"My Father Adalbero of divine memory was both the equipoise and the force in all causes dependent upon the eternal, and now that he has been changed into the original elements one might think that the world is slipping into primordial chaos."

On January 23rd 989, Adalbero, Archbishop of Reims, died.

Things had been looking up for Gerbert. He’d been in a string of perilous scrapes through his entanglement with the politics of France and the Empire, but he’d come out on top when two Kings of France died in quick succession, leaving Hugh Capet on the throne, an ally of the boy-King Otto III and his mother Theophanu - and hence also Gerbert and Adalbero. And according to William of Malmesbury, Gerbert had learned through a device of his own design, capturing the prophetic power of the stars in a brass head, that he was to be the Pope - and that unless he said mass in Jerusalem, he would never die.

But for now he was one of two contenders to replace Adalbero. He’d been effectively the archbishop’s second-in-command at Reims for several years, and he was an ally of both the King of France and Theophanu. The other contender was Arnulf, an illegitimate son of Hugh Capet’s predecessor-but-one Lothair and a former student of Gerbert at Reims. Arnulf had a big advantage over Gerbert in that he had royal blood, while Gerbert was a commoner. That sort of thing really mattered at the time. Gerbert was very unusual as an example of someone with no aristocratic background at all becoming as powerful as he was purely on merit.

Gerbert had a campaign to run, but that didn’t stop Remi of Trier. Remi had been writing to Gerbert for years now, asking him to make him one of the celestial spheres he was famous for using to teach astronomy, and Gerbert had started making one. But he’d had to put it on hold.

"During this great disturbance and, I might add, confusion, you, forgetting to observe proper manners, have inconsiderately taken into account only what you hope for, what you seek. In a crisis of this sort, when the state itself has been abandoned, must recourse be made to the comments of philosophers which at this time are not necessities?"

Is how Gerbert responded to Remi’s requests.

"For only a wooden sphere must we abandon friends who like me had enjoyed the friendship of blessed Father Adalbero?"

But he clearly still had time for some scholarly pursuits. He wrote to a monk called Adam, who he seems to have liked much better than Remi, about timekeeping.

The problem he addresses in his letter is to do with the length of an hour. In early Medieval Europe, and indeed in most societies before the mechanical clock, time was usually measured not in fixed units of consistent length, but in fractions of the time between sunrise and sunset or between sunset and sunrise - so, an hour during the day is one twelfth of the time between sunrise and sunset, and during the night it’s one twelfth of the time between sunset and sunrise. The big advantage of this system is that in a world where mechanical clocks are non-existent and even water-clocks are far from ubiquitous, it’s quite simple to build a sundial that uses this system, and you can roughly estimate what time it is by looking at the sun - and in a time before electric light, the times of sunrise and sunset would determine the structure of your day anyway.

A big disadvantage is that it means that an hour is not a fixed length of time, but can vary wildly depending on how far north you are, the time of year, and whether it’s the day or the night. This made water clocks more difficult to build, because obviously the rate water flows at doesn’t depend on any of these things - you have to have a scale that moves through the year.

And it presumably made it more complicated to plan your day. How many lines of a manuscript can you copy in the three hours between terce and sext? Well, that depends on how long three hours is. And I haven’t found any information on how this worked in, like, Scandinavia, or Northern Scotland. Did they bend the rules a bit? Or adapt them to something like the old Norse system using the direction to the Sun rather than its altitude? Or did they just roll with it and blast through the entire liturgy of the hours in quick succession in the brief winter daylight?

A system that still works like this today is the Islamic schedule of five daily prayers, at specific times defined by the position of the Sun in the sky. At high latitudes this leads to an enormous variation between prayer times throughout the year.

What Gerbert was writing to Adam about was the problem of working out how long the hours actually are in this system in any particular place at any particular time, and converting between unequal hours and equinoctial hours - that is, hours that are equally long, with each day evenly divided in 24, exactly like what we use now.

Some people seem to have approached the problem by treating the day lengths as if they increased uniformly from the winter solstice to the summer solstice and then decreased at a constant rate on the other side of the year. Gerbert doesn’t do this, because it’s wrong. What he does instead is he follows the method used by Martianus Capella - remember him, from the second episode? He said that the day lengths change faster closer to the equinoxes and slower closer to the solstices - in a pattern that basically amounts to how it actually works.

Martianus Capella gives values for the length of the day at the summer solstice for several latitudes from Meroë in present-day Sudan (where, he says, the longest day is 13 equinoctial hours), all the way up to Britain, where he correctly gives the longest day as 17 hours. Gerbert sent Adam whole tables of the variation of day length through the year at the Hellespont - that is, the Dardanelles, the strait between the Balkans and Anatolia South-West of the Bosphorus - and for a latitude further north than Martianus goes, where the longest day is 18 hours (roughly corresponding to Aberdeen, Gothenburg and Riga) - it’s worth noting that the known world to Gerbert and Adam extended considerably further north than the known world to Martianus Capella. Two centuries of Viking raids had made the rest of Europe thoroughly aware of the lands to the north whether they liked it or not.

But remember, Gerbert was doing all this in the middle of an Episcopal election campaign. In the end, he should probably have spent more time schmoozing Hugh Capet or whatever instead of writing about seasonal variations in day length, because, well past the 90 days allowed by canon law to elect a new bishop, Hugh appointed Arnulf as Archbishop of Reims.

Arnulf was the illegitimate son of Lothair, the last-but-one King, and thus the half-brother of Charles, Duke of Lorraine. Charles also claimed the throne, and was engaged in a civil war with Hugh. Hugh thought appointing Arnulf as Archbishop of Reims would work as a gesture of reconciliation towards Charles, to help bring an end to the Civil War.

Gerbert himself had to write the announcement of his rival’s election, and he placed considerable emphasis on the fact that Arnulf had been an enemy of Hugh Capet, and condemned by the French Bishops.

"We speak of Arnulf, son of King Lothair, cleansed by the mystic sacraments of the purifying Church, our Mother, even though his noble blood stained him by a certain contagion resulting from being enmeshed in an anathema."

But, as Gerbert had warned, appointing Arnulf turned out to be a catastrophic mistake. Six months later, Arnulf opened the gates of Reims to Charles’s army, and Hugh lost the city to his enemy. The situation was so chaotic that Arnulf himself was actually taken prisoner by Charles until he pledged allegiance to him.

In the midst of all this Remi of Trier was still asking after his Celestial Sphere, and Gerbert wrote back:

"Your request, sweetest brother, so often repeated, indicates sufficiently well on what billows we are tossed."

And he explained the situation further:

"Violent fortune, taking back everything she had bestowed, added to this misery through those brigands who laid waste the city of Rheims. Now we bewail the captivity of our friends, and we meditate with sleepless worry whether we must change our home. On this account our district sorrows, on that, it mourns. Fear and trembling encompass the walls. Want presses hard upon the citizenry. The clergy of both orders groan over the future desolation."

Arnulf excommunicated the people involved in the sack of Reims, as if it wasn’t him who let them in in the first place, but he still swore allegiance to Charles as King of France. Gerbert had to get out of Reims again, this time to the court of King Hugh.

Hugh tried and failed to take the city back. And, thirty miles away in Laon, Charles and Arnoul received a visitor. Ascelin, the city’s bishop, had been captured when Charles and Arnulf captured his city, but he had escaped and joined up with Hugh. And now he was back. He’d fallen out with Hugh, and wanted to return to his city as an ally of Charles, bringing his knights with him.

When Arnulf, Ascelin and Charles were dining together, so Richer claims, Charles offered Ascelin a cup of wine, mixed with bread, saying

"Drink from this cup as a sign of the faith that you will hold and maintain. If you have no intention of truly keeping faith, leave the cup, lest you emulate the dreadful appearance of the traitor Judas."

And Ascelin downed the wine, and declared

"And I will keep faith, or else I will die like Judas!"

That night, when Charles and Arnulf were sleeping, Ascelin went to each of their rooms. He took away their swords, and he summoned his knights. The King and the Bishop were taken prisoner, locked up in a tower. Hugh had won that war.

Charles died in prison. Arnulf was released, but defrocked. Gerbert was now the Archbishop of Reims.

But was he? While Arnulf had committed treason against Hugh Capet, and that’s not on, he had only been deposed by a council of bishops. The Pope was supposed to be the one who deposed Archbishops.

Enter Arnulf, the Bishop of Orleans. He’s a completely different Arnulf to the one I’ve been talking about so far. Arnulf of Orleans didn’t care what the Pope had to say about the French bishops deposing the similarly-named archbishop of Reims. Gerbert recounted him telling the other bishops, in a speech that could almost have come straight from the mouth of Martin Luther:

"How would it fare with us, if it should happen that the man the most deficient in all these virtues, one so abject as not to be worthy of the lowest place among the priesthood, should be chosen to fill the highest place of all? What would you say of such a one, when you behold him sitting upon the throne glittering in purple and gold? Must he not be the Antichrist, sitting in the temple of God, and showing himself as God? Verily such a one lacketh both wisdom and charity, he standeth in the temple as an image, as an idol, from which as from dead marble you would seek counsel.

But the Church of God is not subject to a wicked Pope; nor even absolutely, and on all occasions, to a good one. Let us rather in our difficulties resort to our brethren of Belgium and Germany, than to that city, where all things are venal, where judgement and justice are bartered for gold."

Not only had the Bishop of Orleans literally called the Pope the Antichrist, he had also explicitly rejected the idea that the Pope was an absolute authority over the Church North of the Alps.

Now we’ve got a situation where there are two people claiming to be Archbishop of Reims. Arnulf of Reims is backed by the Pope, and Gerbert by the King of France. The French bishops, led by Arnulf of Orleans, supported the King, and French monks, led by Gerbert’s scholarly rival Abbo of Fleury, supported the Pope.

And what about the Emperor? Previously, Gerbert had been firmly backed up by the Imperial Court, by Otto II and his wife Theophanu, who had later ended up as regent for her young son Otto III. But on June 15 991, less than a week before Gerbert’s election as Archbishop, Theophanu died. She had been Gerbert’s main backer at the imperial court, and now Adelheid, the Otto III’s grandmother, who notoriously hated Theophanu, and referred to her as that Greek woman, was the regent. There doesn’t seem to be any evidence that Adelheid had any particular beef with Gerbert, but she certainly couldn’t be expected to be as reliably friendly as Theophanu.

And the Emperor ended up backing the Pope. Gerbert basically only had the King of France and the French bishops on his side. The Emperor, the Pope, and powerful monasteries like Fleury and Cluny, were all against him. King Hugh sent envoys to the Pope, but the Pope refused to see them.

And in 995, the Pope called a synod, at Mouzon in the Ardennes. Hugh Capet barred the French bishops from attending, but Gerbert went anyway, but despite him making a long, eloquent speech in his own defence, Gerbert was suspended as Archbishop of Reims, until they could hold another synod, perhaps actually with the French bishops in attendance. Gerbert would have to, at the very least temporarily, give up the position of Archbishop of Reims, or be excommunicated.

It’s quite possible Gerbert himself wouldn’t have minded that too much. It would have meant he could go back to showing his students the wonders of astronomy, and collecting interesting manuscripts from his contacts in the monasteries of Europe, while leaving all the work of actually being an Archbishop to Arnulf. But the King of France was not going to let him.

But Otto was growing up. In 995 he turned 15, and he had already led armies against Vikings and Slavs on the Baltic coast, and had taken over from Adelheid, ruling in his own right as King of the Germans. He admired Gerbert as a scholar, and as an ally of his parents, and in 996, when he went to Rome to deal with the Crescenti, the powerful Roman noble family who had kicked out the Pope yet again, he took Gerbert with him.

They came down from the Alps to Pavia, and while they were there over Easter, and Otto was being crowned King of Italy, the Pope died. The Emperor had his cousin, Bruno, elected as the new Pope, and Bruno proceeded to crown Otto as Emperor. But Otto had to get back to Germany, and Gerbert back to Reims, so they left Bruno, Pope Gregory V, at the mercy of the Crescenti, who promptly forced him out of Rome and installed an Antipope.

Hugh died that same year and was succeeded by his son, Robert. Robert and Gerbert’s relationship suffered almost immediately when Gerbert refused to marry the King to the Countess of Blois, on the grounds that: 1) the King was arguably technically still married to the Countess of Flanders and 2) the Countess of Blois was his second cousin, which meant that, by the rules of the time, the marriage was technically considered incest.

The King married the Countess anyway, getting the Archbishop of Tours to do it instead, and he sent a messenger to try and get the Pope to approve his illegal marriage. And who was the messenger but Abbo of Fleury. Abbo eventually found the Pope hiding out in Spoleto, in the hills of Umbria, nearly 80 miles away from Rome. He didn’t manage to get a Papal endorsement for the King’s dodgy marriage, but, with the help of a forged charter, he did manage to persuade the Pope to grant the Abbey of Fleury complete independence from the Diocese of Orleans.

Now that he’d made an enemy of the King of France, and Abbo’s monks were in the ascendancy over Arnulf’s bishops, Gerbert decided his best option was to leave yet again, this time Eastwards, into the Empire, to Aachen, and then Cologne, and then across the Rhine, leaving behind the Ancient cities, the vineyards, the Roman roads of the West for a sparsely-populated land of thick forests, where towns, roads, writing and Christianity were relatively recent developments, spread by Emperors since the time of Charlemagne, often with extreme violence.

The Emperor Otto III, of Saxon descent himself, was based in Saxony, at Magdeburg on the River Elbe. This river had, until a few decades earlier, marked the Eastern limit of Christendom in Northern Europe. And even now, while technically subject to the Emperor, the lands across the river remained populated largely by Pagan Polabian Slavs, who the teenage Emperor had spent much of his reign as King of Germany struggling to keep under control. In, for instance, England, Christianisation, and the associated incorporation into the literate civilisation of Early Medieval Christian Western Europe, had been largely peaceful, a matter of persuading local rulers one by one of the benefits of adopting Christianity. But that was not the case in Saxony. Christianity had reached Saxony through a process of conquest, colonisation and forced conversion under Charlemagne, and for the Ottonian Emperors, the lands beyond the Elbe were the new frontier of this process.

Otto I had founded an Abbey in this frontier outpost, and developed it into his Imperial Capital, eventually expanding the Abbey into a grand Romanesque Cathedral. His son and grandson continued the tradition. And here, far beyond the limits ever reached by Rome, far deeper into those gloomy German forests than Varus had penetrated when he lost his legions in the Teutoburger Wald, was where Otto III, a son of both Empires that claimed Rome’s legacy, plotted to become the Emperor who at last restored the glories of old as the Year 1000 approached.

Gerbert was fully behind this scheme. As he wrote to Otto:

"Ours, ours is the Roman Empire. Italy, fertile in fruits, Lorraine and Germany, fertile in men, offer their resources, and even the strong kingdoms of the Slavs are not lacking to us. Our august emperor of the Romans art thou, Caesar, who, sprung from the noblest blood of the Greeks, surpass the Greeks in empire and govern the Romans by hereditary right, but both you surpass in genius and eloquence."

An education from Gerbert would be critical in forming a new Roman Emperor. As Gerbert himself put it:

"I do not know what more evidence of the divine there can be than that a man, Greek by birth, Roman by empire, as if by hereditary right seeks to recapture for himself the treasures of Greek and Roman wisdom."

Otto gave Gerbert an estate at Sasbach, at the western edge of the Black Forest, well away from the Eastern frontier and much closer to both France and Italy, and formally offered him a position as his tutor. With his offer of the role, he even included a poem (which the translator of Gerbert’s letters, the appropriately-named Harriet Pratt Lattin, has been good enough to render in English verse):

"Verses have I never made
Nor in such study ever stayed.
When to its practice myself I apply
And can write successfully,
As many men as has Lorraine,
To you, then, songs I’ll send the same."

Assuming the translation is representative of the quality of the original, Otto has very effectively conveyed the message that he needs Gerbert to teach him to write better poetry (although to be fair the original Latin does at least consistently scan).

Otto wanted Gerbert to explain Boethius’s mathematical works to him, and no doubt also astronomy and music theory, and maybe even the Abacus, and to build devices like he’d built in Reims.

At the very least, he seems to have built a device that the chronicler Thietmar of Merseburg, writing in 1018, calls a horologium. There’s some debate about exactly what this is. The word horologium is generally translated as ‘clock’, and some people have claimed that Gerbert d’Aurillac invented the mechanical clock, almost three hundred years before the first other evidence of weight-driven mechanical clocks starts to appear, and it doesn’t help that William of Malmesbury claims that in Reims Gerbert built what he calls a ‘clock constructed on mechanical principles’ (William of Malmesbury was also writing before the first weight-driven clocks, so he was probably talking about a water clock, but more on that later).

But first let’s consider what Thietmar of Merseburg says Gerbert made in Magdeburg.

"And having spent some time with the Emperor, he made a Horologium in Magdeburg, correctly setting it up, having viewed a certain sailors’ guiding star through a tube."

This doesn’t sound like a mechanical clock, or even a water clock. The whole point of mechanical clocks is that you don’t need to use any astronomical observations, yet here Gerbert is, setting up his device by observing a ‘sailors’ guiding star’, by which Thietmar presumably means the Pole Star. No, this is some kind of astronomical instrument, using the movements of the heavens to tell the time.

Maybe it’s a sundial. A sundial has to be aligned in the right direction to tell the time accurately, which you could achieve by sighting the Pole Star, but it would be more obvious to calibrate a sundial during the day, using the Sun. Some people have suggested it might be an astrolabe. You can use an astrolabe for telling the time, and that was one of their most important uses in the Middle Ages - but the Pole Star wouldn’t be much use for timekeeping, because it stays in the same place all night.

But I’ve come across a suggestion from Marek Otisk that I think matches Thietmar’s description a bit better. The 9th Century monk Pacificus of Verona is said to have created some kind of nocturnal timekeeping device. Like Gerbert, Pacificus has been sometimes credited with inventing the first mechanical clock. But, judging by illustrations in manuscripts from the 12th Century onwards, the way this device clearly worked through astronomical observation. What it seems to be is a sighting tube with a circular disk around it with times marked on. You set it up by pointing the sighting tube at the pole star (which wouldn’t have been the current Pole Star), and then through the night you look at the position of another star - which likely would actually have been the current Pole Star, at the time a few degrees away from the actual celestial North Pole, which has the advantage of being very bright. This star would probably have roughly lined up with the inner edge of the circle marking the times. As this star appears to rotate around the North Pole, once per day, due to Earth’s rotation, with regular adjustments to account for Earth’s axial tilt, you can read off the time based on where it is in that rotation.

And building a mysterious device by looking at the stars - doesn’t that remind you of something? Just maybe it’s possible that William of Malmesbury had heard garbled retellings of anecdotes about this device, or maybe the other Celestial Spheres, and those became his tale of Gerbert constructing a device, and setting it up just right by observing the stars, not to tell the present time but to tell the future.

The other thing Gerbert’s sometimes said to have built at Magdeburg, and also actually at Reims, is a pipe organ. Whether he actually ever built any such thing is quite controversial. He certainly knew about them. He wrote a treatise called De Mensura Fistularum - ‘on the measurement of pipes’ - where he explained which ratios of organ pipe lengths produce which intervals - it turns out it’s the same as the ratios of string lengths on a monochord, but adding on the diameter of the pipe, so you can treat the pipe as if it was a string with an effective length of its actual length plus its diameter - the fact these same ratios show up in organ pipes as well as strings is empirical confirmation of the universal Pythagorean mathematical principles underlying reality.

If you just treat an organ pipe as a straight column of air, and account for the fact that the vibrating column doesn’t actually end exactly at the end of the pipe, you actually get an end correction of just under a third of the diameter of the pipe, so what Constantino Sigismondi reckons is that it’s likely Gerbert was actually also including a mouth correction, accounting for the effect of the actual shape of the mouth of the organ pipe. But he must have been working out these ratios empirically, using actual organ pipes, so he must have been getting them from somewhere. So I guess it’s plausible he might have made or at least designed organs in either Magdeburg, Reims or both, even if there’s no particularly good evidence that he actually did.

But for the "ever august Caesar, glorious Lord Otto, emperor of the Romans", there was still work to do in Italy. The Crescenti, after kicking out Otto’s cousin Bruno (Pope Gregory V) had installed John Philagathos as Antipope. He was. Calabrian Greek who had been employed by the court of Otto II and Theophanu. Otto’s job now was to put Bruno back in the Chair of St Peter, and that meant taking Gerbert with him for yet another Winter crossing of the Alps.

Caesar and his troops and his entourage would have taken the Roman Via Claudia Augusta, from Innsbruck over the Brenner Pass to Trento and then Verona, and then headed West, past Lake Garda to the Emperor’s Italian Capital at Pavia. The route is lower than the Great St Bernard Pass, and not as steep, but the Imperial party would still have dealt with thick snow, icy roads, and brutal cold.

The Emperor made it to Rome in time for Easter. Crescentius the Younger, the head of the Crescentii, retreated across the Tiber, to the fortified complex of the Vatican. The Emperor’s German troops scaled the Leonine walls. Crescentius was beheaded, and his headless body hung up by its feet to show the people of Rome that the Ever-August Caesar was back.

As for John Philagathos, the Antipope, he fled the city. But the German troops caught up with him too. They cut off his nose. They cut out his tongue. They gouged out his eyes. They placed him backwards on a donkey and led the blinded, mutilated Antipope through the streets of Rome and even through a church, dressed in his priestly vestments, back-to-front, before stripping him and throwing him in prison.

Nilus of Rossano was 88 years old. He was a Calabrian Greek, from the same town as John Philagathos, and he had spent the second half of the 10th Century wandering through Southern Italy amid repeated Saracen raids, living at times as a hermit, founding several monasteries along the way. And the Emperor faced this holy man’s condemnation for the treatment of Philagathos by troops under his command.

According to a biography by one of the hermit’s disciples, Otto went to Nilus, and Augustus Caesar knelt before this ragged old man, and pleaded with this emaciated Saint to give him a penance to atone for the atrocities committed by his troops against a man who he had, after all, known ever since his childhood in Theophanu’s court. But Nilus refused. After all, Philagathos remained in prison, even if the Emperor had spared his life. And Philagathos would die, imprisoned in a monastery, three years later.

But Otto was back in charge. His cousin was the Pope again, and he could rule the Roman Empire from Rome.

For Gerbert, this was great news as well. Otto and the Pope could finally sort out the Reims situation. There wasn’t much point in the Pope declaring Gerbert to be Archbishop of Reims - the King of France would just ignore him. So Arnulf of Reims was going to be the Archbishop of Reims. But Gerbert was going to get another job, at least as prestigious. He was going to be the Archbishop of Ravenna.

I talked a bit about Ravenna in the last episode. It’s where Gerbert embarrassed Otric of Magdeburg in front of Otto III’s father, Otto II, by flexing his erudition. It had been the Capital of the Western Roman Empire in its last days, and was - and still is - replete with grand monuments from that time, and from the attempts of later Gothic and Byzantine rulers to cling to what remained of that decaying splendour. Gerbert would be one of the most important clerics in Europe. The prophecy of the Head was closer than ever to being fulfilled.

We don’t really know much about what Gerbert actually got up to in Ravenna. We have no letters from him for the entire year of 998. He’s sometimes said to have built a clepsydra there - i.e. a clock that measures time based on water flowing into or out of a container. The problem is that the first source claiming he did doesn’t appear until the late 16th century, closer to the present day than to when Gerbert was in Ravenna. I guess it’s not particularly implausible that he might have made one. Water clocks were certainly a thing at the time. They’d existed since ancient Egypt in fairly simple forms, with water draining out of a vessel with a scale marked on the side, and in Ancient Rome they’d become very sophisticated, with the water driving elaborate sets of dials and alarm mechanisms. Nearly two centuries before Gerbert, the Abbasid Caliph Harun Al-Rashid had sent Charlemagne an extremely elaborate one, with figures on horses that appeared from behind doors every hour, and brass balls that fell onto cymbals as an alarm. And Gerbert describes using a water-clock, in his letter to Adam about timekeeping, where the constant time kept by a water clock is used as a benchmark for comparing the lengths of days at different latitudes.

We have exactly one letter that Gerbert sent while he was Archbishop of Ravenna. He wrote it to Adalbold, a clergyman from somewhere in the low countries, who would later end up as Bishop of Utrecht, and it’s an explanation of a mistake Adalbold had been making about how to calculate the area of a triangle.

To work out the area of a triangle, you need to multiply half of the length of its base by its height. For an equilateral triangle, if you know the side length, you can work out the height using Pythagoras’ theorem (it comes to the side length times half of the square root of three).

Adalbold, though, seems to have been trying to use Boethius’s formula for the triangular numbers - i.e. numbers where if you have that many things you can arrange them in rows to form an equilateral triangle, with each row having one item fewer than the one below. What Adalbold ended up with is an approximation where you imagine covering the triangle with squares, with a side length of one unit. If the side length is n units, the first row will have n squares, the second n-1 squares, and so on until there’s just one square at the top. The number of squares is an approximation for the area, and it’s given by the formula for the sum of an arithmetic series: n squared plus n, all over two. Clearly this is always going to be a bit of an overestimate - the outer squares overshoot the edges of the triangle - but if you make the units small enough compared to the side length of the triangle, it can be quite a good approximation.

Gerbert pointed out that these two methods give different results. In fact, for an equilateral triangle with a side 30 feet long, Adalbold’s method gives you an area of 465 square feet, while half time base times height gives you 390 square feet - more than a 15% discrepancy. He demonstrated again with a side length of 7 feet, treating half the square root of three as roughly 6 sevenths. This time the two methods give you 28 and 21 square feet respectively - as Gerbert puts it,

"Thus, in a triangle of one size only, there are different areas, a thing which is impossible."

He drew Adalbold a diagram to illustrate his point, showing how, in the method with the squares, the squares always extend past the edge of the triangle, so there’s a bit of square left over. Thus, using a diagram - something that really someone should have thought of doing already - he demonstrated that half times the side length times the side length times half of the square root of three is the only correct formula for the area of an equilateral triangle.

You kind of get the impression from the letter that people in the 10th Century generally thought Adalbold’s method was an acceptable way to find the area of an equilateral triangle. Adalbold wasn’t an idiot - later on he actually wrote to Gerbert about calculating the volume of a sphere, and demonstrated that it depends on the cube of the radius. Admittedly Archimedes had also demonstrated this over a millennium earlier, but all that Greek stuff was lost. And Adalbold’s method does actually give you an OK approximation of the area of a triangle, if the units of length are small enough compared to the size of the triangle. Gerbert’s method is an approximation too. He hadn’t actually worked out the value of the square root of three - and that would be very laborious indeed without a calculator - he’s assumed half the square root of three is 6/7, and that means the result of the calculation is going to be about 1% off, and Adalbold’s method, working with exact values, is actually going to end up more accurate if the triangle’s big enough.

But by the time Gerbert wrote this letter, back in Rome, Pope Gregory V had died of malaria at the age of 28. Otto knew exactly who to pick as his successor - an experienced churchman, intelligent, well-spoken and, most importantly of all, loyal to Caesar and to the renewal of the Roman Empire. The first prophecy of the Head was correct.

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