Precision and accuracy in medieval astronomy

What is the difference between precision and accuracy?

In modern English they are used almost interchangeably.  But there is a difference, of course.  I wonder what time it is now, when you are reading this.  Is it about eleven o' clock?  Or is it 09:34?  Of course, I have no way of knowing which of those guesses is more accurate.  But the second is obviously more precise.

Which of these two timepieces
is more accurate? Well,
they've both stopped...

That distinction may be more or less clear to us.  But that wasn't always the case for medieval astronomers.  What if I were to refine my guess, and say you're reading this at 09:34:27?  Is that any better? It's obviously more precise.  But when is it preferable to be more precise?  The answer to that might be more complicated than it appears.  In general, we might say that precision is only preferable when it increases accuracy.  But medieval scholars didn't always see it the same way.

I study astronomical tables.  Take a look at this amazing digitised version for an example.  That link points to a table of the daily precession of the stars and planetary apogees.  It's a lot more exciting than it sounds!

(Here's a brief astronomical explanation: skip it if you want...  Precession is the phenomenon that means that the stars appear to move very gradually around the sky, so that they're not in exactly the same place from year to year.  I don't mean the obvious daily rotation around the North Star that's caused by the Earth spinning on its axis - I mean a much slower change, caused by a "wobble" in the tilt of the Earth's axis.  The stars are moving 1° every 72 years - pretty hard to spot, but it explains why, right now, the Sun is still just about "in" the constellation Sagittarius (i.e. in front of those stars) even though it ought to be passing from Capricorn into Aquarius.  To be clear: the astrologers haven't got that wrong, because when they say it's the cusp of Aquarius, they mean the Sun has gone 120° around the sky (in modern terms, we've completed a third of our orbit) since the last equinox.  It's just that the background of stars has moved since the Ancient Greeks assigned them to their positions between equinoxes and solstices.)

So what?  The point is, precession is a VERY slow motion.  It's obviously almost impossible to observe with the naked eye.  It's impressive enough that ancient astronomers had even noticed it, so we shouldn't be surprised that their estimate of the rate was a bit different from ours.  That's why the table I linked above represents a precession of 1° every 136 years (their theory of precession included a separate, non-linear component that made up most of the difference).

But I said above that that table is a table of DAILY precession.  What's the point of tabulating daily values for something that changes one degree every 136 years?!

Good question! Here's another one: What's the point of tabulating those daily values to a precision of billionths of billionths of degrees?!  I don't even know what a billionth of a billionth of a degree is called, but that is the precision represented by the daily value of 0;0,0,4,20,41,17,12,26,37.  (That's a sexagesimal number: 0°, 0 minutes, 0 seconds, 4 thirds... In decimal terms it's 0.0000201148235466718.)  The 37 in the final column of the table is 3.67 x 10^-15.  To put that in context, that's one 98,000,000,000,000,000th part of a complete circle. It would take approximately 750 billion years for these daily 37s to accumulate to even a degree’s difference.

That level of precision in the tables clearly didn't arise from naked-eye observation of the stars.  No, it's a result of the way the tables were computed.  And astronomers clearly realised that - they understood that such precision was unobservable.  Yet they maintained it when they copied and recomputed the tables.  Why?  Because, I suppose, they reckoned that more precision is better than less.  To put it another way: you say why keep those 37s?  They would say, what makes you so sure you can get rid of them?

Isn't that silly?  Hold on a moment - you may not be much better.  A friend of mine recently posted this on Facebook:

I know how these things work: authors of recipe books work out their recipes in their own ways.  Delia Smith was obviously used to using pounds and ounces.  She used 2 oz of sugar.  2 oz is about 56.75g, but no editor will let that go into the published cookbook.  So it gets rounded down to 50g.  Then when Delia calls for 6 oz (about 170.25g), it gets rounded up to 175g.

Here's the weird bit: I know this is what's happening - I even have a magnetic converter on my fridge door that tells me that 2 oz is 50g and 6 oz is 175g.  But that doesn't stop me measuring out the quantities with exaggerated care, paying attention to the slightest fluctuation on the scales.  And what about the eggs?  I'm precise to the last gram of sugar even in recipes that use eggs, when I'm well aware that the size of eggs can vary widely.  If I can sustain this kind of cognitive dissonance, perhaps I shouldn't be too critical of the medieval astronomers.

Touching, But What's the Point?

by Jenny Bulstrode

Astrolabes and Stuff is proud to present its first ever guest post!  For this we must thank Jenny Bulstrode, a student in the Department of History and Philosophy of Science at Cambridge.

Quadrants, sextants, octants, none of them really meant much to me until last year. They’re all descendants of astrolabes. They’re all based around a circle with a scale marked on it. They’re all just fans of metal with bits on. The only way I could see the Navy using one was as an elaborate bottle opener.

No, I'm still not sure what it is

If anyone asks all you generally need to know is this versatile bit of kit was a tool for measuring the angle between two distant objects. This made it handy for everything from telling the time to surveying.  Last year I found out a bit on how they were used but rather more on how one particular eighteenth century instrument maker, John Bird, made them, and why we should care. It’s these last two points I want to share with you.

‘Now use your hands to warm the metal.. except you Bob,
your hands are too clammy’ (not Bird’s original words)

Metal expands when heated. Enough to get the lid off that gherkin jar by running it under the hot tap, more than enough to mess up the precision process of marking a scale on a circle. John Bird used the warmth of his hands to standardise the expansion of his tools and the circle he was working on. As if that doesn’t seem odd enough, he then wrote about it, in his instruction manual.

Temperature control was so important Bird wouldn't allow a fire to warm the room. Even candles were forbidden because they gave out too much heat. (Actually my mum has a similar policy for controlling utility bills). With little light to work by Bird detected points and traced guideline scratches using his fingertips. Magnifying lenses were widely used by eighteenth century instrument makers, including Bird, but what he really had faith in was his sensitive touch.   

Almost as though fingertips were designed
to move tiny puncture marks in metal…

Finally, when checking back revealed a point slightly out of position, Bird would ‘coax’ it into place. Extraordinary as it sounds, through fingertip pressure on the brass, he would shuffle the point along. His fingertips worked to control, detect, and even correct the scale marking process.

All very touching (see what I did there?) but what’s the point? Three things stand out to me. Firstly, John Bird was one of the last circle dividers working exclusively by hand. Towards the end of his life new, automated machines started taking up the job. That’s the old story of mass-production taking over, right? Except for nearly a century after his death these ‘automated’ machines used handmade circles, including Bird’s, to cut copies from. The action of the machine had to be guided, by hand. In fact even when the machines made errors, which they did, often, the points were still ‘coaxed’ into place. Bird might’ve died but his hands were still very active...

Not a bomb, just a highly controlled loaf

Secondly, handmade does not just mean surprisingly expensive bread. Of course it means those elements of delicacy, care, and exclusivity we associate with a walnut and fig pavé but it can also mean industrial quality control. When Bird used his hands to control the expansion of the metal he was setting an industrial standard. Granted it was one based on his body temperature but the principle for control was the same.

Finally, and above all: Bird achieved an unprecedented level of precision in his instruments. In fact, he divided circles so precisely he was commissioned to make the Mural Arc at Greenwich. This Arc set the standard for British Mean Time. I think if there is a point to be taken from all this it is that a huge idea like ‘Time’ can be traced back, defined even, to something as human as the pressure of a man’s fingertip.

While instrument makers were busy constructing time with their hands,
Time the avatar went to all the good parties