First regarding anthyphaeresis, I wrote a computer program to calculate
the number of passes in a cycle when computing the square roots of
non-square integers. I found that the length of a cycle stays fairly
manageable up to 17, then takes a large jump at 19. (Obviously one
doesn't need to apply it for 18.) That fact made me wonder about the
passage in which Theodorus is said to have stopped at 17. On reflection
I decided it was unlikely that Theodorus stopped AFTER doing 17 because
of the increase in complexity, especially since Wilbur Knorr made such
and elegant argument that he stopped BECAUSE his argument no longer worked
for 17. As Sam and David have repeatedly pointed out, there isn't any
textual trail, so there is little point in speculating. Still, I found
it intriguing. It's no sillier than the "chambered-nautilus" diagram
once suggested as an explanation, which as far as I can see has no
connection at all with the topic. Has anyone suggested this explanation
before?
Second, concerning synthetic geometry (a term I like and still use),
I really love the synthetic approach and often teach hyperbolic
geometry this way before giving models for it within Euclidean
geometry. I like deriving the formulas for the angle of parallelism
and solving triangles the "old-fashioned" way. There is something
very fascinating about hatching a quantitative chick from a
qualitative egg. However, my considered view is that this approach is
a pedagogical dinosaur. Students generally hate it, and I've begun
to see in my old age that the Greek approach to geometry was ponderous
and clumsy. Note in particular that with just a bit of algebra
(analytic geometry and calculus) modern students can solve area and
volume problems that would have baffled Archimedes. Just thought
I'd raise a bit of controversy. How do people here line up on the
following debating proposition? Resolved: The Greek synthetic
approach to geometry is outmoded and clumsy and should no longer
be taught.
Roger Cooke