The Abacus And The Cross Part 3

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.

Finger numbers were not what we dismiss as "counting on your fingers." Known from ancient Greece to the Renaissance and in every culture from Europe to the Orient, the system called for extraordinary flexibility, with each joint of each finger moving independently-impossible for anyone with arthritis. Martia.n.u.s Capella, a fifth-century scholar, complained that numbers above 9,000 called for "the gesticulations of dancers." There was also a great deal to memorize. The left hand was used to express numbers up to 99 (or XCIX). The right hand took care of the hundreds and thousands. Both hands together could account for any number up to 10,000 (MMMMMMMMMM).

The Venerable Bede, an Anglo-Saxon monk in Northumbria, England, describes finger counting enthusiastically in his book On the Reckoning of Time On the Reckoning of Time, written in about 725: When you say one, bend the left little finger and touch the middle line of the palm with it. When you say two, bend the third finger to the same place. When you say three, bend the middle finger in the same way. When you say four, raise the little finger. When you say five, raise the third finger. When you say six, raise the middle finger and bend the third finger down to the middle of the palm. When you say seven, touch the base of the palm with the little finger and hold up all the other fingers. When you say eight, bend the third finger in the same way. When you say nine, bend the shameless finger in the same way.

Yes, the shameless finger is the middle finger-still shameless today. The numbers ten through ninety were also made with the left hand. Some were extremely complicated. For sixty, for example, first bend your thumb toward your palm "with its upper end lowered," like the capital letter for the Greek gamma gamma. Then, keeping your thumb bent, place your forefinger over it, just below the thumbnail.

Similar contortions of the right hand made the hundreds and thousands. You could count higher by placing the left hand in various ways on the chest, back, or thigh. For 90,000, for example, "place the left hand on the small of the back, with the thumb pointing toward the genitals." For a million, join both hands with your fingers interlaced.

In addition to calculating, finger numbers were used as a secret code. Monks a.s.signed a number to each letter of the Latin alphabet. By signaling 3-1-20-19-5-1-7-5, Bede reported, you could tell your companion, Caute age Caute age, "Be careful!" Medieval Arabic poetry, meanwhile, gives useful examples of how you could insult someone by displaying certain numbers: 93 meant the person was stingy, 30 meant he was picking lice off his private parts, 90 meant "a.s.shole."

Gerbert was a finger-counter. Explaining the place-value system to Constantine he had said: "How can the same number be considered ... now a digit, now an article?" Literally, the Latin digitus digitus is "finger," and is "finger," and articulus articulus is "joint." Gerbert was relying on Constantine's familiarity with finger numbers to give is "joint." Gerbert was relying on Constantine's familiarity with finger numbers to give digitus digitus a figurative meaning: A digit was any number placed in the first column of the abacus board, what we still call the digits column. For this reason, Gerbert's abacus is said to be the first calculating device to function a figurative meaning: A digit was any number placed in the first column of the abacus board, what we still call the digits column. For this reason, Gerbert's abacus is said to be the first calculating device to function digitally digitally.

Gerbert would also have been very familiar with Roman numerals, which themselves recalled a basic method of finger counting. One (I) is a single finger. Five (V) is an open palm (think of the thumb as one arm of the V and the four fingers as the other). Ten (X) is two open palms. These three symbols are older than the alphabet; the higher numbers (L for 50, C for 100, and M for 1,000) originally came from Greek letters.

Medieval Greek, Arabic, and Jewish cultures generally used letters to stand for numbers-the first nine letters of the alphabet represented the first nine numbers. Bernelin, describing his thirty-column abacus around the year 1000, says the counters can be marked either with the nine Arabic numerals or with the first nine Greek letters, as if this number system might be more familiar to his readers. Scholars have long used Bernelin to argue that Gerbert used Greek letters-and didn't know Arabic numerals at all, much less introduce them to Christian Europe.

Yet the two copies of Gerbert's abacus board made at Echternach in about 993 prove that he not only taught math with Arabic numerals but was also the first Christian in the West known to do so. Altogether, eight ma.n.u.scripts contain abacus boards based on Gerbert's. One was created at the monastery of Fleury, where Gerbert's student Constantine lived; it now resides in a library in Bern, Switzerland. Incorporated into the design, in the V-shaped gaps between the swooping arches, is a line of Latin hexameter verse: "Gerbert gave the Latin world the numbers and the figures of the abacus." The verse also appears, this time just beneath the Arabic numerals, on an abacus board now in the Vatican.

Many of these abacus boards give the names of the numbers. From 1 to 9 they are: igin igin, andras andras, ormis ormis, arbas arbas, quimas quimas, calctis calctis or or caletis caletis, zenis zenis, temenias temenias, celentis celentis, and for zero, sipos sipos or or rota rota. The same names appear in the poem in Gausbert's notebook and, as we will see, in a poem Gerbert himself wrote while he was abbot of Bobbio in 983. Where these names came from is not known. Arbas Arbas, quimas quimas, and temenias temenias (4, 5, and 8) are likely distortions of Arabic words, while (4, 5, and 8) are likely distortions of Arabic words, while igin igin sounds as if it were Berber. The others may be Greek, Hebrew, or Chaldean. sounds as if it were Berber. The others may be Greek, Hebrew, or Chaldean.

Precisely when the concept of expressing all numbers using only nine symbols and a zero came west from India is also unknown. In Arabic, the symbols were sometimes called ghubar ghubar numbers, or "dust" numbers, because they were easy to write on a board lightly covered with sand. Dust numbers were known in Syria by 662, when Severus Sebokt wrote: "I will omit all discussion of the science of the Hindus, ... discoveries that are more ingenious than those of the Greeks and the Babylonians; their valuable methods of calculation; and their computing that surpa.s.ses description. I wish only to say that this computation is done by means of nine signs." numbers, or "dust" numbers, because they were easy to write on a board lightly covered with sand. Dust numbers were known in Syria by 662, when Severus Sebokt wrote: "I will omit all discussion of the science of the Hindus, ... discoveries that are more ingenious than those of the Greeks and the Babylonians; their valuable methods of calculation; and their computing that surpa.s.ses description. I wish only to say that this computation is done by means of nine signs."

The system was still new in ninth-century Baghdad, when al-Khwarizmi wrote On Indian Calculation On Indian Calculation. Al-Khwarizmi sought, in his words, to "reveal the numbering of the Indians by means of nine symbols." His book was brought to Spain, but the new numbers didn't immediately catch on with merchants or administrators, who preferred to use letters of the alphabet to stand for numbers. Lest we think the Spanish were overly conservative, these numbers were not used by Baghdad's businessmen either. Abul Wafa al-Buzjani is known to historians of mathematics for his advances in trigonometry: He devised the tangent function and found a way to calculate sine tables accurate to eight decimal places. He used Arabic numerals for his theoretical work, but not in his Book of Arithmetic Needed by Scribes and Merchants Book of Arithmetic Needed by Scribes and Merchants, written in Baghdad between 961 and 976. Here, he teaches how to calculate with whole numbers (including negative numbers) and fractions, how to find the volume of a solid body, and how to measure distances. He discusses taxes, exchange rates, maintaining an army, and building dams, all while using letters of the Arabic alphabet to stand for numbers.

The oldest known Latin ma.n.u.script to contain Arabic numerals is the Spanish monk Vigila's copy of Isidore of Seville's encyclopedia, which echoes al-Khwarizmi's description of them as Indian numerals. It is dated 976, six years after Gerbert left Spain. The Book on the Abacus Book on the Abacus Gerbert wrote for Constantine-and the letter in which he complained, it "has now been some years since we have had either a book or any practice in this sort of thing"-cannot be dated precisely. Because he refers to himself as Gerbert wrote for Constantine-and the letter in which he complained, it "has now been some years since we have had either a book or any practice in this sort of thing"-cannot be dated precisely. Because he refers to himself as scholasticus scholasticus ("schoolmaster" or "teacher"), we can a.s.sume he wrote it at Reims between 972 and 980, when these ideas were still very new. ("schoolmaster" or "teacher"), we can a.s.sume he wrote it at Reims between 972 and 980, when these ideas were still very new.

Gerbert's Book on the Abacus Book on the Abacus was very popular: It still exists in thirty-five ma.n.u.scripts. Over the next 150 years, it inspired fifteen other monks to write their own treatises on the abacus. Historians call them "dry" and "dull" and are puzzled when the monks themselves describe their abacus studies as "arduous" and "most difficult," easy to understand on the surface, but hard to those who looked deeper. In the early 1100s, William of Malmesbury mocked his more mathematical peers, saying the rules of the abacus were "barely understood by the perspiring abacists themselves." Yet at the same time, Ralph of Laon wrote about the abacus: "From Gerbert, a man of the highest prudence, whose very name is wisdom, the channel of this science has run down to our own times, even though it is a narrow one." The Laon school can be traced to Gerbert through its influential bishop, Ascelin (a nephew of Archbishop Adalbero of Reims), who had studied with Gerbert and outlived his mentor by nearly thirty years. was very popular: It still exists in thirty-five ma.n.u.scripts. Over the next 150 years, it inspired fifteen other monks to write their own treatises on the abacus. Historians call them "dry" and "dull" and are puzzled when the monks themselves describe their abacus studies as "arduous" and "most difficult," easy to understand on the surface, but hard to those who looked deeper. In the early 1100s, William of Malmesbury mocked his more mathematical peers, saying the rules of the abacus were "barely understood by the perspiring abacists themselves." Yet at the same time, Ralph of Laon wrote about the abacus: "From Gerbert, a man of the highest prudence, whose very name is wisdom, the channel of this science has run down to our own times, even though it is a narrow one." The Laon school can be traced to Gerbert through its influential bishop, Ascelin (a nephew of Archbishop Adalbero of Reims), who had studied with Gerbert and outlived his mentor by nearly thirty years.

In Ralph of Laon's time, a good mathematician was still known as an abaci doctor abaci doctor, an "abacist," or even a "gerbercist." But monastery schools were slowly changing the way they taught arithmetic. Instead of shuffling counters on an abacus board, they were beginning to calculate with pen and parchment or a stylus on a wax tablet. They called this system the "algorithm," since the first complete Latin translation of al-Khwarizmi's On Indian Calculation On Indian Calculation was known by a Latinized form of its author's name, was known by a Latinized form of its author's name, Algorismus Algorismus. The introduction of the algorithm can also be linked to Gerbert and his students. Al-Khwarizmi's translator was the English monk Adelard of Bath, who was in Laon at the same time as Ralph. Adelard also wrote a treatise on the abacus, in which he explicitly credits Gerbert's influence on his work.

The step from Gerbert's abacus to Adelard's algorithm was very small. The main difference between the two ways of calculating was that the pre-drawn column lines of the abacus board disappeared; they were no longer needed once the place-value system was fully understood. In the algorithm, placement on the page alone distinguished a one from a ten or a hundred, and the use of zero to fill the empty s.p.a.ce became standard. For a thousand years, we have added, subtracted, multiplied, and divided essentially the same way Gerbert taught his students at the cathedral school in Reims.

Outside the Church, the "nine signs" Gerbert introduced took much longer to catch on. In 1228, the famous mathematician Leonard of Pisa, known as Fibonacci, used al-Khwarizmi's Algorismus Algorismus, along with other works translated out of Arabic, to write his Book of the Abacus Book of the Abacus. Together with Alexander de Villa Dei's Song of the Algorithm Song of the Algorithm and John of Sacrobosco's and John of Sacrobosco's Popular Algorithm Popular Algorithm, these thirteenth-century works made the use of Arabic numerals standard in teaching math. By 1270, the Algorismus Algorismus had even been translated into Icelandic. had even been translated into Icelandic.

But the new system was not accepted everywhere. In 1299, the Guild of Money Changers in Florence issued an edict banning the new "letters of the abacus": "Be it stated that n.o.body ... dare or allow that he or another write or let write in his account books or ledgers or in any part of it in which he writes debits and credits, anything that is written by means of or in the letters of the abacus, but let him write it openly and in full by way of letters." If this sounds extreme, remember that every time we write a check, we state the dollar amount both in Arabic numerals and then "in full by way of letters."

Fraud and mistakes were common with the new numbers. The versions found in Catalonia in the tenth century, and that Gerbert brought into the Christian West, are different from those used in Baghdad or in Constantinople: Gerbert's 5, 6, and 8 may have come from Visigothic script. But the numerals on the abacus board at Trier don't even exactly match the numerals that go with the mnemonic poem in the same ma.n.u.script. The 3 in the poem is upside down, the 4 has an extra loop, the 9 is lying on its back. Nor are all of the numbers recognizable to the modern eye: The 2 is almost always upside down (the numbers were written on a circular counter, so who could say which way was "up"?). The 3 looks like a backward Z with a hook. The 4 is more like a pair of swimming goggles and a snorkel than anything approaching a number. Worse, the number 5 looks very much like our symbol for 4. Until the printing press arrived in the fifteenth century, the shapes of the symbols were not standardized. It would take nearly five hundred years for Western Europe to realize the vision of a new arithmetic that Gerbert saw when he introduced his abacus board and its counters marked with nine Arabic numerals and zero to Reims before the year 1000.

CHAPTER VI.

Math and the Mind of G.o.d The Church found instant uses for Gerbert's abacus. Churchmen always needed to calculate the t.i.the-10 percent of a farm's proceeds-as well as other taxes, tolls, fines, and fees they could collect.

This "business math" had long been taught through story problems, such as the familiar goat-wolf-and-cabbage exercise: "A man needs to take a goat, a wolf, and a cabbage across a river, but his boat carries only two at a time. How can he get them all across unharmed?" Sometimes, the goat, the wolf, and the cabbage were strangely transformed. In one monk's notebook, the problem reads: "There were three men, each having an unmarried sister, who needed to cross a river. Each man was desirous of his friend's sister. Coming to the river, they found only a small boat in which two persons could cross at a time. How did they cross the river, so that none of the sisters were defiled by the men?"

In the late 700s, Charlemagne's schoolmaster, Alcuin of York, compiled a popular set of these puzzles. Its fifty-six problems are a portrait of monastic life: How many stones will it take to pave the cathedral floor? How many casks of wine will fit in the wine cellar? How many cowls can be sewn from a certain length of cloth? How many eggs will the monks eat for supper? In addition to problems of ordering, like the goat-wolf-and-cabbage one, they include problems with one or more unknowns: Your ox has been used to plow a field all day. How many hoof prints does it leave in the last furrow? You have a hundred silver coins with which to buy a mixed flock of a hundred chickens, ducks, and geese, each of which is priced differently. How many of each kind of bird will you buy? There are also arithmetic and geometric series: For example, a king raises an army by going to thirty villages and from each village brings out twice as many soldiers as went in. Solving such puzzles was something monks did for fun, like we do Sudoku. It was also a way to practice their times tables.

The basic textbook on arithmetic in Gerbert's day had been written in the 450s by Victorius of Aquitaine. Victorius had multiplied every number from 1,000 down to 1/144 by every number between 2 and 50. His ma.s.sive times table was copied, recopied, studied, and memorized (at least partly) for over five hundred years. A popular handbook to the table was in progress at Fleury when Constantine wrote to Gerbert asking for the rules of the abacus. The Commentary on Victorius's Calculus Commentary on Victorius's Calculus by Abbo, then the schoolmaster at Fleury, was finished in about 982. Abbo discusses multiplying whole numbers and fractions, finger counting, and questions of weights and measures, such as why "a torch which is half-burned and thrown into water will surface with the burnt end uppermost," or why "the same quant.i.ty of honey will weigh half as much again as that of oil." Into the section on multiplication, Abbo slips mention of Gerbert's abacus and introduces the place-value system of calculating. That he makes no mention of Gerbert himself is not surprising. by Abbo, then the schoolmaster at Fleury, was finished in about 982. Abbo discusses multiplying whole numbers and fractions, finger counting, and questions of weights and measures, such as why "a torch which is half-burned and thrown into water will surface with the burnt end uppermost," or why "the same quant.i.ty of honey will weigh half as much again as that of oil." Into the section on multiplication, Abbo slips mention of Gerbert's abacus and introduces the place-value system of calculating. That he makes no mention of Gerbert himself is not surprising.

Abbo and Gerbert were lifelong enemies. While Gerbert was schoolmaster at Reims, he sent Constantine a new set of mathematical exercises. He writes, "We are entrusting your sagacity, which has always flourished in the freest honesty of studies, with a prepublication of these axioms designed for the utmost exercise of the mind." The axioms, unfortunately, have been lost, but Gerbert says they pertain to digits and articles, and so to the abacus. "By using them," he continues, "the way for grasping these ideas is immediately opened to those persons of less comprehension who, because this pattern of thinking has either been neglected or completely unknown, exasperate every one of the skilled masters of the subject, moreover, by their habitual loquacity, replete with fallacies."

Who is this loquacious person whose understanding of the abacus is replete with fallacies? He is likely the same person to whom Gerbert was alluding when he sent Constantine his Book on the Abacus Book on the Abacus: that "half-educated philosopher" who "though really still a learner along with me ... pretends that only he has knowledge of it." Abbo of Fleury calls himself abaci doctor abaci doctor in a silly little alliterative verse: in a silly little alliterative verse: His abbas abaci doctor dat se Abbo quieti His abbas abaci doctor dat se Abbo quieti ("With this, Abbot Abbo, abacus expert, rests"). But his writings do not reveal a very deep understanding of the new math. ("With this, Abbot Abbo, abacus expert, rests"). But his writings do not reveal a very deep understanding of the new math.

Abbo was born within a few years of Gerbert in Orleans and was given to the monastery of Fleury as an infant. Like Gerbert, he was not n.o.ble, and like Gerbert he had an inquiring mind. Their curiosities-though not their personalities-overlapped. Abbo served as schoolmaster at Fleury while Gerbert was schoolmaster at Reims. Later, Abbo became abbot of Fleury just before Gerbert became archbishop of Reims-an appointment Abbo fought strenuously, as we will see, acting as the lawyer for Gerbert's rival, sending missives to the pope calling Gerbert a usurper, and trying to insinuate himself between Gerbert and his patrons, the king of France (successfully) and the Holy Roman Emperor (unsuccessfully). He made life miserable for Gerbert's beloved Constantine, who had replaced Abbo as schoolmaster at Fleury for two years while Abbo taught at Ramsey Abbey in England, and who had hoped to become abbot of Fleury himself. Despite Gerbert's string-pulling attempts, Abbo got the post. When he returned from England, Abbo had brought magnificent gifts: a gold chalice and vestments, gold bracelets, necklaces, and a large sum in silver coin. None of this hurt his chances at being elected abbot, though his medieval biographer says "some of the brothers perversely resisted the election."

Fleury under Abbot Abbo drew as many or more wandering scholars as Reims. English, Irish, and German monks visited there and left records of their stays, as well as donating books to the monastery's library and treasures to its altars. One medieval visitor claimed Fleury held three hundred monks. If that is true, it was the largest monastery in all of Europe. Why was it so popular? It housed the relics of Saint Benedict. The bones of the founder of the Benedictine order had been moved from Monte Ca.s.sino, where he had died, to Fleury in the late seventh century-or so the monks of Fleury claimed. The monks of Monte Ca.s.sino denied it, but they were shouted down.

Fleury was a center of the monastic reform movement, with Abbo a champion of monks' rights. He acknowledged only two authorities, the king and the pope. Bishops and archbishops, Abbo believed, should have no rights to the income of a monastery, and he fought hard to separate Fleury from the bishopric of Orleans. A dispute over a vineyard led to a pitched battle; swords were drawn, blood was spilled. Another time, a fight broke out at a church council: Abbo and the entire monastery of Fleury were excommunicated, and Abbo was called before the king to apologize for fomenting rebellion.

To achieve the same goal less violently, Abbo introduced a new way to keep track of the monastery's hundreds of charters and deeds for small pieces of land. Rather than storing the original parchments in boxes, bundles, or rolls, he had them recopied into ledgers-which made it, incidentally, easy to insert forged "doc.u.ments" into a sequence. A forgery of a papal bull supposedly from the 750s, for example, convinced the pope in 997 to make Fleury fully independent from the bishop of Orleans. No bishop could say Ma.s.s, ordain a priest, or even enter the monastery grounds without the permission of the abbot-Abbo had won. Seven years later, he would lose his life in a brawl that broke out in a monastery whose monks did not care to be "reformed" by him. It's likely he was stabbed in the back.

According to his medieval biographer, Abbo studied at Reims before he became schoolmaster at Fleury in 975. This may be the key to his and Gerbert's mutual dislike, for Abbo was a failure as Gerbert's student. Abbo already considered himself an expert in grammar, dialectic, and arithmetic, and "was seeking to add other arts to his talents," wrote his biographer. "So he went to Paris and Reims, to those who taught philosophy. Under their guidance, he did indeed make a little progress in astronomy, albeit not as much as he had hoped." Returning to Fleury, he paid a monk "much money" to teach him music. He remained unsure of himself in rhetoric and geometry (two of Gerbert's best subjects), but with five of the seven liberal arts under his belt, he still considered himself better than most of his contemporaries.

In the eyes of posterity, he does come out ahead of Gerbert: Though both were known as mathematicians, Abbo was proclaimed a saint; Gerbert was said to have sold his soul to the devil. And while Gerbert's genius has to be inferred, Abbo was prolific. Arguing that writing "did most to bridle the l.u.s.ts of the flesh," Abbo wrote not only the book on calculation, but astronomical treatises, texts on logic and grammar and the calendar, a biography of the martyred King Edmund of East Anglia, a book of church law, an abridged Lives of the Popes Lives of the Popes, several poems in which names or double meanings are hidden in acrostics, and a collection of letters (each of them a political tract).

Some sense of what he taught as schoolmaster of Fleury can be gathered from the handbook written in 1011 by his student Byrtferth of Ramsey. There are pa.s.sages on calculating; bits of natural history, grammar, and rhetoric; commentaries on the Venerable Bede and other authorities; and exhortations against the evils of the world. There is nothing new; it is all disappointingly derivative. Byrtferth's Manual Manual summarizes the standard teaching textbooks of the previous centuries-and generally leaves out the more thought-provoking sections. summarizes the standard teaching textbooks of the previous centuries-and generally leaves out the more thought-provoking sections.

Abbo's own scientific works follow much the same pattern. They are clear, well-organized rearrangements of the sources commonly used in monastery schools over the hundred years or so that preceded his day. Rather than introducing a new scientific tradition, as Gerbert tried to do, Abbo was content to create a fine and tidy summation of the old one.

Both Abbo and Gerbert were known during their lifetimes as mathematicians. Why Abbo was made a saint, while Gerbert was called the devil's tool, has to do, not with their facility with numbers, but with how they presented their skill. Gerbert loved math for math's sake-as a way to stretch his mind. He saw G.o.d in numbers. Abbo put math to use in the service of the Church. He saw the abacus as a way to improve computus computus.

This kind of math had been at the center of ecclesiastical matters for hundreds of years. "Take number away, and everything lapses into ruin," complained Isidore of Seville in the 630s. "Remove computus from the world, and blind ignorance will envelop everything, nor can men who are ignorant of how to calculate be distinguished from other animals."

Computus comes from computare computare, "to count on one's fingers." In the Church, the term was even more specific. Computus was the technology needed to make a calendar-and the key to the Christian calendar was Easter. To begin the forty-day fast of Lent at the proper time, the date of Easter needed to be known well in advance. But Easter must fall on a Sunday, during certain days of the Jewish festival of Pa.s.sover, and under the right phase of the moon, after the vernal equinox, but not during the raucous festival celebrating the founding of Rome.

Defining Easter meant integrating the lunar month (twenty-nine days, twelve hours) with the twelve months of the solar calendar, established by Julius Caesar before the birth of Christ. Caesar chose twelve to match the twelve signs of the zodiac through which the sun pa.s.ses in a year. But 29 times 12 is only 354, and the Julian calendar had 365 days-the quarter becoming February 29 every fourth year, or leap year. To get sun and moon in sync, you have to add an extra lunar month every three years.

Dating the vernal equinox was another problem-some churches used March 21, others March 25. And did a day begin at dawn, noon, sundown, or midnight? Then, on top of these natural cycles, governed by sun, moon, and stars, there was the week. The week was created by G.o.d. There is nothing natural about a seven-day cycle.

To make a calendar thus required a great deal of calculation. Monks like Abbo painstakingly calculated Easter tables for 19, 84, 95, 112, or even 532 years, each of which was considered by one school or another to be a full "lunar cycle," after which the date and the phases of the moon repeated exactly. Calculating the date of Easter, ma.n.u.scripts show, was the most common math problem in a monastery school.

The most common error was losing track of the quarter days that add up to a leap day. Before Gerbert introduced Arabic numerals, all the monks' calculations were done in Roman numerals. Even after the abacus began to circulate, they continued to use Roman fractions, as Abbo's Commentary on Victorius's Calculus Commentary on Victorius's Calculus shows. Roman fractions are not decimal fractions, but base twelve-there are twelve parts, each with its own name, to a single unit. If you take an shows. Roman fractions are not decimal fractions, but base twelve-there are twelve parts, each with its own name, to a single unit. If you take an uncia uncia (one-twelfth) from the whole, you are left with a (one-twelfth) from the whole, you are left with a deunx deunx (eleven-twelfths); if you take a (eleven-twelfths); if you take a s.e.xtans s.e.xtans (two-twelfths), you are left with a (two-twelfths), you are left with a dextans dextans (ten-twelfths). To make things worse, each uncia contained twenty-four (ten-twelfths). To make things worse, each uncia contained twenty-four scripuli scripuli, or, literally, "little stones." These clumsy fractions made it even easier to forget that, for every day, both sun and moon had to be accounted for. If you added a leap day to the solar cycle, you also had to add a day to the lunar cycle.