In Milan, Cardano managed to cure several patients from serious illnesses that other physicians could not treat successfully. This brought him new renown and financial success. He was even invited to Scotland to treat the archbishop of St. Andrews, who was suffering from what we now know is asthma. The archbishop recovered, and Cardano was paid two thousand crowns. Now a successful physician, Cardano again turned his attention to mathematics. His ambition was to make his mark on mathematics history. One day he heard about the great mathematical achievements of the man called the Stammerer. In flowery, flattering language, he wrote Tartaglia a letter, telling him he was publishing a book about equations and needed Tartaglia’s help. Moreover, Cardano promised to credit Tartaglia and to enhance his reputation, which would result in better academic and consulting jobs, if only Tartaglia would reveal his secret. Tartaglia refused. Over several years Cardano continued to beseech Tartaglia, always promising better future prospects in exchange for his insight into solving cubic equations. In the meantime, Tartaglia’s economic fortunes began to decline as the novelty of his discovery wore off.
In 1539, while Tartaglia was tutoring private students in Venice, a letter from Cardano arrived in which the latter offered to introduce him to the head of the military in Milan—a powerful man who might offer Tartaglia a lucrative appointment as a consultant on fortifications—if only Tartaglia would pay him a visit. Tartaglia finally took the bait. When he arrived at Cardano’s house, he was disappointed to find that the promised military leader was not there. Tartaglia was angry, but before he turned to leave, he allowed Cardano to offer him a drink, which led to another … and another. Late that night, he gave up his secret.
Cardano’s book was published in 1545 and marked what many historians consider the beginning of modern mathematics. Tartaglia, who was credited in the book, was angry, and throughout the rest of his life tried unsuccessfully to stop later editions of Cardano’s book, the Ars Magna (Great Art), from being published.
Cardano extended the results he conned Tartaglia into providing him, all of which were revealed in the book. Though he thanked Tartaglia three times in Ars Magna, Tartaglia was unappeased. The year following the publication of Ars Magna, Tartaglia published his own book, in which he reported Cardano’s promise and how it had been broken, as well as revealing details of the rest of their conversations. Nonetheless, Cardano’s book enjoyed wide success, and he received much of the credit for solving the cubic equation.
After his response to Ars Magna, Tartaglia published other books as well, including a 1543 work based on a translation of Archimedes that may have been done by Flemish monk Willem van Moerbeke, whom Tartaglia didn’t credit. Also, in Quesiti et Inventioni Diverse (1546) he presented as his own the law of the inclined plane, which had actually been derived by Jordanus Nemorarius.3 Given all these developments, it has been suggested that perhaps Tartaglia himself learned the formula for solving cubic equations from another source. Ultimately, Scipione del Ferro deserves most of the credit for this immense advance in algebra.
Girolamo Cardano’s Ars Magna, the title page of which appears here, was first published in 1545 under the title Artis Magnae.
In 1540 an eighteen-year-old Italian mathematician named Lodovico Ferrari (1522–60) managed to solve the quartic (fourth-power) equation. Ars Magna also included Ferrari’s solution method for quartic equations. When Tartaglia attacked Cardano and his book, Ferrari—who was a disciple of Cardano—defended his teacher and claimed that he had been in Cardano’s house during the fateful night in 1539 when Tartaglia revealed his “secret.” Ferrari argued that it was no secret at all—rather, the method was freely revealed by its possessor. Following this move by Ferrari, Tartaglia challenged him on August 10, 1548, to a contest of solving equations. Ferrari accepted the challenge and later beat Tartaglia at his own game in a public contest in Venice.4
What Cardano was able to do in his book was to show how a single root, or solution, can be obtained for a cubic equation satisfying the conditions under study—i.e., lacking a square term and having positive coefficients. In actuality, Tartaglia, Cardano, and others placed terms on one side of the equation or another so that they would always be positive. Whereas x3 + ax = b and x3 = –ax + b were treated as different equations, today we know that these are the same equation, except that a term in one form is negative, while in the other it is positive. Mathematicians at that time did not consider negative numbers “real,” so Cardano’s method only resulted in one root of the equation. We now know that a cubic equation has as many as three roots. The final solution method, leading to the three possible roots, was provided by the great Swiss mathematician Leonhard Euler (1707–83) in an article published in 1732.5
In 1557 Tartaglia died in Venice an angry man. Cardano did not enjoy a pleasant old age, either. His eldest son, Giambatista, had married Brandonia di Seroni, a woman from a poor family whose intention seems to have been to extort money from their daughter’s wealthy father-in-law. After many failed ploys, Brandonia taunted her husband, saying that he was not the father of her children, and eventually he poisoned her. Giambatista was tortured in jail before his execution in 1560. Another son, Aldo, was a compulsive gambler, like his father, and lost so much money that, at one point, he broke into his father’s house and stole most of his money. Cardano made many enemies, which made it hard for him to stay employed. Late in life he moved to Rome and died of an apparent suicide in 1576.
IMAGINARY NUMBERS
While it still evidences the struggle that sixteenth-century mathematicians had with understanding the role and meaning of negative numbers in mathematics, Cardano’s seminal book is also the origin of a much more abstract concept: complex numbers. These numbers are combinations of real numbers and multiples of the square root of –1, which are called “imaginary numbers.” In many ways—to the mathematician, to the physicist, and to many an engineer—these imaginary numbers are quite real. We denote the base of these numbers—the square root of –1—with i. The term “imaginary,” in fact, was first proposed by Cardano.
Toward the end of his book, Cardano dealt with a problem that involved the search for two numbers whose sum was 10 and whose product was 40: the solutions of x2 – 10x + 40 = 0. He realized that no two real numbers can be found to satisfy these requirements. He then proposed what he called a sophisticated approach, in which he said that he could imagine the number . If that number could be imagined, he said, then the equation and its requirements could be satisfied by noting that the two numbers he was after were 5 + and 5 – :
Thus, imagining a number that satisfies some relations allows us to solve equations that may otherwise be unsolvable. From this one passage in Cardano’s Ars Magna, imaginary numbers—and hence the whole field of complex numbers—were born.
This plate from cartographer Andreas Cellarius’s 1660 tome Harmonia Cosmographica outlines the heliocentric (sun-centered) Copernican system.
SEVEN
HERESY
As research in mathematics was growing in the sixteenth and seventeenth centuries in Europe, mathematicians were better able to address problems of physics—a science born through extensive efforts in applying mathematics to the physical world—as well as astronomy. This inevitably led to the decline and eventual demise of the millennia-old belief that the earth was the center of the universe, bringing criticism from the Catholic Church, which supported this view as consistent with scripture. Some European mathematicians and scholars ran afoul of the Church because of their scientific conclusions about the position of our planet within the universe. As mathematics developed and sometimes became associated with mysticism, the black arts, alchemy, astrology, and Rosicrucianism, some mathematicians were viewed as heretics. Mathematics had become a dangerous endeavor.
FRANÇOIS VIÈTE
François Viète (1540–1603), also known by his Latin name, Franciscus Vieta, was born in Fontenay-le-Comte in western France, where he studied in a Franciscan school. His father was an attorn
ey, and his mother also had jurists in her family, so a career in law seemed the natural course to follow.
Upon earning his law degree in the nearby city of Poitiers, Viète returned to his native town in 1560 and started practicing law. He was involved in a few interesting cases early on, one of which concerned the financial affairs of Mary, Queen of Scots. Four years later he took an opportunity that interested him more than practicing law: he became the tutor of Catherine of Parthenay, an eleven-year-old child of the local Protestant nobility. Viète taught her mathematics and astronomy, as well as other subjects. When the girl’s father died sometime later, she and her mother moved to La Rochelle on the Atlantic coast, and Viète came with them. There he met many influential members of the nobility, including Henri of Navarre, who later became King Henri IV of France.
In 1570, when Catherine was an adult, Viète moved to Paris, although he remained in touch with her and visited often when he was in western France. He practiced law in Paris and studied mathematics and astronomy on his own. Like the famous Pierre de Fermat, who came after him, Viète was a jurist and avocational mathematician whose work produced important results. He often contemplated mathematical problems while leaning his elbow on a desk or windowsill during his leisure time, and he eventually published a book with the abbreviated title Mathematical Canon.
Viète also became a councilor to the Parlement of Brittany. At the time, France was a monarchy, not a democracy, so the Parlement—akin to a parliament—was a body of deputies who advised and served the king. This was the time of the infamous wars of religion in France, and Viète was caught up in the turmoil. He incurred the wrath of the Catholic League when he represented Protestant interests.
François Viète was an attorney and French government official who practiced mathematics as an avocation. He was a fervent advocate of the decimal system, and was the first to use letters to denote unknown quantities in algebra.
Viète became a member of King Henri IV’s council, which gave him the opportunity to view intercepted Spanish communications. It was in this capacity that his mathematical skills became most evident, as Viète had an uncanny ability to decipher coded messages. He was so good at breaking codes, helping his country gain advantage in its international conflicts, that the Spanish, upon learning that their secret codes had been broken by one man, claimed that Viète was in league with the devil.1
Viète made many contributions to all the areas of mathematics known in his time, but one of his key contributions was pushing for the adoption of decimal fractions. The cumbersome Babylonian base-60 system was entrenched throughout the world, from India to Arabia and Europe. Interestingly, it was none other than Viète’s insistence on the abandonment of numerals based on 60 that finally impelled Europe to adopt the full decimal system we know today.
Viète also did important work in algebra. He provided a new way of solving the cubic equation that involved rewriting the relationship between the variables. He introduced concise notation in algebra, using letters to denote unknowns, and thus transformed the “word-based” algebra of the Italian cossists and their Arab predecessors into the modern symbolic language we use in algebra today. In fact, Viète’s notation and methods are said to have catapulted algebra into a new phase in its development.
Viète was especially adept at using trigonometry in algebraic problems. In 1593 a Flemish mathematician named Adriaan van Roomen issued a challenge to mathematicians to solve an equation of the forty-fifth degree:
x45 – 45x43 + 945x41 – … – 3795x3 + 45x = C,
where C is a number. The Low Countries’ ambassador to France flatly declared to King Henri IV that no French mathematician could answer this problem, so the king called on Viète to defend France’s reputation. Viète looked at this problem, leaning on his elbow, and immediately recognized that the equation, in fact, stated the trigonometric relationship between sin(x) and sin(x/45) algebraically. Once he understood the connection with trigonometry, the solution was immediate. He achieved great fame as the mathematician who defended his country’s honor, and his use of trigonometry in algebra increased the scope and applicability of the study of trigonometry. When told of Viète’s feat, van Roomen is said to have saddled his horse and set off for Fontenay-le-Comte, where Viète was at the time. He stayed with him for weeks, and the two became close friends. In 1602 Viète left the service of the king, receiving a compensation of twenty thousand écu. These were found by his bedside when he died in 1603.
JOHN NAPIER
Trigonometry was of great interest to another mathematician, one who lived far from Italy and France. John Napier (1550–1617), Baron of Merchiston, was a Scottish nobleman chiefly engaged in running his vast estate. He had other interests, however. One of them was the book of Revelation. In a commentary on the book, he argued that the pope was the Antichrist.
Napier was an astrologer and necromancer—he delved into the practice of conjuring the spirits of the dead. He was also interested in alchemy and, in his house, performed strange experiments with flasks of boiling liquids. He was said to have always traveled with a black spider he kept in a little box, and he had a black rooster he used as a “familiar spirit” in magical endeavors. For example, he would force each of his servants to sit in a room alone with the rooster and stroke the bird. This trick allowed Napier to determine which of his servants had stolen from him. In actuality, he is said to have covered the rooster with soot and determined the guilt of a servant by seeing which one had clean hands afterward, with the assumption that the guilty person—afraid of being caught—would only pretend to stroke the rooster.
The Scottish astrologer John Napier introduced the concept of logarithms—a way of lowering the levels of calculations—in 1614.
In 1594 Napier began to think about an idea that would change the world of mathematics: how to simplify arithmetic computations by changing multiplication into addition. He was expanding upon the idea that addition of powers in arithmetic is equivalent to multiplication when the base is the same (e.g., 5(2 + 3) = 52 × 53). Napier hoped to apply a variant of this rule to other computations, but he couldn’t find a good way of doing it until his friend John Craig, personal physician to King James VI of Scotland, told him a story. Craig had been with the king when he sailed in 1590 to Denmark to meet his bride-to-be, Anne of Denmark. Just before arriving in Copenhagen, heavy storms in Oresund Strait forced them to come ashore on Hven Island, on which the famous Danish astronomer Tycho Brahe had situated his observatory.
Brahe entertained the king and his entourage, and Dr. Craig learned that, in his calculations of astronomical observations, Brahe was making use of the method of prosthaphaeresis—using trigonometric identities to reduce multiplication to addition (and division to subtraction)—which had been discovered by Ibn Yunus five centuries earlier. This story encouraged Napier to redouble the efforts he had been making to find an even more efficient way of lowering the levels of computations and, in 1614, after finishing the Herculean job of computing logarithms, he published his book, Mirifici Logarithmorum Canonis Descriptio (A Description of the Marvelous Rule of Logarithms).2
For three hundred and fifty years—until the middle of the twentieth century, when calculators became readily and inexpensively available—Napier’s logarithms ruled the world of calculation. The slide rule, which was the mechanical forerunner in the West of the electronic calculator, was based on Napier’s idea of the logarithm. The slide rule was marked with a logarithmic scale of numbers, and it allowed a person to perform multiplication through the actual physical “addition” of two sections of numbers. This was accomplished by sliding the central strip of the ruler back and forth and reading the answer on the fixed strips on the top and bottom.
JOHANNES KEPLER
Tycho Brahe was a very colorful figure. Having lost his nose in a duel while he was a student, he wore a prosthetic nose made of metal. He was a nobleman close to the king of Denmark, who had given him the island of Hven on which to build his observato
ry. There, Brahe carried out the most extensive observations of the sky ever made by one person, as far as we know. His work led him to a theory that fell somewhere between the Copernican model of the universe and the earth-centered model that preceded it. Drawing upon his observations of a supernova, he showed that the stars were far above both our atmosphere and the moon, belonging to a higher “sphere.” This and other observations allowed him to overthrow the Ptolemaic model, which had reigned for almost a millennium and a half. After a quarrel with the king, Brahe left for Prague, where he was given facilities to continue his work under the Bohemian king Rudolph II. The brilliant German mathematician Johannes Kepler became his assistant.
Johannes Kepler (1571–1630) was born prematurely in Weil der Stadt, Germany, to an innkeeper’s daughter and a mercenary. His father, Heinrich Kepler, abandoned the family when Johannes was just five years old and is believed to have perished in the Thirty Years’ War. At an early age Johannes impressed lodgers at his grandfather’s inn with his mathematical ability, and after witnessing the Great Comet of 1577 and a lunar eclipse in 1580, he developed a lifelong interest in astronomy. Childhood smallpox left Johannes with weak vision and crippled hands, but in 1589 he enrolled at the University of Tübingen and quickly gained a reputation as a skilled mathematician and astrologer. Despite his desire to enter the ministry, Johannes was recommended for a professorship at the University of Graz, which he accepted at the tender age of twenty-three.
During his tenure in Graz, Kepler studied the conic sections identified in ancient Greece because he was interested in mirror images of various shapes. In 1595 he married Barbara Müller, a twenty-three-year-old widow (twice over) with whom he had three children. Five years later he was invited by Brahe to assist in calculating planetary orbits at a new observatory he was constructing outside Prague. The study of ellipses and the circle, in conjunction with an analysis of his employer’s vast set of astronomical observations of the planets, allowed Kepler to make one of the most important scientific deductions in history—the laws of planetary motion:
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