This sum is appropriately called a geometric series in honor of its geometric origins. The terms in a geometric series decrease by a constant ratio from one term to the next. In Zeno’s case the ratio was 1/2; in Archimedes’ case it was 1/4. The general rule, as you might have guessed, is this:
Unfortunately, the mathematics of Archimedes’ time would not allow him to take this final step. Instead, he had to resort to an ingenious reductio ad absurdum argument. Just as Zeno would have done, he argues against an imaginary opponent. You think that the parabola’s area doesn’t equal 4/3? Fine. Then you must tell him whether it is greater than 4/3 or less than 4/3. If you say it’s greater than 4/3, then Archimedes will show by his subdivision technique that you have overestimated the area. If you say it’s less than 4/3, he will show that you have underestimated. Either way you lose, and you have to concede that the area is 4/3.
With hindsight, we can see that Archimedes has come a long, long way towards understanding infinite processes*. He is using the infinite to discover new truths—a huge leap forward, which took the ancient Greeks to the brink of mastering the infinite.
* * *
* Actually, Archimedes was far from alone. Eudoxus of Cnidos (408–355 BC) is credited with inventing the “method of exhaustion,” which Archimedes employed to such good effect here.
6
a matter of leverage laws of levers
Archimedes of Syracuse was born about 287 BC and died in 212 BC. In his own time, his reputation rested more on his physical discoveries and engineering inventions than on his mathematics—yet he would surely have considered himself a mathematician. He was proudest of his proof that a sphere has two-thirds the volume of its circumscribed cylinder; or equivalently, V = (4/3)πr3. He even asked for a diagram of a sphere and a cylinder to be inscribed on his tombstone.
Archimedes’ work represents a beautiful unification of applied mathematics with geometry and with the still gestating concept of the infinite. Today, however, most people probably associate Archimedes with the story that he ran down the road naked, crying “Eureka!” (“I have found it!”) The story goes that Archimedes’ friend and patron, King Hieron, wanted to know if a certain crown was made of pure gold. Archimedes supposedly was sitting in his bathtub when the solution occurred to him. If the crown was made of a cheaper alloy, it should be less dense than pure gold. If placed in a container of water, the ersatz crown would displace more water than a gold piece of the same weight.
Often, legends like this are codified versions of real events. One example was the purported drowning of Hippasus by the Pythagoreans, and another will be seen in the discussion of how Isaac Newton discovered the law of gravity. In reality, Archimedes wrote a book called On Floating Bodies, in which he formulates what became known as Archimedes’ principle: the weight of water displaced by an object (either floating or completely immersed) equals the buoyant force exerted by the water on the object.
In a lever, a weight w1 at distance d1 from the fulcrum will balance a weight w2 at distance d2.
From Archimedes’ principle it is possible to deduce a formula for the density of an object immersed in water. If ρbody denotes the density of the object, ρfluid denotes the density of the fluid (which, in the case of water, is conventionally assumed to be 1), wdry denotes the dry weight of the object and wimmersed denotes its apparent weight when fully immersed, then:
This formula makes it possible to compute the crown’s density (or “specific gravity”) directly, so there would be no need to compare it with an actual gold crown of equal weight. Archimedes surely came to this discovery over a period of time, not overnight, and it is pretty certain that he was aware of it before King Hieron came to him. Everything else about the legend is embellishment, including running naked through the streets.
Archimedes’ principle and the specific gravity formula are still used routinely, even today. The rest of On Floating Bodies contained a wealth of information about bodies of different shapes and their stable floating configurations. It was a first step toward making shipbuilding a science instead of a matter of trial and error.
Archimedes also experimented with levers and pulleys. Again, there is a story that King Hieron remarked on the power of Archimedes’ contraptions, and Archimedes replied, “Give me a place to stand and I will move the Earth.” Archimedes understood the lever law, d1w1 = d2w2, which expresses the relationship between the weights of two objects (w1 and w2), if they are balanced on a lever at distances d1 and d2 from the fulcrum. From the formula it follows that a lesser weight can balance a greater weight if it is farther from the fulcrum. For instance, a 150-pound man can lift a 1500-pound safe if he stands on one end of a lever, places the safe at the other end, and places the fulcrum at least ten times closer to the safe than to himself.
Below “Give me a lever and I will move the Earth” – a woodcut showing Archimedes putting his famous saying into action from the title page of The Mechanic’s Magazine London, 1824.
Archimedes frequently used the lever law not only in physical devices, but also in mathematical research. As discussed earlier, he first discovered the area of a parabola “by means of mechanics” and only later proved it “by means of geometry.” That argument, filling the parabolic segment up with triangles, was actually his second proof. His original proof involved an equally ingenious and different way of cutting up the parabolic segment into pieces, and balancing those pieces with rectangles of known area (or weight) on the other side of a lever. It was actually a favorite method of his. However, Archimedes apparently felt that the lever law was too informal or perhaps too empirical to be acceptable as pure mathematics. Thus, after “discovering” a theorem with levers, he felt compelled to confirm it in a way that Euclid would have approved.
ARCHIMEDES WAS FORTUNATE enough to live most of his life during the prosperous and peaceful 54-year reign of King Hieron. Unfortunately, toward the end of his life that period of peace came to an end. Hieron’s son antagonized the growing Roman Empire, and the result was a one-sided war between the Romans and the Syracusans.
Almost single-handedly Archimedes was able to hold off the Roman army, by designing grappling devices and cranes of unprecedented accuracy. According to Plutarch, the Romans became so terrified of Archimedes’ devices that “if they only saw a rope or a piece of wood extending beyond the walls, they took flight exclaiming that Archimedes had once again invented a new machine for their destruction.”
As a last resort the Roman general, Marcellus, laid siege to Syracuse. After two years the Romans entered the city. Marcellus gave orders to spare the life of Archimedes, but, according to legend, a soldier came upon Archimedes kneeling over a mathematical diagram. “Don’t disturb my circles,” Archimedes told him. Enraged by the unknown man’s impudence, the soldier ran him through with his sword.
PART TWO
equations in the age of exploration
On august 10, 1548, the Church of Santa Maria del Giardino in Milan, Italy, was thronged by curious spectators. The event they came to witness was not a church service, but the mathematical equivalent of a duel at twenty paces. Using nothing but their wits, Niccolò Tartaglia of Venice would battle against Lodovico Ferrari, a peasant boy-turned-servant to one of Milan’s most famous citizens: Girolamo Cardano—a physician, gambler, and jack of all intellectual trades.
Curiously, Cardano himself was nowhere to be found. He had precipitated the ruckus three years earlier by publishing a mathematical formula that Tartaglia had given to him in strictest confidence. However, on this day he had found a convenient excuse to be out of town while his servant, who was in all likelihood a better mathematician than himself, defended his honor.
The competitors must have seemed better suited for a back-alley brawl than a contest of minds. Tartaglia had been disfigured as a youth by a deep saber wound to his jaw, received when a French army sacked his hometown of Brescia in 1512. Though as an adult he hid his scar by growing a full beard, the injury had left him with a
permanent speech defect that led to his nickname: Tartaglia, the stammerer. Ferrari, too, bore the scars of a rough-and-tumble childhood, as he was missing some of the fingers of his right hand.
We will never know exactly what happened in the church that day. The planned contest of minds apparently turned into a shouting match. But circumstantial evidence suggests that Ferrari was perceived as the winner. The governor of Milan, who was in the audience, was impressed enough by Ferrari’s talent to hire him as a tax assessor. Tartaglia lost a teaching position he had just gotten in Brescia, for which he never received a penny. He died nine years later as Cardano’s sworn enemy. As for the man who had started it all, Cardano returned home with his reputation intact and continued to enjoy the life of the proverbial Renaissance man.
The battle in the church was the final act of one of the most bitter and bizarre disputes in the history of mathematics, a debate over the rights to the first completely new mathematical discovery in Europe since the fall of the Roman Empire. Until the early 1500s, western Europe had mostly been playing catch-up to the rest of the world, as well as to its own past. The formula that Tartaglia had confided to Cardano—which is now known, rather unjustly, as Cardano’s formula—has been compared to the discovery of America, because it was a new fact about the world that was not even hinted at in any ancient books. It launched an Age of Exploration in mathematics that would transform the map of the mathematical world as profoundly as Columbus’s discovery transformed the map of the physical world.
7
the stammerer’s secret cardano’s formula
The story of Cardano’s formula really begins more than 3000 years earlier. In the period between 1850 and 1650 BC, problems like this one proliferated in Babylonian mathematical tablets: find two numbers whose product is 60, and whose difference is 7. A modern mathematician would call the numbers x and y and note that y = 60/x. Therefore, x – 60/x = 7, or equivalently, x2 – 7x – 60 = 0. Then one would trot out the quadratic formula, which says that the solution to any quadratic equation, ax2 + bx + c = 0, is given by:
Using a = 1, b = –7, and c = –60, said mathematician would obtain the solutions x = 12 and y = 60/12 = 5.
However, the Babylonians did not have the algebraic tools that we do today. Instead, the scribe took a more intuitive approach, which involved drawing a rectangle with the unknown side lengths, x and y, cutting it into pieces, and rearranging the pieces into an L-shaped figure, as shown on page 62. He then “completed” the L-shaped figure by adding a small square of known area in the corner. A similar method of solution—called “completing the square”—is still taught in high-school algebra as a precursor to the quadratic formula, but usually with no reference to its geometric meaning or its historical provenance.
Cardano’s formula for solving a reduced cubic polynomial, x3 + px = q
The other ancient mathematical cultures also “knew” the quadratic formula or else had equivalent methods for solving quadratic equations. Euclid employed a geometrical construction that produced a line segment of the requisite length. In seventh-century India, Brahmagupta, discussed previously in relation to zero, provided a recipe to solve the equation ax2 + by = c that is essentially the quadratic formula written in words instead of symbols.
However, classical mathematics is essentially silent* on the question of how to solve a cubic equation, ax3 + bx2 + cx + d = 0. In 1494, Fra Luca Pacioli, an Italian mathematician, expressed the opinion that cubic equations would never be solved exactly. Pacioli was proved wrong only a generation later!
Left “Completing the square.”
In the early 1500s, a Bolognese mathematician named Scipio del Ferro apparently found a method for solving cubic equations that are lacking the quadratic term: in other words, any equation of the form x3 + px = q. Nowadays, a mathematician who made such a discovery would hasten to publish it. However, in the Italy of that era, mathematicians made their reputations by defeating other mathematicians in problem competitions. Del Ferro therefore kept his method secret, so that he could pose problems that his opponents would not be able to solve. Only on his deathbed did he confide his secret to two of his students, Antonio Maria Fiore and Annibale della Nave.
However, rumors soon spread about del Ferro’s discovery. In the early 1530s, Tartaglia started claiming that he, too, could solve cubic equations. Thinking he could call Tartaglia’s bluff, Fiore rashly challenged Tartaglia to a competition. According to the legend (which is probably a little too good to be true), on the eve of the debate Tartaglia finally figured out how to solve these cubics, and so he thoroughly trounced Fiore.
How, then, did Cardano get his name on what should have become known as Tartaglia’s or del Ferro’s formula? It becomes a little less surprising when you read Girolamo Cardano’s own description of himself: He wrote in his autobiography that he was “hot tempered, single minded, and given to women … cunning, crafty, sarcastic, diligent, impertinent, sad and treacherous, miserable, hateful, lascivious, obscene, lying, obsequious …” He had studied to become a physician at the University of Padua, but perhaps because of his erratic behavior he was forbidden to practice medicine in Milan until 1539. However, even before then he found great success as a public lecturer and writer on a variety of topics, including mathematics.
In 1539 Cardano was composing a mathematical handbook called Practica arithmeticae generalis, and he asked Tartaglia for the secret to solving cubic equations. Tartaglia at first refused, on the grounds that he intended to write a book of his own. But at last, Cardano persuaded Tartaglia to come to his house for a visit. On the night of March 25, 1539, Tartaglia revealed his method under an oath of strict secrecy.
And here is the secret, in an English translation by math historian Jacqueline Stedall. (Tartaglia’s version, in Italian, actually rhymes!) For ease of understanding, Tartaglia’s verse has been interpreted here in algebraic symbols. “The thing” means the unknown quantity x, “the cube” means x3, and “the number of things” is p. The equation Tartaglia wants to solve is x3 + px = q, and the numbers p and q are positive. (This last point is irrelevant to mathematicians today, but was very relevant to sixteenth-century Italians, who were still as skeptical of negative numbers as the nineteenth-century-BC Babylonians.)
Tartaglia
Algebra
When the cube with the things next after
When x3 + px
Together equal some number apart
Find two others that by this differ
= q,
Find u and v such that u – v = q,
And this you will then keep as a rule
That their product will always be equal
And such that uv =
To a third cubed of the number of things
The difference then in general between
Then
The sides of the cubes subtracted well
Will be your principal thing.
= x
Tartaglia’s formula is a well-disguised way of “completing the cube,” but it includes a very clever new step that was not present in the Babylonian process of completing the square: the introduction of two new auxiliary variables, u and v. Here is how it works for an example considered later by Cardano: x3 + 6x = 20. Tartaglia instructs us to find two numbers u and v such that uv = (6/3)3 = 8 and u – v = 20. This pair of equations can be solved by the quadratic formula: u = 10 + 6√3 and v = –10 + 6√3. (These are the two positive solutions.) Now we are supposed to find the cube roots of u and v. In general one would have to do this by approximation, either by hand or with an abacus. But in this particular case the cube roots have a simple exact form:
Below An engraving from frontispiece of Cardano’s work Ars Magna, 1545, the first great Latin book dedicated to algebra.
Finally, we subtract these to get the answer: x = (√3 + 1) – (√3 – 1) = 2
AT FIRST, Cardano honored his pledge not to publish Tartaglia’s method. However, over the next few years several things happened that made him it
ch to see the solution in print. First, he and his protégé, Ferrari, went beyond Tartaglia, by showing how to simplify any cubic equation to del Ferro’s form or one of 12 other basic forms. Second, Ferrari “invented at my request” (as Cardano later wrote) a method for solving quartic or fourth-degree equations. This latter discovery is far more remarkable than Cardano’s offhand comment would suggest. More than 3000 years elapsed between the solution of the quadratic and the first solution of the cubic—but it took Ferrari only four years to move on to quartics! Unfortunately, the solution to the cubic was an intermediate step in the solution of quartics. The promise to Tartaglia was now a major obstacle: Without the method for solving cubics, Cardano could not publish Ferrari’s brilliant solution for quartics.
At this point Cardano found an ingenious loophole. In 1543, he tracked down del Ferro’s other student, della Nave in Bologna, and ascertained that Tartaglia’s method for solving the cubic was exactly the same as del Ferro’s. This fact apparently released Cardano (at least, in his own mind) from his promise to Tartaglia to keep it a secret. Two years later, Cardano published his greatest mathematical work, Ars magna (The great art), with a complete treatment of cubics and quartics, and the secret was out.
The Universe in Zero Words Page 5