ABSTRACT
A few years ago, on a bitterly cold Alberta winter day, near enough to winter solstice that the sun was very low up here, at 52 degrees North Latitude, I (D.J.)1 was out on an elementary school playground with a 12-year-old boy. We had just been inside in a classroom of around 60 students, quarreling in lovely, heated ways, about dropping perpendiculars and bisecting angles with only compasses, pencils, and straight-edges in hand (see chaps. 1 and 5). I was still reeling a bit from the moment at which a group of students turned their page of work on angle-bisection sideways and discovered that the problem of dropping perpendiculars was already somehow solved. Even more disorienting, one girl suggested something that I had never exactly imagined before: that dropping a perpendicular was basically little more than bisecting a 180-degree angle. As soon as she said this, I knew that I had already heard this before, years ago, in my own schooling, but it had, as has so much of those times, faded into forgetfulness and irrelevance.
