Euclid, the famed Greek mathematician and pedagogue, once had a young pupil (most likely one of the several Ptolemys of Alexandria) who asked him if there was any quicker way to learn geometry than to read the *Elements*. Euclid is fabled to have told him, ”οὐκ ἔστι βασιλικὴ ἀτροπὸς ἐπι γεωμετρίαν,” or “there is no royal road to geometry.”

About some 2000 years later, give or take some centuries, D’Alembert, a famed French mathematician, told one of his pupils who was having difficulties understanding the “vanishing and nascent quantities” of the calculus, “Allez en avant, la foi vous viendra,” or “Forge ahead, the faith will come to you.”

I think Calculus is the royal road to geometry. Anyone reading through the *elements* or apollonius’ *conics* will see very soon how greatly it has abbreviated every part of geometry. Sometimes I wonder what has been lost, though. If you liked Calculus, I highly recommend the textbook, *Calculus: the elements*, by Michael Comenetz, from which I stole those epigraphs.

Newton writes in his preface to the *Principia Mathematica* that what he describes there (a strange kind of geometrically-based Calculus), is “rational mechanics,” and that geometry itself is founded upon the axioms of this mechanics, For instance, he gives as an example, the drawing of the circle is a postulate of euclidean geometry, but mechanics can describe how to draw a circle, and so on.

Archimedes found the area under a segment of a parabola by weighing it. Perhaps Calculus is a way of weighing functions.

That is a freaking awesome question to ask on an exam. What did you answer?