European Mathematics Awakens - THE GROWTH OF MATHEMATICS TO 1600 - MATHEMATICS IN HISTORY - Mathematics for the liberal arts

Mathematics for the liberal arts (2013)

Part I. MATHEMATICS IN HISTORY

Chapter 2. THE GROWTH OF MATHEMATICS TO 1600

2.4 European Mathematics Awakens

Europe.1

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History to 1600: An Overview

In the year 1000, Western Europe was a primarily agricultural society. A common organizational pattern in Southern Europe, gradually spreading north over the next couple of centuries, was the manor. The manor was a large estate, ruled over by a lord, and worked by serfs, also called villeins. Serfs were free men, but not free to move; they were bound to the land. Although they were not slaves, their status was not very high; the word villein gave us our word “villain.” This was not a time with strong kings or great empires. There were no large metropolitan cities. Trade was limited. The ruling class was a military elite, the knights.

The great unifying force in Europe was the Catholic Church. It had gradually expanded its range over the preceding centuries, completing the conversion of Europe (or at least the princes of Europe) in the 11th century. The popes were major rulers in their own right, and the Church exercised much influence throughout Europe.

In 1095 Pope Urban II called for a holy war to take the Holy Sepulcher, Jesus’ burial place, in Jerusalem. This led to the First Crusade, which conquered Jerusalem in 1099. The Arabs under Saladin retook the city in 1187. More crusades followed, the last in 1271–72. In the end, the crusades were unsuccessful, as the Muslims regained control of the whole area.

Although the crusades were a failure militarily, they increased contacts between Western Europe and Muslims. This resulted in an expansion of European cultural awareness, and more trade. Also in the 13th century, under Mongol domination, trade increased along the Silk Road, the set of routes across central Asia through which Chinese silk and other goods were transported to the west. The Silk Road was used to head east, as well; Marco Polo and other Europeans traveled to China to learn about that great society. In addition to the Silk Road, there were important sea routes from southern Asia and the Spice Islands to the Persian Gulf and Red Sea along which spices, especially, were traded. These trade goods then traveled overland, were loaded onto ships in the Mediterranean, thence to Italian ports such as Venice and Genoa.

Within Europe itself, the population was gradually increasing, as was the internal trade. Craftsmen in a particular trade—for example, carpenters—formed guilds, associations to promote their interests. In the commune movement, spreading from north-central Italy in the 11th century, people in a town banded together for mutual protection (often against local nobility or church officials). Fairs held in some towns further stimulated trade.

The new towns and merchants asserted themselves politically. Parliaments arose throughout Europe, bodies in which elected representatives of the towns, as well as nobles and clergymen, advised the king. Modem parliamentary systems, in which the center of power is the parliament, were centuries away, but the early parliaments allowed the new merchants to represent their interests to the rulers.

Along with silk and spices, the trade routes were conduits of technology. From China, in particular, came three technologies without which the subsequent European development would be unthinkable: paper, gunpowder, and printing.

The first important European invention from this time was the mechanical clock, invented in the late 13th century. This was probably based on a Chinese invention, the escapement, in which a continuous motion is interrupted, broken into pieces of equal time. (The ticking of a clock arises with the escapement.) The Chinese escapement was invented in the 8th century, when the turning of a water wheel was segmented into discrete steps. In the early European clocks, the escapement acted on a falling weight.

Subsequent European inventions based on Chinese technology proved to be crucial. In particular, we note the cannon from the early 14th century, and Gutenberg’s printing press from the 15th century. It is difficult to overestimate the importance of these in history.

Beginning with University of Bologna 1088, universities began springing up over Western Europe. Notable examples included the University of Paris (c. 1150), the University of Oxford (1167), and the University of Cambridge (1209). These schools fostered the study of Greek and Muslim learning. The scholastic movement, whose most famous representative is Thomas Aquinas, grew out of an effort to meld this imported knowledge, especially Aristotle, with Christian philosophy.

Another result of the renewed interest in classical—Roman and Greek—societies was the artistic flowering of the Renaissance, beginning in 14th century Italy, with such artists are Botticelli, da Vinci, and Donatello.

In 1347 Europe was struck by one of the deadliest plagues in history: the Black Death. The disease was the bubonic plague, endemic (but not so deadly) to rodents in Asia. In 1330 there was an outbreak among humans in China. The disease spread along trade routes, entering Europe originally on Italian trading ships. In the following five years, roughly one-third of the population of Europe, about 25 million people, died. This had profound effects throughout the continent. In particular, there was suddenly a shortage of labor, which improved the lot of the peasants.

Gradually, larger states with more centralized power began to emerge, notably France, England, and Spain. Europe began a long period of expansion. In the 15th century, Portuguese explorers sailed further and further down the west coast of Africa, culminating in the voyage of Vasco da Gama, arriving in India in 1498 after sailing around the southern tip of Africa. In the following century, the Portuguese cemented their grip on the spice trade.

Meanwhile, in Spain, the monarchs had expelled the Muslims, finishing in 1492, and in the same year they financed Columbus’ first trip to America. The subsequent exploitation of the American empire, particularly the silver mines, made Spain the richest country in Europe.

The effect in America was less benign. In 1496 a census was taken in Hispaniola, the island now home to Haiti and the Dominican Republic. At that time, there were 1.1 million people. Another census, taken a mere 18 years later, showed a population of 22,000. The native Americans had no immunity to Eurasian diseases. One result of this disaster was another disaster, the importation of African slave labor.

In the 15th and 16th centuries, in response to massive corruption, there were a number of efforts to reform the Catholic Church. These culminated in the Protestant Reformation, launched by Martin Luther in 1517. Also notable in the movement were the founding of the Church of England in 1532, and the establishment of a theocratic state in Geneva by John Calvin in 1541. Protestantism was adopted by much of Northern Europe.

The 16th century saw several important scientific advances. The Swiss Paracelsus (1493–1541) insisted on observation in medicine, and created medical chemistry. His real name was Philippus Aureolus Theophrastus Bombastus von Hohenheim; his tirades against Aristotle and others gave us our word “bombast.” In 1543 the Flemish physician Andreas Vesalius (Andries van Wesel) (1514–1564) revolutionized the study of human anatomy with his text De humani corporis fabrica (On the Structure of the Human Body). The same year, the year of its author’s death, saw the publication of Copernicus’ blockbuster, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres), which presented his Sun-centered cosmology.

Medieval European mathematicians.

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Transmission

Mathematics never completely disappeared from Western Europe. There was always the practical mathematics of everyday life. In addition, the calendar was important to the Church, especially for determining the date of Easter. (The rule now: Easter is the first Sunday after the first full moon after the vernal equinox.)

Scholars knew that there was an ancient tradition of higher mathematics, but did not have access to much of its literature. The most influential mathematical works were introductory ones, especially those of Boethius (see Section 1.5).

In the 10th century, the scholar Gerbert d’Aurillac (945–1003) reintroduced the teaching of mathematics to the cathedral school at Rheims. In his work is the earliest reference in the west to the Hindu-Arabic numerals, although without a zero and without the algorithms that demonstrated the usefulness of the system. He had studied in Spain, where he presumably learned from Muslin scholars. Gerbert became Pope Sylvester II in 999.

The 12th century saw the translation in Latin of many important mathematical classics. The first Latin translation of Euclid’s Elements, from an Arabic version, was completed by Adelard of Bath (1075–1164) around 1130. Many other works followed, including both Greek and Arabic texts; those of al-Khwimagesrizmimages were particularly influential. The most prolific of the translators were Gerard of Cremona (1114–1187) and his team, who translated more than 80 works, including the first translation of Ptolemy’s Almagest and a new translation of the Elements, from the Arabic version of Thabit ibn Qurra.

The most important center of translation was the city of Toledo in central Spain. It had recently been retaken by Christians from the Muslims and had major Arabic libraries. Toledo was home to three cultures: Muslim, Jewish, and Christian. Many of the translations were done by pairs of translators, from Arabic to Spanish by Jews, then from Spanish to Latin by Christians.

As the Europeans became more familiar with the Greek and Muslim mathematics, they began to publish more works, although it was to be a couple of centuries before they produced important new mathematics. Many of the earliest works were by Jews, who were more familiar with the Muslim culture.

One of these texts (written in Hebrew), the Treatise on Mensuration, by Abraham bar Hiyya (c. 1070–1136), includes an interesting derivation of the formula A = (l/2)C(d/2) for the area of the circle, where C is the circumference and d is the diameter. (See Figure 2.18.) If we think of the circle as being composed of a lot of concentric circles, we can cut this circle along a radius, and unfold its constituent circles into straight lines that form a triangle. The base of the triangle is the length of the longest circle, namely b = C. The height of the triangle is the radius of the circle, h = d/2. Applying the usual formula for the area of a triangle, A = (1/2)bh, gives the result.

Figure 2.18 The area of a circle.

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Leonardo of Pisa (Fibonacci) (c. 1170–1240)

Leonardo was the son of a Pisan merchant. His father lived for a time in the north African city of Bugia (now Bejaia, Algeria), where Leonardo spent much of his youth, studying with Arabic teachers. It was there that he first learned of the Islamic and ancient Greek mathematics. Afterwards, he traveled extensively around the Mediterranean, often meeting with Islamic scholars. He returned to Pisa in northwestern Italy around the year 1200, where he wrote most of his texts.

Leonardo wrote five texts which are still extant. His work was famous in its own time; Pisa acknowledged his contributions with a stipend in 1240.

Leonardo is today known by the name Fibonacci (“son of Bonaccio”), a name he never used. The name was given to him by a 19th century editor of his works, Baldassarre Boncompagni.

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Leonardo’s Liber Abacci (Book of Calculation), published in 1202, was the most influential European work on Islamic mathematics.

The Liber Abaci began with an introduction to the Hindu-Arabic numerals. Leonardo explained the place value system, and gave algorithms for the usual arithmetic operations, on whole numbers and fractions. These algorithms are not much different from our modem ones. He also showed how to use the new system to solve practical problems such as currency conversions and calculations of profit.

Much of the text involves solving a wide variety of problems. Many of these would be familiar to any reader of a modem algebra text. Some problems lead to quadratic equations, which he shows how to solve. Others involve congruences (Chinese remainder problems), indeterminate equations, and systems of linear equations. Some, though not most, of the problems were taken verbatim from Islamic texts.

Here is one of Leonardo’s problems, taken from ibn al-Haytham. Find a number that, when divided by 2 has remainder 1, when divided by 3 has remainder 2, when divided by 4 has remainder 3, when divided by 5 has remainder 4, when divided by 6 has remainder 5, and when divided by 7 has remainder 0.

Here is how Leonardo solved it. Consider first division by 6. If n is any number divisible by 6, then n – 1 will have remainder 5. Similarly, if n is divisible by 5, n – 1 has remainder 4. This sort of argument works for each the first five conditions (excluding dividing by 7). In each case, the remainder is one less than the divisor, so if n is a number divisible by all of 2, 3, 4, 5, 6, then n – 1 will give us the correct remainders. Now, 60 is the smallest common multiple of 2, 3, 4, 5, 6, so 59 = 60 – 1 will have the right remainders for 2, 3, 4, 5, 6. Unfortunately, dividing 59 by 7 leaves remainder 3, not 0. However, if we use the next common multiple of 2, 3, 4, 5, 6, namely 120, we hit pay dirt, since 119 = 120 – 1 also gives us the right remainder upon division by 7.

The most famous problem from the Liber Abacci concerns rabbits. “A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those bom to bear also.”

Let us solve Leonardo’s problem using modem notation. Denote by Fn the number of pairs of rabbits at the start of the nth month. We start out with one pair: F1 = 1. This pair does not produce in the first month, so F2 = 1, but does produce in the second month: F3 = 2. In the third month, the first pair produces another, but the second pair is not yet old enough: F4 = 2 + 1 = 3. In the fourth month, the youngest pair is too young, but the older two produce: F5 = 3 + 2 = 5.

What happens in the nth month? We have all the rabbits that were around in the month before: Fn – 1 (the rabbits never die). Plus, each of the pairs alive two months before will reproduce, giving us Fn – 2 new pairs. Therefore, Fn = Fn – 1 + Fn – was the use of decimal fractions2, i.e., the number of pairs in any month will be the sum of the two preceding months. If we apply this rule, we get the following sequence:

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After twelve months, we have F13 = 377 pairs of rabbits. This sequence is now known as the Fibonacci sequence, one of the most famous in all of mathematics. It has many interesting properties, most discovered long after Leonardo.

There was little new mathematics in the Liber Abacci, and Leonardo did not include all of the latest Islamic developments. Nonetheless, he clearly mastered the earlier Islamic mathematics and demonstrated its power.

Numeration

Before the arrival of the new Hindu-Arabic numerals, Western Europe used Roman numerals. This was a decimal system, but not positional. Different versions occur. Below are the symbols in one version, still occasionally used, as in Super Bowl XLVI.

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Other numbers were represented by repeating symbols, e.g., Ill for 3 or CCXX for 220. The exception was when the number approached the next higher symbol, when the smaller symbol would be put before the larger, e.g., IV for 4 or CD for 400. Here are some more examples.

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Roman numerals were not very convenient for computations. People often did their arithmetic with their fingers or on a counting board, then recorded the results in Roman numerals.

One of the great advantages of the Hindu-Arabic numerals was the ease with which calculations could be made. It was also important that cheap paper was available for the calculations. (Europe had learned to make paper in the 10th century.)

Fibonacci’s Liber Abacci was influential in converting Europe to Hindu-Arabic numerals, but the process was not swift, and not without controversy. A number of cities, including Florence in 1299, outlawed the use of the numerals. One of the perceived drawbacks of the new system was the ease with which numerals could be changed, for example from a 1 to a 4. This is why we still write the amounts on checks in words as well as in numerals. It took several centuries for the new numerals to come into general use.

The main audience for the new mathematics were merchants, especially in Italy. The nature of their business was changing. In the Middle Ages, merchants traveled to distant places, bought their goods, then returned home to sell them. In Italy in the 13th and 14th centuries, a new business model arose. The new merchants stayed at home and hired others to do the traveling. This required different mathematics, to handle, for example, letters of credit, bills of lading, and interest calculations. Double-entry bookkeeping was invented. Not to be confused with keeping two sets of books, double-entry bookkeeping was a system whereby each accounting entry was matched by a second, one entry adding to an account, the other subtracting an equal amount from another account. The system makes it easier to balance the books.

A new class of mathematicians, the Italian abacists (maestri d’abbaco), arose to serve this new need. The abacists ran a number of schools for the children of merchants. One account from the 1320s reports that at least one thousand children were studying the new math in six schools in Florence, a city of about 90,000.

The abacists wrote many textbooks, from which we can tell what they taught. The texts consisted mainly of problems, with solutions but no theory. The problems illustrated the use of the Hindu-Arabic numerals and Islamic algebra, along with some elementary geometry. Here is an example of a problem about simple interest.

The lira earns 3 denarii a month in interest. How much will 60 lire earn in 8 months?

One thing the abacists did not adopt from Islam was the use of decimal fractions (e.g., 3.5 instead of 3images). We saw in Section 2.3 that Al-Samaw’al understood decimal fractions, but the abacists continued to use regular fractions. The European conversion to the easier decimal fractions was largely due to Simon Stevin (1548–1620).

Combinatorics and Induction

The study of how sets of items could be arranged or combined, part of the field we now call combinatorics, has been pursued in many cultures. We have already mentioned the work of Varahamihira in Section 2.2, counting the number of perfumes. In the 11th century, the Spanish-Jewish philosopher Abraham ben Meir ibn Ezra (1090–1167) computed the possible number of conjunctions of the seven “planets” (Sun, Moon, Jupiter, Saturn, Mars, Venus, and Mercury), in a work on astrology.

In 1321 Levi ben Gerson published the Maasei imagesoshev (The Art of the Calculator), in Hebrew, in which he proved a variety of combinatorial identities, and gave applications of them. The theorems were not necessarily new, but some of his proofs were. In particular, he mastered the application of what we now call mathematical induction.

Levi ben Gerson (1288–1344)

Levi ben Gerson was bom in Bagnols-sur-Cèze near the city of Orange in what is now France. Orange was not in France when Levi lived there, which was fortunate, since the king of France in 1306 expelled all Jews and confiscated their property.

Levi himself had good relations with the Christians, in fact dedicating one of his works to Pope Clement VI. He was also aware of much of Islamic and Greek science.

He was a philosopher, astronomer, and Talmudic scholar, as well as a mathematician. He invented the Jacob Staff, a device used to measure the angular separation between celestial objects, which was an important navigation instrument for sailors for several centuries.

Levi ben Gerson is also known by several other names, including Gersonides and Levi ben Gershon.

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We saw in Section 2.3 how Abimages Bakr al-Karajimages used induction to prove the formula for the sum of cubes: l3 + 23 + … + 103 = (1 + 2 + … + 10)2. Here is how Levi used induction to count the number of permutations of n objects.

A permutation of n objects is a linear arrangement. For example, the permutations of the letters a, b, and c are abc, acb, bac, bca, cab, and cba. Let us denote by Pn the number of permutations of n elements, e.g., P3 = 6. Levi proved the formula Pn = n!. (Recall that n! = 1 · 2 · 3 · … · n.)

Note first that the formula holds trivially for n = 1, since there is only one permutation of 1 object. Now suppose that we know that Pn = n! for some fixed n. Consider Pn+1, the number of permutations of n + 1 objects.

Suppose we have a set of n + 1 objects. Pick one, say a. One way to arrange the objects is to start with a, then finish with any permutation of the remaining n objects. So the number of permutations of the larger set which start with a is just Pn. But there is nothing special about a: the number of permutations of the larger set which start with any fixed element is just Pn. Since there are n + 1 choices for the first element, we have Pn+1 = (n + l)Pn.

Recall that we started out assuming that Pn = n!. Then

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so the formula works for n + 1.

So what? Well, suppose that we wanted to know that the formula worked for n = 10. We started out noting that it works for n = 1. But the last argument shows that, since it works for n = 1, it works for n + 1 = 2, so P2 = 2!. But then since P2 = 2!, the argument shows that P3 = 3! We can repeat this until we get P10 = 10!. Nothing is special about 10, so the formula is true for any positive integer. Levi called this process “rising step by step without end.”

Geometry

The geometry of Western Europe at this time continued to be dominated by the ancient Greek classics. Euclid’s Elements was widely taught in the universities. After the conquest of Constantinople by the Ottomans in 1453, a number of scholars fled to the West, bringing along with them a deeper knowledge of Greek geometry, and in some cases Greek manuscripts. In the 16th century, Euclid’s work became more widely available, with its first translations into Italian, German, French, and English.

One area of geometry which the Europeans developed, starting in 15th century Italy, was the mathematics of perspective. This new work was done primarily by artists, in some cases artist-mathematicians. The basic problem was to develop techniques that gave the illusion of three-dimensional depth to a two-dimensional drawing. For example, parallel lines receding from the viewer have the appearance, in a painting, of converging to a point in the distance (Figure 2.19).

Figure 2.19 Parallel lines in perspective.

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The Italian artist and goldsmith Filippo Brunnelleschi (1377–1446) created a number of paintings illustrating perspective. The first text on the subject was produced by Leon Battista Alberti (1404–1472) in 1435. In it, he showed how to render a checkerboard in perspective. Figure 2.20 is an example of a 6 × 4 checkerboard in perspective.

Figure 2.20 Checkerboard in perspective.

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Piero della Francesca (1420–1492) extended Alberti’s work, showing how to use perspective in drawing a variety of three-dimensional figures. Finally, we mention the work of Albrecht Dürer, who instructed artists in Northern Europe in the new techniques. Along the way, Dürer produced the first geometry text in German.

This new work on perspective produced some stunning pictures but was not particularly theoretical. A sound theoretical basis for the mathematics of perspective came later.

Algebra

It was in algebra that the new Europe produced its first major advances over the mathematics it inherited.

One crucial development was the widespread introduction of a better notation. Compare these problems.

A thing and its square, decreased by three times the thing, gives eight. What is the thing?

Solve x + x2 – 3x = 8.

It is difficult to imagine modem algebra without its notation. This notation developed in Europe, beginning in the 15th century (although not completed for a couple of centuries).

The early developments came from Italy, like most of the important changes, economic and mathematical, taking place in Europe in this time. By the early 15th century, the abacists had started to abbreviate words in algebra. For example, the word cosa (thing) was their word for the unknown, what we now would write as x. In the 15th century, this was sometimes shortened to c. Similarly, censo (square) was written ce. Later in that century, Luca Pacioli (1445–1517) used the symbols images and images for plus and minus, from the Italian più and meno.

By the late 15th century, the new math began to spread northward first to France, then Germany, then England. Christoff Rudolff (1499–1545) wrote the first important algebraic text in German, the Coss. (In Germany, algebraists were called Cos- sists, from the Italian cosa.) In it, he introduced our symbols + and – for addition and subtraction. The Englishman Robert Recorde (1510–1558) created our symbol = for equality.

The second major algebraic advance of this time was the solution of the cubic. Recall that a cubic equation is one of the form ax3 + bx2 + cx + d = 0, where a ≠ 0. Cubic equations had been studied as long ago as Babylonian times, and later by the Greeks, Chinese, and Muslims. As mentioned in Section 2.3, cubics had been classified, and solved geometrically, by Al-Khayyimagesmimages in the 11th century. But no one was able to find an algebraic solution similar to the quadratic formula for degree two equations. This problem was finally solved in the 16th century.

The first important step was taken by Scipione del Ferro. He was able to solve equations of the form x3 + cx = d, sometime in the first two decades of the 16th century. He did not, however, publish his results. It was common at this time to keep discoveries secret. They could be useful in winning public challenges. In such a challenge, each of two contenders would present the other with a list of problems to solve. The contenders would later meet publicly to present their solutions to the problems. These challenges were sometimes used to fill university professorships. There might also be considerable sums of money riding on the outcome.

Scipione dei Ferro (1465–1526)

Scipione del Ferro was bom in Bologna, in northern Italy. His father Floriano was in the paper making business. Scipione probably studied at the University of Bologna, a major center of learning in his time. He was appointed a lecturer in arithmetic and geometry at the university in 1496. He was also a businessman. None of his writings have survived.

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Del Ferro did reveal his solution to his student, Antonio Maria Del Fiore. In 1535, after Niccolò Tartaglia announced that he had solved the cubic of the form x3 + bx2 = d (keeping his solution secret), Fiore challenged Tartaglia to a public contest. He apparently hoped to win using del Ferro’s work, since all thirty of the problems he submitted were of the form x3 + cx = d that del Ferro had solved. In response to the challenge, Tartaglia managed to figure out the solution to this form as well, and handily won the contest. He prudently declined the prize of thirty banquets prepared by the loser.

Niccolò Tartaglia (1499–1557)

Niccolò Fontana was bom in Brescia, in northern Italy. His father Michele earned his living making deliveries on horseback between Brescia and neighboring towns. Michele Fontana was killed when Niccolò was six years old, and the family descended into poverty. In 1512 a French army invaded Brescia and massacred an estimated 45,000 citizens. During the massacre, Niccolò received several saber cuts, including one to his jaw and palate that left him with a speech impediment. This was the source of his nickname Tartaglia (stammerer), which he later used in his publications.

Tartaglia studied in Padua, although his formal training was minimal. He taught mathematics in Verona and, after 1534, in Venice, where he gave lessons in local churches. Even though he had a good reputation for his mathematical ability, he was poor throughout his life.

Tartaglia was an engineer as well as a mathematician. He published many works, including the first Italian translations of Archimedes and Euclid.

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Working in Milan, Gerolamo Cardano heard about this contest. He wrote Tartaglia, asking permission to include Tartaglia’s work in a text that he was writing. Tartaglia refused, since he intended to publish this work himself. But Cardano was persistent, and in 1939 Tartaglia gave him a cryptic version of the solution of three types of cubic equation, without proof, phrased as a poem. Cardano pledged not to publish Tartaglia’s work.

Gerolamo Cardano (1501–1576)

Gerolamo Cardano was bom in Pavia in northern Italy, the illegitimate son of Fazio Cardano, a lawyer and friend of Leonardo da Vinci. Gerolamo was trained as a doctor, practiced medicine for several years in a small town near Padua, then moved to Milan in 1534. His early career was difficult; he was repeatedly denied admission to the Milan College of Physicians due to his illegitimate birth.

Eventually, Cardano overcame his origins, and became one of the most celebrated physicians in all of Europe. In 1543 he became professor of medicine at the University of Pavia, where he was a popular lecturer. In the 1550s he traveled extensively throughout Europe. In one incident, he reportedly consulted with the archbishop of Scotland about a worsening case of asthma. After observing the archbishop’s habits for a month, he recommended replacing his feather bedding. This worked, and won Cardano an influential supporter.

Cardano’s later life was more difficult. In 1560 one of his sons was convicted of poisoning his wife, and executed. In 1570 Cardano was accused of heresy for computing Jesus’ horoscope, and was tortured and imprisoned by the Inquisition. His influential friends got him released from prison after a few months. He then traveled to Rome where he was astrologer to the Pope. Cardano committed suicide in 1576. One report claimed that he predicted the exact date of his death, and made sure his prediction was correct.

Cardano published over 200 works on a wide variety of topics, including mathematics, science, medicine, astrology, music, philosophy, religion, and gambling.

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Cardano, unlike Fiore, was a first-rate mathematician. Equipped with Tartaglia’s poem, he and his student Ludovico Ferrari managed to obtain solutions and proofs for all of the cubic forms. Before publishing anything, he and Ferrari, hearing rumors of del Ferro’s discovery, traveled to Bologna, where Fiore’s successor allowed them to inspect del Ferro’s work.

Cardano now considered himself free to publish Tartaglia’s solution, since it was essentially the same as del Ferro’s. In 1545 he produced his masterpiece, Artis Mag- nae, Sive de Regulis Algebraicis (The Great Art, or The Rules of Algebra), now simply called the Ars Magna, with solutions to the cubic. Tartaglia felt himself cheated, although Cardano had mentioned Tartaglia’s role in the discovery. After various exchanges of public insults, another contest was held, although Tartaglia’s opponent was not Cardano, but his student Ferrari. This time Tartaglia lost.

Let us take a closer look at the solution of a cubic of the form x3 + cx = d, using our modem notation. Tartaglia’s poem handled this case, telling the reader to find numbers u and v such that u3v3 = d and 3uv = c. If we can find such numbers, we let x = uv. Using the identity (r + s)3 = r3 + 3r2s + 3rs2 + s3, we get

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so x is a solution to the cubic.

How do we find u and v? Cardano gives the following.

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Therefore

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As an example, Cardano solved x3 + 6x = 20. In this case, the formula gives

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Cardano notes that this reduces to x = 2, but doesn’t tell us how he knew that.

You may have noticed that the last cubic was not of the most general form; in particular, there was no x2 term. A simple substitution, however, allows us to handle that case. For example, consider x3 + bx2 + cx = d. If we plug x = yb/3 into this equation, we get a cubic in y that has no y2 term, and it can be solved by Cardano’s formula above. Details are left to the exercises.

Ludovico Ferrari (1522–1565)

Ludovico Ferrari was bom into a poor family in Bologna, Italy. At the age of 14 or 15, he went to work in Milan as a servant of Cardano, who soon realized Ferrari’s talent and taught him mathematics. At the age of 20, after defeating another job applicant in a debate, Ferrari assumed Cardano’s old position as lecturer in geometry at the Piatti Foundation in Milan.

After he bested Tartaglia in debate in front of a huge audience, at the age of 26, Ferrari received many job offers. The one he accepted was as tax assessor to the governor of Milan. He became rich in this job, retired young, and moved back to Bologna where he became a mathematics professor at the university. He died the following year, of white arsenic poisoning. There has been some speculation that he was poisoned by his sister, who stood to inherit his wealth. This did not turn out well for her. She married two weeks after her brother’s funeral, but her new husband took all of her money and left her in poverty.

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After the cubic was conquered, it was natural to tackle the case of the quartic ax4 + bx3 + cx2 + dx + e = 0. In fact, the quartic was solved by Ferrari, in time to be included in the Ars Magna.

After Ferrari’s work, mathematicians naturally started studying the quintic (fifth degree) equation. This was rather more difficult, and wasn’t handled successfully until the 19th century (see Section 3.3).

Cardano’s Ars Magna was very influential, but was not easy for a student to read. In 1557 Rafael Bombelli began work on a textbook aimed at students, but included the new work on the cubic and quartic. He completed only the first three of the five planned parts of this work, called Algebra; they were published just before his death in 1572. In addition to the new algebra, Bombelli included many problems from Diophantus’ Arithmetica, newly translated, and intended to include geometrical work on polynomials, similar to that of Al-Khayyimagesmimages. Despite being incomplete, Bombelli’s Algebra was the culmination of Italian Renaissance algebra.

Rafael Bombelli (1526–1572)

Rafael Bombelli was bom in Bologna, Italy, the son of a wool merchant. He did not attend the university, instead being trained by the engineer and architect Pier Francesco Clementi. Bombelli became one of the leading engineers of his day, working to reclaim marshlands in several regions of Italy.

It was during a pause in one of his reclamation projects, in 1557, that he began his Algebra. He worked on his mathematics until the engineering project resumed in 1560. As mentioned above, he was never able to finish his text.

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Bombelli did more than repeat his predecessors’ work, however. In particular, he addressed a conundrum that had puzzled Cardano.

Recall the quadratic formula for solutions to ax2 + bx + c = 0:

images

In the case where a = 1, b = 2, and c = 2, this simplifies to – 1 ± images. Before Bombelli, mathematicians had merely discarded such values, arguing that a negative number can have no square root. (Many rejected negative roots as well.)

What Cardano noticed was that, for x3 = 15x + 4, his formula gave the root

images

instead of the obvious root x = 4. He naturally wanted to treat the expression above as nonsensical, but could not reject 4 as a root. He wrote to Tartaglia about this problem, but got no solution (Tartaglia misunderstood the problem).

Bombelli investigated this case, deciding to treat the square roots of negative numbers as actual numbers. In particular, he defined an arithmetic on numbers that included these square roots, arguing by analogy with regular square roots. For example,

images

To address Cardano’s problem, Bombelli used his new arithmetic to calculate

images

This gives the solution to x3 = 15x + 4 as

images

Cardano’s difficulty was resolved, at the cost of expanding the notion of number. In subsequent centuries these new numbers, which we now call complex numbers, have played a crucial role in the advance of many areas of mathematics.

Bombelli was the last great Italian algebraist of the Renaissance; the center of mathematical research had moved north. The most important figure of the late 16th century was François Viète in France.

François Viète (1540–1603)

François Viète was bom in Fontenay-le-Comte in western France, the son of a lawyer. He graduated from the University of Poitiers in 1560. Like Bombelli, Viète was not a professional mathematician. Instead, he followed his father’s career, working first in Fontenay, then in Paris. His law career was very successful. He worked on mathematics in his spare time.

In 1589–90, during a period of conflict between France and Spain, Viète was able to decipher coded messages sent from agents in France to the Spanish emperor Philip II. One code he broke was so complex that Philip complained to the Pope that it must have been done by sorcery.

In 1593 a Belgian, Adriaan van Roomen, set a challenge to all European mathematicians to solve a particular polynomial of degree 45. The ambassador from the Netherlands (ruled by Spain at this time) reportedly claimed to the French king that there was no French mathematician capable of tackling the problem. The king thereupon summoned Viète, who produced two solutions within minutes. The following day, he presented the full set of twenty-three positive roots. (Negative roots were not then considered solutions.)

images

The Islamic mathematicians helped free algebra from its dependence on geometry, which ultimately led to the advances of the 16th century Italian mathematicians. It was Viète who started to free algebra from its dependence on arithmetic.

To begin, Viète employed an improved algebraic notation, although not the notation we now use. He designated unknown quantities (what we would call x, y, …) by vowels, and constants (what we would call a, b, …) by consonants.

Even though we have written formulas to explain the solution to various polynomials, before Viète solutions were always given as rules. Here, for example, is the beginning of a solution from Cardano.

Let us divide 10 into equal parts and 5 will be its half. Multiplied by itself, this yields 25. From 25 subtract the product itself, that is 40, …

Viète was the first to write general formulas for solving equations, instead of giving rules of solution. Thus he was the first to actually write down a version of the quadratic formula.

Viète made an important distinction between algebra and arithmetic. He considered algebraic equations to be separate entities, which could be manipulated symbolically without regard to any arithmetic interpretation. Thus, one could write (x + 2)(x2 – 1) = x3 + 2x2x – 2 without insisting that x actually stand for anything.

This point of view led naturally to a study of the relationship of the coefficients of polynomials to their roots. For example, Viète considered the equation bxx2 = c, which has two roots. If we call the roots x1 and x2, we have images (they are both equal to c), so that images. Factoring, we get b(x1x2) – (x1 + x2)(x1x2). Dividing by x1x2 yields b = x1 + x2. Thus the coefficient b is the sum of the two roots. Substituting this result into images and solving tells us that the product of the two roots is c (details left to the exercises). Viète also wrote equations relating the coefficients of cubics to their roots.

EXERCISES

2.44 Abraham bar Hiyya derived the formula A = (l/2)C(d/2) for the area of a circle, where C is the circumference and d the diameter of the circle. Show that this is equivalent to our usual A = πr2. (Hint: π is defined to be C/d.)

2.45 One of Leonardo’s problems is about two men with money, measured in denarii. The first says to the second: if you give me 7 denarii, I will have five times as much as you have left. The second says to the first: if you give me 5 denarii, I will have seven times as much as you have left. How much does each man have now? (Hint: the answer uses fractions.)

2.46 Find a number that, when divided by 7 has remainder 6, when divided by 8 has remainder 7, when divided by 9 has remainder 8, and when divided by 10 has remainder 9.

2.47 What numbers do the following Roman numerals represent?

a) VI

b) IX

c) DCC

d) MMCM

2.48 Write the following in Roman numerals.

a) 14

b) 326

c) 1999

d) 2014

2.49 What is XXII times IV?

2.50 What is the answer to the abacist’s interest problem quoted in this section?

2.51 Use Alberti’s method (and a ruler) to draw a 4 × 3 checkerboard in perspective.

2.52 Check that Cardano’s values of u and v satisfy u3v3 = d and uv = c/3.

2.53

a) Apply Cardano’s formula to find a solution to x3 + 3x = 36.

b) Use a calculator to simplify your answer.

2.54 Plug x = yb/3 into the equation x3 + bx2 + cxd, and show that the result is a cubic in y with no y2 term. You may wish to use the identities (r + s)3 = r3 + 3r2s + 3rs2 + s3 and (r + s)2 = r2 + 2rs + s2.

2.55

a) Use the substitution of the last exercise on x3 + 3x2 + 2x = 6.

b) Find a solution to x3 + 3x2 + 2x = 6.

2.56 Complete the proof in the text that the product of the two roots of bxx2 = c is c.

2.57 Modem theory of equations tells us that if the two roots of the quadratic equation ax2 + bx + c = 0 are x1 and x2, then we can rewrite the quadratic as a(xx1)(xx2) = 0. Use this form to show that the sum of the roots is –b, and their product is c.


1 Some Arabic names contain “al-.” The word al means “the.” In names, it often refers to the person’s origin, e.g., Muhammad al-Khwimagesrizmimages is from Khwimagesrizm, or some characteristic, e.g., Harimagesn al-Rashimagesd is literally “Harun the Just.” Other common elements of Arabic names include ibn, son of, and abu, father of.

1 From “Earth at Night.” C. Mayhew and R. Simmon (NASA/GSFC), NOAA/NGDC, DMSP Digital Archive.