Mercury's meandering deferent

This week I'm back with my equatorium, trying to pin down precisely how well it models Ptolemy's planetary theories (and, in turn, finding out the strengths and weaknesses of those theories in predicting the locations of the planets in the night sky).  Today it's the turn of the Winged Messenger, Mercury.

In my last "medieval craftsmanship" post I described the procedure for using the Equatorie of the Planetis to find the longitude of most planets.  I mentioned that the procedures for the Sun, Moon and Mercury vary to a greater or lesser extent from this basic procedure.

So here's how it works for Mercury.

Let's first discuss why the Ptolemaic model doesn't work as well for Mercury as the other planets.  Note: the simplest way for me to explain this is by referring to modern astronomical theories.  Before I do, I should stress that the success or failure of medieval theories was obviously not judged by how well they approximated modern astronomy; instead, their success was measured, in good empirical fashion, by how closely they matched observed data.  Ptolemy's genius - building on the work of earlier, less celebrated men such as Apollonius and Hipparchus - was to devise theories for each planet that produced predicted locations that would match observations to an astounding degree of accuracy.

To recap: deferent=big dashed circle
centred on X; equant=big black dot;
epicycle=little dashed circle.

So, that said, what's up with Mercury?  Well, as you may have read in one of my previous posts, the Ptolemaic system of planetary motion is based on two circles for each planet, the deferent and epicycle.  Together these model the movement of the planets as observed from the earth.  In other words, [ANACHRONISM ALERT] they account for both the earth's and the planets' motion around the sun.  But if you're familiar with the basics of modern astronomy you may be aware that, according to Kepler's first law of planetary motion, the planets' (and earth's) motion around the Sun is elliptical, not circular.  How can we make circles into ellipses?

We get a pretty good approximation of one ellipse from the fact that (a) the deferent is not centred on the earth, and (b) the centre of the epicycle moves around the deferent at a constant speed relative to another point, the equant.  But the epicycle is still a circle, and the planet is assumed to move around it at a constant speed.  Problem, no?

Well, no, not really.  Why not?  Precisely because Ptolemy was not trying to model modern systems: the deferent and epicycle are not pretending to be two ellipses; they're just accounting for what we see in the sky.  As it happens, in the case of Venus, Mars, Jupiter and Saturn, the location of the deferent and equant can effectively factor in a bit of extra eccentricity, so the epicycle doesn't need it.  The resulting level of (in)accuracy will vary depending on the orbits of the various heavenly bodies, but this system gives us a maximum inaccuracy of about 9 minutes of arc (three-twentieths of a degree) for Mars, and is even better for the other planets.

But not for Mercury, though - with this system the locations for Mercury might be almost a degree out.  You might think that's still pretty accurate (remember that we're talking about naked-eye observation here), but it wasn't good enough for Ptolemy.  Here's his solution:

You may remember I mentioned before that we have to mark an aux, or apogee, for each planet; I drew a line (sometimes called the line of apsides) through earth, the deferent centre and equant, out to the rim of the equatorium.  Because the deferent centre is on this line, the aux is the point on the deferent circle that is furthest from earth.  With all the other planets, as the diagram above shows, the centre of the deferent is halfway between earth and the equant point.  But Mercury's deferent centre (D) moves: it can be any point on a circle which straddles the line of apsides, goes through the equant (E), and has a radius equal to the distance from earth to E.  You can see this little circle in the photo below: the pin is at the centre of earth, and E is on the right of the little circle, nearest the pin.

OK then.  We start, as we did before, by finding the mean motus (let's call it λ) of Mercury in our handy astronomical tables.  As usual, we find the point on the rim of the equatorium that is λ° round anticlockwise from the vernal equinox.  Now we subtract the value of the aux from the mean motus; if we get a negative answer, we add 360°.  Let's call the result ψ (technically, it's known as the mean centre).  Now, to find your deferent centre, count clockwise ψ° from the point on the little circle opposite E (on the left on the picture above, furthest from the pin, where I've unhelpfully marked a tiny D).  You might be able to see a little dashed line on the picture above, showing a value of 79° for ψ (as it was on 31 December 1392).

That's it - once you have the location of your deferent centre, you can go ahead with the method as I described before, laying out your parallel threads, placing the common centre deferent of the Epicycle on D, and so on.

I hope all that makes sense!  Next time I'll explain the (much simpler) procedure for the Sun.  Then it's the Moon, which will raise the question of latitude for the first time.  Please visit again soon!

Croissants and Communication at the Catalan Congress

"my" globe

I have just returned from the XII Trobada (twelfth congress) of the Societat Catalana d'Història de la Ciència i de la Tècnica (SCHCT - Catalan Society for the History of Science and Technology).  This conference was held in Valencia, Spain, and I was honoured to be able to attend and present a paper on a globe in the collection of the Whipple Museum here in Cambridge.

I thought I would post some thoughts on this conference while it is still fresh in my mind.  It was my first academic conference, so I can make no direct comparisons, but I still came away with some strong impressions.

First and foremost, I thought it was a well organised, wide-ranging conference.  The venues were ideal and the hospitality arrangements were excellent: good accommodation and plentiful refreshments (hence the title of this post) certainly made a favourable impression on me! The talks were well scheduled and, although the focus was naturally on Iberian (and especially Catalan) subjects, impressively wide-ranging.  Most speakers managed to produce succinct, interesting 15-minute talks, though a disappointing number of people chose to read from pre-prepared scripts and PowerPoint was not always well used!  Here and there there were stimulating questions that led to interesting discussions.  Overall, then, I had a fantastic time and got a great deal out of the experience.

The striking (and controversial) Ciutat de les Arts i les Ciències

All that said, here's what this blog post is really about: the issue of language, which is undoubtedly where the conference made its strongest impression on me.  Since this was the Catalan Society of the History of Science and Technology, you probably won't be surprised by the following breakdown of languages: of 104 presentations at the conference, 66 were in Catalan, 33 (including mine) were in Spanish, and 5 were in English.  Also probably unsurprising is the fact that 40 of the presentations were given by representatives of just two universities: the University of Barcelona and the Autonomous University of Barcelona.

Now, I'm really not sure what I think about this.  Of course I'm in favour of cultural diversity and I'm aware that languages are becoming extinct at an alarming rate (check out the World Oral Literature Project, based at Cambridge and Yale, if this is a subject that interests you).  I'm also aware that Catalan language and culture, in particular, were suppressed for forty years during the dictatorship of Francisco Franco.  It is surely positive for people to be able to present papers at academic conferences in their native language, and for a language to be used in the full range of possible registers and settings (rather than narrow domestic, non-literary settings, as might happen).

But the fact is that there was no-one at that conference who spoke Catalan but not Spanish; all 66 of those talks could have been in Castilian.  Now, most people who speak Spanish can probably follow the general gist of a presentation in Catalan (especially the relatively comprehensible Valencian dialect; not so much the Barcelona accent, and much less the dense Balearic version), but it is certainly going to limit their ability to grasp its finer points, and particicipate in discussions.  If the organisers are proud of the growing international nature of their conference (there were speakers from the Czech Republic, Mexico, Portugal and the UK) and want that trend to continue, wouldn't they be better off encouraging presentations to be given in Castilian?  Wouldn't it better serve the purpose of breaking down barriers between academics, promoting the history of science and publicising their impressive research achievements?

Still, members of the Catalan Society are rightly proud of their achievements: by all accounts it is thriving, more active than its Spanish counterpart.  If they can't give presentations in Catalan at their own biennial conference, when can they?  At a time when Catalan independence is very much in the news, this is clearly a live political issue - apparently the hostel I was due to stay in cancelled the Society's booking when they heard who had made it.

What do you think?  Is it bigoted of me even to raise the issue?  I should emphasize that in general I was content to listen to talks in Catalan (I tweeted from many of them; they're collated here), though I obviously got more out of those in Spanish.  There was just once I found it a little irritating: after a talk in Catalan, I asked a question in my competent but obviously accented Spanish.  The speaker replied in Catalan.  To me, that was just plain rude, but it was the exception: most conference-goers were happy to speak Spanish or even English, and there were some native speakers of Catalan who gave their presentations in Castilian.

In the Friday evening plenary, Charles Withers of Edinburgh University presented a talk entitled "Place, Space, Nation, Globe: Thinking about the Geographies of Science."  Using the examples of the Scottish Enlightenment, the development of Scottish identity through map-making, and the late-nineteenth-century prime meridian debates, he argued that we need to understand the importance of place and space (in both literal and figurative senses) to the development of knowledge; not only how knowledge is made in certain places and situations, but also the transactions between them.  These ideas are hardly revolutionary - Withers acknowledged his debt to a range of thinkers, from Bruno Latour to Jim Secord - but the talk certainly got me thinking about the ways that science and its history are communicated, not to mention the multi-layered geographies of Spain.

That's enough from me - I look forward to your comments.  I'm off to listen to the new BBC Radio 4 series "The Invention of Spain".  Its webpage quotes historian Felipe Fernández Armesto: "I can't imagine Spain ever cohering.  If it did, it wouldn't be Spain."

Tweets from the Catalan Congress

At the weekend I attended my first ever academic conference, the XII Trobada (twelfth congress) of the Societat Catalana d'Història de la Ciència i de la Tècnica (Catalan Society for the History of Science and Technology).  I had a wonderful time, enjoyed making a presentation about this globe, met some very friendly fellow historians of science, and learned a lot.

I will post more of my thoughts about the conference soon, but for now, here are some tweets.  I've just joined Twitter (as @astrolabestuff) and thought I would tweet my thoughts on the conference (or at least summaries of the talks) as it progressed.  Here's what I wrote...

Medieval Craftsmanship, Part 3

[Drumroll] Ladies and gentlemen, I shall now demonstrate the use of a medieval planetary equatorium!

In the first two posts in this series (here and here) I described how I made the physical structure of an equatorium according to the instructions in a fourteenth-century manuscript, and explained how I marked out the equatorium so that it could be used to find the location of the sun, the moon and the planets.

(Technical point: skip this paragraph if you're not interested.)  To be precise, the "location" is what medieval astronomers called the true place, which is more or less the same as the planet's longitude.  Celestial longitude is measured as an angle along the ecliptic (the path traced against the background of fixed stars by the sun in the course of the year) between the planet or star you're measuring and the vernal equinox (traditionally the first point of the constellation Aries, though a phenomenon called precession means this no longer corresponds to the true spring equinox).  We can also measure the celestial latitude, the angular distance above or below the ecliptic, but since the planets orbit broadly in the same plane as the sun, it doesn't vary much.

As I've suggested, an equatorium is a sort of analogue computer: you input information and it uses that information to give you another piece of information.  Unlike, say, an astrolabe, an equatorium can't be used for observation.  What it does is simply and speed up calculations; you could calculate the planets' positions geometrically, but it would be much more difficult and time-consuming.

In the last post I explained how Ptolemy modelled the planets' motion using two circles: the deferent and epicycle.  So in order to use the equatorium, we need to input two pieces of information: the planet's position on the epicycle, and the position of the centre of the epicycle on the deferent.  These were called the mean argument and mean motus, respectively.

There's one more piece of information, which I should have mentioned last time, as it involved marking my apparatus.  Of course we need to know the relative sizes of the two circles.  Fortunately the standard tables used by medieval astronomers (about which more later) included lists of the radii of the planets' epicycles.  Following the manuscript instructions, I marked the radius of each planet on the label [pointer] that spins around the Epicycle (I'll use a capital E to distinguish this bit of wood from the theoretical epicycle I've just been talking about).

Now to the fun stuff.  Here's how you find a planet:
1.  Look up the mean motus and mean argument of a planet in your tables.
2.  Lay a black thread in a straight line from the centre to the edge of the face, at the point equivalent to the mean motus.
3.  Lay a white thread parallel to the black thread, with one end at the equant point for the planet.
4.  Place the common centre deferent (a sort of reference point at the top) of the Epicycle on the deferent centre of the planet whose location you want to calculate.
5.  Turn the whole Epicycle so its centre lies over the white thread

This is what your equatorium will look like after step 4.  I couldn't find black thread, so I used green twine instead.

Almost there - just a few more steps...
6.  Starting from the white thread, turn the label anticlockwise the number of degrees equivalent to the mean argument.
7.  Now move your black thread to line up with the planet’s mark on the label
8.  The place where the black thread crosses the edge of the face is the planet’s true place.

This is what it will look like after stage 8.  I used it to work out the location of Mars, for a mean motus of 40° and mean argument of 100°, and got the result 84°.

That's all there is to it.  I've simplified the method slightly, but the whole process takes just a few minutes, even allowing plenty of time to get the threads nice and straight and parallel.  I think you'll agree that, once you've got your head around the terminology, it's remarkably straightforward.  It's also a very accurate representation of the theory (and the theory does a great job of modelling planetary motion).  The biggest causes of inaccuracy are undoubtedly the way I've marked the degrees on the equatorium, and my (in)ability to get the threads straight and parallel.

So that's how you find the location of Venus, Mars, Jupiter or Saturn.  The method varies somewhat for the Sun, Moon and Mercury, but I probably won't post about that for a while, as all this has raised plenty of questions for me!  Many of the questions are about the astronomical tables, which I keep referring to rather breezily, but about which I know very little.  I'll blog about them as I learn more.

It's been fun making this model, and I've learned a lot, starting from a position of almost total ignorance about medieval astronomy in general and equatoria in particular.  There's a long way to go!  Thanks for reading.

Medieval Craftsmanship, Part 2

In my first post I described how I began making an equatorium according to the instructions in the fourteenth-century manuscript I'm researching.  Since writing that post, I discovered a full-size replica made in the 1950s (you can read about that here).  Needless to say, that replica is far more attractive and probably more accurate than anything I can make at home with my saw and pencil, but I never said beauty or accuracy were priorities of mine.  So I went ahead with my own model anyway.  In this post I'll describe how I finished making it; the process of using it will have to wait for a future post.

Here's what I made earlier: (1) the "face" - a three-foot disc of MDF; (2) the "epicycle" - a circle three feet in external diameter and 34 inches in internal diameter with a half-inch bar across the middle (patched up with some gaffer tape); and (3) the "label" - a pointer three feet long, fixed to the middle of the epicycle but free to turn around its circle.

My first task was to divide both the face and epicycle into 12 (zodiac) signs, 360 degrees, and 21,600 minutes.  Yes, you read that correctly: the Middle English manuscript can be vague, but here it is quite explicit that "everi degre shal be devided in 60 mi."  This is staggeringly ambitious: even if the the equatorium is made on the six-foot scale demanded by the manuscript, its circumference would only be 18' 10" (5.745m), which would demand almost four minute marks to be squeezed into every millimetre around the instrument (95 minutes to each inch).  It's that kind of unrealistic demand that makes you suspect that the author of the manuscript was describing an ideal instrument, and never expected his instructions to be followed literally.

For me, of course, making a half-size instrument, it's quite enough to divide the face and epicycle into degrees (which works out at 3 degrees to the inch).  But even that is a tricky proposition: dividing a circle into equal divisions is notoriously difficult and time-consuming.  (It was a problem that exercised instrument-makers, astronomers and navigators right up to the eighteenth century - the dividing engine devised by Jesse Ramsden to mark circles mechanically is arguably one of the key inventions of the Industrial Revolution.)  I had a head-start: I bought a cheap protractor and used it to mark out the degrees.  But that circle of dots was obviously quite small; when I tried to extend it by drawing lines from the centre, through each dot, to the rim of the circle, I realised how hopelessly inaccurate this method was.  A tiny error in placing my home-made ruler at the centre of the face could make my markings on the rim as much as a centimetre out.

Accepting that my divisions were bound to be pathetically inaccurate, I proceeded to mark the face of the equatorium for the sun, the moon and each planet.  Because, from our perspective, the sun does not move at a constant speed around the zodiac throughout the year (the number of days between solstices and equinoxes is not equal), the sun's path has to be marked as a slightly eccentric circle (i.e. a circle whose centre is not quite at the centre of the face of the equatorium).

The procedure for the planets is a little more complicated.  If you track the progress of the planets through the sky, they all move gradually eastward around the zodiac, night after night.  But there are some nights when they appear to move back towards the west, in what is called retrograde motion.  (If you believe that the earth is going round the sun, and not the other way around, you can explain this by thinking about the different relative speeds of the earth and planets.)  Medieval astronomers modelled this, and the changing apparent speeds (as viewed from earth) of the planets, using two circles: a deferent and an epicycle.  According to the theory, which medieval astronomers took from the ancient Greek mathematician, geographer, astronomer and general genius Ptolemy, each planet moves at constant speed around the epicycle, but the centre of the epicycle itself moves around the deferent circle (see the diagram on the right).  Ptolemy added a refinement to earlier theories: not only is the deferent eccentric to the earth, but the centre of the epicycle moves round the deferent at a constant speed relative to a point that is not at the deferent's centre; instead, it is the same distance as the earth from the centre of the deferent, but in the opposite direction.  In other words, the earth, the centre of the deferent, and the centre of the epicycle's motion (known as the equant point) lie on a straight line, with the centre of the deferent exactly in the middle of the other two.  (The situation is a little different for Mercury, but we won't worry about that for now.)

Is that all clear now?  It doesn't really matter: what's important for our fourteenth-century instrument maker is to know that for each planet he has to mark an equant point, and midway between the equant point and the centre of the face he has to mark the centre of the deferent.  The placement of these marks depends on two bits of astronomical data: the constants of eccentricity for each planet, which tell our instrument maker how far from the centre of the face to put the centre of the deferent; and the direction of the aux (roughly similar to the planet's apogee in modern terms), which tells him in which direction to place the deferent and equant.  The first of these figures was constant and had been calculated pretty accurately by Ptolemy; the second shifts slowly (by about a degree every 136 years) and would have to be kept up to date.  But it was not necessary for our instrument maker to do any calculation: he could lift the necessary data directly from the tables which accompanied the manuscript.  These list auges for each planet, for 31 December 1392.

Having marked up the face and epicycle with signs and degrees, deferents, equants and auges, we are now ready to start calculating the positions of the planets.  The picture on the right will give you a flavour of how it works; for a full description, watch this space!