Euclid’s postulates
One of the characteristics of mathematics is that a few assumptions can generate
extensive theories. Euclid’s postulates are an excellent example, and one that set the
model for later axiomatic systems. His five postulates are:
- A straight line can be drawn from any point to any point.
- A finite straight line can be extended continuously in a straight line.
- A circle can be constructed with any centre and any radius.
- All right angles are equal to each other.
- If a straight line falling on two straight lines makes the interior angles on
the same side less than two right angles, the two straight lines, if extended
indefinitely, meet on that side on which the angles are less than two right angles.
It is the style of Euclid’s Elements that makes it noteworthy – its achievement
is the presentation of geometry as a sequence of proven propositions. Sherlock
Holmes would have admired its deductive system which advanced logically from
the clearly stated postulates and may have castigated Dr Watson for not seeing it
as a ‘cold unemotional system’.
While the edifice of Euclid’s geometry rests on the postulates (what are now
called axioms; see box), these were not enough. Euclid added ‘definitions’ and
‘common notions’. The definitions include such declarations as ‘a point is that
which has no part’ and ‘a line is breadthless length’. Common notions include
such items as ‘the whole is greater than the part’ and ‘things which are equal to
the same thing are also equal to one another’. It was only towards the end of the
19th century that it was recognized that Euclid had made tacit assumptions.
The fifth postulate
It is Euclid’s fifth postulate that caused controversy over 2000 years after the
Elements first appeared. In style alone, it looks out of place through its
wordiness and clumsiness. Euclid himself was unhappy with it but he needed it to
prove propositions and had to include it. He tried to prove it from the other
postulates but failed.