Some apologies for matters that are beyond my control: First, I've posted a message from the system managers about the sluggish response of WebCT, which they hope will soon be remedied. Second, it has surely not escaped your notice that the online text is full of misprints. This is not surprising if Prof. Allen had to produce one chapter per week during the first semester he taught this course. I would like to add at least one chapter of my own about more modern aspects of mathematics, and I am wondering when I will find time to write even one chapter. (You see, I have this compulsion to proofread :-) Anyway, in the Greek chapter I noticed some misprints in the mathematics itself, or in the text in places that threaten to make it unintelligible. Fortunately, I have Heath's History of Greek Mathematics, so I should be able to clarify some points. (See below.)
One student, and I, tried to prove the validity of the algorithm and almost succeeded. It is easy to show that the rows and columns add to the right number. The hard part is showing that each number appears in the square exactly once. We thought we had this, and that the generalization to composite n merely requires that p and q be relatively prime to n; but we discovered that the algorithm does not work right in the latter case. I hope to get back to you on this later, although it's not central to our syllabus.
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.
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.