A Greek mathematics lesson

[We know a great deal about ancient Greek mathematics but very little about the Greek mathematicians who produced it. In his text, the Elements, Euclid organises an extraordinary amount of beautiful mathematics into a deductive wholef, but we can only guess how he expected students to learn from it.

We do have one passage from Plato, the philosopher who founded the intellectual school to which Euclid belonged. It takes the form of an imaginary dialogue between Socrates (Plato's teacher), Meno (a young aristocrat) and a slave. (I shall use the translation [180] by W. K. Guthrie in the splendid Penguin Classics series.) The reader may sometimes wonder how we can ever understand anything new in mathematics. For if we can understand it, then it cannot be essentially new, and if it is essentially new, we have no way of understanding it. Meno repeats this argument in a different context to show that we cannot learn the nature of virtue.]

MENO But how will you look for something when you don't in the least know what it is? How on earth are you going to set up something you don't know as the object of your search? To put it another way, even if you come right up against it, how will you know that what you have found is the thing you didn't know?