Euclid’s Point

Monday morning dawned bright and early for the New England High School Summer Program students! After a rainy day of rest and relaxation, they were ready to bounce back into the classroom immediately after breakfast. In their one and only class of the day, students began studying Euclid’s Elements, the seminal text of Western mathematics. Today’s slender reading assignment covered Euclid’s definitions, postulates, and common notions, together no more than a few paragraphs. They seemed simple — almost too simple, some thought.

In this short reading, Euclid defines simple geometric objects such as lines and triangles, and he lays out some basic operating principles, such as claiming that “the whole is greater than the part.” With these few assumptions as a foundation, he builds a whole geometry such that each argument — called a “proposition” — follows indisputably from the previous ones. As he goes along, his propositions get more and more complex, slowly constructing the beautiful and intricate work that won Euclid his luminous reputation. Tomorrow, students will begin demonstrating some of those propositions…

… but can you have a good conversation about a few paragraphs on the definitions of basic concepts such as lines and triangles, or innocuous claims about the whole and the part? You’d be surprised — the students certainly were! They poured from classrooms still trying to wrap their heads around what they had discussed. The tutors’ questions slowly revealed that the definitions, which seemed so simple on paper, are actually quite mystifying. For example, Euclid defines a point as “that which has no part,” and a line as “breadthless length.” But what do those statements really mean? What does it mean to have “no part?” Is a line made up of points, or is it something else entirely?

“I was surprised!” exclaimed student Vitor S. “I read it thinking it was simple geometry, but we had an amazing conversation!”

After midmorning Mass, students headed to the Gould Commons for a lunch of orange chicken, rice, potstickers, and broccoli, making sure to eat their fill before their afternoon outing. They then raced to their residence halls to change into swimsuits and sandals, thrilled to head to a nearby lake for an afternoon of sunshine and kayaking.

Come back to the Summer Program Blog tomorrow morning to read about our students’ adventures on the water, as well as their evening practicing their first Euclid propositions!