Part 4 (1/2)
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.