Greek Mathematics: Calculating the Circumference of the Earth

Even if you've only taken high school geometry, you've probably heard of Euclid. He is responsible for some of the most fundamental and foundational pieces of modern geometry, among other topics. In many ways, Euclid's work The Elements forms the basis for all modern mathematics, not just geometry.

About Euclid himself little is known. According to Proclus, one of the only surviving Greek mathematical authors, Euclid worked at the Library of Alexandria under Ptolemy I, sometime between 323 and 285 BCE. However, since Proclus was writing in the 5th century CE, this information is somewhat circumspect. What is lacking in knowledge about Euclid himself is made up for in part by his works, which we have translations of many because they were considered so important. Some have even argued that Euclid's Elements is the most-published work of all time, with over a thousand editions in various languages.

The Elements is often considered a geometry textbook, but this is a simplification. Students have used it to learn for hundreds of years, but its ten books contain results far outside the scope of geometry. At the time, numbers weren't considered in the abstract, so results about multiplication, fractions, and other properties of numbers were expressed as geometric principles. In addition, Euclid's Elements draws heavily on several previous works, now lost. It is as much a compendium and history of the known mathematical knowledge of the time as it is anything else.

However, the most important thing about the Elements is the way in which Euclid presented the mathematical results within it. Instead of presenting real-world problems and solutions that the Greeks were using, Euclid focused on general results. In addition, he presented every result as a logical conclusion based on solely statements that were already known to be true, either because they had already been proven or because they were assumed to be true. The five statements he assumed to be true are known as Euclid's postulates, and all attempts to prove them based on other statements have failed. It truly seems as though there is no way to establish their truth without assumption.

The idea of proving general results based on known statements may seem obvious to modern mathematics students, but this idea was quite new in classical Greek times. Most of the older mathematics we have could best be considered "special cases" of various problems - a solution or two was known, but the underlying mathematical relationships were still completely obscure. Euclid's work is the oldest example we have of someone clearly laying out not only results, but how he got to them and why that method always works.

The preceding sections discussed people who dedicated their lives to mathematics, but math played an important part in Greek life more generally as well. Sundials were likely invented as early as 3500 BCE, but they became increasingly common in the 3rd century BCE.

The Greeks made several different types of sundials, but they all relied on the properties of a sphere. As you can see at left, the center piece (called a gnomon) cast a shadow onto a curved surface. When the sun is higher in the sky, the shadow falls deeper in the bowl while when it is lower (such as in the winter) it will be closer to the rim. This works well, since the total amount of daylight and the length of the path traveled by the shadow both get shorter in the winter. This allows the 12 solar hours of the Greek day to be demarcated easily by straight lines.

The third image shows possible markings for path of the sun on the summer and winter solstices. Lines going from the gnomon to the edge demarcated the hours.

In addition, mathematics was used in all sorts of other fields, such as architecture and astronomy.

Eratosthenes may have used sundials or wells to do his experiment, but neither are particularly common. Jake walks through how he and Miriam attempted to replicate Eratosthenes' experiment and find the circumference of the Earth.

This is a great video from ViHart, a math YouTuber. In it she discusses Pythagoras, another famous Greek mathematician. He may have codified the Pythagorean theorem, but he was also afraid of beans and irrational numbers (perhaps irrationally so).