"Lecture" for Week 3
Early Greek mathematics
The main point
Greek mathematics is strikingly different from what came before.
The Greeks introduced three new elements, which are logically distinct
but nevertheless somehow related:
We must be careful not to oversimplify and overgeneralize.
Over a thousand years passed between Thales (c. 600 BCE) and
Proclus (c. 450 CE),
the last great name in Greek geometry (see Allen's timeline).
Therefore, to say "The Greeks believed ..." is like saying
"The French believed ..." without saying whether you have in mind
Charlemagne or Marie Curie.
- 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).
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.
(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
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
The point is that the prescription is
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)!
- 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