# Euclid's Life: A Short Biography of Euclid

Any biography of Euclid must be short as we know little about him.

Much of what we know about Euclid comes from
a summary by the Greek philosopher Proclus in 450AD, which states that Euclid, "put together
* the Elements*,
collecting many of Eudoxus' theorems, perfecting many of Theaetetus',
and provided indisputable demonstration for things which were only somewhat
loosely proved by his predecessors. This man lived in the time of the first
Ptolemy. For Archimedes, who came immediately after the first Ptolemy,
makes mention of Euclid: and, further, they say that Ptolemy
once asked him if there was in geometry any shorter way than that of
the Elements, and he answered that there was no royal road to geometry. He is then
younger than the pupils of Plato but older than Eratosthenes and Archimedes
for the latter were contemporary with one another, as Eratosthenes
somewhere says."

The "first Ptolemy" is Ptolemy I, Alexander the Great's general and ruler of Egypt. From the clues in this passage it can be surmised that
Euclid flourished around 300 B.C. It is
most probable that Euclid
received his mathematical training in Athens
from the pupils of Plato—mathematicians on whose works *The Elements*
were
based. He may himself have been a Platonist, but this does not follow from the text
by Proclus quoted above.

The mathematician Pappus of Alexandria, in about 320AD, is the only source of our knowledge that Euclid founded a school in Alexandria. He also gave us an insight into his character. According to Pappus, Euclid was scrupulous in giving credit to minor mathematicians, whose techniques he might easily have passed off as his own. We might expect such practices to be the norm today (perhaps naively), but in the ancient world many scholars were not so scrupulous about "quoting their sources".

There is a suggestion that Euclid had a wry sense of humour. This comes from Stobaeus of Macedonia, who lived about 500AD. Stobaeus was no great philosopher, but had the habit of noting down interesting passages that he read. We are indebted to him for providing fragments of the lost works of around 500 Ancient Greek writers. He relates a story of a student asking Euclid what gain he would make by studying geometry— Euclid responds by giving him a penny.

Euclid's five*postulates*are central to his plane geometry. These are:

- There is a unique (straight) line segment connecting any two points
- Any straight line segment can be extended as far as you want.
- Any point can be the centre of a circle of any radius.
- All right angles are equal.
- The parallel postulate (for any line b and point P, there is only one straight line through P that is parallel to b).

Since the publication
of Euclid
's elements, in about 300BC, many
mathematicians have tried to prove the fifth postulate from Euclid's other axioms and other postulates. In
the early 18^{th} century Girolama Saccheri attempted a *proof by contradiction*,
or *reduction ad absurdum*, to show that the fifth postulate is false. *Proof
by contradiction* is a powerful method of mathematical reasoning often used today.

*hyperbolic geometry*. Although Carl Friedrich Gauss was the first person to accept this as a valid geometry.