"Why do we have to learn this?" is a question so frequently heard in the required Latin classes in the Middle School that we Classics teachers are forced to confront the issue. My response lately has been to ask the question, "why this curriculum at all?" Students often point to mathematics as the most useful and used of our disciplines; however, when was the last time you factored a polynomial expression, found the integral of a sine function, or formally proved that the number of chairs that fit around a table in the cafeteria is not a rational number?
One of the legends that surround Euclid, a figure whose identity is even more shrouded in mystery than Shakespeare's, is as follows: he was asked by a student who had just begun to work through the Elements what good he would derive from his work; Euclid is said to have responded to his assistant, "give the man three obols, since he must profit from what he learns." Cute story, but what does it mean?
A fifth century philosopher and mathematician, Proclus, head of the philosophical academy at Athens, described mathematics as follows:
Mathematical science must be considered desirable in itself, though not with reference to the needs of daily life. If it is necessary to refer the benefit arising from it to something else, we must connect that benefit with intellectual knowledge, to which it leads the way and is a propaedeutic, clearing the eye of the soul and taking away the impediments which the senses place in the way of the knowledge of universals.
When we hold debates about Christopher Columbus in our history classes, people might learn debate more than anything else; when we teach people to synthesize lab data on chemical reactions, they might learn synthesis; when we teach people to sing in harmony, they might learn focus and timing. In each case I say "might" because I grow less sure each year about what people learn here and surer that it is important whatever it is. Content, the "disciplines," seems less significant than people coming together and hashing it out. The Latin word disciplina means merely "training," which is exactly what we are offering. I say only "training" and not "training for..." because I think that to add a teleological tale to our immediate and urgent acts of propaideusis seems presumptuous. We are preparing them for something or other, probably other more than some thing.
The etiology of any curriculum is profound and intricate. We teach with what we were taught and against what we were taught, just as artists must respond to the anxieties of their own influences. With luck, we produce from the only resources we have that which is vibrant, green and compelling to our own age.
Geometry has been taught in its current form even before Euclid wrote The Elements sometime in the third century B.C.E. Euclid lived in Egyptian Alexandria not long after the city was founded by Alexander. The Elements is defended by many as the most brilliant, significant and widely published textbook in the history of the world. It systematized the works of generations of mathematicians before Euclid and, equally important, it was preserved by later generations, passing from the hands of the Greek empire to the pagan Roman empire, the Byzantine empire, the Islamic empire, across North Africa to Spain, to Medieval European universities, to the Colonies. You may find it here cloaked in a modern geometry textbook by Weeks and Adkins.
My aim is to illuminate for our geometry students the source and inspiration for all subsequent geometers, The Elements. However they respond to Weeks and Adkins, they will see, by studying The Elements and its history, how geometry has seemed indispensable to some people for over two millennia.