"The responsibility for this flight lies first with history and with the giants of science who have preceded this effort. Next with the American people, who have through their will, indicated their desire. Next, to four administrations, and their Congresses, for implementing that will. And then, to the agency and industry teams that built our spacecraft, the Saturn, the Columbia, the Eagle, and the little EMU; the space suit and backpack that was our small spacecraft out on the lunar surface. We'd like to give a special thanks to all those Americans who built those spacecraft, who did the construction, design, the tests, and put their - their hearts and all their abilities into those craft. To those people, tonight, we give a special thank you, and to all the other people that are listening and watching tonight, God bless you. Good night from Apollo 11." 

- Neil Armstrong, July 23, 1969

On July 24th, 1969, Apollo 11 command module Columbia employed a computer-controlled reentry and splashed down in the Pacific ocean. The crew was recovered first, leaving Columbia floating in the water for some time before it was lifted aboard the aircraft carrier USS Hornet with a crane.

To accomplish this, several things had to be known: the value of pi, gravity, the spherical nature of Earth, how objects float, ways of pumping liquid, ways of lifting heavy objects using as little energy as possible, and infinitesimals.

Hi, I’m David Grider, and welcome to How We Went to the Moon. In this episode, we're going to explore the life and times of one of the giants of science that Armstrong wished to thank while silently falling back to the Earth, the person responsible for all those things that needed to be known.

In the year 467 Ab urbe condita, corresponding to 287 BC, the Roman Senate is forced by the plebeian soldiers to pass a new law giving the plebeian Assembly more power than the Senate. However, in practice, the patrician-lead oligarchy remained in place, and the new powers given to the Assembly were mostly ignored or overridden. Meanwhile, military rebellion was also influencing the goings-on in Greece, when the Macedonian army deserted after building resentment for their leader, Demetrius I. Eventually, Demetrius was defeated by Pyrrhus I, who then became king of Macedonia.

And a future scientist and inventor named Archimedes was born in Syracuse, Sicily, Magna Graecia.

At the time, Syracuse was an independent colony, but during Archimedes time it was under constant threat from the expanding Roman Republic and the neighboring Carthaginians who controlled about half of the island of Sicily. 

Not a lot is known about the life of Archimedes. The only known biography of him was lost to history. Even his birthdate is not known for certain; it is calculated by his known death date and the statement of his age at death by John Tzetzes, a Byzantine historian in the 12th century.

What is known is some of what he invented and the mathematical and scientific achievements he made.

Everyone is familiar with story of Archimedes and the gold crown of the king, his bath and subsequent discovery of hydrostatics, causing him to run through the streets in the nude shouting, "Eureka!"

Unfortunately, like the story of the plague of Delos, many stories that we'll run across throughout this long tale of science and engineering are largely apocryphal. The Eureka moment is one more.

He did however discover the principle of hydrostatics, sometimes called Archimedes' principle. As written in his treatise, On Floating Bodies, it states: "Any object, totally or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object."

For those more scientifically-inclined out there, when he says weight, he actually means weight, not mass. Therefore, in metric, we're talking newtons, and in imperial, pounds-force. As a result, the buoyancy of an object is different in different gravitational fields.

Archimedes' principle is a core part of the field of hydrostatics, which describes how fluids at rest react in relation to objects immersed within them. Archimedes' principle is itself a mathematical framework to describe how objects float and sink, and it is used to design boats, ships, submarines, and even space capsules that must float after splashdown.

A lot of Archimedes engineering work was done out of pure necessity. According to Athenaeus, Tyrant Hieron II commissioned Archimedes to design a ship that is likely to have been the largest ship in antiquity, named the Syracusia. Its size meant that it was not perfectly watertight, and water seeping into the ship collected in the bilge, the bottom-level of the ship, a problem that still exists today in large ships. Archimedes solved this problem by developing the bilge pump. The pump design involved a screw-shaped blade inside a cylinder that would be turned manually. This design became known as the Archimedes' screw. In the modern day, bilge pumps are still in use on large ships like aircraft carriers, but they don't rely on the Archimedes' screw anymore. This isn't to say that the Archimedes screw isn't still in use. It is still used to pump liquids and to move granular materials like grain. After the Syracusia was built it took only one known trip, sailing to Alexandria, Egypt, where it was gifted to Ptolemy III and renamed Alexandreia.

Archimedes also focused some of his time on understanding how levers work, which became a treatise called On the Equilibrium of Planes. The lever had existed for sometime, but its principles were not as heavily studied until this time. His studies on leverage lead him to design a block-and-tackle pulley system that we still use today, which was designed to allow sailors to load and unload materials that would have otherwise been to heavy to move. This shipboard technology eventually evolved into the crane, such as the crane used the lift the command module aboard the USS Hornet.

Aside from his inventions, more of which I'll talk about in a few minutes, he invented a method of using infinitesimals in a way functionally similar to that of integral calculus, which wasn't to be invented until 17th century by Newton and Leibniz. Using a technique called the method of exhaustion, he was able to accurately calculate values that either couldn't be calculated before, or couldn't be calculated with accuracy, such as the value of pi, or the value of the square root of 3.

His calculation of pi, described in his treatise, Measurement of a Circle, involved drawing a polygon around a circle, and then calculating the length of the sides of the polygon, then doing the same with a polygon drawn inside a circle. The value of pi would then fall somewhere between the two values. By making a 96-sided polygon, he was able to calculate the value of pi to be between 3 1/7 and 3 10/71, or between 3.14286 and 3.14085. The true approximate value of pi is 3.14159, which is within 0.02% of the lower limit, and within 0.04% of the upper limit. The average of the two numbers is 3.14185, placing Archimedes approximation within 0.008% of the true value, a remarkably accurate estimation for the period, especially when you consider the Phoenician estimate of pi as 3 only 300 years prior. This method for calculating pi was the only one employed until the 17th century. An approximation for the value of pi was programmed into the Apollo Guidance Computer and was used for many parts of the flight, including the computer-controlled reentry.

In the same work, he gives the value of the square root of 3 with high amounts of accuracy, though without explaining how he obtained the value.  He was also able to, in the treatise, The Quadrature of the Parabola, calculate the area under a parabola using an infinite geometric series.

The work he was most proud of, On the Sphere and Cylinder, describes the volumetric relationship between the sphere and the cylinder. In it, he was able to discover that, if there is a sphere and a circumscribed cylinder of equal height and diameter, the sphere's volume is two-thirds the volume of the cylinder and that the sphere's area is two-thirds that of the cylinder.

By the late third century BC, Syracuse was at war with the Roman Republic, whose expansion into Magna Graecia was in full swing. During the First Punic War, Archimedes increased the power and accuracy of the catapult. During the Second Punic War, Archimedes developed a heat ray, which consisted of an array of mirrors pointed out to sea, which, under the right weather conditions, would cause the Roman ships to burst into flames, or at least burn. This defensive method was supposedly attempted during the Siege of Syracuse from 214 to 212 BC. Since then, several teams of researchers (and MythBusters) have tried to replicate Archimedes' heat ray with varying degrees of success. It is more likely that if it was tried against the Roman navy, it would have served more to blind and confuse the attacking Roman soldiers. It is also said that a device called the Claw of Archimedes was developed, a large crane-operated hook that could literally grab a ship and pull it out of the water then drop it back down, causing it to break and sink.

In the end, the Romans managed to enter the city of Syracuse while the inhabitants were distracted, and the city was sacked. Roman proconsul Marcus Claudius Marcellus, who led the Roman forces during the siege, ordered that Archimedes be spared, knowing of his usefulness as an inventor and mathematician. Archimedes was working on a mathematical problem when he was encountered by a Roman soldier. Unwilling to leave his work until it was finished, the soldier became enraged, and slew Archimedes with his sword. He was 75 years old.

His tomb was built according to his request, which included a sculpted sphere and cylinder to represent his discoveries in the treatise, On the Sphere and Cylinder. During the sack of Syracuse, the Romans did discover one last invention: an analog calculator that showed the position and movement of the seven classical planets: the sun, moon, Mercury, Venus, Mars, Jupiter, and Saturn.

After some time, the tomb of Archimedes was lost. In 75 BC, Cicero was serving as questor of Sicily. Of his own accord, he uncovered the tomb and had cleared of the overgrown foliage. Since that time, the location of the tomb has once again been lost to history.

Next time, we will examine other scientific advancements made during the Hellenistic period before it ended in 31 BC, including the first known computer: the Antikythera mechanism.