Launch of the Digital Equatorium

I wrote this post for the Peterhouse Perne and Ward Libraries blog.  It is cross-posted from there with their kind permission.

28 May saw the launch at the Whipple Museum of the online Peterhouse Manuscripts Collection, housed in the Cambridge Digital Library. The collection aims to present highlights from the College’s collection of 276 medieval manuscripts, and will be developed as time and funding allow. High-quality images are presented alongside searchable transcription, commentaries and critical apparatus, making the Peterhouse manuscripts accessible to scholars around the world. Initial work on the collection has been made possible by generous funding from donors to the College, particularly Dr Joe Pesce.

The launch focused on the first manuscript to be digitised, the fourteenth-century Equatorie of the Planetis (MS 75.I). This manuscript has been at Peterhouse since at least 1538, but it was first brought to the world’s attention in the 1950s by the historian of science Derek (de Solla) Price. Price was a PhD student, conducting research into “the history of scientific instrument making”, and came to the Perne Library expecting to examine an unexceptional astrolabe treatise. He found something quite different, as he later recalled:

As I opened it, the shock was considerable. The instrument pictured there was quite unlike an astrolabe – or anything else immediately recognizable. The manuscript itself was beautifully clear and legible, although full of erasures and corrections exactly like an author’s draft after polishing (which indeed it almost certainly is) and, above all, nearly every page was dated 1392 and written in Middle English instead of Latin. [Science Since Babylon, enlarged edition, 1975, 26-27]

What Price saw, one cold December day in 1951...

Price realised straight away that the manuscript might be by the poet and astronomer Geoffrey Chaucer (c.1343-1400), whose Treatise on the Astrolabe, probably written in 1391, is a very early example of scientific writing in English. He quickly changed his PhD to focus exclusively on this manuscript, and the resulting thesis (published in 1955) included an edition and translation of the instrument treatise that takes up nine folios of the manuscript (alongside seventy folios of astronomical tables).

The instrument Price could not at first identify turned out to be an equatorium, a device designed to compute the positions of the planets. Few equatoria survive today, but they were popular tools of astronomy and astrology in the later middle ages. They were based on the models of planetary motion explained by the Greek astronomer Claudius Ptolemy (c.90-c.168) – essentially three-dimensional diagrams with moving parts. Medieval astronomers took pride in adapting and refining their predecessors’ designs, and the Peterhouse equatorium, whose construction is explained in detail in the manuscript, represents an improvement on the equatoria of notable astronomers such as Campanus of Novara (c.1220-1296) and Richard of Wallingford (1292-1336). Because the manuscript is a draft, we can see the author-translator working out and refining his ideas, learning new techniques and devising improvements as he goes.

Equatorium made at Cavendish Laboratory for
Derek de Solla Price, 1952. Now at Whipple Museum
of the History of Science, Cambridge (Wh.3271).

Price decided to build the equatorium, following the manuscript’s instructions. In an era when historians favoured intensive textual scholarship and did not particularly value reconstruction, this was unusual. So why did Price do it? The answer perhaps lies in his biography. He was from a working-class, Jewish background in the East End of London, and had taken his first PhD in metal physics at the South-West Essex Technical College in 1946. He came to Cambridge from the University of Malaya, where he had been teaching applied mathematics. He arrived in Cambridge in 1951, the year that the University set its first exams in History of Science, and the Whipple Museum of the History of Science opened. The discipline of history of science was in its infancy, and scientists and historians were competing for authority as its boundaries were laid out. In this context, Price clearly felt he needed to establish himself; the publicity surrounding the discovery of a manuscript that might be written in the hand of Chaucer allowed him to do that. Price had worked in the Cavendish Laboratory, helping organise its archives and historic apparatus, and had a good relationship with the Cavendish Professor Sir Lawrence Bragg. Bragg helped him organise a full-scale model of the equatorium to display at an event at the Royal Society in 1952. The model, pictured above, is now at the Whipple Museum. An account of its construction, later loss and rediscovery, has just been published in the Royal Society journal Notes and Records, written by current Petrean Seb Falk.

Now, though, another model of the equatorium has been made – but this one is virtual. Produced by programmer and designer Ben Blundell, in collaboration with Scott Mandelbrote and Falk, the model is embedded in the Digital Library website alongside the manuscript. It allows users to gain the full experience of using the equatorium, giving results for the longitudes of the planets very close to those achieved by modern astronomical computation. In order to produce the model, Blundell needed to create his own calendar that transitioned seamlessly between the Julian and Gregorian systems, and to write new programming language to simulate movement of the equatorium’s silken threads!

Virtual equatorie created by Ben Blundell for the Peterhouse
collection at the Cambridge Digital Library

It is hoped that visitors to the website will gain a new understanding of how the equatorium works and might have been used. It is based on a simplified version of Ptolemy’s planetary models, ignoring the planets’ motions in latitude, and by scaling the parameters of the different planetary models to give them all equally sized deferent circles, their motions in longitude can all be modelled on a single disc. A single epicycle is used, its radius corresponding in size to the common deferent radius; a rotating rule is fixed at its centre and marked with the radii of the planets’ epicycles, which are thereby traced out as it rotates. The longitudes of the planets are found by taking easily calculated linear components of their motion from pre-prepared tables, and transferring those values to the equatorium by laying threads on the scales on the circumference of the disc and epicycle. (For more information, see the explanation on the Digital Library website, and try the model there!)
Study of the manuscript has not been confined to its technical content. At the launch, Professor Kari Anne Rand explained how linguistic and palaeographic evidence has been used to locate the manuscript’s production to the periphery of London, and to cast doubt on its attribution to Chaucer. She showed how certain characteristic features of the scribe’s practised, informal hand appeared in another manuscript that she has found, raising the possibility that an alternative candidate for the authorship of MS 75.I may soon be identified.

Detail from British Library MS Burney 275, f.390v (early 14th century). The illustrator of this copy of Ptolemy's Almagest clearly had some understanding of the use of astronomical instruments.

Whoever wrote the manuscript was part of a thriving astronomical culture, based in but not restricted to the growing universities of Oxford and Cambridge. Instruments like this equatorium were used not just for astrology, or to model the movements of celestial bodies with greater ease, but as a route to greater comprehension of the cosmos. As the picture above indicates (and as Dr Catherine Eagleton reminded us at the launch), devices like astrolabes were familiar features of literate culture. The equatorium was undoubtedly a more complex device but, as the references to the Treatise on the Astrolabe in the Equatorie suggest, it might be a suitable next challenge for someone who had already mastered that more commonplace tool. If the fox could learn from nature with the help of his astrolabe, so too could the medieval English readers who, for the first time in the Peterhouse manuscript, had the opportunity to learn about equatoria in their mother tongue.

How to cast a medieval horoscope

I wrote this post for the blog of the 24th International Congress of History of Science, Technology and Medicine (iCHSTM), which takes place in Manchester on 21-28 July 2013.  Loyal readers of this blog won't find much new here, but it's a fair summary of my research so far.

I have modified my views slightly since writing this, mainly about how sophisticated an astronomer the equatorium's creator was, and how sure we can be about Schöner's purposes.  I'm looking forward to discussing these issues with people at the conference.

Posted on 20 June 2013 by admin

In preparation for iCHSTM 2013, I’ve spent the last few weekends indulging my creative side.  Sawing and filing wood and brass into a disc, ring and pointer may have disturbed the peace of my neighbours’ Saturday afternoons, but it has meant I will be able to demonstrate a particularly ingenious, user-friendly medieval device: a planetary equatorium.

I have recently begun PhD research into a unique fourteenth-century manuscript.  Known as The Equatorie of the Planetis, it describes how to construct an equatorium.  This makes it one of the earliest pieces of writing about a scientific instrument in the English language.  The first person to study it, Derek de Solla Price, was convinced not only that it was written by Geoffrey Chaucer, but that it was a draft in Chaucer’s own handwriting.  The authorship debate still rages; meanwhile, I am looking at some of the other fascinating aspects of this manuscript.

The equatorium nears completion

Much like their better known cousins, astrolabes, equatoria were medieval calculating devices.  These devices made use of astronomical theories and models that were long-established, having first been refined around 150 CE by the Greek astronomer Ptolemy.  In both cases, they existed in something close to their complete form in the late Classical period, before being further developed in the Islamic world from around the tenth century, and refined still further in western Europe between the thirteenth and sixteenth centuries. While astrolabes could be used for a range of functions, from telling the time to measuring the height of a building, equatoria just did one thing: modelled the motions of the planets.

They did this by recreating the essentials of Ptolemy’s planetary theories as a kind of diagram with moving parts.  These became progressively simplified, so that a single device could model the motion of the Sun, the Moon and the five known planets.  After an initial investment of time making his equatorium, an astronomer could then predict the location of the planets to a high degree of accuracy, far faster than by the alternative method – trigonometric calculation.  Using this basic computer, planetary astronomy could be as simple as looking up a couple of values in a table, and using them to place some pieces of brass, wood and string.  The question is: why?

For early modern astronomers such as Johannes Schöner, who included cut-out-and-build equatoria in his 1521 Aequatorium Astronomicum, they had a largely educational purpose: they could be used to demonstrate the fundamentals of the Ptolemaic theories, just as many classrooms today use globes (another favourite device of Schöner’s) to teach children about latitude and longitude. [I'm no longer so confident about this claim: Schöner’s equatoria could be used for practical astrology, though it's hard to be sure that they actually were.]

But equatoria also had practical importance.  Although nowadays we are dismissive of astrology, and think of horoscopes as a simple matter of making (up) predictions about people’s future fortunes based on the month of their birth, it wasn’t always that way.  In the medieval period there was no hard distinction between astronomy and astrology, and the calculations that could be made using personal and planetary information were complex and varied.  They had a range of possible uses, too, guiding anything from political decision-making to the timing of medical procedures.

In the case of The Equatorie of the Planetis, the simplifications made by its designer make it less suitable as a demonstration device, but much easier to make, transport and use to calculate planetary positions.  The designer has shown great imagination in paring the instrument down to its bare essentials.  It could be argued that by simplifying the Ptolemaic model, he demonstrated a lack of understanding and precision, but I think it is the reverse: he showed great sophistication in understanding where approximations could be made for the sake of greater usability, without sacrificing too much accuracy.

It’s sometimes suggested that these medieval “instruction” texts were not really designed to be followed except in the reader’s imagination. Certainly it’s true that it would be expensive and rather unwieldy to make it at its full six-foot scale! (Though that is precisely what Derek de Solla Price did in 1952.)  But with my newly built equatorium I’m looking forward to showing people at iCHSTM that these six-hundred-year-old instructions can be followed to produce a user-friendly, and useful, little computer.

This blog post is based on the paper , “Putting classical astronomy to work: the design and use of a medieval equatorium,” which [I am] due to give as part of symposium T157, “Pre-modern astronomy and cosmology,” on Saturday 27th July at ICHSTM.

Making a brass equatorium

I'm flying to Chicago on Wednesday.  There'll be an unusual object in my luggage (is it wise to put that kind of statement online?).  Yes, you guessed it: it's another equatorium.

Loyal readers will remember my early series of posts (here, here and here) in which I described how I made, and learned to use, a medieval equatorium.  Although I am no great artist, I managed to create a passable replica of the instrument described in the manuscript I'm studying.  In the process, I also had to think a little about the tools and techniques that medieval craftsmen would have used.

But I didn't learn much about materials - I made my replica out of MDF, the cheapest and most manageable material I could find in my local DIY store.  So, when I arranged to speak about the equatorium at a conference in the USA, and I realised that my first replica was too big to fit in my suitcase, I decided to make another one using a more authentic - and challenging - material: brass.

The word "more" is crucial in that last sentence.  When I went online, researched a few different suppliers, ordered a 360 x 400 x 1.2 millimetre sheet of brass, and paid for it using my credit card, I was hardly recreating the experience of the medieval instrument-maker.  It's important to note that even the basic material is quite different: brass is an alloy of copper and zinc, but the ratio of these two elements can vary substantially.  The melting together of metallic copper and zinc, which is necessary to produce the high-zinc brass that can be rolled into thin sheets, was a seventeenth-century innovation.  (Before then, copper was heated with zinc oxide and charcoal.  This produced zinc gas, which diffused into the melted copper.)  And of course medieval metalworkers didn't roll their brass into sheets either - they hammered it out.

For the face, I used an off-cut from the Epicycle of the
last equatorium - so you can see the difference in scale.

But I still felt there was something to be learned (and a prettier end result could be achieved) by working in brass.

Whereas the full-size equatorium would be six feet in diameter and my first one was half that, I had to scale it down again.  In order to avoid making the calculations and measurements too complicated, I decided the easiest thing would be simply to use a 1:1 ratio of inches to centimetres.  The effect is of course to reduce the dimensions by a factor of 2.54.  So my new equatorium is 36 cm in diameter.

The manuscript instructions specify a face (see the earlier posts for what this is) of wood, and all that brass is expensive so I was happy enough to follow that.  But the Epicycle and label (pointer) had to be cut out of brass.  With only a basic hacksaw and file, this was quite a challenge, especially when it came to cutting out the semicircles from the inside of the Epicycle.  Although I bought a nice new workbench and some clamps, it was still a tough job drilling the starter hole - I broke two or three drill bits during this stage of the process.  (If there's a next time, I'll be tempted to invest in a drill press.)

Still, it was very satisfying filing the brass down to precisely the right shapes, and I'm pretty pleased with the final result:

So, what have I learned?  Well, working in brass is hard!  And although I'm no artisan, it must still have been hard for craftsmen in the Middle Ages, whose tools were even more basic than my cheap set.  This raises a number of questions for my research.  For example, does the difficulty of working in brass make it more likely that an astronomer collaborated with an artisan to make this, rather than doing it himself?  Frankly, having had to make it myself, I am more sympathetic to the suggestion that this instrument wasn't actually made in the fourteenth century...  Or, at least, that it wasn't made at full size.  Working with a thin sheet of brass is hard enough when it's a 36 cm ring, and the brass has been rolled on an industrial machine.  I can't begin to imagine how tough it would have been not only to hammer out a 72" ring of brass, but to work with it without it bending out of shape.

It's all fodder for future research.  For now, I'm just curious to see what the friendly folk at United States customs will think of my unusual item of baggage...

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!

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!