# X-Ray Vision and Ancient Math

So, for the past 11 days, scientists at Stanford University’s Linear Accelerator Center have been using powerful X-rays to read as many as 15 pages of Archimedes’ work that no one has read in nearly 800 years. A medieval monk, not realizing that he had in his hands the only known copy of several of Archimedes’ texts, erased the one-of-a-kind pages around the year 1229. He scraped off the ink, cut the parchment in two, and used it to record prayers. Some 20th-century forgers later erased more pages, when they painted over them to make the manuscript look more valuable.

Despite this mathematical mutilation, the scientists’ X-rays – a million times more powerful than the ones used to see your bones – can still illuminate what’s left of the ancient ink. The technique is called X-ray fluorescence, and scientists say that it’s allowing us to re-read “one of the greatest figures of western civilization.”

We don’t have a superpowerful X-ray, but we can strap you into a time machine for a little visit with the old math Greeks themselves. Meet them and their mathematical achievements:

Pythagoras and the Triangle of Truth

Pythagoras used math mainly to make a point about religion. Not much is known about his life, except that he was born around 580 BC on the island of Samos. After early philosophical study, sources say he traveled in Egypt and Babylonia, learning what he could of their science and math. Later, he founded a religious school in the city of Croton in southern Italy. There he began preaching that numbers are the building blocks of existence.

To prove his point, Pythagoras and his followers examined numerical phenomena closely. Along the way, they discovered that a triangle’s angles always add up to 180 degrees. They also discovered that, in a right triangle, the square of the hypotenuse (the long side) is equal to the sum of the squares of the other two sides. That, of course, is the famed Pythagorean Theorem.

Pythagoras’ favorite discovery, though, might have been the numerical basis of musical harmony, as harmonious arrangements were critical to Pythagoras’ thought. If you take two harp strings of equal thickness, a ratio in length of 2:1 will produce the musical interval of an octave (if the strings are held under equal tension). A ratio in length of 3:2 will produce the musical interval of a fifth.

Euclid and the Timeless Textbook

As little as we know about Pythagoras, we know even less about Euclid. We know that he lived around 300 BC, probably studied in Athens at the Academy founded by Plato, and worked in the Library of Alexandria. Otherwise, all we know is that he wrote one truly excellent textbook – so good that he’s now called “the father of geometry.”

Euclid’s textbook, the Elements, was in many ways unoriginal. He organized and systematized the work of previous mathematicians, filling in holes where he found them and arranging geometrical concepts and proofs from simple to complex. He did his job so well, though, that afterward no one bothered making copies of those earlier geometrical works, which are now completely lost. Euclid’s Elements was so influential that we now use the phrase “Euclidean geometry” to refer to both two-dimensional (or “plane”) geometry and three-dimensional (or “solid”) geometry.

Archimedes and the World-Lifting Lever

Pythagoras and Euclid were great, but Archimedes was greater. In fact, he’s often cited as one of the three greatest mathematicians of all time (along with Isaac Newton and Carl Friedrich Gauss). The son of an astronomer, Archimedes was born around 290 BC in the Sicilian city of Syracuse. He became famous in his day for inventing marvelous machines. For Archimedes, though, such machines were useful mainly as a means to discover mathematical theorems.

Archimedes worked in practically every area of mathematics. His bathtub study of buoyancy is the foundation of modern hydrostatics (though the story about him running naked down the street shouting “Eureka!” is likely embellished). His work on determining the area of shapes bounded by a curve anticipated modern calculus by 2,000 years. He did ground-breaking work on pulleys and levers, allegedly saying, “Give me a place to stand and I will move the world!” He’s even credited with inventing the “Archimedes screw,” a device for raising water still used in various ways all over the world.

Unfortunately, much of this came to nothing. Archimedes died in 212 or 211 BC, when the Romans sacked Syracuse (though his war machines made things difficult for Rome’s legions). Later generations mostly forgot he had ever lived. The West rediscovered his genius only centuries later, via Arabic translations of his work.