Euclid's Elements

School begins this month, perhaps this week. Teachers again will introduce students to questions on math, science, English, social studies, and foreign languages. I wish them all well. No subject is easy, but some say math is the hardest.

I remember geometry as my hardest. Theorems spooked me. The following year I found algebra more enlightening. Graph coordinates with x’s and y’s, as well as logarithms seemed to click in place.

Jordan Ellenberg, a math whiz from Wisconsin, just published a new book, Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. 

In an early chapter he describes Abraham Lincoln’s admiration for the Greek mathematician Euclid, who wrote Elements of Geometry, “the most successful textbook of all time,” around 300 BCE, in Alexandria, in Egypt, in northern Africa.

Lincoln’s law partner, Billy Herndon, remembers watching Lincoln struggle for two days to square a circle, “almost to the point of exhaustion.” He “was trying to construct a square with the same area as a given circle, using two tools, a straightedge and a compass.” He gave up. He found the job too difficult.

For centuries, “squaring the circle” has been synonymous with “completing an impossible task,” or like “passing an act of Congress.” Yet, Lincoln tried.

The job involves first finding the area of a circle, using the formula πr². Then, finding the square root of that product will produce the length of each of the four sides of a square. Both the circle and the square will have the same area.

It sounds easy, and it is with computers and hand calculators, but only to an approximation, and the reason is because π is an irrational number. It is 3.141 . . . , without any recurring pattern.

Lincoln though took to heart Euclid’s talent for making sense out of the bewildering. Euclid started with 35 definitions, a handful of postulates, a dozen or more axioms, a series of postulates, and argued for a set of theorems, all about triangles, lines, angles, squares, and circles.

In so doing, he created a complete body of mathematical work, 2300 years ago. 

As a young man, Lincoln was reading a stack of law books, hoping someday to practice law. He said, “I constantly came upon the word demonstrate.” He looked the word up in the dictionary. He read, “Certain proof,” and “proof beyond the possibility of doubt.” But then, “what is proof?”

Finally, he gave up, and said to himself, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means. I went home to my father’s house, and stayed there till I could give any propositions in the six books of Euclid at sight.

“I then found out what ‘demonstrate’ means, and went back to my law studies.”

Lincoln read a lot of Thomas Paine, the American revolutionary writer, who in his The Age of Reason, declared his unwavering admiration for Euclid. 

“I know, however, but of one ancient book that authoritatively challenges universal consent and belief, and that is Euclid’s ‘Elements of Geometry,’ and the reason is, because it is a book of self-evident demonstration, entirely independent of its author, and of everything relating to time, place, and circumstance.”

In Shape, Jordan Ellenberg points out that Euclid understood the “extreme and mean ratio.” Form a line, A to B. At a certain point on that line, between A and B, but closer to B, place a point, called C. If you divide the length of AB by the length of AC, you will get a number about 1.6.

If you divide the length of AC by the length of CB, you also will get 1.6. If the point C was placed correctly, the actual ratio will be 1.6180339887 . . . , another irrational number, but less popular than π. 

Mathematicians since Euclid now call this the Golden Mean, identified by the Greek letter φ, that that pronounce as “fee.” They find this number when they count leaf arrangements in botany, when they measure the swirl of galaxies, and when they look into number theory.

For example, consider the Fibonacci numbers, a series in which the next number is the sum of the previous two numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The ratio of 3 to 2 is 1.5, 5 to 3 is 1.666, 8 to 5 is 1.6, 13 to 8 is 1.625, 21 to 13 is 1.61538, and 144 to 89 is 1.61797.

The ratio keeps getting closer to 1.618 . . . .

Ellenberg writes, “In Lincoln, we find a more appealing character: enough ambition to try, enough humility to accept that he hadn’t succeeded.” He further says, “The ultimate reason for teaching kids to write a proof is not that the world is full of proofs, but that it is full of non-proofs.”

Assertion is not evidence. To assert is not to demonstrate a proof.

I wish all this year’s crop of math students the best. You might ask your teacher about Euclid.