of the mathematics department there, leading me to wonder if that situa-
tion was something like what happened during the Great Mathematics
Teacher Drought of the 1970s, where the paucity of math teachers at the
junior and senior high school level sometimes resulted in shop or PE
teachers becoming algebra instructors. A good friend of mine had ma-
jored in political science while she was in college. When she became a
middle-school teacher (the West Coast equivalent of junior high) in the
1970s, someone was needed to fill an algebra staffing gap; she did so and
had a satisfying and successful career as an algebra teacher.
Anyway, not much was heard from Saccheri until 1733, when his bomb-
shell Euclides ab Omni Naevo Vindicatus (variously translated, I’ll go with
“Euclid Freed from Every Flaw”) was published. It was more of a time
bomb, not being recognized for its value until substantially later. In it,
Saccheri was to make the first important moves toward the development
of non-Euclidean geometry.
Saccheri did what others before him attempted to do—prove the parallel
postulate from the other four postulates. He started with a line segment
(the base) on which he constructed two line segments of equal length (the
sides), each making a right angle with the base. He then connected the
endpoints of the two line segments (the top), making a figure that is now
known as a Saccheri quadrilateral—and which you, when you do this,
will immediately recognize as a rectangle.
However, you know it’s a rectangle because each point of the top is the
same distance from the base (the sides of equal length make it so), and
you have accepted Proclus’s version of the parallel postulate. Saccheri
didn’t assume the parallel postulate. By using the other postulates, he
was able to show quite easily that the vertex angles, which are the two
angles made by the top with the sides, had to be equal. There were then
three possibilities: the vertex angles could be right angles (which would
then demonstrate that the parallel postulate could be proved from the
other four), the vertex angles could be obtuse (greater than 90 degrees), or
the vertex angles could be acute (less than 90 degrees).
Saccheri first developed an indirect proof in which he showed that the
hypothesis that the vertex angles were obtuse led to a contradiction. He
then attempted to show that the hypothesis that the vertex angles were
acute also led to a contradiction—but after much work was unable to do
so without fudging the proof by assuming that lines that met at a point at
infinite distance (this is called “a point at infinity”) actually met at a point
on the line. At this juncture, Saccheri had two choices—go with the
fudged proof in order to show the result in which he had an emotional
investment, or admit that he was unable to show that the hypothesis that
104 How Math Explains the World