The Abacus And The Cross Part 4

Easter tables were laboriously copied from ma.n.u.script to ma.n.u.script: 1,200 such copies, made from the eighth to eleventh centuries, still exist. Display models were carved in stone. The bishopric of Ravenna owned a sixth-century Easter table that Gerbert might have seen in later years. It was made of white Grecian marble, three feet square and over an inch thick, on which five cycles of nineteen years were laid out in the form of a nineteen-spoked wheel, radiating from a central cross. Yet in spite of such efforts, errors in calculation (or copying) often led to different dates for Easter at different churches in a given year.

While he was compiling an Easter table covering five future nineteen-year cycles, a monk named Dennis the Humble invented the concept of Anno Domini (A.D.)-the "Year of Our Lord"-the root of our modern dating system. In Dennis's time, years were dated by reference to the reign of the Emperor Diocletian. Using rules of thumb to calculate backward, Dennis decided Christ was born two hundred and forty-seven years before Diocletian took office, making the dates of his new Easter table A.D. 532 to 626.

Dennis's Anno Domini scheme did not immediately catch on. But two hundred years later, the Venerable Bede in England incorporated it into his book On the Reckoning of Time On the Reckoning of Time. From there it was picked up by Charlemagne, for whom it solved a pressing problem. According to the Church, the world would end six thousand years after the Creation. But 6000 Annus Mundi ("Year of the World") was the year Charlemagne was to be crowned emperor in Rome. He found it much preferable to change the date of his crowning to 800 Anno Domini. His change was codified in a new computus textbook by his court mathematician, Hraban Maur. (It begins beautifully: "Time is the motion of the restless world and the pa.s.sage of decaying things.") Like Charlemagne, Hraban Maur left it to his successors to deal with the End of the World predicted to arrive a thousand years from the birth of Christ, in A.D. 1000.

As that date approached, Abbo of Fleury took up Hraban Maur's challenge-to move the problem of the Apocalypse into someone else's lap. He checked the calculations of Dennis the Humble and the Venerable Bede (possibly using a Gerbertian abacus) and found mistakes. If Saint Benedict had really died on an Easter Sunday-and a.s.suming Abbo's Easter tables were correct-Bede's calculations were off by twenty years. And where was Christ's infancy? Dennis the Humble had followed Roman practice, counting Christ's birthday as the first day of the year A.D. 1. Abbo counted from Christ's birth as we count birthdays today: The birth is zero; on the first birthday, the child is one. According to Abbo's figures, then, the year 979 was actually a thousand years after the birth of Christ. Abbo spread the good news: A.D. 1000 had already pa.s.sed without mishap; the predicted Apocalypse had not come. Abbo devised a new calendar, but no king or emperor or pope promoted his discovery, and our modern dates are still based on those of Dennis the Humble.

For all its practicality, math for Abbo was infused with spiritual purpose. Contemplating "what is unchangeable and true," he says in the introduction to his Commentary on Victorius's Calculus Commentary on Victorius's Calculus, "reforms the image and likeness of the Creator in man's soul." It provides "a defence against evil and error" and "leads men to G.o.d, who is himself Wisdom, by drawing them from the visible through the invisible to the unity of the Trinity." Arithmetic was thus a form of wors.h.i.+p, leading one to recognize that "all number and mutability" derived from "unchangeable unity." Unity, he continued, "is a term of number from which 'one' is derived." And, rather less clearly, "What is 'one' 'is,' and what is, is one." One, for Abbo, was a symbol for G.o.d.

Abbo got these mystical ideas about the number one from Boethius, but they originated with the pagan thinker Pythagoras in the sixth century B.C. In addition to devising a theorem for the length of the long side of a right triangle, Pythagoras believed numbers had spiritual properties. Plato, in The Republic The Republic, picked up on the Pythagorean idea two centuries later, saying it was "good for the soul" to contemplate the "eternal verities as expressed by the properties of numbers."

Nicomachus of Gerasa, in the second century A.D., provided the next step. Nicomachus saw numbers in two ways: as both purely mystical (as in his book Theologumena Theologumena) and more worldly (as in his Introduction to Arithmetic Introduction to Arithmetic), but still not in any way approaching practical business math, what the Greeks called "logistics." Nicomachus's "arithmetic" is what we would call number theory: even and odd numbers, prime numbers, perfect numbers, numerical ratios or harmonies, polygonal and polyhedral numbers, and the three means (arithmetic, geometric, and harmonic). His book-well organized and clearly written-brought him fame. The name Nicomachus was to arithmetic as Euclid was to geometry.

In the sixth century A.D., Boethius translated Nicomachus's Introduction Introduction . He left "nothing essential" out, but, he says in his introduction to the work, "I did not restrict myself slavishly to traditions of others, but with a well formed rule of translation, having wandered a bit freely, I set upon a different path." He added quite a bit of geometry (Book Two is almost entirely taken up with triangles, squares, pentagons, hexagons, pyramids, cubes, spheres, and their proportions) as well as dwelling on the concept of unity, or how every "inequality proceeds from equality" and "every inequality can be reduced to equality." . He left "nothing essential" out, but, he says in his introduction to the work, "I did not restrict myself slavishly to traditions of others, but with a well formed rule of translation, having wandered a bit freely, I set upon a different path." He added quite a bit of geometry (Book Two is almost entirely taken up with triangles, squares, pentagons, hexagons, pyramids, cubes, spheres, and their proportions) as well as dwelling on the concept of unity, or how every "inequality proceeds from equality" and "every inequality can be reduced to equality."

Found in over two hundred medieval ma.n.u.scripts, Boethius's On Arithmetic On Arithmetic was taught in cathedral schools and universities throughout the Middle Ages and into the Renaissance. It had nothing to do with calculating the t.i.the or taxes, with Easter tables or anything practical. Why was it so popular? Because Boethius had Christianized Pythagoras. "Everything," Boethius writes, "which has been built up from the first substance of matter seems to be found in accord with the science of numbers. Therefore this was the original pattern in the mind of the creator." His biblical source is the Book of Wisdom, chapter 11, verse 21: "Thou hast ordered all things by number, measure, and weight." was taught in cathedral schools and universities throughout the Middle Ages and into the Renaissance. It had nothing to do with calculating the t.i.the or taxes, with Easter tables or anything practical. Why was it so popular? Because Boethius had Christianized Pythagoras. "Everything," Boethius writes, "which has been built up from the first substance of matter seems to be found in accord with the science of numbers. Therefore this was the original pattern in the mind of the creator." His biblical source is the Book of Wisdom, chapter 11, verse 21: "Thou hast ordered all things by number, measure, and weight."

This one line-ill.u.s.trated by G.o.d leaning down from the clouds with his compa.s.s in hand (a visual tradition that led to William Blake's famous etching Ancient of Days Ancient of Days)-justified the study of math and science in monastery and cathedral schools for hundreds of years. There was no clash between science and faith: Science was was faith. Then Martin Luther took the Book of Wisdom out of the Bible in the sixteenth century, relegating it to an appendix. It was deleted altogether in Protestant Bibles of the nineteenth century-which may be one reason why many Americans today consider science and religion ant.i.thetical: No longer does math reveal the mind of G.o.d. faith. Then Martin Luther took the Book of Wisdom out of the Bible in the sixteenth century, relegating it to an appendix. It was deleted altogether in Protestant Bibles of the nineteenth century-which may be one reason why many Americans today consider science and religion ant.i.thetical: No longer does math reveal the mind of G.o.d.

When it came to Boethius's On Arithmetic On Arithmetic, and that mystical concept of "unity," Abbo's and Gerbert's curiosities overlapped. Yet where Abbo found a path to virtue, "a defence against evil and error," Gerbert found wonder and joy.

To him, On Arithmetic On Arithmetic provided a treasure-trove of thought experiments. He owned a spectacular copy-written in gold and silver inks on purple parchment-which he would later present to Emperor Otto III. When the teenaged emperor asked him to explain it, he replied happily, "Unless you were not firmly convinced that the power of numbers contained both the origins of all things in itself and explained all from itself, you would not be hastening to a full and perfect knowledge of them with such zeal." provided a treasure-trove of thought experiments. He owned a spectacular copy-written in gold and silver inks on purple parchment-which he would later present to Emperor Otto III. When the teenaged emperor asked him to explain it, he replied happily, "Unless you were not firmly convinced that the power of numbers contained both the origins of all things in itself and explained all from itself, you would not be hastening to a full and perfect knowledge of them with such zeal."

Gerbert delighted in the logical problems Boethius posed-and particularly in those that baffled more practical people like Abbo. For example: Take a ratio of three numbers, 16, 20, 25, and reduce it to the series 1, 1, 1. To Constantine, Gerbert writes, "This pa.s.sage, which some persons think is insolvable, is solved thus. ..." Following Boethius's rules for superparticulars (that is, "a number compared to another in such a way that it has in itself the entire smaller number and a fractional part of it"), through several pages of argument, Gerbert transforms 16, 20, 25 into 1, 4, 16, then 1, 3, 9, then 1, 2, 4. To resolve this so-called sesquiquarta into three equal terms, he explains, "take away the lesser from the middle, that is, 1 from 2, and place this 1 as the first term, and the remainder place second, that is 1. From the third term, that is from 4, take away unity, that is, 1, and twice the second term, that is two unities, and the remainder will be for you one unity." Gerbert concludes, "Therefore, you see how the whole quant.i.ty of the sesquiquarta has been changed into three equal terms, that is, unities: 1, 1, 1, not confusedly but in definite order, just as it was procreated in the beginning. This, therefore, is the true nature of numbers."

Baffling indeed. The problem seems meaningless. Yet to Gerbert, reducing 16, 20, 25 to three ones revealed the mathematical principle behind the creation of the universe. In the beginning all was One. From One, all of creation arose, logically and mathematically, and so-logically and mathematically-anything, no matter how complex, could be reduced once again to one. Figuring out how to do so was like reading G.o.d's mind. Gerbert's answer to this problem became so well known it was given a name, Saltus Gerberti Saltus Gerberti, or "Gerbert's Leap," and sparked two of his contemporaries to try their own solutions: Notger of Liege and, of course, Abbo of Fleury.

This Boethian quest for unity can also be seen in a treatise on the physics of organ pipes, found in a twelfth-century ma.n.u.script alongside some of Gerbert's correspondence and Boethius's works on arithmetic and music. Called Rogatus Rogatus from its opening words, from its opening words, Rogatus a pluribus Rogatus a pluribus, "having been asked by many," it is, like all of Gerbert's technical writings, a response to a student's question. It uses Arabic numerals (mixed with Roman numerals), well-honed Latin rhetoric, and quotations from Boethius, Pythagoras, Macrobius, Calcidius, Plato, and others. It is the work of an excellent mathematician, not only someone who understood the casting of metal pipes and the construction of organs. For these reasons, it has been attributed to Gerbert.

The Rogatus Rogatus explains the physics of organ pipes in comparison to the more familiar monochord. The monochord is a single string stretched over a sounding box, rather like a one-stringed guitar. A movable bridge-like the guitarist's kapo-let the string be shortened to alter the pitch. When the string was "open" (no bridge) and plucked or bowed, it played a certain note. When the string was halved, using the bridge, the note was an octave higher. explains the physics of organ pipes in comparison to the more familiar monochord. The monochord is a single string stretched over a sounding box, rather like a one-stringed guitar. A movable bridge-like the guitarist's kapo-let the string be shortened to alter the pitch. When the string was "open" (no bridge) and plucked or bowed, it played a certain note. When the string was halved, using the bridge, the note was an octave higher.

The monochord was a favorite visual aid of Gerbert's when he taught the quadrivium. Music was part of the mathematics curriculum because all sequences or harmonies were translated into numerical ratios based on the monochord: The octave was 2:1. Music was a matter of numbers-even today we speak of fifths and fourths and diminished sevenths, time signatures, tempos, and rhythms. But to Gerbert and his peers, certain ratios of notes or rhythms were not just more pleasing to the ear, they were more sacred.

This theory was also based on Boethius. Sound, Boethius wrote in his equally mathematical On Music On Music, was caused by percussion-the force of a vibrating string hitting the air. If the string was taut and beat the air rapidly, the sound would be high-pitched; if the string was loose and vibrated more slowly, the sound would be low. Sound traveled as a wave: Just as a pebble dropped into a pond causes rings of waves to spread out from the spot, so sound waves spread and grew fainter the farther they traveled from the vibrating string.

From this accurate scientific beginning, Boethius turned mystical. Sound-as music-was all around us. We heard the musica instrumentalis musica instrumentalis , music of the human voice and other instruments. But it was only a faint echo of the Music of the Spheres, the , music of the human voice and other instruments. But it was only a faint echo of the Music of the Spheres, the musica mundana musica mundana, produced by the turning of the invisible spheres that held the stars (or perhaps by the spirit blowing through them). It was music, too, the musica humana musica humana , that held body and soul together. Because of this, we could be seduced to evil by immoral music, and restored to health, physical or spiritual, by music that was modest or simple (considered masculine), not violent or fickle (feminine). , that held body and soul together. Because of this, we could be seduced to evil by immoral music, and restored to health, physical or spiritual, by music that was modest or simple (considered masculine), not violent or fickle (feminine).

Composing modest, moral music was central to Gerbert's world. In a monastery's seven services a day, much of each service was sung, or rather, chanted, for the only sacred music of the time was the Gregorian chant. Originally chant had no harmony: It was pure melody. With rich and subtle variations of the melodic line, it required the hearer to listen horizontally, across time, to recognize patterns and notice when they s.h.i.+fted. When harmony began to be added, composers wondered why some intervals made a "sweet mixture" and others sounded harsh. The author of a musical tract from A.D. 860 concludes that we will never understand the "deeper and divine reason" for this, since it "lies hidden in the remotest recesses of nature." Tellingly, he cites Boethius, "in which it is convincingly shown ... that the same numerical proportions by which different tones sound together in consonance also determine the way of life, the behavior of the human body, and the harmony of the universe."

Gerbert had no interest in writing music. But he was fascinated by the "deeper and divine reason" why some harmonies sounded sweet and others harsh. According to Richer of Saint-Remy (who himself was a cantor cantor, or choirmaster), Gerbert was already a master of musica musica when he left Spain. What Richer means by when he left Spain. What Richer means by musica musica becomes clearer in his next anecdote: On his way to Reims for the first time, Gerbert instructed the schoolmaster Gerann in mathematics. "But the difficulties of that science so discouraged him that he renounced completely the study of music." Gerann did not, of course, renounce singing in choir-no churchman could do that. But not everyone needed to know the underlying mathematics of the chant. becomes clearer in his next anecdote: On his way to Reims for the first time, Gerbert instructed the schoolmaster Gerann in mathematics. "But the difficulties of that science so discouraged him that he renounced completely the study of music." Gerann did not, of course, renounce singing in choir-no churchman could do that. But not everyone needed to know the underlying mathematics of the chant.

Describing Gerbert's teaching methods, Richer writes, "Music, previously unfamiliar to the Gauls, he made very well known. Arranging its notes on a monochord; dividing the consonants and symphonies of the [notes] into tones and semitones, also di-tones and quarter-tones; and dividing rationally the tones into sounds, he rendered [music] fully accessible."

At some point, it occurred to Gerbert to try to do the same with a pipe organ. The string of a monochord could be divided into halves, thirds, fourths, fifths, and sevenths. Plucked, it would produce the note of the scale the musician expected. But organ pipes, Gerbert noticed, were not like strings. A pipe half as long as another did not produce a tone a full octave higher. Pipes built using the Boethian ratios applicable to a string would produce acoustical distortions-the music would sound "off." Instead, the pipe maker had to add a little bit to the length of each pipe to tune it.

In his Rogatus Rogatus, Gerbert explains mathematically how to compute the length of organ pipes for a span of two octaves. His solution was ingenious, though labor intensive, and stands up to the scrutiny of modern acoustics theory.

Yet Gerbert was not writing for organ builders who wanted their pipes to sound sweet. Instead he was searching for a mathematical truth: a law for computing the dimensions of an organ pipe that would sound the same note as the string of a certain length on the monochord. He came up with such an equation using what physicists call "opportune constants" (or "fudge factors") that allowed him to switch, mathematically, from the monochord to organ pipes and back. His treatise shows an extraordinarily modern perspective. He did not simply theorize-or search out authorities. He did experiments. He collected data and made practical acoustical corrections.

But behind his modern scientific approach remains a very medieval urge. Gerbert was searching for another form of "unity," like the 1-1- 1 he had reached in the Saltus Gerberti Saltus Gerberti, proof of the unity of Creation. His scientific goal, as always, was to reveal the single mathematical order in nature, given by G.o.d.

Gerbert also took a modern scientific approach to geometry. At the same time, he saw its utility in terms of religion. "It is full of accurate observations," he wrote, "for the purpose of... contemplating, admiring, and praising the wondrous meaning of nature and the wisdom of its Creator."

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A page from Gerbert's geometry textbook, in a twelfth-century copy. The most advanced geometry book in the West at the time, it contains more Euclid than we would expect a tenth-century monk to know. This ma.n.u.script also contains a treatise on the astrolabe and a copy of one of Gerbert's mathematical letters.

The textbook Gerbert wrote for his students at Reims was the most advanced geometry book in the West. It was not supplanted until centuries later, when Euclid was fully translated from Arabic (the Greek had long been lost) in the twelfth century. Gerbert's book contained, however, more Euclid than we would expect a tenth-century monk to know. One of his sources is known as Geometry I Geometry I. It is attributed to Boethius, though the version that has survived is a clumsy concoction drawn from several works. Only one of them was Boethius's (now lost) Latin translation of Euclid. The copy Gerbert used still exists, in the library of Naples, with corrections made by Gerbert himself or one of his students. It contains the complete text and diagrams for numbers one through three of Euclid's definitions, postulates, axioms, and propositions; enough for a student to learn how to verify a theorem. There's also a good deal of the rest of Euclid's Book One, as well as lengthy excerpts from his other four books.

Gerbert drew from other sources as well. To Euclid, Gerbert added the geometrical bits of Boethius's On Arithmetic On Arithmetic and his commentary on Aristotle, practical examples from the Roman surveyors' tradition (whose straight roads and arched aqueducts were used and admired in Gerbert's day), spiritual explanations by Saint Augustine, and introductory material from the standard quadrivium texts by Calcidius, Macrobius, and Martia.n.u.s Capella. He compared and contrasted their approaches in a sophisticated and well-thought-out volume that proves Gerbert had a better grasp of geometry than anyone else in his time. and his commentary on Aristotle, practical examples from the Roman surveyors' tradition (whose straight roads and arched aqueducts were used and admired in Gerbert's day), spiritual explanations by Saint Augustine, and introductory material from the standard quadrivium texts by Calcidius, Macrobius, and Martia.n.u.s Capella. He compared and contrasted their approaches in a sophisticated and well-thought-out volume that proves Gerbert had a better grasp of geometry than anyone else in his time.

Gerbert begins by showing how any solid body was composed of parts. Taking a body-one that, he stresses, can actually be seen by the "eyes of the flesh"-he reduces it to its surfaces, the surfaces to lines, and the lines into points. The point, the most basic component of a solid, was equal to the number 1 in arithmetic: It represented the unity behind all creation. Geometrical unity, as in the 1-1-1 exercise in arithmetic, is perceived by the "eye of the soul." It, too, was a glimpse into the mind of G.o.d.

The triangle, likewise, was enticing because it was the basis for all other shapes. Gerbert writes, "The triangle exists for this reason as the origin and, as it were, the element in angled figures, in that every one of these figures is composed from it and resolved back into it again."

In his last letter before being elected pope, on the eve of the year 1000, Gerbert wrote to his friend Adalbold of Liege about this "mother of all figures." He was following up on an earlier letter, now lost, discussing how to find the area of an equilateral triangle. Gerbert's original example, with sides measuring 30 feet long, would have an area of 390, calculated using one standard method, he remarks, but 465 if calculated another way. "Thus, in a triangle of one size only," Gerbert says, "there are different areas, a thing which is impossible. However, lest you are puzzled longer, I shall reveal to you the cause of this diversity."

First Gerbert notes that area is calculated using square feet, not linear feet or cubic feet. Then, using a triangle of more manageable size, with sides measuring 7 units long, he explains the two formulas in use at the time: One was arithmetical, the other geometrical. The arithmetical method was the one taught by Boethius; the result it gave for the area of an equilateral triangle with sides 7 units long was 28. The geometrical answer gave 21, which was correct. Boethius, whose authority on most subjects Gerbert would believe, was wrong because, when "the triangle touches only a part, no matter how small," of a square foot, "the arithmetical rule computes them as a whole." Gerbert even drew a figure to make his point.

Adalbold and Gerbert must have often talked about math. We know Adalbold's bishop, Notger, was mathematically inspired by the Saltus Gerberti Saltus Gerberti, and that Gerbert sent Adalbold his copy of Geometry I Geometry I after he had used it to put together his textbook for Reims. Their mathematical correspondence would continue after Gerbert became pope and Adalbold the schoolmaster of the monastery of Lobbes (after Gerbert's death, he would become the bishop of Utrecht). In the only other letter that still exists, Adalbold asks the pope about finding the volume of a sphere. after he had used it to put together his textbook for Reims. Their mathematical correspondence would continue after Gerbert became pope and Adalbold the schoolmaster of the monastery of Lobbes (after Gerbert's death, he would become the bishop of Utrecht). In the only other letter that still exists, Adalbold asks the pope about finding the volume of a sphere.

The triangle letter is important not because the math is insightful, but because it exists at all. Churchmen-monks, clerics, bishops, even a pope-in the Dark Ages were investigating geometric puzzles, and they were doing so experimentally. Gerbert and Adalbold had noticed that their textbooks gave two different solutions to the same problem, which made no sense. So they experimented; Gerbert found the right answer-21-and explained the difference to his friend. The two schoolmasters did not simply accept the authority of their books, they questioned them and worked the answer out.

The beginning of Gerbert's letter to Adalbold of Liege on how to find the area of a triangle, from a twelfth-century ma.n.u.script containing Gerbert's geometry book. Mathematicians have found fault with the drawing, which may be different from Gerbert's original.

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Gerbert found his solution by drawing a triangle with equal sides, each 7 units long. Then he cut out little squares of parchment, each 1 unit square, and laid them on top of the triangle. It took 28 squares to completely cover the triangle-as the arithmetical rule given by Boethius predicted. But parts of many squares stuck out over the lines. Copying this model into his letter, Gerbert explained to Adalbold, "The skill of the geometrical discipline, rejecting the small parts extending beyond the sides, and counting the halves about to be cut off and the [squares] remaining within the sides, computes what is shut in by the lines thus. ..." After giving the formula (written out at length in words, not as we now write mathematics, as equations), he concluded, "To comprehend this more clearly, lend your eyes to it."

This concept of geometry as an experimental science caught on, at least in Liege. A famous series of eight letters exists from about twenty years later in which Rodolf of Liege and Ragimbold of Koln discuss the interior and exterior angles of a triangle; the definitions of linear feet, square feet, and "solid feet"; and how to find the correct ratio of the diagonal of a square to its side. They tried to understand theorems about the equality of angles or the sums of angles by cutting the angles out and laying them over each other.

A few years later, Franco of Liege tried to solve the famous puzzle known as "squaring the circle"-finding a square with the same area as an existing circle. He, too, began by cutting up a circle of parchment and trying to rearrange the pieces into a square.

The puzzle had been solved long ago by Archimedes. Franco knew that, but he did not know how: The Greek texts had never been translated into Latin. He had never heard of the idea of pi, without which he couldn't solve the problem. Nevertheless, he made a good effort. He was persistent, and set down his reasoning systematically, rather than just working at random. He also tried to solve related problems, such as finding a circle with the same area as an existing square. He came up with a powerful iteration procedure for finding square roots and showed that the square roots of 2, 3, and 5 could not be calculated as fractions, but could only be found geometrically. Though his work contains nothing new mathematically, it tells us that, in the early eleventh century, there was a vibrant school of geometry in Liege. This school existed as a direct result of Gerbert's teaching.

And yet, Franco could could have known Archimedes' work. In 1999, a small thirteenth-century prayer book sold at auction for over $2 million. The ma.n.u.script was begrimed and moldy, some pages charred, some water-damaged, others stuck together. The prayers were almost illegible, the illuminations not very pretty (and later revealed to be forgeries painted after 1938). But prayers and pictures were not the point-the book was a palimpsest. The parchment had been reused-soaked in whey, the ink sc.r.a.ped off, the pages shuffled and turned ninety degrees to make a new half-size book. The erased text could still be partly discerned. It was a book of Archimedes in Greek. have known Archimedes' work. In 1999, a small thirteenth-century prayer book sold at auction for over $2 million. The ma.n.u.script was begrimed and moldy, some pages charred, some water-damaged, others stuck together. The prayers were almost illegible, the illuminations not very pretty (and later revealed to be forgeries painted after 1938). But prayers and pictures were not the point-the book was a palimpsest. The parchment had been reused-soaked in whey, the ink sc.r.a.ped off, the pages shuffled and turned ninety degrees to make a new half-size book. The erased text could still be partly discerned. It was a book of Archimedes in Greek.

The Archimedes codex contains the mathematician's well-known theorem on squaring the circle, explaining the concept of pi, along with his treatises on balancing planes, sphere and cylinder, measurement of the circle, spiral lines, and a game, known as Stomachion Stomachion ("Bellyache"), in which fourteen cut-out shapes have to be rea.s.sembled into a square. It also contains works by Archimedes that modern mathematicians had never seen before 1999, such as Archimedes' letter to Eratosthenes, in which he explains his method. The ("Bellyache"), in which fourteen cut-out shapes have to be rea.s.sembled into a square. It also contains works by Archimedes that modern mathematicians had never seen before 1999, such as Archimedes' letter to Eratosthenes, in which he explains his method. The Method Method combines calculus ("the mathematics of infinity") and physics. It was Archimedes' greatest achievement. combines calculus ("the mathematics of infinity") and physics. It was Archimedes' greatest achievement.

Based on the style of handwriting, scholars say that the Archimedes codex was written in Constantinople in about 975, possibly as late as 988-a time when numerous scientifically minded people in Gerbert's circle, from Reims, Barcelona, Rome, and even Cordoba, had contacts with the Byzantine Empire, and when books on science were being translated from Greek to Arabic to Latin. Gerbert was then the most influential mathematician in the West and one of the most avid book collectors. What would modern science look like, if the Archimedes codex had reached him?

CHAPTER VII.

The Celestial Sphere The school at Reims was famous for astronomy as well as math. This science Gerbert also pursued experimentally, creating instruments to observe, measure, and model the brilliant chaos of the starry sky.

That sky was not the few twinkles we're used to. Thanks to smog and light pollution, most of us have never seen the full s.h.i.+mmering panoply of stars, planets, and Milky Way. Modern astronomers a.s.sociated with the Dark Sky project like to tell of the blackout following the Northridge earthquake in 1994. Los Angelenos lit up the hotlines, fearful of the "giant silvery cloud" over the city. To Gerbert, that cloud was a clock and a compa.s.s; its regularities (and irregularities) gave a lesson in divine harmony, a way to reach G.o.d by studying his Creation.

At its most practical, studying the night sky-what medieval astronomers called the Celestial Sphere-was a way to tell time. The Rule of Saint Benedict prescribed prayers at the first hour of the morning; the third hour of the morning; the sixth hour, or midday; and the ninth hour-all times that, in Rome, had been announced by the changing of the guard. Other prayers were said at sunrise and sunset, dawn and dark. These last four were not difficult to identify, but what was "the third hour of the morning" to a monastery in France, where guards were not changed with loud clas.h.i.+ng of sword on s.h.i.+eld and-this was the greater difficulty-the normal concept of an hour of daytime and an hour of nighttime were not even? An hour of daytime was not sixty minutes, but one-twelfth of the time the sun was up; an hour of nighttime was one-twelfth of the darkness. A night-hour in the summer was thus significantly shorter than a night-hour in the winter.

In the late 500s, Gregory of Tours came up with a new way to know when to pray. The ordinary, unequal hours he called the temporal temporal hours. But astronomers knew another measure of time: They divided the circular motion of the stars into twenty-four equal hours, which they called hours. But astronomers knew another measure of time: They divided the circular motion of the stars into twenty-four equal hours, which they called equinoctial equinoctial hours, because only at the two equinoxes are days and nights (and thus hours) the same length. Gregory came up with a formula for calculating the average day-length-how long the sun shone in equinoctial hours-for each month. He converted that sum into temporal hours (dividing by twelve), rounded up, and adapted his sundial each month accordingly. hours, because only at the two equinoxes are days and nights (and thus hours) the same length. Gregory came up with a formula for calculating the average day-length-how long the sun shone in equinoctial hours-for each month. He converted that sum into temporal hours (dividing by twelve), rounded up, and adapted his sundial each month accordingly.

But what happened at night, when a sundial was useless? Gregory calculated the length of each month's average temporal night-hour, then counted how many psalms went into that hour. The monk on night watch would dutifully chant the required number of psalms, then ring a bell to wake his brethren-the original alarm clock (the word "clock" comes from glocke glocke, German for "bell").

Gerbert taught a much more precise way to keep time. With the astronomical instruments he made, Gerbert could calculate, during the day, the time that a certain star would rise or set. Conversely, seeing a star rise (or set) at night, he could tell the time-in either temporal hours or equinoctial hours-to within a quarter of an hour. He could trace the path of any star and say where in the sky it would be at any time of night or day-and as seen from any place on earth. He could calculate the length of daylight (in equal hours) or the length of a daytime hour (in unequal hours) for any day in any place. He could find the height of the sun at noon for any day in any place, or, given the height of the sun at a certain place, and knowing the day of the year, he could tell the time.

Gerbert's instruments are known as celestial globes or celestial spheres, because they are models of G.o.d's Celestial Sphere. Richer of Saint-Remy, in his history of France, sketches four of them. After so tangling himself up in technical details that it isn't clear what any of them actually looked like, Richer-no astronomer, he-figuratively throws up his hands: "It would take too long to tell here how he proceeded further."

In his letters, Gerbert provides very little more. He talks mainly about the difficulty of making these spheres. Remi, the schoolmaster at the important cathedral of Trier, for example, had written to ask about a fine point of mathematics on the abacus, which Gerbert succinctly explained. Remi also wanted a celestial sphere. Gerbert replied: We have sent no sphere to you, neither have we any at present; nor is it an object of small work.... If, therefore, you are eager to have this that involves so much work, send to us a carefully written volume of Statius's Achilleidos Achilleidos in order that, unable to have the sphere gratis because of my excuse of its difficult construction, you may be able to wrest it from us as your reward. in order that, unable to have the sphere gratis because of my excuse of its difficult construction, you may be able to wrest it from us as your reward.

Four months later, having received the first half of the poem about Achilles, he wrote again to Remi: Your good will, beloved brother, was overburdened by the work of the Achilleidos Achilleidos which, indeed, you began well, but left incomplete because your copy was incomplete. Since we are not unmindful of your kindness we have begun to make the sphere-a most difficult piece of work-which is now both being polished in the lathe and skillfully covered with horsehide. So, if you are weary from the excessive anxiety of antic.i.p.ation, you may expect it, divided by plain red color, about March 1st. If, however, you are willing to wait for it to be equipped with a horizon and to be marked with many beautiful colors, do not shudder over the fact that it will require a year's work. As for giving and receiving among our followers, how true the saying that he who owes nothing need return nothing. which, indeed, you began well, but left incomplete because your copy was incomplete. Since we are not unmindful of your kindness we have begun to make the sphere-a most difficult piece of work-which is now both being polished in the lathe and skillfully covered with horsehide. So, if you are weary from the excessive anxiety of antic.i.p.ation, you may expect it, divided by plain red color, about March 1st. If, however, you are willing to wait for it to be equipped with a horizon and to be marked with many beautiful colors, do not shudder over the fact that it will require a year's work. As for giving and receiving among our followers, how true the saying that he who owes nothing need return nothing.

Gerbert was irked. Even Constantine had never asked for a celestial sphere, only the instructions for making one. But Remi could not be turned down. The cathedrals of Reims and Trier were closely linked-among Gerbert's letters are nineteen he wrote, under his own name or for Archbishop Adalbero, to Archbishop Egbert of Trier. One concerns the monk Gausbert, whose notes on Gerbert's abacus may now rest in the Trier archives. Other letters show the warm friends.h.i.+p between Adalbero and Egbert, both n.o.blemen from Lorraine, both politically astute, both engaged in a ma.s.sive program to enlarge and embellish their churches and to claim "primacy"-the right to sit down in the presence of the king or emperor before the other bishops.

Set amid the vine-covered hills of the Mosul wine region, Trier had been a Roman capital. Roman gates opened onto the medieval city, which was the oldest archbishopric north of the Alps. Egbert made much of the fact that Saint Peter himself had sent the first bishop there. Egbert owned a relic-a chip of wood-from the staff Saint Peter gave that bishop. He encased it in a six-foot-tall crozier coated with gold foil and encrusted with jewels, coins, enamels, and ivory work showing a distinct Byzantine influence. When the region suffered a drought, Egbert proceeded from church to church, brandis.h.i.+ng Saint Peter's staff. The metalwork of Trier was so fine that Adalbero commissioned a jeweled cross for the Reims cathedral; he arranged to pick it up while visiting a holy place along the Rhine.

Gerbert accompanied Adalbero on that pilgrimage, during which they were forced to detour. Gerbert wrote to Egbert, saying, "Continual torrents are in possession of the slopes of the mountains. Ever-flowing waters so clothe the fields that, with villages and their inhabitants submerged and the herds destroyed, they bring terror of a renewal of the Flood. The hope for better weather has been shattered by the phisicis phisicis. Accordingly, we are fleeing to you just as Noah to the Ark."

The word "phisici" could refer to experts in physics, such as meteorologists-or to astrologers. Perhaps while they were waiting at Trier for the floodwaters to recede, the two schoolmasters, Gerbert and Remi, talked about astrology. In those times, the term encompa.s.sed both the scientific study of the stars and fortune-telling. A celestial sphere was useful for both.

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A fifteenth-century wood carving from Ulm Cathedral showing what one of Gerbert's simplest celestial spheres might have looked like. The astronomer pictured is Ptolemy.

Remi may once have been Gerbert's student. That would explain why Gerbert was miffed at being asked to do the man's work for him. But Remi was also well known as a scholar: He wrote a book on the abacus, following Gerbert's, along with hymns and sermons and a Life of Saints Eucharius, Valerius, and Maternus Life of Saints Eucharius, Valerius, and Maternus, the founders of Trier, written in rhymed prose. He was named abbot of Mettlach before Egbert's death in 993-which, curiously, is just when the English monk Leofsin left that monastery, taking a copy of Gerbert's abacus with him to Echternach.

The celestial sphere Remi wanted was not Gerbert's invention. Such instruments were known since ancient times. Cicero mentioned them in his Republic Republic, a book Gerbert asked Constantine to bring with him once when he visited Reims. Gerbert may also have read about them in Plato's Timaeus Timaeus, through the commentary by Calcidius, a third-century author well known to him. In Martia.n.u.s Capella's fifth-century handbook of the liberal arts, of which Gerbert owned a copy, Lady Geometry bears a globe on which "the intricate patterns of the Celestial Sphere, its circles, zones, and flas.h.i.+ng constellations, were skillfully set in place."

While celestial globes or spheres are referred to in books that Gerbert knew, instructions on how to make them are rare. Ptolemy gives explicit instructions on making these devices in his Almagest Almagest, a work that circulated in tenth-century Spain-but only in Arabic, as far as we know. A seventh-century Byzantine writer, Leontius, made such spheres out of wood, smoothed over with plaster and painted a dark blue. The Syrian astronomer al-Battani, whose father was an instrument maker, made an elaborate one of metal that he called "the egg"; his description was translated into Latin in Catalonia before the year 1000. In 1043 an instrument maker wrote of seeing a silver celestial sphere in the library of Cairo. It had been made by the Persian astronomer al-Sufi, who died in 986, and weighed as much as 3,000 silver coins. The oldest extant celestial sphere, dated to about 1080, is uniform and precise, evidence of a long history of sphere-making preceding it. Like most remaining spheres, it is made of metal, in this case bra.s.s, formed as two joined hemispheres.

Gerbert may have learned to make his celestial spheres of wood and horsehide in Catalonia, from Arabic sources (written or oral). Or he may have concocted them out of cla.s.sical learning, hearsay, and his own ingenuity. From the description by Richer of Saint-Remy it is clear that nothing like them had ever been seen at Reims.

No one else writing in Gerbert's lifetime describes a celestial sphere. But two later treatises give practical tips on making them. First you need wood that will not warp, split, or rot. "It must be gathered when the moon is waning in the last days of the lunar month," says an Arabic text translated in Spain in the thirteenth century. Soak it in hot water for two days and dry it in the sun; if it warps or splits, start over. Turn the block of wood on a lathe until it is perfectly round. Carefully cut off a small circle and hollow it out, gluing the circle back as a plug. Cover it with leather "of the sort used for s.h.i.+eld covers, but cut thinner." Then coat it with plaster and paint it as dark as night. The stars, says a Latin text printed in 1518, are made of wire pushed into the wood, flush with the surface.

How the constellations are laid out depends-literally-on your point of view. Since antiquity, there had been two ways of thinking about the stars. In each case, the stars were imagined as attached to a hollow sphere rotating around the earth at its center. You could stand on the earth and look up at the stars. (This is the viewpoint of a modern planetarium.) Or you could imagine yourself hovering above the sphere of the heavens, like G.o.d, looking down through the stars to the earth. Ill.u.s.trations in medieval astronomy books show both points of view: If you looked up, the figures of Orion or the Great Bear faced you; if you imagined yourself looking down, they would turn their backs to you and become reversed, right to left. All extant celestial spheres take the G.o.d's-eye view.

To fix the stars in their proper places, according to al-Battani, the sphere needed three great circles: the equator; a circle perpendicular to the equator and pa.s.sing through both poles; and the ecliptic, the path the sun seems to take through the sky, inclined at an angle of 23 degrees from the equator. To draw the circles, Gerbert used a pair of compa.s.ses. He divided each circle into 360 degrees. To set each star, he observed (or calculated) its celestial coordinates: its distance in degrees from the equator, the north-south line, and the ecliptic.

Once the stars were in place, other circles could be painted on. Parallel to the equator were the so-called climate circles: the Arctic Circle, the Tropic of Cancer, the Tropic of Capricorn, and the Antarctic Circle. The Arctic Circle marked the lat.i.tude where, for observers in the Northern Hemisphere, such as Gerbert, the same stars were visible all night; the stars that circled the Antarctic were always invisible. The tropics-from the Greek tropos tropos, or "turning"-are the circles made by the zodiac signs Cancer (the northernmost sign on the ecliptic) and Capricorn (the southernmost sign) as they turn around the earth each day.

The zodiac is a narrow band to either side of the ecliptic. It was divided into twelve signs, each spanning 30 degrees of s.p.a.ce and marked by a constellation. The degrees were counted off from the vernal equinox, at the beginning of Aries, where the zodiac crossed the equator. The width of the zodiac band is defined by the travels of the moon. When the moon crosses the ecliptic-as the name implies-an eclipse can occur. The five "wanderers"-Mercury, Venus, Mars, Jupiter, and Saturn (the other planets were not discovered until after the telescope was invented in the 1600s)-are also confined by the band of the zodiac. Some medieval astronomers preferred to call the planets "confusers," from the Greek planontes planontes, instead of "wanderers," from planetai planetai. Like all of G.o.d's creation, planetary motions are by definition orderly. "There is no inconstancy in divine acts," said one early astronomer. If the planets appear to wander, our eyes have simply failed to perceive G.o.d's pattern.

Another circle was necessary to make the celestial sphere useful for telling time, but it couldn't be painted on: the horizon. On earth, the horizon is the line between ground and sky. In the heavens, the horizon divides the stars you can see from those that have not yet risen. To mimic the rising and setting of the stars, Gerbert let his wooden sphere rotate within a ring-probably bra.s.s-that represented the horizon.

To master the celestial clock, a monk first needed to know his constellations. To teach them, Gerbert made a second, simpler sphere, Richer says, with no circles, on which the constellations were clearly mapped in iron and copper wire. A sighting tube through the poles acted as the axis. By sighting on the North Star, the sphere could easily be aligned with the night sky. This sphere, says Richer, "has something divine in itself, as even those ignorant in this science, if they were shown one of the constellations, could then recognize all the other constellations thanks to the sphere and without the aid of a master."

The concept of the climate circles also needed to be taught. For this, Gerbert made another instrument, using a hollow sphere cut in half along the north-south axis. He added seven sighting tubes, one for the north pole, one for each of the five climate circles, and one for the south pole. Each tube was half a foot long. "They differ from organ pipes by being all equal in size, in order not to distort the vision of anyone observing the circles of the heavens," Gerbert explained in a letter to Constantine. To keep them steady, they were attached to an iron semicircle. To use the device, Gerbert wrote, aim through the two pole tubes at the North Star. Turn the hemisphere round side up and fix it in place. "You will be able to determine the North Pole through the upper and lower first tube, the Arctic Circle through the second, the summer circle [or Tropic of Cancer] through the third, the equinoctial circle [or equator] through the fourth, the winter circle [or Tropic of Capricorn] through the fifth, the Antarctic Circle through the sixth. 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."

This instrument, says Richer, "was so well contrived that ... it brought to light circles which were new to the eyes and securely fixed them deep in the memory." In addition to teaching the concept of the climate circles, it also allowed Gerbert to measure the height of a star above the horizon, at least approximately, in relation to those circles.

Gerbert's fourth and most sophisticated astronomical instrument was an armillary sphere made of seven open rings (in Latin, armilla armilla). Two rings pa.s.sed through the north and south poles, spread at a 90-degree angle to create the basic sphere shape. Perpendicular to these, Gerbert placed five rings, graduated in size, for each of the five climate circles. To the outside of this sphere of seven rings, Gerbert attached the oblique band of the zodiac. Inside the sphere, writes Richer, "he suspended the circles of the planets by a very ingenious mechanism. It could show, in a surprising manner, their absides, their alt.i.tudes, and their respective distances."

These three measurements revealed how the universe worked. In Gerbert's day, the sun, moon, and planets were thought to circle the earth in eccentric, not perfectly round, orbits. Earth was not at the center of most of these orbits. A planet could thus seem to speed up when it came closer to Earth and slow down as it drew farther way. The planet's...o...b..t itself was called an "apsis" (plural "apsides"). A planet's "alt.i.tude" was its apogee, or farthest point from the earth. Its "distance" defined the interval between its...o...b..t and its neighbor's.

Drawings from the ninth through eleventh centuries ill.u.s.trate this theory-and its problems-clearly (see Plate 6). The circle of the moon is generally centered on the earth, while some of the other circles are more eccentric than others. One image shows the sun dead-center in the zodiac, with Earth well off to the side (though the lines show the sun still managing to circle the earth). Mercury and Venus also inspired serious debate in Gerbert's time: Several ill.u.s.trations show these two inner planets circling the sun, which itself circled Earth. In a ma.n.u.script made at Fleury around the year 1000, the astronomer, clearly intrigued, has drawn three different diagrams to try to perceive how such epicycles could work: Are they two concentric circles? Intersecting circles? Arcs?

Gerbert explained the planets' wanderings to his students through three-dimensional models, not two-dimensional drawings. First, he made observations. Then he built an armillary sphere, like the one Richer described, to model the planets' movements. That his students picked up on his experimental technique is proved by a drawing discovered in 2007. As a theoretical model it is almost unreadable. To the uninitiated, it looks more like a ball of twine than a model of the heavens. The orbits of the planets are drawn, not as circles, but as wide bands overlapping each other in confusing ways-until you realize the picture is not a diagram but a drawing of an actual three-dimensional object. It is a picture of one of Gerbert's armillary spheres.

It does not exactly match Richer's description. The seven armillary rings themselves are missing. But these rings are structural; they do not affect the absides, alt.i.tudes, and distances of the planets. And even if he had wanted to add them, the artist was not technically skilled enough to do so: The concept of perspective drawing, which allowed artists to give the illusion of depth to flat representations of three-dimensional objects, was not invented until the 1400s.

The ma.n.u.script containing this ill.u.s.tration is a copy of a student's notebook-a mishmash of notes and drawings. The copy was made in Augsburg, Germany, in the twelfth century, and now resides in the Vatican archives. The teacher, unnamed, had devised a far-ranging course of study that matches exactly what we know about Gerbert. He used visual aids to explain mathematics (the abacus), astronomy (celestial spheres), and music (the monochord). He quoted cla.s.sical authors such as Boethius and Pythagoras. Some of the same terminology and topics occur in the letters Gerbert sent to his students. Finally, the ma.n.u.script contains two different diagrams explaining the various subjects that make up philosophy. In one diagram, physics is shown as a subdiscipline of mathematics. In the other, the two are separate and equal sciences. Two diagrams like these, as we will see, sparked a famous debate in 980 between Gerbert and another schoolmaster who was jealous of Gerbert's fame. The original of this notebook could have belonged to the spy Gerbert's rival had sent to infiltrate the Reims school.

With his armillary sphere, Gerbert could demonstrate the relative movements of the sun, moon, and "wandering stars." He could explain eclipses, rotating the small wooden b.a.l.l.s that represented sun or moon around on their metal rings until they lined up with the earth and then, with a candle, showing how blocking the sun's light caused the shadow.

Kings always needed someone to explain eclipses. Medieval historians unanimously called them "terrible" and "terrifying." The eleventh-century French historian Ralph the Bald described "a terrible event, an eclipse or obscuring of the sun from the sixth to the eighth hour. Now the sun itself took on the colour of sapphire, and in its upper part it looked like the moon in its last quarter. Each saw his neighbour looking pale as though unto death, everything seemed to be bathed in a saffron vapour. Then extreme fear and terror ripped the hearts of men."

The bishop of Liege was with the emperor on campaign in southern Italy in 968 when there was an eclipse. Called on to calm the soldiers' fears, says the Life of Heraclius Life of Heraclius, the bishop quoted "Pliny, Macrobius, Calcidius, and many others, both astrologi astrologi and computists," to explain the phenomenon scientifically. and computists," to explain the phenomenon scientifically.

Ralph the Bald must have been exposed to such a lecture. Later in his history he writes about "an eclipse of the moon which terrified men. At the eighth hour of the night either G.o.d placed some wondrous thing between the sun and moon, or the sphere of some other star intruded into that position. What really happened is known only to the Creator. At first the whole moon took on a foul and b.l.o.o.d.y aspect, which went on gradually disappearing until dawn next day." Still later, after having spoken with an archbishop of Reims, Ralph has learned a little more: "Eclipse means a failing or a want, but it is not the result of any failing within the heavenly body itself, but occurs because it is hidden from us by some obstacle."

As long ago as 612, when Isidore, the bishop of Seville, sent his first draft of On the Nature of Things On the Nature of Things to his patron, the king of the Visigoths replied with a diagram explaining a lunar eclipse. He labeled the obstacle that keeps the sun's rays from reaching the moon as the to his patron, the king of the Visigoths replied with a diagram explaining a lunar eclipse. He labeled the obstacle that keeps the sun's rays from reaching the moon as the globus globus of the earth. In fact, every explanation of an eclipse Ralph could have heard around the year 1000 required him to know already that the earth was round. No one conceived of the cosmos as a huge, vaulted arch with the earth below, flat as a plate and surrounded by ocean. of the earth. In fact, every explanation of an eclipse Ralph could have heard around the year 1000 required him to know already that the earth was round. No one conceived of the cosmos as a huge, vaulted arch with the earth below, flat as a plate and surrounded by ocean.

Gerbert's teaching that the earth was a globe was not heresy-as later interpreters would have it-but orthodox Catholicism. Saint Augustine himself, the most influential of the Church Fathers, noted in A.D. 401 that it was both "disgraceful and dangerous" if a Christian was heard "talking nonsense on these topics." How could pagans believe us on "matters concerning the resurrection of the dead, the hope of eternal life, and the kingdom of heaven," Saint Augustine said, if they think our holy books "are full of falsehoods on facts which they themselves have learnt from experience and the light of reason? Reckless and incompetent expounders of Holy Scripture bring untold trouble and sorrow on their wiser brethren," he concluded, "when they are caught in one of their mischievous false opinions."

We do not know which reckless and incompetent fools Augustine was thinking of. The argument over the shape of the earth has a long history. A thousand years before Augustine, Thales of Miletus had suggested the earth was a flat plate floating on the ocean. Thales's contemporary, Anaximander of Miletus, argued instead that it was a cylinder hanging in empty s.p.a.ce. Pythagoras, born a hundred years later, around 530 B.C., decreed the earth was a sphere, because a sphere was a perfect shape. It and the other heavenly bodies-including the sun-circled around a central fire, he thought. Plato proposed that the earth was at the center of the universe, but later decided Pythagoras's central fire made more sense. Aristotle moved the earth back to the center and listed scientific observations to prove it was a globe, one being that the shadow of the earth on the moon during an eclipse was circular. By 240 B.C., Eratosthenes had devised how to calculate the spherical earth's circ.u.mference.

Few thinkers challenged Aristotle's model of the universe over the next thousand years (though Pliny, in the first century A.D., thought perhaps the earth was an irregular sphere, more like a pinecone). Most churchmen found the shape of the earth irrelevant; in the words of Saint Basil of Caesarea (A.D. 330 to 379), it is "of no interest to us whether the earth is a sphere or a cylinder or a disk, or concave in the middle like a fan."