"Lecture" for Week 2

Katz concentrates almost entirely on Euclid, and with Euclid he concentrates on two of the biggest theorems, the Pythagorean theorem and the construction of the pentagon. What I found most interesting was an implied criticism of the traditional way of writing mathematics. He writes (pp. 36, 38, 49):

"Euclid's Elements ... to the modern reader is incredibly dull. There are no examples; there is no motivation; there are no witty remarks; there is no calculation. There are simply definitions, axioms, theorems, and proofs.... If one reads Book I from the beginning, one never has any idea what will come next. It is only when one gets to the end of the book, where Euclid proves the Pythagorean theorm, that one realizes that Book I's basic purpose is to lead to the proof of that result.... As usual, Euclid does not show how he arrived at the [pentagon] construction ..."

It seems like Prof. Katz has been reading one of my favorite books, "Proofs and Refutations: The Logic of Mathematical Discovery", by I. Lakatos (Cambridge UP, 1976). (That would be a good choice for a book review, come to think of it.) Lakatos sees Euclid as the source of the Original Sin of mathematicians, our careful and proud presentation of ideas in the most logical order, which leaves our students spending most of their time asking, "Why are we doing this?"

By skipping from Pythagoras to Euclid, Katz misses out the Greek alphabetic system of numerals, such major mathematicians as Anaxagoras and Eudoxus, and two quirky, thought-provoking figures, Zeno and Hippias (of whom more below).

Here are some things that were unclear to me [after reading Allen in 2003] until I read up on them in Heath's book.

• Gnomons and polygonal (figurate) numbers. Although the original meaning of "gnomon" is a stick used to cast a shadow (as in a sundial), the relevant early Greek meaning was an L-shaped instrument for drawing right angles, called a "carpenter's square" in English. When the Pythagoreans built up the square numbers (1,4,9,...) as square arrays of dots, they progressed from n² to (n+1)² by adding an L-shaped border of 2n+1 dots. (See Allen, "Pythagoras ...", page 16.) This L-shaped region reminded them of a gnomon, so it was called that. Eventually the word was applied to any region that was added to another region to produce a new region of the same shape as the latter; so if the Greeks had got around to calculus and done the classic "evaporating raindrop" problem, they would probably have called the surface layer of volume 4 Pi r² dr a gnomon.

  Having constructed square numbers and triangular numbers (1,3,6,10,...) as expanding arrays of dots, the Pythagoreans or their successors tried to do the same with pentagons, etc. This does not work out quite so neatly, since the dots can't form an equally spaced, regular lattice. Nevertheless, the generalization of the triangle and square constructions is this (stated for the pentagon, for definiteness): Start with one point and add 4 more points to make a pentagon. Now leave two adjacent sides of the pentagon alone, but build out the pattern in the other three directions -- thus adding successive gnomons that consist of 3 straight segments. At each step the number of dots on each segment increases by 1, so that the total number of dots in the new gnomon is larger by 3 than the old gnomon. (This generalizes the fact for the square that the size of each half of the gnomon (not counting the point at the corner) increases by 1 at each step.)

  More generally, for n-gonal numbers the successive gnomons differ by n-2. Thus the total number of dots in a polygonal number is the sum of a finite arithmetic progression, and in this way the Greeks were led to some of the summation formulas that we now encounter in calculus books as examples of functions whose Riemann sums can be evaluated exactly, or in other textbooks as examples of proofs by mathematical induction.

  The diagrams of pentagonal and hexagonal numbers in Allen (p. 16) and Heath may look like 3-dimensional objects (pyramids or Christmas trees), but they are not intended as such. Later Greeks (principal author Nichomachus, c. 100 AD) did, however, associate numbers with 3-dimensional shapes, notably cubes (of course) and pyramids (take a point, add a triangle of 3 points, then a triangle of 6 points, etc.; similarly for the sequence of n-gonal numbers for any n). This game further extended the collection of finite series whose sums were known.

• The trisectrix, and the three famous unsolvable problems. The curve called the "trisectrix" by Allen is called the "quadratrix" by Heath. Remarkably, if one is in possession of such a curve, one can use it to solve two of the three famous problems, trisecting an angle and squaring a circle. (The third problem is constructing a cube of volume twice a given cube.) The problems are "unsolvable" in the sense that they can't be solved by square and compass alone. The Greeks (or some of them) were interested in "high-tech" solutions as well as in the issue of whether elementary solutions existed, just as today one will happily approximate an integral by Simpson's rule while continuing to wish for a formula for the antiderivative. Like the numerical integration, the trisectrix solution is only approximate, because in practice the construction of the trisectrix itself could only be approximate. Heath (pp. 229-230) argues that even with modern precision machinery one cannot construct a quadratrix without committing a circular fallacy. The point is that the prescription is
  • Let a radius of a circle rotate uniformly from vertical to horizontal position. In exactly the same time, let a horizontal line move uniformly from the top of the circle to the horizontal diameter. Then the intersection point of the radial and horizontal lines traces out the quadratrix.
  It is impossible to get the two speeds right without in some sense knowing the numerical value of Pi, and according to the Euclidean rules of the game, that means that you have already solved the problem of constructing a square with the same area as a given circle (which is equivalent to constructing two straight line segments with lengths in the ratio Pi)!

  Here is a more modern analogy: A slide rule allows you to calculate (or "look up") the logarithm or square root of any number. But you can't construct an accurate slide rule without being able to calculate those functions. (If you are too young to know what a slide rule is, we'll get back to it in the Renaissance.)

  How the curve is used to trisect angles and to square the circle is described by Heath on pages 227-229. Rendering that discussion in HTML is beyond my powers, so we now branch off to a TeX/PDF continuation.