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...

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!