A Strange Wilderness

by Amir D. Aczel

  How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair, which becomes productive from the second month on?1

  Fibonacci had an interesting relationship with the Holy Roman emperor. Frederick II had been crowned king of Germany in 1212. Eight years later, the king was elected Holy Roman Emperor. In November that year, the pope crowned him in a majestic ceremony at the Cathedral of St. Peter in the Vatican.

  A page from Fibonacci’s monumental work Liber Abaci (1202), which advocated the use of Hindu-Arabic numerals.

  As Holy Roman Emperor, Frederick sided with Pisa in its rivalry with Genoa and was thereafter very popular with the Pisans. He became familiar with the mathematical work of Pisa’s native son because Fibonacci had an extensive correspondence with the scholars and mathematicians already in Frederick’s court. The emperor’s imperial court met in Pisa in 1225, and some of the city’s dignitaries, including the Fibonaccis, were invited to a banquet. Court mathematicians issued problems as a contest, and Fibonacci correctly answered a number of them, gaining the increasing respect of the emperor. Soon afterward Frederick invited the brilliant Fibonacci to join his court. Fibonacci accepted and eventually became the emperor’s favorite mathematician. In addition to being paid by the emperor, the mathematician’s native city established a salaried position that paid him a stipend for consulting on issues of administration and taxation, and on any problem requiring mathematical analysis. Furthermore, it was through Fibonacci’s influence that Frederick II founded the University of Naples to promote his vision of a good Italian education. This university is still referred to by Italians as Federico II, in recognition of the thirteenth-century emperor’s generosity.

  Fibonacci wrote several books on mathematics. In Flos (The Flower), published in 1225, he provided solution methods for sophisticated equations that resembled those of Diophantus of Alexandria and, hence, went far beyond the practical-minded work of al-Khwarizmi and other Arab and Hindu mathematicians. The book also provided groundbreaking methods for the solution of cubic equations. For example, he gave a very good approximation to a solution of the equation x3 + 2x2 + 10x = 20, a problem that had been issued as a challenge by a mathematician named Johannes of Palermo but that originated in the work of Omar Khayyam. One of his books, now lost, was a commentary on Euclid’s Elements and included a treatment of irrational numbers, which the Europeans had not yet addressed following their discovery by the ancient Greeks. Fibonacci discussed them from a purely computational viewpoint without addressing the philosophical difficulties involved.

  Fibonacci’s seminal work, Liber Abaci, was reissued many times and became a major European text in mathematics well into the nineteenth century. Through his famous sequence and his introduction of the efficient Hindu-Arabic numerals to European mathematics, Fibonacci achieved immortality as the leading mathematician of his time.

  THE INVENTION OF PERSPECTIVE

  As Euclid’s Elements—one of the first books ever to be printed on the new printing presses in Venice after their invention in 1440—and al-Khwarizmi’s Algebra were finding wide distribution and success in Europe during the fifteenth and sixteenth centuries, their ideas were further pursued in the same region of northern Italy that was flourishing culturally and economically under the influence of the Venetian mercantile empire. It has also been asserted that the fall of Constantinople to the Turks in 1453 led to mass migrations of educated Byzantine citizens to Italy and that these refugees brought with them many intellectual works of the East, including Arab mathematical writings. Thus, with the Byzantine decline and the mechanization of bookmaking, Italian and other European mathematicians came to possess the priceless works of their predecessors and were in a position to expand and extend these earlier achievements.

  In the sixteenth century the pursuit of geometry in Italy and Germany was buoyed by the invention of perspective in Renaissance painting. Medieval artists had a general idea of perspective, representing distant objects as smaller than objects in the foreground, but there was no governing system of measurement that determined the positions and sizes of various elements. In the fourteenth century, however, the geometrical portrayal of vision in Alhazen’s Book of Optics caught the attention of Renaissance painters such as Filippo Brunelleschi (1377–1446) and Leon Battista Alberti (1404–72).

  In Germany, Johannes Werner (1468–1522) and artist Albrecht Dürer (1471–1528) pioneered the technique. Werner worked on conic sections and made futile attempts to solve the persistent doubling-of-the-cube problem of classical Greece. His work on conic sections, however, led to results that foreshadowed the idea of perspective.

  Albrecht Dürer’s symbolic 1514 engraving Melencolia I provides ample evidence of the artist’s interest in mathematics. Note the tools of geometry and architecture at the feet of the seated woman, the polyhedron with an unknown number of sides, and the magic square at the top right.

  Intrigued by the concept of fixing elements to a mathematical grid in order to convey the illusion of depth, Dürer identified a fixed point on a circle and then let the circle roll along the circumference of another circle, which generated an epicycloid. He was not a mathematician, so he lacked the tools to produce a precise analysis, but he was able to make projections of helical curves onto a plane, producing spirals, which he incorporated into his art. He also produced an approximation of a nonagon—a polygon with nine sides.

  In many ways Dürer was ahead of his time, as mathematics was not yet advanced enough to allow him to create art with sophisticated mathematical tools. Instead, like many mapmakers of that era, Dürer used projections that were not understood by mathematicians.

  LUCA PACIOLI

  Franciscan friar Luca Pacioli (1445–1517) explored algebra and geometry, including the idea of linear proportion, which was increasingly being incorporated into works of art. Pacioli was born in the town of Sansepolcro, which lies in the Apennine Mountains of central Italy, about halfway between Perugia and Florence. For some reason he was not raised by his parents; rather, he was raised by the Befolci family, who lived in Sansepolcro. Not far from their home was the studio of the famous artist Piero della Francesca. This painter pioneered the use of perspective in art, and Pacioli is believed to have spent time in the artist’s studio, learning from the master about perspective as a child. When he became a young man, Pacioli tutored the three sons of a wealthy Venetian family named Rompiasi, who were living on the island of Giudecca—one of Venice’s best neighborhoods. In Venice the young tutor continued his mathematical education on his own, reading mathematics books and deriving his own results. He wrote a book on arithmetic and dedicated it to the three young boys he was tutoring. When he finished his job educating the boys, Pacioli left Venice for Rome, where he made strong contacts in Catholic circles and joined the Franciscan order, becoming a friar.

  This 1495 portrait of Luca Pacioli, attributed to Jacopo de’Barbari, depicts the mathematician demonstrating one of Euclid’s theorems. The figure in back is Pacioli’s student Guidobaldo da Montefeltro. A rhombicuboctahedron (which has eight triangular and eighteen square faces), half filled with water, hangs from the ceiling.

  Pacioli loved to travel. In 1477 he left Rome and traveled to Perugia, where he taught mathematics at the local university until 1480. Then he moved again, teaching at the University of Naples and later the University of Rome. Through his ecclesiastical connections, he met Federico da Montefeltro, whom the pope had recently made the duke of Urbino. An enlightened ruler interested in education and science, Montefeltro invited the wandering friar to tutor his son Guidobaldo, who would become the last ruler of Urbino from the Montefeltro family. When this position ended, Pacioli again moved to Rome, before eventually returning to Sansepolcro. He had achieved fame as a mathematician by that time, inciting jealousies from lesser but more powerful individuals in his hometown.

  In 1496 another invitation arrived, from an unexpected source. Ludovico Sforza, the enl
ightened duke of Milan, was building up his city’s cultural institutions to rival those of all other European cities. Sforza had brought to Milan the great painter, sculptor, inventor, engineer, architect, Renaissance man, and genius Leonardo da Vinci (1452–1519) after Leonardo had written him his now-famous letter in which he offered to effect several great engineering projects in Milan and mentioned, as an aside, that he could also paint. In Milan, Leonardo painted some of his best works of art, including the Virgin of the Rocks and The Last Supper. One of his many important projects in Milan was the design of the dome for the Milan Cathedral. It has been surmised that Leonardo, with his interest in mathematics, had suggested to Ludovico that he invite Luca Pacioli to his court. When Pacioli arrived in Milan, he and Leonardo became very close friends. They spent many hours together discussing the two topics that consumed them both: art and mathematics.

  Two years earlier Pacioli had completed a major book on mathematics, Summa de Arithmetica, Geometrica, Proportioni et Proportionalita, a collection of his own and his predecessors’ arithmetical, geometric, and algebraic work. It contained few attributions but was clearly influenced by al-Khwarizmi’s Algebra, which had been in print for three decades by that time. Since he had received instruction in the terminology of commerce, his book also included work on double-entry bookkeeping, which was an Italian invention of the time that revolutionized business practices and inaugurated the modern field of accounting.

  Like Omar Khayyam, Pacioli believed that cubic equations could only be solved geometrically and that an algebraic solution was something that mathematicians of the future might achieve. But that future was near, and it belonged to Pacioli’s own countrymen. In Milan he spent time on geometry and produced a book, De Divina Proportione, that expounded on the golden ratio. The book revealed the relationship between polygons and three-dimensional solids and the ways in which various proportions follow the golden ratio, often called phi. The beautiful figures within the book were drawn by none other than Pacioli’s friend Leonardo da Vinci. Leonardo himself used mathematical concepts in his writing. In his notebooks we find constructions of regular polygons and ideas about centers of gravity. In art he was a pioneer of the mathematical use of perspective.

  In 1498 the French monarch Louis XII declared that Milan belonged to France, but the reigning duke refused to abdicate, so the French army attacked the city the following year. Ludovico had to flee his duchy, and in December 1499 Luca Pacioli and Leonardo da Vinci escaped together after French troops began occupying Milan. They stopped at Mantua, where they were the guests of Marchioness Isabella d’Este, and from there they continued to Venice. After a period of time in Venice, Pacioli and da Vinci continued to Florence, where they shared a house. Both men remained in Florence for several years, spending time away from the city teaching at various universities. Around this time, Pacioli began to work with another Italian mathematician, Scipione del Ferro, who plays a major role in the next significant development in Italian mathematics.

  After further travels to Perugia and Rome, Pacioli returned to his native Sansepolcro, where he died in 1517, leaving behind an unpublished book about recreational mathematics that made frequent references to his famous friend, Leonardo da Vinci.

  TARTAGLIA

  With the transition from medieval to Renaissance Europe, the mathematics of the ancient Greeks, Indians, Arabs, Egyptians, and Mesopotamians converged in Europe—Italy, in particular—through the work of Fibonacci and his contemporaries. Finally, European mathematics came of age. With the solution of equations, algebra was mastered by European mathematicians, and new advances were on the horizon. Four Italian mathematicians played key roles in the development that followed.

  Niccolò Fontana, known as Tartaglia (ca. 1500–59), was born in Brescia to an unmarried mother who lived in poverty. In 1512, when he was twelve years old, Niccolò’s fortunes went from bad to worse. French troops led by Gaston de Foy attacked Brescia, and the boy was severely deformed after a French soldier in the invading army slashed his face with his sword. Niccolò’s mother nursed him back to health, but his lips were so badly cut that he stuttered throughout the rest of his life and was thereafter known as Tartaglia (the stammerer).

  As a young adult, Tartaglia moved to Venice and pursued the life of an aspiring mathematician. Tartaglia translated the treatises of Aristotle, Euclid, and other Greek mathematicians into Latin. In the meantime, a professor of mathematics at the University of Bologna named Scipione del Ferro (1465–1526), with whom Pacioli had worked, made a stunning discovery: he found a way to solve cubic equations. Nasir Adin al-Tusi had previously made some progress in understanding some of these equations, but no general formula had been known—the Arabs who had studied the cubic-equation problem could not come up with a good general way of solving it.

  Niccolò Fontana, known as Tartaglia (the stammerer), wrote his solution for solving cubic equations as a poem.

  In 1526, while Tartaglia was working on this same problem in Venice, del Ferro died in Bologna. Just before his death, however, he revealed his big secret to a mediocre student named Antonio Maria Fior. What del Ferro had discovered was a way to solve cubic equations that contain no x2 terms and in which all the coefficients are positive numbers.

  Fior understood that he possessed a very powerful formula that could allow him to make large sums of money in competitions, so he moved to Venice and challenged other mathematicians to solve cubic equations. In 1535 Fior challenged Tartaglia. Each contestant was to provide his opponent with thirty problems to solve. All the equations Fior gave Tartaglia were in a form that he thought he knew how to solve using del Ferro’s method—equations of the form x3 + px = q—but he was ultimately unprepared for the wide variety of cubic equations that Tartaglia gave him.

  Tartaglia, who managed to solve some of the cubic equations by educated trial and error, beat Fior because the latter had misunderstood del Ferro’s formula and obtained wrong answers. Having pondered the problem of finding a general formula for the solution of cubic equations, Tartaglia gleaned from Fior’s errors something about the mysterious formula.

  During the night of February 12, 1535, Tartaglia managed to derive his own general formula for solving cubic equations—he had figured out del Ferro’s secret. Tartaglia wrote his solution method as a poem, and instead of the x we use for an unknown quantity in mathematics today, he used cosa, the Italian word for “thing” (plural: cose). It began as follows:

  Quando che’l cubo con le cose appresso

  Se agguaglia a qualche numero discreto …

  [When the cube with the “things” is equal to a number …]

  Because the word cosa represented the unknown in an equation, eventually all mathematicians concerned with solving equations in sixteenth-century Italy became known as cossists. This terminology originated with Luca Pacioli, who abbreviated cosa as co; censo—the square of the unknown cosa—as ce; and aequalis—the equal sign—as ae. Using Pacioli’s notation, for example, the equation 5x2 = 7x might be written as 5ce ae 7co. But Tartaglia, who favored verse, did not abbreviate cosa and other elements of his analysis.2

  Having derived a powerful method of solving cubic equations, Tartaglia was able to make a good living by taking part in public competitions of equation solving, which were held in piazzas and other locations in Venice. His discovery also brought him many pupils who wished to learn his methods, and he was offered a number of university positions. Soon word of his success reached a man named Cardano, who lived in Milan.

  GIROLAMO CARDANO

  Girolamo Cardano (1501–76), a mediocre Italian mathematician with exceptional drive and greed—which he apparently bequeathed to his progeny—was the illegitimate son of a prominent Milanese lawyer named Fazio Cardano, who taught geometry at the University of Pavia. When Fazio was in his fifties, he had an affair with a woman in her thirties named Chiara Micheria, who had become widowed and was forced to care for three young children alone. She became pregnant by Cardano, but before sh
e gave birth, a plague ravaged Milan and she traveled to Pavia, where her lover was teaching, leaving her children behind. Girolamo Cardano was born in Pavia, and when he and his mother returned to Milan, she was horrified to find that all three of her older children had died from the plague. She raised the boy alone, and only late in life did Fazio Cardano marry her.

  As a young man, Girolamo worked as his father’s legal assistant, but he wanted a lot more out of life, so his father taught him mathematics and enabled him to enroll at the University of Pavia. After considering a career in mathematics, he decided to study medicine, which he thought would be more financially rewarding. When the war with the French broke out, the university closed its doors and Cardano moved to Padua to continue his education. He joined the faculty and even tried to get elected rector of the university. Despite his lack of popularity, which was probably due to his unpleasant disposition, he won the election by one vote. But he did not hold the post for long.

  Fazio died that year, and when Girolamo got hold of his inheritance, he became a compulsive and aggressive gambler. At times he was even violent. It was said that his understanding of probability helped him win large sums of money; but when he lost, he would get angry. On one occasion he slashed the face of an opponent with a knife he kept in his pocket whenever he went gambling. This incident brought him infamy and cost him his life’s ambition: membership in the College of Physicians in Milan.

  Lacking this necessary professional recognition, Cardano had a hard time making a living as a physician and fell into poverty. Forced to move to the small village of Sacco, outside Padua, he married Lucia Bandarini and had three sons with her. They struggled financially and were forced to move to another village and then to Milan, where Cardano was able to teach mathematics and practice medicine on the side.