- Proof (not just examples).
- Idealization. (To Euclid, lines are perfectly straight, infinitely thin, and (potentially) infinitely long. The last of these is particularly surprising, since Aristotle (who slightly preceded Euclid) and almost all other Greek thinkers taught that the universe must be finite! But without the consensus on indefinite extendability, the distinction between parallel and intersecting lines would become a hopeless muddle.)
- Math for math's sake (not primarily practical applications).

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. (Stillwell's figure on p. 39 is mystifying at first glance, since the points inside the pentagon seem to have random gaps.) 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
(*A History of Greek Mathematics*)
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.

- 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.

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.