called the ‘triangle inequality’ and is important in abstract mathematics. The
Epicureans, with their down-to-earth approach to life, claimed this required no
proof, for it was evident even to an ass. If a bale of hay were placed at one
vertex and the ass at the other, they argued, the animal would hardly traverse
the two sides to satisfy its hunger.
Pythagoras’s theorem
The greatest triangle theorem of all is Pythagoras’s theorem, and is one which
features in modern mathematics – though there is some doubt about Pythagoras
being the first to discover it. The best known statement of it is in terms of
algebra, a^2 + b^2 = c^2 but Euclid refers to actual square shapes: ‘In right-angled
triangles the square on the side subtending the right angle is equal to the squares
on the sides containing the right angle’.
Euclid’s proof is Proposition 47, in Book 1 of the Elements, a proof which
became a point of anxiety for generations of school pupils as they struggled to
commit it to memory, or take the consequences. There are several hundred
proofs in existence. A favourite is more in the spirit of Bhāskara of the 12th