equation of time offset for the current date.
  7. Correct for longitude. For every degree east of the astrolabe's calibration, subtract four minutes (for every degree west, add four minutes). 
Of course a limitation here is that we only have altitude lines every 5° so this is going to limit our precision. As a second example, on 18th March  (28° Pisces) at 3:05 pm the Sun's altitude is 27°. Guestimating the rete alignment as best I can (particularly difficult because 28° Pisces is near a strut, which gets in the way), the astrolabe time reading is 2:40 pm. Adding 8 minutes for equation of time and 4 minutes for longitude gives 2:52 pm, 13 minutes out. So there are definite limitations here, though there's no reason we couldn't have a finer altitude scale.

3) Finding the time from the stars

Made with Stable Diffusion.

So far the astrolabe seems to be a sort of elaborate sundial. We can use it to estimate the current time to within a couple of minutes under optimum conditions, and also predict sunrise and sunset with about the same precision. Wonderful, but there are some obvious disadvantages to this at night.

That's where the star markers come in. The rete contains little pointers that mark the positions of various stars : mine has 17. This includes some in the most obvious constellations : Rigel and Betelgeuse in Orion; Alioth in the Plough. Even I can find those ones unaided, so this doesn't require any particularly esoteric knowledge of the night sky.

The method here is basically identical to the case of finding the time from the Sun. We use the alidade to find the altitude of a star (the position of major stars can be checked here), then align its marker with the corresponding altitude line. And again this gets us to within a couple of minutes of the correct time, under optimum conditions and applying the standard corrections.

4) Finding the stars

We can also reverse this. If we already know the time, which of course we can get from the stars anyway, we can use the astrolabe to find their position instead. We read off their elevation directly from the altitude lines. Again, we have to guestimate if a pointer doesn't lie neatly on a line, but I was able to get typically to within 2 degrees despite this. 

The astrolabe encodes two dimensional information about the star's position, meaning we can also get its azimuth. This too is straightforward. We align the pointer with the star marker, then from this we read off using the scale just interior to the time on the outermost edge. The only slight complication is that the modern convention is to give the angle in degrees to the east from the line due north, whereas the astrolabe's values are in degrees to the west from the line pointing south. This conversion is trivial :

  • If the astrolabe's value is > 180°, subtract 180 to get the modern convention.
  • If the astrolabe's value is < 180°, add 180 to get the modern convention.

The azimuth scale has ticks every degree, though I was able to get agreement to typically within 5 degrees of the actual value. Which in terms of finding the stars by eye is way more than sufficient.

5) What else ?


This is looking pretty impressive now. We can calculate sunrise and sunset times, find the altitude and azimuth of the Sun at any time, or use the current elevation of the Sun to tell the time. We can locate the stars to within a few degrees and use them to tell the time to within a couple of minutes, all with just a bit of wood (and a correction table). I find this really ingenious, and I can't imagine the sort of mentality needed to come up with the idea for such a device in the first place.

There are a few other fairly obvious things we can do with this. Simply rotating the rete and watching the positions of the stars tells us which ones remain low on the horizon (and thus are potentially difficult to find) and which traverse higher altitudes. Similarly, we could calculate the time when any star would be at its highest altitude. We can also see at a glance which ones are below the horizon line.

But there are many other markings I don't know the meaning of. One guide I found suggests that the lines below the horizon line mark twilight (their are different conventions for how this is defined, hence multiple lines), so one could calculate the hours of true darkness. Some of the other circles might mark the Tropics of Cancer and Capricorn, which I think are used for calculating the exact dates of the solstices and equinoxes. The dashed lines and Roman numerals on the front are apparently something to do with mapping unequal hours when not on the equinox, and apparently the nested circles on the back serve the same function. Though exactly what one does with these, I really don't know.

The back also has a shadow square, a surveying instrument used for measuring the size of distant objects. I suppose if you're going to have a device for measuring altitude anyway, and you've got the space for this, why not ? But what the arc below this is, with its February-October scale, again I don't know. Nor do I understand the 0-60 scale on the alidade itself, or the irregular -20 -> +50 scale on the pointer. ChatGPT* suggests the former are for using the shadow square as an alternative way of estimating altitude, while the latter are degrees above and below the celestial equator and could be used as another way of aligning the rete. But its answer isn't clear enough to properly explain how to use them. Another document** suggests it might be a way to calculate the equation of time correction without needing to look this up elsewhere, or maybe involved in calculating the right ascension and declination of the stars. It's even possible to use the thing to estimate planetary orbits.

* This is something which thinks that Poland is a landlocked country and has strenuous moral objections to satirising The Lord of the Rings, so we shouldn't take it seriously... but on the other hand, a standard Google search is no help at all.

** This one is the most thorough guide I've found. I suspect from this that the markings on the pointer are to do with ecliptic longitude and used for finding the azimuth of the Sun; it can also be used for finding the declination of a star.

Field Tests

I'm not exactly sure what's going on here, but it pretty much sums up the perils of real world conditions. 

Right, so the astrolabe is pretty bloody awesome. Under ideal conditions, given the accuracy of the device and the difficulties in taking a reading, we can use it to get the time to within a couple of minutes and the position of a star to within a couple of degrees. But how does this play out when conditions are not ideal, in the messy conditions of the real world ? In, say, an actual field ?

Badly. Very, very badly.

I am under no illusions about my quite shocking lack of any practical skills whatsoever. Even when experimenting in comfort, I could see exactly what would go wrong in practise and was quickly proven right. So at least I can't be accused of a lack of self-awareness in that respect.

The astrolabe is a veritable Swiss Army knife of medieval astronomical instrumentation. What makes it especially powerful is that it needs only a single direct connection between the astrolabe and reality, one primitive "sensor" if you will : the alidade. Once you've got your altitude measurement with this, everything else follows with deterministic precision, with no other free parameters to adjust. The problem is that while the device is undeniably very sophisticated, actually using it – even given its total simplicity – is bloody f*"#ing difficult.

The first problem is that it's very light. This makes it highly susceptible to even a light breeze, so trying to sight something through the holes is extremely difficult for this reason alone. And this is made much, much worse because of the friction of the alidade with the rest of the device. Every adjustment made necessarily moves the whole astrolabe, starting it swinging and thus making each adjustment nearly useless. 

I started by trying to line it up with the Sun. Carefully guestimating an initial pointing, holding it up to quickly check by eye if the Sun was visible though the hole... didn't work. The Sun is just too damn bright. It's impossible to look through the hole towards the Sun (even quickly) because the rest of the solar disc is just feckin' blinding. And trying to look at the shadow, to see if the hole is especially bright when it's lined up correctly, just doesn't work either.

What about stars ? In principle, the holes on the alidade are such that measuring an angle to the 1° precision of the measurement scale shouldn't be a problem. Now, holding the astrolabe horizontally, I found I was indeed able to see a star quite clearly though both of the holes. This takes some care because in the dark it's surprisingly easy to not see the second hole at all, to see a star though just the one hole and think it's all fine. But it can be done.

... not while holding it vertically though. Even without any sort of breeze, it's nigh-on impossible to keep the azimuth of the astrolabe fixed; trying to hang it vertically on your finger high enough above your head to find a star is just so much nope. It really just doesn't work. Never mind two minutes precision, I couldn't take a reading at all. It sways, it's so dark it's hard to see the hole, you have to hold it at a very awkward angle... it's just bonkers to think that people really used to do this.

In principle I think a mounting system would be able to overcome these difficulties. You'd need something to keep it vertical, with the capability to adjust the height and smoothly rotate the azimuth. In that case I firmly believe you could take a reading and apply the corrections so quickly that your estimated time would still be accurate; once you've got the reading, the adjustments to the astrolabe are 30 seconds work. But those drawings of people holding them ? Naah. Not unless you're an actual magical ninja wizard.

It's just not going to happen people.


Even if it has all the practical advantages of the proverbial chocolate teapot, the astrolabe is still a ridonculously impressive piece of kit. Its versatility is crazy, matched only by its sheer maddening  uselessness as a practical instrument.

My guide calls it a medieval computer, but this is not right. It doesn't do any computations in the modern sense; it can't take arbitrary input values, much less perform arbitrary operations on them. At most you could liken it to a specific computer program rather than a computer itself. When you need to do certain specific operations it's incredibly useful, and far, far simpler than doing the calculations by hand – and better by far than a gigantic lookup table. One can certainly see the glimmers of computational logic here, even though the device itself is a long way from a computer in the modern sense.

It's likely an exaggeration to say the astrolabe had over a thousand uses, unless you count every minor variation of every single task for every single star. But it's probably not crazy to say it had dozens. I certainly haven't figured them all out. I'd certainly like to continue investigating at some point, but only if I can figure out how to take attitude readings in a way that isn't likely to have me hurling it across the room (or field) in frustration.

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