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Triumph of the Greeks

Wednesday, April 18, 2018

Daniel South has been studying the history of Greek mathematics. Having completed calculus BC, Junior Daniel South is currently studying Advanced Topics Mathematics, including proof writing, linear algebra, and math history. He wrote the following on Greek mathematics for his independent study. Daniel also qualified this year for the United States of America Mathematical Olympiad (USAMO)​, a highly selective high school mathematics competition held annually in the United States. Roughly 260 students qualify per year to take a six-question test that lasts nine hours. 

Some of the most important mathematical results that we know and use today originated in ancient Greece. Most famous is the Pythagorean theorem, which, though perhaps not originating in Greece, was named after and proved by Greeks. The famous number π which we celebrate on March 14 was approximated by Archimedes, a Greek. Greek mathematics is so amazing because it provides a basis for modern thought which is still applicable today.

With very limited tools, Greek mathematicians established amazing results. For instance, Hippocrates (a different one) showed how to construct a square with the same area as a crescent-moon shape called a lune. What’s so remarkable is that his complex proof only uses circles and straight lines, the building blocks of geometric construction. Other mathematicians including Euclid and Archimedes constructed triangles, hexagons, dodecagons, and even 768-sided figures using only a compass and unmarked ruler! Eratosthenes also managed to calculate the circumference of the earth using only the sun and a stick.

Greek mathematics encompassed not only geometry but also number theory, the study of whole numbers. They had a firm grasp of prime numbers, which are the foundation of any number system, including Eratosthenes’ sieve for finding primes. Notably, Euclid cleverly proved that the prime numbers are infinite. This is a hugely significant result, which led to more questions. For example, twin primes, like 5 and 7, or 11 and 13 are only 2 apart and both prime. It is still unknown today whether there are infinitely many twin primes!

Perhaps the most remarkable achievement of the ancient Greeks was the first mathematical textbook, Euclid’s Elements. It not only provided stunning results, including a beautiful proof of the Pythagorean Theorem, but did so using only five basic facts. These facts, unproved but taken as axioms, provide the foundation for all the geometry we know. Euclid’s Elements is second only to the Bible as the most popular book of all time. In fact, famous figures such as Thomas Jefferson studied this book. In writing the Declaration of Independence, Jefferson referred to “self-evident” truths, which are known as axioms in the mathematical world. Jefferson created the axioms of democracy, just as Euclid did to geometry. The Greek mathematicians affect us today in more ways than we can realize.