Mind The Gap: A Geometrical Mystery

If I delight in Greek geometry and even study it for fun, it is not because I have a good mathematical mind or because I have a temperament that enjoys solving problems and constructing complex figures. The opposite is true: I am slow at mathematics; with me even the most basic mathematical insights are earned through painstaking labor. I am also not a natural problem solver. Some of my friends obtain deep satisfaction from solvng problems: you can see it in how their faces beam and their eyes sparkle when they complete a difficult jigsaw puzzle, or when they succeed in calculating the force needed to throw a basketball through a hoop on a planet the size of Jupiter. I, on the other hand, might be able to reach the same solutions if given enough time, and if I solve them I feel at most a mild satisfaction, but nothing strong enough to make me want to do it again. I was never good at puzzle books, whodunnits, or chess — where the problem-solving is at least spiced with competition.  Of course if I had a life-or-death reason for ascertaining how much gunpowder I’d need for my 97 cannons to hit an enemy target  500 yards away with ten-pound cannonballs, then an accurate calculation might be satisfying — but still not as satisfying as actually beating the enemy. Nonetheless, the delight I take in geometry is not for its usefulness to personal goals and triumphs, but rather for the almost philosophical, almost poetic riddles it confronts me with — and in places where I least expect to encounter the outer edges of my own mind. Here are two examples.

The third book of Euclid’s Elements is about that apparently simple figure, the circle. In Proposition 16, Euclid shows that in a given circle with diameter AB, a straight line EA drawn at right angles to the diameter at A will fall outside the circle (that is, be a tangent to the circle at A),

(Diagram credit: Green Lion Press edition)

and that between the circumference and EA no other straight line can be interposed. In real life this makes sense: if you have a wheel on the ground, and you try to push a flat stick into the space between the wheel and the ground, the stick will eventually bend under the wheel — in other words, to fit between the wheel and the ground it cannot remain straight. In Euclidian geometry, however, this is a conundrum: a line is defined as a breadthless  length, and right up to the moment (which “has no part”) when circle meets tangent at a single point there is always some breadth or space — so surely a breadthless length will be able to fit into some amount of space? How could a breadthless length not fit in there? A Euclidian geometer would not use a word like “infinitesimal” because such a thing cannot be directly conceived, but even if he did, a breadthless length could still fit into an infinitesimally tiny space. Euclid will go on in this proof to say something about the angle CAE, made by the tangent EA and the circumference of the circle, and the angle CAD, made by diameter BA and the circumference of the circle — namely, that the latter is smaller than any angle made by two straight lines, and the latter is greater than any angle made by two straight lines that is smaller than a right angle. The first time we read this our naive reaction is to wonder how this thing that is a fusion of curved line and straight line can be discussed intelligibly as a magnitude that can be measured or compared. Thus what goes on in this hole, this gap between straight line and curved line, becomes very mysterious: there is something here that we can’t think.

In Book 2 of Apollonius’ Conics, there is a similarly striking moment. He has shown that two straight lines, called asymptotes, can be constructed that will flank a hyperbola and never touch it. In the following diagram these are AM and AB. He is then presented with a short, even minuscule, magnitude K: as we go further down the hyperbola and its asymptote, can we find a distance between them smaller than K? — no matter how microscopic K might be. Apollonius shows, building on previous propositions, that on any line EF drawn parallel to a tangent it will always be possible to cut off a length EL smaller than K, and a line drawn from L parallel to AM will meet the curve. Thus, no matter how small K is, even “infinitesimally” small, we will always fit between curve and asymptote an even smaller line.

(Diagram credit: Green Lion Press edition)

We moderns would say something like this : that as hyperbola and asymptotes increase infinitely, the gap getween them becomes infinitesimally small — but a limit or end-point can never be reached. Again, the Greek geometers would never use words like “infinite” or “infinitesimal,” because they were more honest: such fancy words make no sense. Instead, they express the question operationally: is it possible to fit a smaller straight line in there or not? The mystery is encountered: we find this thing we cannot think — the gap between asymptote and hyperbola, way down there. As with Euclid 3.16, we cannot grasp what happens at the end of the hole, where a curve has the audacity to meet a straight line. But isn’t it wonderful that we can, with our own puny minds, uncover the unthinkable and know — not just religiously assert — that it is unthinkable?

 

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