sides extended indefinitely. Look at the side you consider to be the base
of the triangle. The “interior angles” referred to above are the angles the
base makes with the other two sides; the sum of those two angles is less
than the 180 degrees, that is, the sum of the two right angles. If one
changed the orientation of the sides by a sufficient amount that they in-
tersected on the other side of the base, the “interior angles” here would
again sum to less than 180 degrees. So what happens when the interior
angles sum to precisely 180 degrees? The two other lines don’t meet on
either side of the base, so they either have to meet on the base (but that
would happen only if the base coincided with the sides) or not meet.
As you can see, this formulation of the parallel postulate is not easy to
work with, and even back in ancient Greece suggestions were made to
revise it. It was Proclus who suggested a version that we frequently use
today (two parallel lines are everywhere the same distance apart), but it
was the Scottish mathematician John Playfair who gets the credit, as he
wrote a very popular geometry text at the turn of the nineteenth century
incorporating it. Then as now, credit accrues to the individual with the
best public relations department.
Postulate 5 (Playfair’s Axiom): Through each point not on a given line,
only one line can be drawn parallel to the given line.^4
This was the form in which I learned the parallel postulate. It has two obvi-
ous advantages. The first is that it is much easier to understand, visualize,
and use than Euclid’s original formulation. The second advantage is more
subtle—it leads one to ask the question, is it possible to create geometries in
which more than one line can be drawn parallel to the given line?
Certainly, such a geometry cannot exist on the plane, as that’s the habi-
tat of Euclidean plane geometry with the five postulates. However, if we
move into Euclidean three-dimensional space, we can have infinitely
many lines through a given point parallel to a given line—parallel, that is,
in the sense that both lines, when extended, do not meet. Simply take a
line and a plane parallel to that line but not containing it. If one fixes a
point in that plane, any line through that point will obviously not meet
the given line—although all but one of these are called skew lines in
modern terminology (there is one that is genuinely parallel to the given
line because it lies in a plane with the given line).
Girolamo Saccheri
Not all Italian mathematicians were as colorful as Tartaglia, Cardano,
and Ferrari. Girolamo Saccheri was ordained a Jesuit priest and taught
philosophy and theology at the University of Pavia. He also held the chairNever the Twain Shall Meet 103