I love Euclid.
Most of the benefit of my year at St. John's was social rather than academic, but the first semester of freshman math was a joy. Second semester was Copernicus and Kepler and Ptolemy, and astronomy using the geocentric model, which I couldn't bring myself to care about. But for the first few months, all we did was work our way through Euclid's Elements, proposition by proposition.
Elements was written as a textbook for ancient Athenian schoolboys. It starts with definitions of concepts like point, line, plane, right angle, and so forth. Working from these postulates, proofs are constructed, each of which builds upon the last. So Book I, Prop. 1 shows how to construct an equilateral triangle on any given straight line. Now that we've proven it can be done, that proposition goes into our mental toolbox, and can be used whenever handy on subsequent proofs.
We met two or three times a week, and were assigned two or three proofs for every session, or sometimes just one if it was a real bitch. We didn't have to solve them, exactly, since the answer's right there on the page -- we had to understand how and why they worked and be ready to demonstrate them on the blackboard. Even though we weren't doing anything even remotely new, I got a real thrill of discovery every time I grasped a new one. I felt richer, as though I had gained something of real value.
Of course, nobody actually gives a shit if you can demonstrate Euclid III.16. But it's fun, anyway.
Elements was written as a textbook for ancient Athenian schoolboys. It starts with definitions of concepts like point, line, plane, right angle, and so forth. Working from these postulates, proofs are constructed, each of which builds upon the last. So Book I, Prop. 1 shows how to construct an equilateral triangle on any given straight line. Now that we've proven it can be done, that proposition goes into our mental toolbox, and can be used whenever handy on subsequent proofs.
We met two or three times a week, and were assigned two or three proofs for every session, or sometimes just one if it was a real bitch. We didn't have to solve them, exactly, since the answer's right there on the page -- we had to understand how and why they worked and be ready to demonstrate them on the blackboard. Even though we weren't doing anything even remotely new, I got a real thrill of discovery every time I grasped a new one. I felt richer, as though I had gained something of real value.
Of course, nobody actually gives a shit if you can demonstrate Euclid III.16. But it's fun, anyway.