Angles that Euclid Would Know

Next up on the list, as promised, is Euclid's Elements of Geometry. This is a dated but still definitive translation of Euclid's foundational work on Geometry done by Sir Thomas Little Heath. This particular edition is a slimmed down version his translation with the notes removed so that it can fit in a single volume. This volume is used at St. John's College and at a few other places with Great Books programs since Heath's full three-volume edition is considered too cumbersome. The same publisher, Green Lion, also provides a further condensed pocket version that has only the propositions and diagrams but no proofs called The Bones. If you would prefer all of Heath's unabridged commentary on top the translation, there is also a three volume paperback edition from Dover. You can also find the three volume version online. However, Heath's commentary adds substantially to the work's size and provides unnecessary distraction for the first time reader. That said, now that I've read through The Elements, I am interested in seeing what Heath had to say.

Euclid's Elements provides a fascinating look into a world of math without numbers. It is like learning some strange and beautiful alien language. To figure out this alternate system of math, I found it very helpful to buy a compass and straight edge so that I could replicate Euclid's geometric constructions. Euclid does not calculate. Euclid draws and "measures". I put "measures" in quotes because, once again, there are no numbers. The scale is completely arbitrary. A 5" circle and a 5' circle are identical in Euclidean geometry. The only thing that changes is the basic unit of measurement. Some operations are easier in this system. Others are harder. Cutting a line exactly in half or doubling it is very simple with a compass and straight edge. It is not, however, very useful for anything involving non-flat planes. Still, it should give the diligent reader a completely different way of looking at many areas of math.

Euclid is credited with being the origin of mathematical proofs. While it is true that he uses rigorous proofs in every proposition, it is inaccurate to saythat most modern proofs are quite on the same level as Euclid's. First, Euclid builds his whole geometry piece by piece. He starts with a few definitions and propositions and then builds from there. Most proofs depend on previous proofs. It's a self-contained system. Second, and more importantly, Euclid's focus on geometry means that he is not limited to just proving things mathematically or logically. No, Euclid is able to physically construct his objects on paper, in wax, or in the sand. In other words, much of what Euclid says can be tested empirically. Physical construction is a degree of proof beyond what symbols and numbers can provide.

There are many modern mathematicians who find it a remarkable coincidence that so much of the universe appears to be "mathematical". But when one reads Euclid, part of the earliest foundation of math, it is clear that these mathematicians have the arrow of causality backward. Mathematics is primarily descriptive. Its original purpose was to count, measure, calculate, and model very real things. And, historically, any math that did not mirror reality was not valid math. Reality was the arbiter of all things, no matter how much some new math may appear to be internally consistent. To say that the universe is mathematical is like saying that a beautiful landscape or portrait has the qualities of a painting. The model is not the thing, but we do not keep around defective models.

In a similar vein, there are those who claim that geometric forms and proportions show up in art, architecture, and nature as some sort of microscopic reflection of the universe's true nature. In the case of art and architecture, the fact that symmetry and ratio are pleasing to look at has been well-known and established since antiquity. While today these methods have become something of a lost art that people stumble upon intuitively, Ancient, Medieval, and Renaissance architects and artists exploited them knowingly and deliberately. As for nature, it's more a matter of things tending toward simplicity and equilibrium, in the long run. However, things tend to be much more chaotic and less clearly mathematical in the short term or on small scales. This wide-spread tendency may actually point to some fundamental truth about our universe. I have no way of knowing. However, for those interested in how natural things tend to shift between the ugly and chaotic to the beautiful and orderly depending on scale, I highly recommend D'arcy Thompson's On Growth and Form. Thompson was, incidentally, a friend and colleague of Heath's at Cambridge.

One other thing struck me while reading Euclid's discussions of ratio. We are so obsessed with numbers these days that it is easy to miss that numbers really have two functions: to count and to measure. These are related but not quite identical things. With regard to counting, the numbers correspond to the fact that objects in the real world are often discrete things, either existing or not existing. There is a natural and obvious connection. However, with measurement, our units are entirely arbitrary. Ratios can be obscured by simply choosing the wrong size for the unit when represented by numbers. For example, the ratio between one meter and three meters is immediately obvious. On the other hand, the ratio between 3.281 feet and 9.843 is not quite so immediate. In Euclid's realm of arbitrary units, there is no real distinction. 1s are simply not equal if the units discussed are different. But one apple is numerically equivalent to one orange. We have combined two different things into a single idea of "number". This is why we end up with the strange distinction between "natural numbers", "rational numbers", "real numbers", and "irrational numbers". Even computers have to make a distinction between "counting numbers" and "measuring numbers", i.e. integers and floats, for performance reasons. We are forcing numbers to do more than one job. Our system definitely works and I am no mathematician. However, I can not help but wonder if there is anything we have failed to see because of our blurry distinction between the uses of numbers.

Did I mention that Euclid can also help you form geometrically perfect pentagrams for all your demon binding needs?