On Loving Geometry

There is a strange kind of book originating in the ancient Greek world and found in no other of the great ancient civilizations: the geometry book, as exemplified by those of Euclid, Apollonius, and Archimedes. I emphasize the word book: other cultures later had loose collections of mathematical insights, or mathematical writings geared to the solution of specific problems, but from Euclid on the mathematical book of the Greeks and Romans was conceived as a whole, with narrative threads laying out questions and exploring answers, and with sometimes dizzying climaxes that could make one see all that went before in a different light. Such books would build upon a small set of axioms, and then by inference and imagination reach conclusions about simple but very interesting objects. They would not be primarily interested in solving practical problems; for example, Euclid’s preoccupation with perfectly regular solids such as the dodecahedron and icosahedron, and with perfect numbers as well as the many kinds of irrational numbers, seems mainly motivated by love of beauty and by the satisfaction of understanding. Before the blossoming of applied algebra in the 16th century, when for example parabolas acquired utility in the calculation of the trajectories of cannonballs, geometers had happily studied the subtleties of conic sections for two thousand years without needing to find uses for them. It was enough to contemplate, behold, wonder, question, and understand. The geometrical endeavor must have spanned the whole of the Mediterranean: Euclid lived in Alexandria, Apollonius in what is now Turkey, Archimedes in Sicily.

In ancient India and China there was no book like Euclid’s Elements or Apollonius’ Conics. I do not say that to disparage those civilizations, because of course they also had their unique genres of book. Greece, India, and China all had philosophical dialogues, treatises, anthologies of poems, collections of aphorisms, dramas, medical studies, and biographies. China and Hellas had great books of  histories, India did not; Hellas and India had epic poems, China did not. But only Greece had the geometry book — and moreover, not only was the geometry book held as central to education (so central that the study of geometry could be considered essential for philosophy) but it became, and remains, at the heart of the civilization. People still study Greek geometry for delight and mental refreshment. The great 20th century mathematician Hardy remarked in his autobiography that the continuing study of the Greeks was like going to the college next door to see what the Fellows there were currently working on: for a mind that is mathematically alive, Euclid remains ever contemporary and a bottomless well of insight.

I am a mediocre mathematician who discovered the joys of geometry late in life, thanks to my work at an unusual liberal arts college where Greek geometry is a vital part of the culture for all students and faculty members. In this short essay I want to celebrate a few of the delights of one particular masterpiece of Greek geometry, Apollonius’ Conics, one of the greatest mathematical works of all time but also one that few people will ever read — understandably, because the book is hard and requires familiarity with all thirteen books of Euclid. However, the delights of Apollonius are not essentially mathematical, but can be felt by a student who knows the delights of poetry, drama, and philosophy. To me they have the kind of playful, dreamy, yet rigorous abstraction of a story by Borges or Calvino. In the following few examples I am going to abbreviate and simplify to get to what interests me, but if the mathematical exposition starts to get tedious just skip to my conclusion.

To get a conic section, you cut a cone with a plane. The line made by the intersection of the surface of the cone with the cutting plane is a section. If you cut the cone through the vertex,  the section will be a triangle; if parallel to the base, a circle; if through both sides of the axial triangle (such as ABC in the following diagram, where BC is the diameter of the base circle) but not parallel to the base circle, an ellipse; if at an angle not parallel to the base circle or to one of the sides of the axial triangle or cutting both sides of the triangle, a hyperbola; and if parallel to the side of the axial triangle, a parabola. This is an oversimplified summary, but for the purpose at hand I’d like to dwell on proposition 1.11, where Apollonius shows us how to get a parabola.

In this cone there are three planes of interest: the one making the axial triangle ABC, the plane of the base circle, and the plane DFE that cuts the plane of the base circle along the line DE perpendicular to diameter BC. The line FG is the common section of the plane of the axial triangle and the cutting plane DFE, and is the diameter of DFE because all lines drawn across the section parallel to DE will be bisected by FG; thus the section will be symmetrical. It is necessary that FG be parallel to one of the sides of the axial triangle, here AC. Because of this, if FG were continued past A it would go on forever, and if the cone were continued indefinitely at the bottom the section DFE would also expand indefinitely. This section is a parabola (literally, “along-thrown”).

Apollonius is not content merely to cut a cone in this manner to get a new curve, because as I have described it so far this curve is unintelligible and cannot be related to anything we know in any specific way. What would it mean for a curve to be made intelligible? To understand this we need to see clearly that all the conic sections — triangle, hyperbola, parabola, ellipse, and circle — are made in the same way, by cutting a cone. Therefore they have something in common. We know a lot about circles from Euclid, and one of the things we know is that if we drop a perpendicular from the circumference to the diameter the square on that perpendicular will equal in area the rectangle made by the two parts of the diameter cut off by it. In the diagram above, because lines KL and MLN lie in a plane parallel to the base circle, this plane will itself be a circle; and the square on KL will equal the rectangle made by ML and LN. This is a remarkable defining property of circles and expresses something crucial about the way circles curve. The line KL, because it is bisected by the diameter MLN, is called an ordinate to the diameter of the circle; but as we can see in the diagram it is also an ordinate to the diameter FG of the parabola — and therefore is the means by which Apollonius relates something we don’t yet understand, the parabola, to something we  understand a little, the circle. Just as in the circle the square on KL equals the rectangle formed by ML and LN,  in the parabola the square in ML is equal to the rectangle formed by FL (which is analogous to ML) and…what? He demonstrates that if you set up a mystery line FH such that  FH:FA (line between vertex of parabola and vertex of cone):: square on BC (base of axial triangle): rectangle made by AB and AC (sides of axial triangle), then the rectangle made by FL and FH will equal the square on KL — and this is one defining property of a parabola. It is noteworthy that FH is not arbitrarily concocted, but seems to be determined by structural relations in the cone. He does something analogous with hyperbolas and ellipses, so that now the conic sections all can be related to each other through specific proportions.

These proofs come early in the Conics, and the soup will thicken considerably and gain in flavor as it simmers. But even though we are not very far in the game, the mind is stirred by the beauty of the conception: these different curves can be made intelligible, because they all follow a kind of law that can be related back to the laws of the circle — and, moreover, may be variations of those circular laws. Does the fact that they are all obtained by slicing a cone suggest that underlying their apparent differences they might in fact be the same — obeying laws not accessible to sense but to the intellect alone? Later in the book we even abandon the cone altogether and reach a point where we are able to discuss a conic section without needing to specify which one it is. And all these sections come from something as idiotically simple as a cone, which is basically a hybrid of circle and triangle, and thus partakes in both straight and curved: paradox in simplicity.

A friend of mine regularly issues gasps of amazement that a curve should have any symmetry at all: why should it? — and yet Apollonius proves that these conic curves all have diameters. Moreover, just as a circle has an infinite number of diameters, so do the other sections. In 1.46 Apollonius demonstrates that in a parabola a line parallel to the original diameter is another diameter, bisecting all lines drawn to it at a given angle. In the following diagram, where AD is the original diameter, HM is the new one, bisecting all lines drawn parallel to the tangent CA, such as LF:

Shockingly, it turns out that not only are these curves symmetrical, but they are infinitely so! Yet the symmetry here is not the same as the symmetry in a circle: there, you can fold the figure along the diameter, and both halves will coincide; here, that will happen only along  one diameter, but along all the others the folded halves will lie at a subcontrary angle and yet be equal in area. To encounter such things and understand — not see — that they are so is as amazing as standing on the rim of the Grand Canyon: an endless landscape of patterns opens out before one. And to encounter it in something as simple as a cone is as wonderful as watching water boil or frost crystals take form.

The cone is full of mysteries. One recurrent one is that two forms as seemingly opposite as ellipse/circle and hyperbola — one closed, the other endlessly open — might actually be the same: that is, spatially they seem opposites, but noetically they are identical. Thus in 1.21 Apollonius demonstrates that if you cut the diameter at points E and G, as in this diagram,

then the square on FG : square on DE :: rectangle made by AG and GB : rectangle made by EG and GB ( that is, rectangles made by the lines from the cutting point to each end of the diameter). The big miracle in this proof is that Apollonius’ words apply simultaneously to hyperbola and ellipse/circle; in his argument, it does not matter that these shapes look like polar opposites. How can it be that two things so evidently different can also be so evidently identical?

Greek literature and philosophy are rooted in the dilemma, the insoluble contradiction of the soul, the knot of life itself. In dramas and epic poems, the heart of the conflict is Right versus Right — Hector and Achilles, Odysseus and Ajax, Antigone and Creon — because Right versus Wrong is not only uninteresting but also inadequate to our experience. The philosophical masterpieces are also centered on the dilemma: in the Republic, justice or eros?  Just as the principal strategy of Socrates is to bring us to aporia, the knowledge that  we are well and truly stumped, by exposing the contradiction in two lines of our thinking, so Apollonius and Euclid both rely on the reductio ad absurdum, in which two lines of thought are made to collide. The essence of dilemma literature is the laughter and despair of perplexity. This is generally not so in the philosophies and literatures of civilizations that seek useful ends, even noble useful ends such as the cessation of all suffering; but even in those civilizations there are always the great dissidents who stand up for the play of perplexity and for the relishing of what cannot be solved, which will necessarily be useless. Zhuangzi would have poked fun at the good-for-nothing tree of the Conics and yet thoroughly enjoyed lying under it and marveling at its branches.

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