*Editor's note: The following is an excerpt from *They Called Me Mad: Genius, Madness and the Scientists Who Pushed the Limits of Knowledge* by John Monahan (on sale December 7 from Berkley)*. *In it Monahan takes the reader from Archimedes archetypical "Eureka!" moment to J. Robert Oppenheimer's fraught findings.*

His genius shone like a beacon throughout the Hellenistic world, and his dazzling mathematical insights and wondrous inventions continue to fascinate us to this day. Unfortunately much of his actual life is obscured by the mists of time. In the absence of facts, a body of legend has grown, punctuated by secondhand and thirdhand accounts of varying accuracy. Galileo venerated him. The Fields Medal, one of the most prestigious prizes for mathematicians bears his image. The tenth-century Islamic geometer Ab Sahl al-Kh was so impressed by his works that he called him the “imam of mathematics” (Hirshfeld, 2009). He is credited with calculating pi and the volume of the universe, discovering principles of buoyancy, inventing water pumps, and building war engines capable of grinding the Roman army to a halt. Not to mention inventing what may have been the world’s first death ray. The name of this legendary genius, perhaps the greatest mathematician and inventor of all time, is Archimedes.

His life began on the sun-drenched shores of the island of Sicily, in the city-state of Syracuse. Originally a Greek colony, sitting at the nexus of Mediterranean trade, it was one of the most influential cities of the ancient world, described by Cicero as, “the greatest Greek City and the most beautiful of them all.” Its harbor was filled with Egyptian, Greek, and Phoenician trading vessels bearing all manner of oils, wines, and exotic spices. Unlike most of the other city-states of the time, the leaders of Syracuse had managed to safely navigate the treacherous political waters between Rome and Carthage, the super-powers of the day. So at the time of Archimedes’ birth, Syracuse enjoyed an unusual period of peace and prosperity.

Exact dates are difficult, but it is believed that Archimedes was born around 287 b.c.e. His father was the astronomer Pheidias, who passed on to the young Archimedes his love for the stars, the planets, and the other wonders of the universe. Starting around the age of seven, the boy Archimedes would have received the formal education typical of Greek males, including lessons in Greek grammar, literature, and music as well as training in sports such as running and throwing the discus and javelin.

When Archimedes was a teenager, something occurred that would have important implications for the young man. Around 270 b.c.e., Hieron, a military commander and illegitimate son of a Syracusan nobleman, seized power and became king of Syracuse. Archimedes and Hieron were friends; some have even suggested they may have been related. Whether or not that was true, the two men would form a long-lasting relationship that would serve them both well.

Shortly after Hieron took the throne, Pheidias sent the young Archimedes to continue his education across the sea in the storied city of Alexandria. Founded in Egypt near the Nile Delta by the legendary Alexander the Great, the city had been built by one of Alexander’s generals, Ptolemy. When he succeeded Alexander and became king of Egypt, Ptolemy I dedicated the city to the pursuit of culture and learning, turning it into the greatest intellectual center in the ancient world.

The city was home to the temple of the Muses, the origin of our word museum. This wasn’t one building, but a complex of buildings, including lecture halls, dissection rooms, botanical gardens, and even a zoo, in addition to accommodations for visiting scholars from all over the known world. Next to the museum was the famous Library of Alexandria. At the time, the library was the greatest repository of knowledge civilization had ever known, containing over half a million works.

At the museum, Archimedes studied with the disciples of the renowned mathematician Euclid. One of his notable teachers was Conon of Samos, not only a brilliant mathematician in his own right, but also an accomplished astronomer who studied eclipses and discovered the constellation Coma Berenices. He and Archimedes established a lifelong friendship and would frequently write letters to each other even after the latter had returned to Syracuse.

In addition to Conon, Archimedes established enduring friendships with many of his fellow students, including Eratosthenes. In later years, Eratosthenes went on to become head of the library and accurately calculate the circumference of the Earth using the length of shadows cast on the summer solstice. He and Archimedes established a long correspondence, telling each other of their latest discoveries.

While in Alexandria, Archimedes began his career as an inventor, developing a device for moving water uphill. What came to be known as Archimedes’ screw consisted of a hollow cylinder. Inside of this was a central shaft around which was wrapped a long spiral blade, similar to the threads on a screw. One end of the device was placed in the water, and the other end was placed above at an angle. As the central shaft was rotated, either by hand or by draft animals, it turned the blade, which in turn pushed the water up the hollow cylinder and out the upper end. By use of Archimedes’ machine farmers could easily irrigate their fields, and sailors could pump out the bilge of their ships. Modern versions of Archimedes’ screw are still in use today in water-treatment plants and for moving grain; a miniaturized version is used maintain blood flow in heart patients.

Eventually, Archimedes completed his education and returned to Syracuse, where he spent the remainder of his life. Not long after he came home, King Hieron put his newly returned friend to work. It is one of the most famous stories about Archimedes and involves a golden crown in the shape of a laurel wreath.

The crown had been given to King Hieron and was supposed to be made of pure gold, which the goldsmith had been given for the project. However, it was not uncommon for goldsmiths to adulterate or dilute gold with cheaper metals such as silver. The king was suspicious that the artist had done just that and pocketed the difference. The problem was that the crown was not only a beautiful work of art but was supposed to be used in a religious ceremony, and so was considered a holy object. If the true composition were to be discovered, it had to be done without damaging the crown itself in any way.

The king posed the question to all of his advisers, but they were unable to solve the mystery. Archimedes took up the problem. He pondered it for quite a while, but according to the story, the answer came to him in a flash of inspiration. He was visiting a public bath, and as he stepped into the tub of water, it began to overflow. The more he immersed himself in the water, the more it overflowed. He realized that the amount of water being spilled was proportional to the volume of the body being placed into the water. He further made the connection that if he placed the crown into a given amount of water, and it displaced more water than the same weight of pure gold, then the crown and the pure gold must have a different volume. In other words, if they had a different volume, then the crown could not be pure gold.

He was so excited by his discovery that he leaped from the tub and ran home through the streets, still wet and naked, yelling, “I have it! I have it!” The word for this in ancient Greek is Eureka! The king was delighted by the brilliance of Archimedes’ insight. The goldsmith was not. Archimedes’ test worked. The crown was not pure gold, and the hapless smith was executed.

When not helping out his friend the king and giving catch phrases for countless inventors and geniuses to come, Archimedes was immersing himself in mathematics. He was particularly interested in circles. Other Greek mathematicians had noted that there was a relationship between the circumference of a circle and its diameter. Today we represent this relationship with the Greek letter pi. Archimedes knew that the circumference of the circle worked out to a little more than three times its diameter, but he wished to calculate it more precisely.

First, Archimedes started by drawing a circle. He knew how to calculate the perimeter of polygons, or multisided shapes, like hexagons and octagons, so inside the circle he drew a polygon that exactly fit within it. Next he drew a polygon with the same number of sides just outside the circle so that it touched the circle within. By calculating the perimeter of both polygons, Archimedes could deduce that the circumference of the circle lay somewhere between the two.

Initially this didn’t offer a very precise figure, but then Archimedes began to increase the number of sides of his polygons. He started with triangles, then went to hexagons (six-sided figures), then dodecagons (twelve-sided figures), and so on. As the number of sides increased, Archimedes came closer and closer to the actual value of pi.

Eventually, he worked his way up to a ninety-six-sided figure, an enneacontahexagon. When he was done his calculations he wrote, “The circumference of any circle is three times the diameter and exceeds it by less than one seventh of the diameter and more than ten-seventy-oneths.” In other words, he had established that pi was between 3 1/7 and 3 10/71. Archimedes didn’t have the benefit of a decimal system, but if he had, his value for pi would have been between 3.1428 and 3.1408. Hitting the pi key on a modern calculator yields a value of 3.1415926, amazingly close to Archimedes’ figure.

While Archimedes was wrestling with mathematical problems like this, his concentration was so intense that it may have contributed to our image of the absent-minded professor. The Greek historian Plutarch describes him as, “continually bewitched by a Siren who always accompanied him.” He goes on to write that Archimedes was, “possessed by a great ecstasy and in truth a thrall to the Muses” (Hirshfeld, 2009). This possession often manifested itself in Archimedes’ forgetting mundane tasks like eating or changing his clothes. He would also make use of almost any available surface to do his written calculations, including ashes from the fire, the ground outside, and even oil anointed to his own skin. Plutarch further noted that “he placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.”

Another example of Archimedes applying his powers of mathematical genius to ordinary objects involves the surface areas of spheres and cylinders. He envisioned a sphere that would just fit inside a cylinder, so that the sides of the cylinder would always be in contact with the side of the sphere. The top and bottom of the cylinder would likewise be in contact with the sphere. Archimedes then calculated the surface area of both objects, the sphere and the cylinder that contained it. He was amazed to find that the surface of the sphere was always two thirds the surface area of the cylinder. It didn’t matter how large or how small they were, as long as the sphere and cylinder fit snugly together, the ratio of their surface area always worked out to 2:3. The same ratio held true when he calculated their volume. The sphere always had two thirds the volume of the cylinder. This might seem like a trivial discovery, but to Archimedes, it was wondrous, because it confirmed his belief in a universe that could be understood because it obeyed regular mathematical principles.

Math was where Archimedes took his greatest joy. He viewed his many inventions and feats of engineering brilliance as simply distractions from his mathematical accomplishments. Ironically, it was one of these distractions that lead to another famous story. Archimedes once claimed, “Give me a place to stand and I will move the Earth.” King Hieron heard of the boast and told him to prove it. The king’s shipbuilders had recently completed construction on what was at the time the world’s largest ship. Dubbed the Syracusia, and intended as a gift to the Egyptian ruler Ptolemy, it was a monstrous three-masted vessel that weighed over two thousand tons. Now the great ship needed to be launched. The king told Archimedes that if he could move that single-handedly into the water, then he would be believed.

Archimedes quickly set to work. He constructed a complex series of ropes and pulleys and attached them to the ship. The other end of the ropes was fastened to a rotating helix, a type of large screw-like device that was attached to the dock. Word of the challenge spread, and when it came time for Archimedes to prove his boast, people from all over the city came to witness it. With the king and the populace looking on, Archimedes quietly took a seat next to the helix and began turning the handle. As the ropes grew taught, the crowd held its breath, and suddenly, the mighty ship began to move. Slowly, but steadily the genius of Syracuse pulled the hulking vessel into the water. The crowd erupted into wild cheers, and the king was so impressed that he declared, “I order, from this day on, that Archimedes is to be believed in anything he says.”

His fame assured, Archimedes returned to his mathematical pursuits. Having proved he could move the world, now he set his sights on the entire universe. Specifically, he claimed that he could not only calculate the volume of the universe, but could also calculate the number of grains of sand needed to completely fill it. He started by using estimates of the Earth’s diameter and the diameter of its orbit, which had been found by other mathematicians, including his friend Eratosthenes.

Next, unlike the other Greek philosophers who were using a geocentric, or Earth-centered, model of the universe, Archimedes used the heliocentric, or sun-centered, model that had been developed by a fellow Greek named Aristarchus of Samos. This was approximately eighteen hundred years before the sun-centered model of Copernicus would take center stage. Archimedes liked this model because it gave him a much larger universe in which to play his mathematical games. Based on the ratio between the Earth’s diameter and its orbit, Archimedes calculated that the universe had a radius of roughly ten trillion miles. This is substantially smaller than modern estimates, but surely big enough to boggle the minds of the ancient Greeks.

Next, Archimedes further inflated the numbers he was dealing with by choosing the smallest possible size he could imagine for the sand grains needed to fill the universe. By the time he was done, Archimedes calculated that the number of sand grains needed to fill the universe was one thousand trillion trillion trillion trillion trillion grains. In our modern notation that’s 1063, or a one with sixty-three zeros after it. Archimedes, however, didn’t have our modern notation. He was forced to work with the Greek numeral system. In that system the largest number was a myriad, equal to ten thousand, and the largest possible number, a myriad of myriad was equal to 108. To express his beautiful calculations, Archimedes needed to develop a new way of expressing large numbers. He presented his calculations and his new system in a work called The Sand-Reckoner, which Archimedes concluded with:

*These things will appear incredible to the numerous persons who have not studied mathematics; but to those who are conversant therewith and have given thought to the distance and the sizes of the earth, the sun, and the moon, and of the whole of the cosmos, the proof will carry conviction. It is for this reason that I thought it would not displease you either to consider these things.*

*Excerpted from* They Called Me Mad: Genius, Madness, and the Scientists Who Pushed the Limits of Knowledge* by John Monahan. Reprinted from by arrangement with Berkley, a member of Penguin Group (USA) Inc., Copyright (c) 2010 John Monahan. *