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Last time, we saw how Gerbert, a boy from an obscure rural backwater in South-Central France joined the local monastery, and turned out to be intellectually impressive enough that when Borrell II, Count of Barcelona, visited the monastery at Aurillac, him and the Abbot decided that it was in everyone’s best interests that he should go back to Catalonia with Borrell, to further his education there, at the Cathedral of Vic, rebuilt after the region was conquered by the Franks from the Umayyads, and the Monastery of Ripoll, founded by Frankish monks moving south to settle this depopulated frontier territory.

Frankish Catalonia’s position adjacent to the border between Christian and Muslim civilisations meant that it started to function as a bit of an interface between them, and as the point where ideas from the Islamic World could enter Christendom. The translation movement of the Abbasid Caliphs gave the Caliphate’s scholars access to ideas from Greece that had been lost in Western Europe since the fall of the Western Roman Empire, and ideas from Persia and India, all translated into Arabic, and these scholars had built on these foundations to produce all sorts of remarkable achievements in astronomy, medicine, mathematics and philosophy. And the fact that there was a common government, language and religion all the way from Central Asia to North Africa meant that it was fairly straightforward for these ideas to spread across that distance. Even Spain, a very long way from the Abbasid Capital at Baghdad, and part of the Emirate of Cordoba that had broken away from the Abbasid Caliphate more than 200 years before Gerbert reached Vic, was plugged into this civilisation stretching from the Indian Ocean to the Atlantic. 

In 976, not long after Gerbert was in Catalonia, at another monastery in the Christian parts of the Iberian Peninsula, this time at San Martin de Albelda, in La Rioja, in the Kingdom of Pamplona, a team of three monks completed a book. Their leader was called Vigila, hence the book became known as the codex vigilanus, and he was assisted by another monk called Sarracinus and his pupil Garsea, the three of them depicted on a page at the end of the book, beneath six Spanish Kings, past and present. 

It’s a huge, spectacular volume, spanning 429 folios, which I think comes to a total of one thousand, seven hundred and sixteen beautifully decorated pages. As far as content goes, it’s an assemblage of legal, historical and theological texts, including sermons by the Church fathers, excerpts from the Etymologies, an encyclopedic work by the 7th-Century Visigothic bishop Isidore of Seville (that featured in Episode 9 of this podcast - The Forbidden Experiment - Part 2), the law code of the 7th-Century Visigothic Kings (extremely sophisticated by Early Medieval standards, and that remained the basis of the laws of Christian Spain in Gerbert and Vigila’s day), a biography of Mohammed, and the rulings of various Church councils, all interspersed with poems, acrostics and illustrations. And among this sprawling array of content, there is the paragraph

“We must know that the Indians have the most refined skill, and other nations should give way to them in arithmetic and geometry and the other liberal disciplines. And this is manifest in nine figures which each designate the value of each number, of which these are the forms.”

And then these nine figures are represented below. The shapes are a bit different from what we’re familiar with now, but this is the very first known representation from Christian Europe of what we now know as Arabic numerals (or Hindu-Arabic numerals, to emulate Vigila in giving proper credit to the nation where this ingenious system originated).

For well over a thousand years by this point, Europeans had written numbers using the system invented by the Romans. You’re probably familiar with it already. One is the letter I. Then you string them together like tally marks until you get to five, which is V (the Romans eventually realised you could save a bit of space by writing four as IV, rather than IIII). Then you go VI, etc. up to ten, which is X. XI, XII, XIII, XIV, XV, etc. Fifty is L, one hundred is C, five hundred is D, one thousand is M, and you just kind of string them together. The system works OK for representing small whole numbers, for example when Roman numerals are used as the indices of numbered lists. But for large numbers it becomes somewhat bonkers. You’ll have seen dates written in Roman Numerals. The current year is MMXXVI. That’s not too bad, but consider, for instance, that I was born in MCMXCVII, and the longest year we’ve had so far to write in Roman Numerals is 1888 - MDCCCLXXXVIII. 

And then imagine doing something like long division with them. Absolute chaos.

But the notation from India was more sensible. There were 9 different symbols, each representing a whole number from 1 to 9, and you can think of them as being arranged in columns. If you move a digit one column to the left, you multiply its value by 10. In the column to its right (the ones column, to the right of the tens column), you put in the digit you want to add on to it. And when you want to leave an empty column, you mark a little dot, to show that you’ve left that gap on purpose. Gradually the dot began to be treated as a number in its own right, rather than just a placeholder, and it would eventually morph into a hollow circle.

In about the 8th Century, the expanding Abbasid Caliphate came into increasing contact with India, and with Central Asia, where Buddhist monasteries had been established on the trade routes between India and China and become important centres for the transmission of Indian scholarship.

As I’ve already mentioned, the Abbasid Caliphs sponsored translations of all sorts of high-brow content into Arabic, and this included texts from India, on things like Mathematics, Astronomy and Medicine, from Sindh and from the Buddhist monasteries of present-day Afghanistan. 

In the last episode, I mentioned Al-Khwarizmi, whose Concise Book of Calculation by Restoration and Balancing was the origin of algebra as we know it, and whose name gives us the word Algorithm - well, he was from the far east of the Abbasid empire - from the Central Asian region of Khwarezm, in fact, hence his name, the same part of the world where ideas were moving between India and China, and westwards to Persia, so he’d probably have come into contact with Indian numerals there, and once he was in Baghdad, some time in the first half of the 9th Century, he wrote a book on them, the oldest known Arabic text on the subject (although the original Arabic text no longer survives, and nor does the 12th Century Latin translation, so all we’ve got to go on are two 13th-Century adaptations of the 12th-Century translation.)

And as I’ve also already mentioned, a common language and religion throughout the Islamic world meant ideas could spread pretty quickly from India to Spain. So it’s hardly surprising that by the closing decades of the First Millennium, monks at the Christian end of Spain were aware of this clever system, and heaping praise on its creators for producing something so obviously superior to Roman numerals.

The other product of the intellectual environment of the Islamic world that was filtering through to the Christian parts of Spain while Gerbert was there was astronomy. 

There was another important manuscript produced in the Spanish march, probably a few decades after the Codex Vigilanus. It’s smaller, and admittedly less visually impressive, but as an indication of the kind of ideas trickling through from the Islamic world it’s extremely important. It was made at the monastery of Santa Maria de Ripoll, where Gerbert spent a lot of time. Like the Codex Vigilanus, it’s an assemblage of miscellaneous texts, although this time with a more scientific and mathematical focus, rather than legal and historical, and a lot of it is Latin translations of Arabic texts. Among these, are a few sections describing how to use an astrolabe.

The first written references to these devices come from Roman Egypt, and by about the 6th Century, astrolabes seem to have been relatively common throughout the Eastern Mediterranean, where they eventually caught on in the Islamic world, and by the 10th Century knowledge of this technology had reached Al-Andalus. Meanwhile, in the West, knowledge of the Astrolabe had not survived the fall of the Western Roman Empire (in fact as far as I can tell all the evidence of astrolabes being made and used before this time comes from the eastern half of the Empire anyway).

The simplest astrolabes were simply devices for measuring angles, mostly useful for navigation. It’s a circle of wood or metal with angles marked around the outside, and a movable bar fixed across its diameter. On a clear night, you hold up the astrolabe so that a line down the middle of it is level with the horizon, and then you rotate the bar until you can look along it to the pole star, and then you read off the angle and that tells you your latitude.

The more complex version is very sophisticated indeed. It has a movable bar that you can use to take the altitude of the Sun or a star in the same way, and on the other side it has another movable bar, and a pair of circles, one inside the other but off centre, representing the path of the sun through the sky, and these move together around the centre and have bits of metal sticking out to mark the positions of various important stars. All this is fixed on top of a map of the sky from wherever you are, projected onto a flat disc (these only work for a specific latitude, and can be swapped out depending on where you are). By taking the angle of the Sun or of a particular star, and rotating these circles, and a ruler fixed to the centre, to match the altitude of the sun or the relevant star on the projection of the sky, you can tell the time, or by doing the process in reverse, moving the circles to correspond to a particular date and time, you can see what stars were where at the time (useful if you want to, for example, cast a horoscope).

The oldest known astrolabe with Latin inscriptions dates to some point in the 10th Century, and is indeed from Spain. Degrees are marked with a unique numbering system with Latin letters representing numbers, and it includes plates for several latitudes, from 36 degrees North, roughly the latitude of the coast of the Maghreb, all the way up to 47 and a half degrees, roughly the latitude of Vienna, Basel and Nantes, far outside the Dar ul-Islam, where Muslims were generally discouraged from travelling. One of the plates is labelled “ROMA ET FRANCIA” - ‘Rome and the Frankish lands’, and this is roughly the latitude of Zaragoza, the northernmost major city in Al-Andalus, Barcelona (the southernmost major city of the Frankish-ruled Hispanic March) and in fact also of Rome. Traditionally it’s been thought to have been made in Catalonia, at around the sort of time Gerbert was there, but the historian Thomas Freudenhammer reckons it would actually make more sense for it to have been made by or for a Christian in Muslim Al-Andalus, who was planning on travelling to the Christian world.

It’s not completely clear whether Gerbert himself actually ever used one of these devices. Later on, after he’d moved on from his Catalan days, he asked a contact in Barcelona to send him an astronomical manuscript. It was traditionally thought that that was the manuscript from Ripoll with sections on the astrolabe, but nowadays the consensus seems to be that this manuscript was probably made too late for it to be the one. And there’s an astrolabe manual that was traditionally said to have been written by him, but most people now seem to think it actually wasn’t, and was from a bit later. So there’s actually no smoking gun linking Gerbert himself to the earliest appearances of astrolabes in Western Christian Europe. Nonetheless, the fact that he requested an astronomical treatise from Catalonia does generally suggest that it was his experiences in this region that inspired his later astronomical exploits.

And later writers (writing at a time when the astrolabe was becoming more well-known in Christian Europe) certainly seem to have thought this was something he learned about in Spain. William of Malmesbury, in his lurid tale of Gerbert’s Andalusian adventures, writes that “There, he surpassed Ptolemy with the astrolabe”, although this is the same William of Malmesbury who also claims that Gerbert fled his monastery for Muslim Seville, where he stole a book of magic, evaded its owner’s attempts to track him down by means of astrology, and sold his soul to the devil in return for being supernaturally transported across the sea, so I don’t think we need to take anything he says remotely seriously.

Anyway, Gerbert spent three years studying in Catalonia, at the cathedral of Vic, and at the monastery of Ripoll in the foothills of the Pyrenees. But in 970, Borrell, the Count of Barcelona, had a problem. Barcelona didn’t have its own archbishop. The Frankish conquest of Catalonia had started from Narbonne, on the South Coast of France, and as a result, for ecclesiastical purposes, Barcelona, and indeed Vic, were subordinate to Narbonne, all the way across the Pyrenees. 

Borrell would have to talk to the Pope in person, and who better to accompany him than his genius monk from Aurillac. 

That December, Borrell, Atto and Gerbert, and presumably a whole entourage, set out from Barcelona to meet the Holy Father, Pope John XIII himself.

Easily the fastest route from Barcelona to Rome would have been directly across the Tyrrhenian Sea, via Corsica or Sardinia, but what with the pirates and the fact that ships just weren’t very reliable, this would have been extremely risky - plus, it was winter, and nobody sailed in winter. Instead, the Count and the monk and the bishop would almost certainly have travelled overland, passing from one monastery to the next, probably visiting religious sites along the way, as Borrell had been doing at Aurillac when he met Gerbert.

They’d probably have started out that December the same way Gerbert entered Catalonia three years earlier, crossing the eastern end of the Pyrenees at the Col de Panissars. From Barcelona they would most likely have followed the Roman roads to Arles in Provence, and then either followed the Via Julia Augusta along the Riviera, skirting the Southern end of the Alps, and passing the isolated Saracen outpost at Fraxinetum, now St Tropez. Or given that this road seems to have fallen into disrepair by this point, perhaps they’d have taken the less direct Via Domitia, climbing from the rolling hills of Provence into the depths of an Alpine winter, through Briancon to the Col de Montgenevre, probably paying a substantial fee to the Saracens from Fraxinetum who also controlled the passes between France and Italy - and would probably have abducted these illustrious travellers and demanded a large ransom if given the chance, as happened to the Abbot of Cluny two years later. Then they would have descended out of the snows to the broad plains of the Po Valley. They’d have joined the Via Francigena, the pilgrim route from most of the rest of Western Europe, through the rugged Apennines and the olive groves and bare vineyards of Tuscany, before passing through the Aurelian walls into Rome.

Rome, by the end of the first Millennium, had changed a lot in the past thousand years. At its peak, in the first century CE, it was home to hundreds of thousands of people, potentially over a million. But as the Empire collapsed, and amid the war and plague of the 6th Century, the City’s population had declined, and then plummeted, and wouldn’t surpass its ancient peak until the 20th century. The Capital of Western Christendom was now home to the Pope, and probably around 30,000 other people, living among great ancient ruins that they gradually plundered for stone.

Entering the city from the North, Gerbert would have passed through the Campus Martius - the ‘Field of Mars’. In ancient times this had been a district on the outskirts of the city where Emperors had built theatres and baths and temples, but when the aqueducts supplying water to the seven hills had been cut off, much of the population had migrated to this malaria-infested floodplain among these decaying monuments.

Rome was certainly a mere shadow of what it had been in antiquity, and about the size of a small modern town, but it was still impressively large by the standards of Christian Western Europe at the time, and would have been by far the largest city Gerbert had ever visited in a life split between the rural backwater of Auvergne and the rugged frontier country of Catalonia (that is, assuming he didn’t actually travel to Muslim Spain, where Cordoba was probably more than ten times the size of Rome). 

Borrell, Atto, and Gerbert would have passed right over the Capitoline Hill next, through the ancient city centre, past the crumbling, half-buried columns of the forum, and the dilapidated shell of the Colosseum.

The Pope at this point didn’t actually live in the Vatican, where a church had been built on the site of St Peter’s execution, across the Tiber from the city centre and outside the ancient city limits, but in the Lateran Palace, in the South-East of the city, adjoining the Archbasilica of St John Lateran, the Cathedral of the City of Rome. At the time it was a huge  fortified complex, built against the Aurelian walls, on the site of a Basilica from the time of Constantine, but rebuilt after an earthquake in 896, surrounded by imposing walls to defend against raiders from North Africa. 

Anyway, Borrell met the Pope, and, from his perspective, it went brilliantly. He came away with exactly what he wanted. Atto was to be made Archbishop of Tarragona, completely independent from Narbonne. The problem was that Tarragona, further down the Catalan coast than Barcelona, was ruled by the Umayyads. So Atto would have to be entirely theoretically Archbishop of Tarragona while actually running the Archdiocese from Vic. He seems to have been also referred to as Archbishop of Ausona, after the County of Ausona, the inland region of Catalonia that Vic was in. 

Gerbert had managed to impress the Pope himself with his knowledge of astronomy and music theory. The Pope, of course, had an impressive list of contacts, and sent a message to the Holy Roman Emperor himself, Otto I, who was looking for a tutor for his 16-year-old son (also called Otto), and who actually lived in Rome at this point. This trip had turned out very well for Gerbert too. 

Borrell and Atto, according to Richer of Reims, writing not that long afterwards, took some persuading to let Gerbert stay, but this was the Pope they were dealing with, and he promised Gerbert could return to Catalonia in due course. So the Count and the Archbishop left Rome, to make the hazardous journey back to Atto’s new Archbishopric, and Gerbert stayed behind, preparing to start his most prestigious assignment yet, at the Imperial Court. 

Meanwhile, shortly afterwards, likely as he was on the way back to Vic, Atto was murdered. We don’t really have any detail of what exactly happened, but the Archbishop of Narbonne can’t have been very happy with Atto’s promotion, which had meant he lost a big chunk of his territory. Anyway, if he did do it it seems to have worked, because the Archdiocese of Narbonne remained in control of Catalonia for several decades more. Interestingly, Froia, Atto’s replacement as Bishop of Vic was also assassinated, 20 years later, by supporters of a rival claimant to the bishopric.

The following year, 972, Otto Jr. married Theophanu, the niece of the Byzantine Emperor, who brought her sophisticated Greek habits with her to the court of these German barbarians, supposedly including eating with a fork, a utensil previously unknown in Western Europe. And this was also when Gerbert met Gerannus, the archdeacon of Reims. Like just about everyone who met Gerbert, he seems to have been absolutely blown away by this man’s mathematical ability. With Gerbert’s services presumably no longer needed by Otto, Gerannus took Gerbert away to Reims, back to his French homeland, to teach at the Cathedral school.

Reims was an important place. The Cathedral had been there since Roman times, and the Archbishop of Reims was usually the one who crowned the King of France. And the Cathedral school at Reims was quickly becoming very important, too.

The Cathedral schools at this time were intended to train priests - and at this time the job of a priest went far beyond just religious matters. Priests and monks were the people you could rely on to be able to read and write, and to understand Latin, so all the admin involved in running a Kingdom, that was all being done by priests. There’s a reason the words clerk and cleric are related. Pretty much anything that involved writing was generally a job for the clergy, so at the Cathedral school, much like at the monastery school of Aurillac where Gerbert had studied, students had to teach the Seven Liberal Arts - Grammar, Logic, Rhetoric, Arithmetic, Geometry, Music, and Astronomy. 

The Bishop, Adalbero, had a plan to turn this place into one of the greatest educational establishments in Christian Western Europe, and Gerbert would be the perfect person to have on board.

At Reims, Gerbert seems to have had quite a hands-on approach to teaching mathematics, astronomy and music. Most of what I’ve described already in early medieval education was focused entirely on learning from books - and as you should by now be aware, the books themselves were very limited - but Gerbert became known for the practical demonstrations he used in his teaching.

Musical theory, he could demonstrate using a monochord - that is, a single, taught string, stretched across a board, with a weight hanging from the end, and a bridge that could be moved along to produce different lengths of string either side of it. This wouldn’t be a particularly useful instrument for actually playing music, but that wasn’t what medieval students were learning when they were learning music as part of the Quadrivium. This was all about the Pythagorean idea of musical harmonies as expressing the deep mathematical structure of reality, and to demonstrate that, the monochord was perfect.

By swapping in different weights you could vary the tension in the string, and by moving a bridge along the board you could vary the ratio between the lengths of string either side of it - effectively giving you two vibrating strings that you know have exactly the same tension because they’re being pulled by the same weight. Then you can find the positions of the bridge, and thus the ratios of string length, that produce harmonious intervals. If you put the bridge half way along, both sides of the bridge will give exactly the same note when you pluck them, and that will be an octave higher than without the bridge there at all. Likewise if you put the bridge a third of the way along, so that one side is half the length of the other, the two sides of the bridge make notes an octave apart. Putting the bridge three-fifths of the way along, so there’s a three to two ratio between the lengths on each side, gives you notes exactly a fifth apart. Putting the bridge four sevenths of the way along - a 4:3 ratio - gives you notes a fourth apart. A 5:4 ratio gives you a major third, a 6:5 ratio gives you a minor third, 9:8 is a whole tone, and so on. These tidy, whole-number ratios are the ratios that give you harmonious combinations of notes. 

As far as content goes, this was actually pretty standard for early medieval education. These concepts are covered pretty thoroughly by authors like Boethius, whose works Christian Western Europeans at the time absolutely did have access to in Latin, and who, as I’ve mentioned before, was the main route for transmitting knowledge from Ancient Greece to Medieval Western Europe at a time when almost nobody (except presumably Theophanu) could understand Greek.

It was in arithmetic and astronomy that Gerbert’s teaching methods really were unusual. Early Medieval astronomy was traditionally taught as a largely theoretical matter. Students would read Calcidius’s Latin translation of Plato’s Timaeus and his commentary on it, and various other miscellaneous fragments of ancient Greek astronomical texts that had happened to survive the fall of the Western Roman Empire.

Gerbert, though, used several powerful teaching tools. His student, Richer of Reims, described them in his chronicle, and we can piece together more detail from letters Gerbert sent. The first that Richer describes was a solid sphere, representing the night sky, with stars marked on it. The local horizon was also marked, with a circle around the middle of the globe. It seems to have been possible to rotate the globe to visualise stars rising and setting on the horizon.

Second was a hemisphere - half a sphere - with sighting tubes to align it with the stars. On the edge of the hemisphere, there was a tube that was supposed to point to the North pole, and you could line it up by making sure you could see the pole star through it. In one of Gerbert’s letters, he described how you could tell if it was lined up correctly by looking through the North Pole tube all night, and if the star remained in view then you knew it actually was the pole star and not some other star. This might have been more difficult than it is now, because Earth’s axial precession, a slight drift in the orientation of Earth’s axis of rotation, means that the positions of stars in the sky changes noticeably on a timescale of centuries, and in the 10th Century the extremely bright and obvious Polaris, while it was already close enough to the north pole to be useful for things like navigation, likely wasn’t the star he was talking about. He may well have been referring to a star like HR 4893 which is much fainter, but at the same time was closer to the North Pole. Nowadays you need to be somewhere with very dark skies to even see this star, but in the Middle Ages the sky was dark everywhere, so they’d have been used to seeing far more stars than most of us.

Once you’ve got the North Pole tube lined up with the celestial North Pole, and you’ve made sure the setup is actually vertical (maybe you hang it from something), then the other tubes are going to be lined up correctly too. One is going to correspond to the Arctic Circle, and you’ll see in turn all the stars that are above the Arctic Circle. The next does the same for the Tropic of Cancer, then the Equator, then the Tropic of Capricorn, then the Antarctic Circle, then a tube directly opposite the North Pole, pointing to the stars above the South Pole. It’s worth noting that Reims is very much in the Northern Hemisphere, so the South Pole tube would have been pointing straight into the ground. In the words of Gerbert, writing to his fellow mathematician-monk Constantine of Fleury, 

“As for the south polestar, because it is under the land, no sky but earth appears to anyone trying to view it through both tubes.”

An obvious use for this would be identifying the solstices and equinoxes, and demonstrating to your students how they work. On the summer solstice, at noon (if you point the hemisphere in the right direction, the sun should shine directly down the Tropic of Cancer tube. At the Winter Solstice it shines down the Tropic of Capricorn tube, and at the Equinoxes it shines down the Equator tube.

The third of Gerbert’s devices was probably the most intriguing of them all. Essentially, it was another sphere, this time demonstrating the positions of the planets in the zodiac. This seems to have been hollow, made up of a network of metal rings. There were rings representing each of the main parallels of latitude, and another representing the zodiac - the constellations that the sun appear to pass in front of over the course of the year. And since all the planets orbit the sun (or, according to Gerbert and everyone else at the time, the Earth), in roughly the same plane, they’re all moving in the same circle as the Zodiac, so that was where he put markers representing the planets, too, so he could move them to show his students how they moved through the zodiac over time, with at least some of the complicated motions necessary in the Ptolemaic model. Richer tells us that

“In this oblique circle, he suspended the wandering stars with wonderful ingenuity”.

But then, annoyingly, he stops short of telling us exactly what this wonderfully ingenious method was.

“How this was, because of its length, is not convenient to describe here, lest we seem to stray too far from our purpose.”

And finally there’s the fourth, possibly the simplest of them all. It’s a celestial globe, a solid sphere with the shapes of the constellations marked on it with wire, to form a map of the night sky. Like the second device, it had a tube passing through it that you could use to find the pole star, and line it up with the actual sky, presumably meaning he could take his students out at night and point out the constellations to them on the globe.

Let’s talk a bit more about that third sphere, with the interlocking circles and the planets that move in an unspecified way through the Zodiac. What Richer seems to be describing here is an armillary sphere. These were very widespread in Western Europe later on, and were popular in Ancient Greece. In Gerbert’s day, though, in Christian Western Europe at least, there’s no other evidence that they existed. At the very least, Richer’s description makes it clear that he’d never heard of them before.

There are a few possibilities for how Gerbert found out about these devices. First, Martianus Capella, who Europeans of the 10th Century were familiar with, kind of briefly mentions that they were a thing in his allegorical work on the liberal arts, bizarrely titled The Marriage of Philology and Mercury, but he doesn’t in any way actually describe how they work. It’s possible Gerbert had never heard of these before and was just the first person smart enough to figure out how to make these things that Martianus Capella describes.

However, these things were much better-known in the Greek East and the Islamic world. After all, Ptolemy’s Almagest goes into much greater detail about them than Martianus Capella, and if you’ll recall from the last Episode (which, to be fair, was a while ago now), Ptolemy’s Almagest was entirely unknown in the Latin West at this point, but it was known in both the Byzantine Empire and the Islamic world. The Syrian Astronomer Al-Battani seems to have been the leading expert in these devices, making them and refining Ptolemy’s design almost a century before Gerbert d’Aurillac was demonstrating them in Reims, and like the Astrolabe, we know for a fact that by the eleventh century these things were being made in Al-Andalus, but again there’s nothing conclusively linking Gerbert to them. 

But one thing that does seem to pretty decisively demonstrate that Gerbert had been picking up ideas from the Islamic world when he was in Catalonia, is a device he used to teach arithmetic. This seems to have consisted of a board, divided into columns, along with a thousand counters. Richer is infuriatingly vague about how it actually worked, but he seems to have been absolutely blown away by how quickly Gerbert could perform complex calculations just by arranging the counters in various ways. But what’s really fascinating, is that Richer says the counters were marked with 9 different symbols. Richer doesn’t say what the symbols are, but it’s certainly suggestive.

We also get Gerbert himself explaining how it works, and he goes into much more detail, in a treatise he wrote for his friend Constantine of Fleury. After a prologue addressed to Constantine, that he begins by declaring

“The power of friendship reduces the almost impossible to the possible. For how could I attempt to explain the calculation of numbers on an abacus unless you were encouraging me, O, my sweet consolation of labours, Constantine.”

Then he proceeds to explain how you can use it for division and multiplication. It’s difficult to follow, because he’s trying to explain all this entirely verbally to someone unfamiliar with the concept of place-value notation, and his vocabulary is borrowed from finger-counting systems (although the way he figuratively uses digit would end up sticking), but what he’s describing, at least for the multiplication, is actually just the modern long multiplication method we all know and love.

These things spread, pretty quickly. From Bern, in Switzerland, probably towards the end of the 10th Century, we have a drawing of one of these counting boards. It has 27, columns, labelled with Roman Numerals, and at the top a line of Latin verse saying “Gerbert gave the Latins the numbers and the figures of the Abacus”, indicating that this was generally seen at the time as something introduced to the West by Gerbert d’Aurillac specifically, rather than just a general trend that he happened to pick up on.

But we still haven’t been told what the symbols on the counters are. For that, we have to turn to Gerbert’s student, Bernelinus.

Some time around the turn of the 11th Century, Bernelinus wrote a book called Liber Abaci - “the book of the Abacus”. This is another explanation of how to use this device, but unlike the explanations written by Gerbert himself, Bernelinus actually shows us the shapes of the symbols Gerbert was using. And guess what? They’re Arabic numbers. The two is upside-down, the seven is backwards (but it’s worth remembering that these were inscribed on counters, and the orientation of the counters on the board wasn’t necessarily consistent), and the three, four, and five are completely unrecognisable, but the one, six, seven, eight and nine are exactly what you’d expect.

And from about the same time as Bernelinus, or maybe even slightly earlier, around the end of the 10th Century, at Echternach in Luxembourg, we have two separate drawings of an abacus, one on a scrap of parchment that ended up being recycled a few decades later as part of the binding of a Bible, and wasn’t discovered until 2001, and another that was actually used in a proper manuscript, along with other texts on arithmetic - but nobody seems to have noticed it until after they discovered the one from the Bible (and with this one they had no excuse). These illustrations also illustrate the Arabic numerals, at the top of the Abacus, with similar shapes to the ones from Bernelinus. And also, in the second of these two manuscripts (which Nancy Marie Brown, in The Abacus and the Cross, argues might actually be the notebook of one of Gerbert’s students by the name of Gausbert), there’s actually a page with the numbers written out, and a mnemonic poem for remembering their names.

The names are another interesting thing. A couple of them clearly correspond to the modern Arabic names of the numbers. Arbas for four matches ‘arba’ah. Quimas for five matches khamsa, maybe confused with Latin quinque.Temenias for eight matches thamaniyah. The others are more mysterious. Igin for one? Nancy Marie Brown reckons this might be Berber. Tashelhit and Central Atlas Tamazight are both Yan, and Kabyle is Yiwan, and this would certainly make geographical sense given it was mostly Berbers settling in Al-Andalus. Andras for two? Ormis for three? Calctis for six? These seem to have no clear explanation, nor does Zenis for seven, or Celentis for nine.

And then, last in this list, is something we haven’t come across before. Not in the numbers in the Codex Vigilanus, and not in the numbers mentioned by Bernelinus. Presumably Gerbert didn’t use it much, either, because Richer only mentions nine figures. The last symbol in the list looks like a sort of wheel, with a circle inside another, connected by five spokes. And the line alongside it says “From here follows Sipos, for that’s what the wheel is called.”

Sipos, from the Arabic sifr - “empty”. That’s right, as far as I can tell, this is the earliest known appearance of zero in any Western document. 

I absolutely do not have time here for a detailed exploration of the development of the concept of zero. Maybe that could be another series in itself. The point is that the ancient Babylonians used a place value system to write numbers, a bit like what we use now, or what Gerbert was doing on his abacus, although they did it in Base 60. The problem was originally, they had nothing to show when there was supposed to be a gap between the digits. For example, 1, 60, 3600 or even 216,000, would all look exactly the same. So they started using a little mark to show that there was supposed to be a gap there. For the Babylonians, though, it only seems to have been a placeholder, not a number in its own right. This concept never spread west - the Greeks kept using a Roman Numeral style system (but with Greek letters), so they had no need of a placeholder character, and given that they thought of maths very much in terms of geometry and ratios, the idea of zero as a number would have made no sense. But it does seem to have spread east, if over a very long time. By the fifth century CE, Indian mathematicians were using a place value system, and the oldest known written example of zero in the Indosphere comes from a date in an inscription from 683, from Cambodia of all places, written in Khmer. 

And that same century, the Indian mathematician Brahmagupta wrote about zero’s mathematical properties, and in doing so became the first person known to have written about zero as an actual number, rather than just a placeholder. He even made a bold yet ultimately doomed attempt to understand what happens when you try to divide things by it. 

From here zero spread to the Islamic world, along with the Indian number system more generally, with the Sanskrit word Sunya, meaning void, being translated to the Arabic Sifr. 

And from there, at some point, it must have ended up in Echternach. 

It’s not particularly clear what the monk at Echternach was using this Sipos for. It would be centuries before there’s any evidence of westerners thinking of zero as a number, that you can actually do maths with. The corresponding line of the mnemonic doesn’t tell us anything about what the monk thought it did, only that it looks a bit like a wheel. He’s put it at the end of the sequence as well, which kind of suggests he’s not thinking of it as a number that comes before one, but as some kind of separate thing (the reason it’s like that on a modern keyboard is apparently because the first typewriters saved space by not having keys for zero or one, and typists just using I and O. They started introducing a dedicated zero key before the dedicated one key, and having zero on the end looks less weird than having it immediately before 2). 

The reason Gerbert had little use for zero is pretty clear, and that’s that he was doing everything on a physical abacus board, with columns marked out. You don’t need a special character to mark when you’re leaving an empty space, because that’s just a column with nothing in. And all these numbers were used for was simplifying a calculation process - if you were writing up a nice pretty document with the results of your calculations in, you’d put the final numbers in Roman numerals. Parchment was very expensive. Nobody was wasting it on scribbling out rough calculations, so that all happened on these boards. Maybe the Echternach monk just found Sipos made it easier to keep track of how many empty columns he was leaving, or maybe it helped him make sure the rows didn’t get mixed up when there was a gap in one but not another. Maybe he didn’t actually have any sipos counters, but thought he needed to complete the set in his mnemonic.

But for all Gerbert’s success at Reims, he was in a powerful position now, with connections to the Emperor and the Pope, and he wasn’t going to be able to stay out of the chaotic politics of 10th-Century Europe.

But that’s a story for another episode. You’ve been listening to Science: A Peculiar History. You can find the podcast almost anywhere you get podcasts, or at scienceapeculiarhistory.co.uk, and donate to help support the podcast at buymeacoffee.com/joebath. Remember to follow the podcast on Facebook, Twitter, Instagram and Bluesky. The website also has episode transcripts, as well as images and sources for the earlier episodes. If you have any questions comments, corrections or suggestions, you can email admin@scienceapeculiarhistory.co.uk, message any of the social media channels, or use the contact form on the website.