Part 3 (1/2)

Yet friends.h.i.+p was also useful, Gerbert noted to the abbot of Tours. ”Because I am not the sort of man who, with Panetius, sometimes separates the honorable from the useful, but rather with Cicero would add the former to everything useful, so I wish that this most honorable and sacred friends.h.i.+p may not be without its usefulness to both parties.” How could the abbot best demonstrate his friends.h.i.+p? Gerbert attached a list of books he could have copied and sent to Reims.

Given his own reticence, our best window into Gerbert's school is the description by Richer of Saint-Remy in his History of France History of France, written between 991 and 997. This was twenty years after Gerbert had come to Reims and coincides with a time when his political troubles were at their height.

Richer begins by outlining Gerbert's teaching of the trivium. To learn Latin grammar, his students studied Cicero; the poets Virgil, Statius, and Terence; the satirists Juvenal, Persius, and Horace; and Lucian the historiographer. ”Once his pupils were familiar with these and acquainted with their style, he led them on to rhetoric,” writes Richer. ”After they were instructed in this art, he brought up a sophist on whom they tried out their disputations, so that practiced in this art they might seem to argue artlessly, which he deemed the height of oratory.” Concluding the study of dialectics, Gerbert read aloud from ”a series of books”-most of them by Boethius-”accompanied with learned words of explanation.”

Moving on to the quadrivium, Richer depicts Gerbert as a master of visual aids. ”He demonstrated the form of the world by a plain wooden sphere, thus expressing a very big thing by a little model,” according to Richer. ”This is how he produced knowledge in his pupils.” He made celestial spheres for observing the stars, to explain the motion of the planets, and for learning the major constellations. He had a s.h.i.+eldmaker construct an abacus: ”Its length was divided into twenty-seven parts, on which he arranged nine signs expressing all the numbers.” Using a thousand counters made of horn and marked with these ”nine signs,” he could multiply and divide with such speed that ”one could get the answer quicker than he could express it in words.” To teach music theory, Gerbert used a monochord, a simple one-stringed instrument.

Richer does not go into much detail. In many cases his descriptions of Gerbert's methods are unclear-he seems not to have been mathematically inclined himself. About the abacus, he says, ”Those who wish to understand fully this method should read the book which he wrote to the scholasticus Constantine, where one will find this subject fully treated.” While discussing Gerbert's celestial spheres, he simply breaks off: ”It would take too long to tell here how he proceeded further; this would sidetrack us from our subject.”

Nor can his account be entirely trusted. History for Richer was a literary art: He saw nothing wrong with putting into the mouth of Charles of Lorraine, who fought Hugh Capet for the French throne from 987 to 991, a speech by King Herod from the fourth-century Latin translation of Flavius Josephus's The Jewish War The Jewish War. Another of Charles's moving speeches comes directly from Sall.u.s.t.

Moreover, Richer fiddled with facts. He took episodes from the annals of Flodoard of Reims, who died in 966, and changed such things as the size of an army, the number of casualties, the locations of battles, and who won and who lost. He saw the changes as improvements: ”I think that I have done well enough by the reader,” he wrote, ”if I have arranged all things credibly, clearly, and briefly.” We should not expect him to be accurate or complete. His definition of ”history” was not the same as ours.

Richer knew Gerbert, but he was not Gerbert's student or admirer, as some historians have claimed. He was a monk at the monastery of Saint-Remy a few miles outside of Reims and the same age or older than Gerbert. The two had serious political disagreements, Richer being a partisan of the last Carolingian, Charles of Lorraine, while Gerbert, as we will see, was central in placing the challenger Hugh Capet on the French throne. Richer had reasons to flatter Gerbert, and reasons to distance himself. While Richer was writing his history, Gerbert, threatened with excommunication by the pope, was struggling to hold onto his position as archbishop of Reims. There are hints that Gerbert may have commissioned Richer to write about him in hopes of salvaging his reputation: Some pa.s.sages in the History History seem to have been cribbed from Gerbert's own letters. Yet, as he revised his book-and as it became clear that Gerbert would be evicted from his post at Reims-Richer put a subtle twist on events, calling Gerbert's actions and character into question. seem to have been cribbed from Gerbert's own letters. Yet, as he revised his book-and as it became clear that Gerbert would be evicted from his post at Reims-Richer put a subtle twist on events, calling Gerbert's actions and character into question.

When Richer died, with the history unfinished, Gerbert apparently got hold of the only copy-and hid it away. It did not circulate in the Middle Ages. No copies were made. All we have is Richer's very messy rough draft. It was discovered in the 1830s among the books of Gerbert's last student, Emperor Otto III, in the library of the cathedral of Bamberg. With its scribbles and cross-outs, tipped-in pages and marginal notes, asterisks and erasures and several colors of ink, the ma.n.u.script is evidence of a complicated writing process, and a writer trying to make up his mind.

As a close reading of the ma.n.u.script shows, Richer wrote the story of Gerbert's school separately and struggled to add it to his work-in-progress. He erased and rewrote the text that preceded it to make a better transition, but the section remains jarringly different from what surrounds it. And while the entire History History is dedicated to Gerbert, the dedication seems contingent on Gerbert's keeping the archbishopric. Although the king of France had appointed him to the position, the pope refused to consecrate him, arguing that another candidate had a better claim. Describing that seven-year-long dispute between the king and the pope, Richer sides with Gerbert's enemies and subtly contradicts the official record that Gerbert wrote. Gerbert comes off so poorly that it seems Richer never meant for him to read the final, much-revised account. is dedicated to Gerbert, the dedication seems contingent on Gerbert's keeping the archbishopric. Although the king of France had appointed him to the position, the pope refused to consecrate him, arguing that another candidate had a better claim. Describing that seven-year-long dispute between the king and the pope, Richer sides with Gerbert's enemies and subtly contradicts the official record that Gerbert wrote. Gerbert comes off so poorly that it seems Richer never meant for him to read the final, much-revised account.

Some of what Richer says about Gerbert can be corroborated. We know Borrell and Ato went to Rome-two of the five papal bulls still exist in Vic-and the cathedral records show Ato died before reaching home. From Gerbert's letters we know he met the emperor, briefly taught his heir, and then went to Reims to teach. Gerbert wrote about the abacus, the celestial spheres, and some other visual aids, but his descriptions are vague: They a.s.sume that his correspondent has seen the object under discussion.

Since he did not begin keeping copies of his correspondence until ten years after he had left Spain, it's hard to say what Gerbert taught when he first arrived at Reims. What did Gerbert know of musica musica, astronomia astronomia , and , and mathesis mathesis that had so impressed the pope, the emperor, and the archbishop in Rome? What science had he learned in Spain? Gerbert left no scientific ma.n.u.scripts, firmly dated before 970, to prove he learned anything extraordinary at all. that had so impressed the pope, the emperor, and the archbishop in Rome? What science had he learned in Spain? Gerbert left no scientific ma.n.u.scripts, firmly dated before 970, to prove he learned anything extraordinary at all.

Yet Gerbert's Catalan friends were well placed to learn more, as translations were made from Arabic and new scientific instruments-and the knowledge needed to make them-seeped north. And Gerbert's letters prove they kept in touch. Whether he learned Islamic science in Catalonia between the years 967 and 970 or he learned it later-by correspondence course, as it were-we can see from the excitement his school aroused; by the new directions taken by Gerbert's students, and by their their students; and by the criticisms of Gerbert's peers that the students; and by the criticisms of Gerbert's peers that the mathesis mathesis, astronomia astronomia, and musica musica Gerbert taught at Reims were unlike anything the Christian West had seen before. Gerbert taught at Reims were unlike anything the Christian West had seen before.

PART TWO.

GERBERT THE SCIENTIST THE SCIENTIST.

I shall, if life continues, explain these matters to you more clearly, as much as is necessary for you to attain the fullest understanding.

GERBERT OF AURILLAC, C. 979.

CHAPTER V.

The Abacus.

The rumors began seventy years after his death. Over time, they grew more and more surreal. By the twelfth century, a courtier could ask, ”Who has not heard of the fantastic illusion of the notorious Gerbert?” He sacrificed to demons and summoned up the devil himself. He was a wizard, a necromancer. Through a ravis.h.i.+ng witch, or a golden head, or a magical book he had stolen in Cordoba, or simply through ”the stars,” he foretold the future. He grew fabulously rich. He obtained everything he desired.

Ironically, both the honors Gerbert enjoyed in his lifetime and the contumely heaped on him later were due to one thing: what he taught at Reims. But Gerbert's genius is hard to pin down. His science has to be inferred, and the evidence is scanty. We have a few letters, the account in Richer of Saint-Remy's History of France History of France, the fresh approaches of his students, and a handful of artifacts. One such artifact came to light in 2001: an actual copy of Gerbert's abacus board. It makes clear that when Richer described Gerbert's abacus, with its counters marked with ”nine signs,” he was, in fact, recording the introduction of Arabic numerals to France.

The abacus found in 2001 is a stiff poster-sized sheet of parchment; it was trimmed down and reused as a pastedown in the binding of the Giant Bible made for the abbot of Echternach sometime between 1051 and 1081 (see Plate 5). The Bible is owned by the national library of Luxembourg, which had unbound it in 1940 in order to photograph the pages. The abacus sheet was taken out of the binding and stored in a box, where it remained, unrecognized, for sixty-one years.

Shortly after this find was announced, a librarian at the state archives of Trier identified a matching copy, smaller, but written in the same handwriting; it also came from the scriptorium at Echternach. This second abacus was bound into a very interesting ma.n.u.script. It could be the notebook of one of Gerbert's students. Following the abacus is a mnemonic poem on the names of the nine Arabic numerals and zero. The ma.n.u.script holds miscellaneous notes on multiplication and division, the use of Roman fractions, and the etymology of the word digit digit. These notes contain many corrections and erasures. They do not match any known source but seem to be the messy jottings of a single scholar, penned over a number of months or years. This student copied Gerbert's poem on Boethius into his notebook, along with other texts that reflect Gerbert's curriculum at Reims, as Richer describes it.

We can even guess at the student's name. The ma.n.u.script can be dated to 993 by its similarity to other large-format books made by a scribe historians have named ”Hand B.” One of these books is dedicated to the monastery of Echternach by an English monk named Leofsin. Leofsin moved to Echternach in 993. Before that, he had lived at Mettlach, alongside one of Gerbert's favorite students, a monk named Gausbert.

Gausbert is mentioned in several of Gerbert's letters. Some of these are addressed to Abbot Nithard of Mettlach, who himself had been Gerbert's student in the early 970s. Nithard's relations.h.i.+p with his former teacher was intimate-and a little p.r.i.c.kly. ”You think you alone bear burdens but you do not know what the overwhelming trials of others are,” Gerbert wrote him in 986. At issue between the two men was a ”treasure” belonging to Nithard that Gerbert refused to bring or send back to Mettlach. ”Since men are tossed about by an uncertain fate, ... why do you lay up a treasure for a bad turn of fortune by leaving it with me for so long a time? And, inasmuch as I, a trustworthy man, am addressing a trustworthy man, make haste. For, either the imperial court will summon me quickly, or, more quickly, Spain, which has been neglected for a long time, will seek me again.”

The treasure might have been a book or an abacus-or a monk. Nine months earlier Gerbert had complained to Nithard: ”Suddenly, with no consideration for the shortness of the time, you force Brother Gausbert to return with everything belonging to him. ... You have said that he is unwilling to return to the tedium of the monastery. If this is so, how are you going to hold him after he has been returned?”

Gausbert is also mentioned in two letters to Nithard's superior, the archbishop of Trier. ”We have never tried to hold the monk Gausbert against your wishes,” Gerbert claims, though it is clear he is sorry to see him go. ”We ask only this of your customary good will, that you exhibit kindness to him on account of our recommendation, and ... let him not lack the studies to which he has arranged to devote fuller attention.”

It's tempting to think Gausbert, bored by the tedium of the monastery and worthy of continuing his studies, was the messy scholar who brought his notes from Gerbert's cla.s.ses to Mettlach. There, Gausbert shared them-including the abacus and the poem on Arabic numerals-with Leofsin, who carried them on to Echternach and, ultimately, to us. For until the two abacus sheets made at Echternach were identified, scholars did not agree that Gerbert had used Arabic numerals or that his abacus was really anything out of the ordinary. With copies of Gerbert's abacus now in hand, every history of mathematics will have to be revised.

Sometimes called the first calculator or even the first computer, an abacus can take many forms. The ancient Chinese developed the version that springs first to mind: colored beads strung on wires set in a vertical or slanted frame. Gerbert's abacus did not look like this. Nor was it like the ancient Roman abacus: a palm-sized rectangle of bronze or clay with seven vertical grooves divided in half by one horizontal line. The Roman abacus used b.a.l.l.s, not beads, to represent numbers. A ball parked in a groove below the line stood for one-one unit, one ten, one hundred, and so on, up to one million. A ball placed above the line meant five of the same.

In Latin abacus abacus means ”table”; it may refer to a sideboard, a game-board, or a counting board like Gerbert's abacus, which was a simple grid of twenty-seven columns, scored or painted on a flat surface. Later, merchants and bankers found having a counting board so useful that they drew them on tabletops, which they called ”counters”: That is why we now do business ”over the counter.” means ”table”; it may refer to a sideboard, a game-board, or a counting board like Gerbert's abacus, which was a simple grid of twenty-seven columns, scored or painted on a flat surface. Later, merchants and bankers found having a counting board so useful that they drew them on tabletops, which they called ”counters”: That is why we now do business ”over the counter.”

Gerbert's abacus board introduced the place-value method of calculating that we still use today. Each column on Gerbert's abacus represented a power of ten. The ”ones” column was placed farthest to the right, and the numbers increased by a multiple of ten, just as we read numbers now, in each column to the left. The twenty-seven columns were grouped in threes, linked by swooping arches, just where we would divide a large number by commas. Each group of three was labeled with an Arabic numeral, from 1 to 9. With twenty-seven columns, Gerbert could add, subtract, multiply, or divide an octillion (1027). There was no practical reason for octillions-Gerbert was merely showing off. Witness the air of braggadoccio with which he writes to the emperor in later years, ”May the last number of the abacus be the length of your life.” That's 999,999,999,999,999,999,999,999,999. Don't even think of writing this in Roman numerals.

But the twenty-seven-column counting board caught on. A monk named Bernelin wrote a Book of the Abacus Book of the Abacus while Gerbert was pope. Protesting that it was presumptuous of him to try to better Pope Gerbert's brief and subtle work, he nevertheless designed an abacus with thirty columns (Gerbert's twenty-seven plus three for fractions). He suggested it should be drawn on a polished table. while Gerbert was pope. Protesting that it was presumptuous of him to try to better Pope Gerbert's brief and subtle work, he nevertheless designed an abacus with thirty columns (Gerbert's twenty-seven plus three for fractions). He suggested it should be drawn on a polished table.

Gerbert had his counting board constructed by a s.h.i.+eldmaker, according to Richer of Saint-Remy. To make a s.h.i.+eld, a piece of prepared skin is stretched over a large wooden frame. A s.h.i.+eldmaker would know how best to get a wide, smooth surface that could be painted on. Such a counting board would be light, st.u.r.dy, and more portable than a tabletop. For a teacher, it would make a fine visual aid.

A s.h.i.+eldmaker also had the tools to cut a thousand apices apices, or ”counters,” out of cow's horn. These counters (some modern translators use the term ”markers” to avoid confusion with the counting board itself) looked rather like checkers, with one important difference: Each was marked with an Arabic numeral, from 1 to 9. To calculate, Gerbert placed the counters on the counting board and shuffled them around. The speed with which he did so, said Richer, was astonis.h.i.+ng.

To reconstruct the process takes some imagination. Gerbert's own Book of the Abacus Book of the Abacus provides little help. Answering the request of his friend Constantine of Fleury, Gerbert claimed it was ”nearly impossible” to explain the rules of the abacus in writing and, moreover, he was out of practice: ”Since it has now been some years since we have had either a book or any practice in this sort of thing, we can offer you only certain rules repeated from memory.” Apparently he had been criticized for teaching this new math at Reims and saw the need to justify himself, at least to sympathetic ears, for he continued: provides little help. Answering the request of his friend Constantine of Fleury, Gerbert claimed it was ”nearly impossible” to explain the rules of the abacus in writing and, moreover, he was out of practice: ”Since it has now been some years since we have had either a book or any practice in this sort of thing, we can offer you only certain rules repeated from memory.” Apparently he had been criticized for teaching this new math at Reims and saw the need to justify himself, at least to sympathetic ears, for he continued: Do not let any half-educated philosopher think [the rules of the abacus] are contrary to any of the arts or to philosophy. For who can say which are digits, which are articles, which the lesser numbers of divisors, if he disdains sitting at the feet of the ancients? Though really still a learner along with me, he pretends that only he has knowledge of it, as Horace says. How can the same number be considered in one case simple, in another composite, now a digit, now an article?

Here in this letter, diligent researcher, you now have the rational method, briefly expressed in words, 'tis true, but extensive in meaning, for the multiplication and division of the columns [of the abacus] with actual numbers resulting from measurements determined by the inclination and erection of the geometrical radius, as well as for comparing with true fidelity the theoretical and actual measurement of the sky and of the earth.

The letter itself is hard to interpret. The person Gerbert calls a ”half-educated philosopher” can possibly be identified, as we will see. The ”actual numbers resulting from measurements” are also tantalizing. They could refer to measurements taken with an astrolabe or other scientific instrument, while the idea of ”comparing with true fidelity the theoretical and actual measurement of the sky and of the earth” is a foretaste of the scientific method. Those who believe there was no experimental science in the Dark Ages, only memorization and appeals to authority, have never read the letters of Gerbert.

But the rules for the abacus that Gerbert appends to this letter provide no details on what it looked like, and very little help on how to use it. Constantine must have already seen an abacus board, for Gerbert does not explain how to make one. Instead he merely lists which column the result should be placed in if one multiplies a unit by a ten, a ten by a ten, a ten by a hundred, a hundred by a hundred ... on up to a million by ten million. The rules for division are equally boring. He omits the rules for addition and subtraction, those being too elementary.

Gerbert does give some explanation of the twenty-seven columns. These were a sticking point for many learners, for whom the place-value system of arithmetic was radically new. What did the columns mean mean? Gerbert refers to them as ”intervals,” alluding to Boethius's use of the word in music theory. An interval was the distance between a low-pitched note and a higher-pitched note: the s.p.a.ce between the two points. Elsewhere, Gerbert calls the columns ”the seats of correct figures,” a.n.a.logous to Cicero's idea that topics are ”the seats of argument.” When preparing to debate, students were taught to organize their stock phrases, allusions, and other rhetorical flourishes in the rooms of their ”house of memory.” When preparing to calculate, they first arranged their numbers in their proper s.p.a.ces on the abacus board. Three hundred sixty-five would be converted into a 3 counter placed in the hundreds column, a 6 counter in the tens column, and a 5 counter in the ones column. The same three counters, placed in different columns, could make 536 or 653. This was the key to the place-value system: The place place where the counter sat determined the where the counter sat determined the value value of the number written on it, whether it meant five or fifty or five hundred. of the number written on it, whether it meant five or fifty or five hundred.

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The copy of Gerbert's abacus board in his student Gausbert's notebook is followed by a mnemonic poem on the names of the Arabic numerals. At the end of each line, the numeral itself is given. Significantly, the poem includes a symbol for zero. Some of the nine numbers, as shown here, are recognizable to modern eyes (if sideways or upside down), while others look very different.

A zero counter was not strictly necessary-to make 10, they just put a 1 counter in the tens column and left the ones column blank. For 100, the 1 counter was simply placed one column further to the left. But larger numbers, such as 10,001, might be confusing; the student's eye might not automatically connect the two 1 counters as parts of the same, composite number across so many blank columns. The mnemonic poem in the Trier ma.n.u.script includes a zero, looking like a spoked wheel, which Gerbert thought of as a placeholder. It filled the empty s.p.a.ce, showing the column was in use. The idea that zero was an actual number number would not arrive until much later. Ralph of Laon, who wrote an abacus treatise in about 1110, vaguely explains the zero by saying, ”Even though it signifies no number, it has its uses.” would not arrive until much later. Ralph of Laon, who wrote an abacus treatise in about 1110, vaguely explains the zero by saying, ”Even though it signifies no number, it has its uses.”

Gerbert did not invent the counting board. Before his time people drew a grid on a flat surface and calculated with calculi calculi-that is, pebbles. The word ”calculate” originally meant nothing more than ”move pebbles around.” To add nine plus eight, you put a pile of nine pebbles in the units column. You put a pile of eight pebbles beneath it, in the same column. You smushed the two piles together and picked out ten pebbles. Discarding nine of them, you put one pebble in the tens column (or two pebbles in the fives column, depending on how your abacus was set up). Then you counted what was left in the units column-seven pebbles-and wrote the answer in Roman numerals: XVII. For small numbers, this system was not very useful (you could do it faster in your head). For large ones, it wasn't convenient-you needed a huge bag of pebbles. To calculate t.i.thes and taxes, a monk would more likely calculate in his head, using ”finger numbers” to record the intermediate stages in a sum.