How short is the shortest day?

Happy Yule!  As you probably know, Yule (Jól) was the pagan Nordic celebration of the bleak midwinter.  And it's the winter solstice today - the shortest day of the year.

Ring of Brodgar (from @VisitScotland via @HistoryNeedsYou)

What does that mean?  The word solstice means "stopped Sun" in Latin - so it's not really a whole day, but the moment when the Sun appears to stand still at its most southern point (so, furthest away from us in the northern hemisphere), before returning back towards the north. To put it another way, it's the moment in our annual journey around the Sun when the Earth's axis is tilted directly away from it (or directly towards it if you're reading this in Madagascar).

Notice that the first explanation I just gave - and the word solstice - is a geocentric view of things: we're talking about the Sun moving, rather than the Earth.  But that's what makes most intuitive sense (and of course it doesn't make any practical difference, since these motions are only relative).  Still, I was interested to see that that was the explanation the Met Office used in the "6 facts about the winter solstice" they published this morning.

One of their 6 facts was that the shortest day is "nine hours darker" than the longest day.  That made me wonder: Where?  Of course you know that the winter days are shorter in Aberdeen than they are in Aberystwyth - so at what latitude is it true that, as the Met Office say, the shortest day is 7 hours and 50 minutes long?

Peterhouse, Cambridge MS 75.I, "The Equatorie of the Planetis", f. 63v

That was a question that interested medieval astronomers too.  In the days before electricity, they were naturally more aware than us of the receding and returning daylight.  And there's lots of evidence of their scientific approach to the matter.  To take just one example, my favourite manuscript (from Peterhouse in Cambridge) features this table (right).

The title (in Latin) is "Table of the increase of the longest day over the equinoctial day, for all the habitable earth".  As you can see, there are two columns, which repeat 3½ times.  They are headed "altitude of the pole" and "half addition".

That's simpler than it may sound!  The altitude is that of the celestial pole - the height of the pole star above the horizon.  That height - an angle on the sphere of the sky - is equal to your latitude.  When the pole star is directly overhead, you're at the north pole.  So the "habitable earth" in this table is from 1° to 60° North (sorry, Icelanders).  The "half addition" tells you that what we are actually being given is the difference in the length of the afternoon (or morning) of the longest day - from noon to sunset, compared with the equinox.  It's given in degrees. To find the difference in hours, you divide by 15 (360° ÷ 24 hrs = 15).

The fact that we are given altitude rather than latitude, and additions in degrees rather than hours, reminds us that these medievals were astronomers and mathematicians.  They were interested in scientific questions, not just mundane practicalities.  And those questions involved some complex science.  To draw up a table like this, showing the different day lengths at different latitudes, you don't just need to know spherical trigonometry.  You also need to have an estimate for the axial tilt of the Earth.

And medieval astronomers did. (Though because they didn't think the Earth spun on an axis, it was called the obliquity of the ecliptic - the angle between the celestial equator and the Sun's annual path between the tropics of Cancer and Capricorn.)  Estimates varied - the obliquity itself has too, over time - between about 23½° and 24°.  So historians trying to find the sources of science through the ages can check these tables to see what parameters are being used, and work out where they came from.  The table above uses an obliquity of 23° 35', which was a value popularised by the 9th-century Arab astronomer Al-Battani, and used by a number of Europeans in the middle ages.

So can we use this table (produced by a 14th-century monk, John Westwyk) to work out where in the UK the Met Office's numbers are true?  You bet!  Their blog post gives 16 hours 38 minutes as the length of the longest day.  On the equinox, of course, it's exactly 12 hours.  So we're looking for a difference of 4 hours and 38 minutes.  Of course the table gives half-additions, in degrees; 4h38 ÷ 2, x 15 = 34.75, or 34° 45'.  The table above gives a value very close to that (34° 43') for a latitude of 52° 30' N - the latitude of Birmingham.  The Met Office headquarters is actually in Exeter (latitude 50° 43') - so maybe they didn't want to rub in the fact that they have a few minutes more light today than the rest of us...

May your days be merry and bright!

Medieval finger-counting on the BBC

"Hand of Bede" from Bodleian Library, Oxford,
MS Digby 56, f. 165v (12th century)

Yesterday I had an opportunity to present some medieval science on the BBC.  I had been asked to contribute to "Free Thinking" on Radio 3.  The theme of the programme was "Hands", so I decided to fill my allocated 4 minutes by talking about early medieval finger-counting.

You can listen to the full programme here - my short essay and interview start at 22 minutes. (You can also download Free Thinking as a podcast on iTunes.)

I also recorded this spur-of-the-moment clip, showing some numbers on my fingers.  A bit clumsy, but I only got one try at it!


If you want to read some more about finger-counting and hand-diagrams in the Middle Ages, I recommend this blog post by Irene O'Daly.  You can also read Bede's The Reckoning of Time in a great edition by Faith Wallis.

The start of summer?

Facebook greeted me today with this little graphic:

Given that this is the radar picture over the UK this morning, it's not surprising that several Facebook users felt a certain irony.

But in what sense is it the first day of summer? As the Met Office explain in a lovely clear blog post, it depends whether you're talking about the meteorological summer or the astronomical summer.  For our forecasting friends, summer began three weeks ago on 1 June.  (Though as my wife often reminds me, summer in Ireland is often thought of as beginning on 1 May.)

But of course I'm interested in the astronomical calendar.  There, summer begins at the summer solstice.  That's today.  In astronomical terms, the solstice occurs when the Earth's axis is most inclined towards the Sun.  Or to put it another way, it's when the Sun stops in its passage north, pausing directly over the Tropic of Cancer before beginning its return journey back towards the south.

Since our calendar years are of unequal length (remember 29 February?), the solstice doesn't occur at exactly the same time every year.  Normally it's on 21 June, but this year, because it's a leap year, it's on the 20th.  To be precise, the moment of solstice will be at 23:34 BST tonight.  After that the Sun will head back south and the days will get gradually shorter.

Medieval astronomers understood this very well.  In his Treatise on the Astrolabe, Chaucer explains that

the Head of Capricorn [start of the zodiac sign of Capricorn] is the lowest point that the Sun goes in winter, and the Head of Cancer is the highest point that the Sun goes in summer. And therefore understand well that any two degrees [of the ecliptic] that are alike far from either of these two Heads, are of alike declination, be it southward or northward; and the days of them are alike in length and the nights also, and the shadows alike, and the altitudes alike at midday forever. (II.16)

This makes the connection between the Tropic of Cancer and the zodiac sign of Cancer quite clear.  (Not the constellation Cancer though; the Sun won't get there for another month, because of the precession of the equinoxes.)  Chaucer points out that any two places that the Sun passes in the zodiac, which are equally far from one of the Heads/solstices, will have the same length days (and nights).

Making use of shadows - if there are any.

They'll also have the same declination - that is, the Sun will be equally far from the equator (which it crosses at the equinox).  The Sun's declination, combined with its noon altitude, allows Chaucer to find his latitude.  I'm sure you've noticed that the Sun gets higher in the sky in summer.  If you measure its altitude at noon, subtract the Sun's declination at that date, and then subtract the whole lot from 90°, you have your latitude.

Because the Sun is higher in the summer than in the winter, a shadow cast by any object at noon will be shorter.  Astronomers (and even architects) had, since the ancient Greeks if not the Babylonians, been well aware of the usefulness of shadows for marking the seasons; for finding local time, the direction of north, and the direction of sunrise and sunset.

Chaucer wasn't shy of dropping his astronomical knowledge into his poetry, of course.  In the Canterbury Tales, the Merchant describes the weather a few days before the solstice.  It's rather better than today's:

Bright was the day, and blue the firmament;
Phebus [the Sun] hath of gold his streams down sent
To gladden every flower with his warmnesse.
He was that time in Gemini, as I guess,
But little from his declination
Of Cancer, Jove's exaltation [an astrological reference to Jupiter].

Today the Sun will set further north than any other day this year.  But I can't guarantee the weather will be summery enough for you to see it.

Leap years and astrolabes

Since today is 29th February, a leap-year-themed post is in order.  This one answers the question you've all been asking: how are leap years represented on astrolabes?

Astrolabe-equatorium at Merton College, Oxford

First, a word about the Julian calendar.  Most astrolabes were made before the Gregorian calendar reform (1582), and that made life a bit simpler for instrument-makers.  In the Julian calendar, leap years happen every four years, without exception.  On the other hand, the Gregorian calendar got rid of 3 leap days in every 400 years, by decreeing that centurial years (1700, 1800, 1900...) would not be leap years, unless they were divisible by 400.  That's why 2000 was a leap year, but 2100 won't be.

Still, astrolabes have to deal with the fact that one year in four has an extra day.  And astrolabes basically only map celestial motions over a single year.  So how did makers handle the irregularity?

This astrolabe at the Oxford Museum of the History
of Science says it has 28 days in February, but there

seem to be 29. A mistake?

They certainly knew about it.  For the most part they made their instruments to be correct 2 years after a leap year, thus averaging out the errors (which were insignificant anyway).  But that approximation didn't satisfy everyone.

Jean Fusoris, the Parisian craftsman - and alleged English spy - whose trial for treason was taking place exactly 600 years ago, wrote in detail about astrolabe calendars.  He argued that

"Their major defect is that they assume that the Sun on its deferent circle traverses the entire zodiac in precisely 365 days, which is not true."

Fusoris proposed that marks could be added to an astrolabe's alidade (the rule used to read information between the solar and Julian calendars), so that the calendar could be read differently for different years in the leap cycle.

But this still doesn't solve the problem of the Julian calendar.  Fusoris was well aware that one leap day every four years was too much - it meant the Sun effectively moved 1 minute and 46 seconds too far every four years (there are 60 minutes in a degree).  So he suggested you could customise your astrolabe to keep it up to date.

How?  Simple.  Just file down the alidade a tiny bit:

"In this way the instrument will show the true place of the Sun precisely for the lifetime of a man and more, so it is a good way of putting the motion of the Sun on the back of an astrolabe.  It can be done just as the zodiac of the rete of an astrolabe is commonly filed down."

It's important to remember that instruments were frequently customised in this way - they weren't kept in pristine condition as museum pieces, but were designed to be working objects, to be altered and added to just as you might buy a new case for your smartphone.  (Though it may be fair to say that most medieval astrolabe-owners were about as capable of performing these kinds of upgrades as most people today are of repairing their phones.)

However, some instruments were designed to make leap year calculation easy.  The instrument pictured at the top of this post is a combination astrolabe-equatorium from Merton College, Oxford.  It was made around 1350, when Merton was Europe's centre of astronomical and mathematical learning.  The picture just above shows a segment of the same instrument's solar and Julian calendars.  (They're usually on the back of an astrolabe, but they're on the front of this instrument in order to make space on the back for a planetary equatorium.)  Above where it says "Pisces" in the middle of the picture, you can see there are four curves arcing across the photo from the top-left corner to the lower-right side.  They're crossed at an angle by more-or-less vertical lines.  Those allow the calendars to be read differently in different years.  Depending on which year you were at in the leap cycle, you simply read from the calendar to the solar longitude (or vice versa) using a different one of the four circles.

It's an ingenious solution to what was a pretty complex problem.  Of course the results weren't exact, but they never were with these instruments.  That wasn't the point.  Astrolabes - not unlike like your smartphone today - were designed to be quick and clear, convenient and user-friendly.  And attractive of course.  This one's designer succeeded admirably.

Medieval (g)astronomy: my PhD in biscuit form

The Equatorie of the Planetis, from
Peterhouse, Cambridge MS 75.I, f. 74r

I submitted my PhD thesis last week (and now have a little more time to post on this blog).  In large part it's a study of this fascinating instrument.

If you've read this blog before, you'll know I've studied a 1950s replica of this equatorium, made my own replica - and then made it again in a smaller form and (slightly) more authentic materials.


But until now, I'd never made an edible equatorium.

I made the face out of chocolate shortbread (an adaptation of Jamie Oliver's recipe, with 3 tbsp of cocoa added per equatorium).  The epicycle was gingerbread.

It was a bit tricky to get everything the right shape, and the gingerbread expanded more than I had expected in the oven, but it all came out pretty well...

Add a screw and nut to hold the rule to the epicycle, position them correctly, and here's your complete equatorium!

Masculine Mars? Planetary degrees in medieval astrology

I handed in my PhD thesis earlier this week, so I finally have time for a new blog post.  It's another small step towards the blogging task I've been putting off for months: using my son's horoscope as a way in to understanding medieval astrology.

This chart has been pinned above my desk for some months:

Horoscope from Peterhouse 75.I, f. 64v

Much of my research investigates how medieval astronomers found the locations of the planets, using instruments and tables.  I explained in an earlier post how, in order to cast a nativity (an astrological analysis of the moment when someone was born), the first step was usually to find the locations of the planets in the 12 astrological houses.  The chart above is a traditional layout (here's the same layout used in a 9th-century horoscope, copied in the 14th-century manuscript at the centre of my research). It shows the cusps (boundaries) of the houses, and the locations of the planets within them.  They start in the middle on the left, and go round anticlockwise.  So in my chart, the first house starts at the 4th degree of Capricorn, the second house starts at the 16th degree of Aquarius, and Mars was at the 29th degree of Capricorn.

Now, the location of the planets in the zodiac was thought to determine the strength and nature of their influence.  But this basic astrological axiom could be interpreted in many ways.

The Declarations, a brief manual written for "The Queen" (probably Philippa of Hainault, the wife of Edward III) by the great astronomer Richard of Wallingford, who was Abbot of St Albans 1327-36, begins thus:

If there be a question made of the nativity of a man, and the planets be in masculine degrees, that shall be to him a strength.  And if there be a question made of the nativity of a woman, and the planets be in feminine degrees, that shall be to her a strength.

What are masculine and feminine degrees?  Ptolemy (whose Tetrabiblos is as important a text in astrology as his Almagest is in astronomy) had written that the stars were masculine when they rose and set before the Sun, and feminine when they followed it.  But here we see a different doctrine, in which certain degrees within each sign are assigned one or other sex.

Here, as on so many other topics, medieval astrologers were following the authority they knew as Alkabucius or Alchabitius.  This was (Abu as-Saqr 'Abd al-'Aziz ibn Uthman ibn 'Ali) al-Qabisi, a 10th-century Syrian who, along with the 9th-century Persian Albumasar (Abu Ma'shar), wrote the works of astrological theory that were most popular in the Middle Ages.  Al-Qabisi stated that the first 11° of Capricorn were masculine; the next 8° feminine; and the last 11° masculine again.  Each sign was divided in a similar way (but always in different proportions) into between 3 and 7 groupings of masculine and feminine degrees.

Opening of the Declarations, in Wellcome Library 8004, f. 31v

You'll already have worked out that for my little boy, Mars was in a masculine degree on the day of his birth.  (Matters weren't always this easy: in a table found with one copy of Richard of Wallingford's Declarations, individual hours of the week were assigned sexes.  On the other hand, the 11th-century Persian scholar al-Biruni thought the whole idea of masculine and feminine degrees was confused and lacking in substance.)

Anyway, if I use al-Qabisi's layout for little ADJF's horoscope, Mercury is also masculine; all the others (the Moon, Jupiter, Saturn, the Sun and Venus) are feminine.

This could be interpreted in a number of ways, depending on what we're interested in: are we investigating the subject's health, wealth, chances in life and love?  And how do we balance this information against other data in the horoscope concerning the Signs and planets?  I'll explain some of this in the next post (coming soon!), when I talk about the very important doctrine of planetary dignities, which considers the locations of the planets in the Signs and their relationships with each other.

For now, though, we can say that Mars and Mercury are strong in my son's nativity.  Mars, according to al-Qabisi,

indicates tyranny, bloodshed, conquering, highway-robbery, wrongful seizure, the leadership of armies, haste, inconstancy, smallness of shame, journeys, absence, indulgence in love-making, miscarriages, middle brothers, and the management of riding animals. (translation by Burnett, Yamamoto and Yano)

 Meanwhile Mercury suggests

public address, rhetoric, and activities which arise in mathematics like business, calculation, geometry, philosophy, taking omens, sorcery, writing, poetry, and all kinds of calculation . . . It indicates fear, fighting, killing, enmity, tyranny, opposition, prosperity, craftsmanship, kindness in deed, investigation, and everything else concerning commerce and contentions.

Does this mean that little A is going to be a tyrannical accountant? Well, it does run in the family.  But the more important point is that it took an experienced astrologer to interpret all the data in a horoscope.  This whole post is based on just the first two sentences of Richard of Wallingford's Declarations, and already we have a bewildering array of options.  What I thought was a simple little square diagram turns out to be surprisingly complex - in my attempts to read it, I'm beginning to understand why astrology was thought to be such an advanced science in the Middle Ages.