284 10. EUCLIDEAN GEOMETRYpoint ahead of the faster. Hence the race can be thought of as beginning again at
that instant, with the slower runner still having a head start. The race will "begin
again" in this sense infinitely many times, with the slower runner always having a
head start. Thus, as in the dichotomy, infinitely many things must be accomplished
in a finite time in order for the faster runner to overtake the slower.
The Arrow. An arrow in flight is at rest at each instant of time. That is, it does
not move from one place to another during that instant. But then it follows that
it cannot traverse any positive distance because successive additions of zero will
never result in anything but zero.
The Stadium. (In athletic stadiums in Greece the athletes ran from the goal, around
a halfway post and then back. This paradox seems to have been inspired by imag-
ining two lines of athletes running in opposite directions and meeting each other.)
Consider two parallel line segments of equal length moving toward each other with
equal speeds. The speed of each line is measured by the number of points of space
it passes over in a given time. But each point of one line passes twice that many
points of the other line in the same time as the two lines move past each other.
Hence the velocity of the line must equal its double, which is absurd.
The Pythagoreans had built their system on lines "made of" points, and now
Zeno was showing them that space cannot be "made of" points in the same way
that a building can be made of bricks. For assuredly the number of points in a
line segment cannot be finite. If it were, the line would not be infinitely divisible
as the dichotomy and Achilles paradoxes showed that it must be; moreover, the
stadium paradox would show that the number of points in a line segment equals
its double. There must therefore be an infinity of points in a line. But then each
of these points must take up no space; for if each point occupied some space, an
infinite number of them would occupy an infinite amount of space. But if points
occupy no space, how can the arrow, whose tip is at a single point at each instant
of time, move through a positive quantity of space? A continuum whose elements
are points was needed for geometry, yet it could not be thought of as being made
up of points in the way that discrete collections are made up of individuals.
1.6. Challenges to Pythagoreanism: incommensurables. The difficulties
pointed out by Zeno affected the intuitive side of geometry. The challenge they
posed may have been an impetus to the kind of logical rigor that we know as Eu-
clidean. There is, however, an even stronger impetus to that rigor, one that was
generated from within Pythagorean geometry. To the modern mathematician, this
second challenge to Pythagorean principles is much more relevant and interesting
than the paradoxes of Zeno. That challenge is the problem of incommensurables,
which led ultimately to the concept of a real number.
The existence of incommensurables throws doubt on certain oversimplified
proofs of geometric proportion. When two lines or areas are commensurable, one
can describe their ratio as, say, 5 : 7, meaning that there is a common measure such
that the first object is five times this measure and the second is seven times it. A
proportion such as á : b :: c : d, then, is the statement that ratios a : b and c : d
are both represented by the same pair of numbers.
This theory of proportion is extremely important in geometry if we are to have
such theorems as Proposition 1 of Book 6 of Euclid's Elements, which says that the
areas of two triangles or two parallelograms having the same height are proportional