27 The parallel postulate
This dramatic story begins with a simple geometric scenario. Imagine a line l and a point
P not on the line. How many lines can we draw through P parallel to the line l? It appears
obvious that there is exactly one line through P which will never meet l no matter how
far it is extended in either direction. This seems self-evident and in perfect agreement
with common sense. Euclid of Alexandria included a variant of it as one of his postulates
in that foundation of geometry, the Elements.
Common sense is not always a reliable guide. We shall see whether Euclid’s
assumption makes mathematical sense.
Euclid’s Elements
Euclid’s geometry is set out in the 13 books of the Elements, written around
300 BC. One of the most influential mathematics texts ever written, Greek
mathematicians constantly referred to it as the first systematic codification of
geometry. Later scholars studied and translated it from extant manuscripts and it
was handed down and universally praised as the very model of what geometry
should be.
The Elements percolated down to school level and readings from the ‘sacred
book’ became the way geometry was taught. It proved unsuitable for the
youngest pupils, however. As the poet A.C. Hilton quipped: ‘though they wrote it
all by rote, they did not write it right’. You might say Euclid was written for men
not boys. In English schools, it reached the zenith of its influence as a subject in
the curriculum during the 19th century but it remains a touchstone for
mathematicians today.