# Here's looking at Euclid

When people say that the ancient Greeks invented mathematics, they do not mean that they were the first to count on their fingers, measure how much flour was needed for 10 loaves or calculate how many loaves would feed a thousand. The ancient Egyptians, Chinese, Indians and Mesopotamians also developed those maths skills needed for everyday life, commerce, engineering, astronomy and taxation, but it was the Greeks who crystallised these skills into one theoretical discipline.

That system of mathematics was - and remains - underpinned by two philosophical ideas: that mathematical statements are general (not just this triangle, but if this triangle then all triangles) and by a logical structure of proof (if this is true, then that must be true).

Pythagoras of Samos (c 580-500BC) was one of the earliest Greek mathematicians, although that was only one strand in his amazing life and thought. Before Aristotle and Plato came along to classify and distinguish intellectual disciplines - science, art, religion and so on - for a couple of centuries the Greek-speaking world was a ferment of ideas about the nature of existence, the mechanics of earth, air and water, the existence of God and whether pigs have wings. Dynamic thinkers like Pythagoras gathered schools of eager disciples (the word comes from discipline) around them.

Maths was only part of Pythagorean teaching. escaping from the tyranny of Samos, Pythagoras founded a mystical school in Croton. Some of his ideas are still crucial to Western thought - that music is essentially numerical, that the world is created by a dialogue ("dialectic") between polarities like black-white, hot- cold, solid-liquid, that the soul or personality is located in the brain. Some were once influential but now have passed away - that harmonies between the heavenly bodies influenced earthly affairs (this idea still hangs on in astrological charts). And some now seem positively barmy - believing in the transmigration of souls, the Pythagoreans would not eat beans, which contain the germs of life.

Pythagoras and the Pythagoreans are probably the discoverers of irrational numbers. Certainly they found that whole numbers can never express the ratio between the side of a square and its diagonal: these numbers are "incommensurable", said the Greeks. They went on to discover (by 400BC) that square roots of prime numbers are irrational.

But, of course, it is the good old right-angled triangle theorem for which Pythagoras is chiefly known, that and the sets of numbers (for instance, 3,4,5 - "Pythagorean triples") whose squares are in fixed relation to each other.

Pythagoras himself believed that in proving mathematical theorems he laid the foundations for a completely coherent view of the universe. Whether for this or their non-eating of beans, his followers were mocked and persecuted, the last of them being wiped out in a brutal massacre in about 350 BC.

"There is no royal road to geometry." Thus, supposedly, Euclid (who flourished about 300BC) rebuked Ptolemy, ruler of Egypt, who asked if there were any shorter way to master geometry. For 2,000 years, the 13 books of Euclid's Elements (his most famous work) on plane and solid geometry, the theory of whole numbers, primes, cubes and irrational lines have dominated the teaching of maths.

Euclid was a synthesiser (he joined ideas together) but not an analyst (he did not break solutions down into their separate ideas); an editor but not an originator. A populariser who was not always consistent, Euclid boiled down the mathematical thinking of the previous two centuries into 10 apparently self-evident axioms or postulates, from which 465 theorems or propositions could logically be derived. His way of setting out proof is still used today.

Euclid is supposed to have flung a coin at a student who wanted to know how much he would gain materially by learning geometrical propositions, saying, "Give him a farthing since he needs must gain by learning." He clearly didn't work at the DFEE. Interestingly, one of the flaws in his thinking has led to one of the most exciting developments in maths today.

One of Euclid's postulates - that through a given point P not on a line L, there is only one line which can be drawn parallel to L - was always controversial. It was not until the 19th century that it was demonstrated that replacing this postulate with either of its alternatives - that there is either no such line, or, that there is more than one - instead of generating a mass of contradictions actually led to the formation of two completely consistent but mutually contradictory new geometries. But which of the three describes the "real world"? Mathematicians are still arguing the toss. Some have abandoned the search for "real world" maths, others are pushing it into more arcane realms of thought.

Mathematics is rigorous, yes, but only within itself. It's a harmonious model - rather like art. Or perhaps religion. Which would have pleased Pythagoras.

"Eureka", cried Archimedes (who lived in the third century BC in Syracuse), leaping from the bath. Or did he? His renowned principle, of the forces which ensure that a body in fluid displaces its own volume, is the basis of all naval engineering.

Archimedes was a superstar in his own time as well as thereafter. His inventions - the discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder, the Archimedes principle and a device for raising water, the Archimedes screw - were so startling and useful that many stories, like his dazzling the enemy Roman army with mirrors, grew up around his name. One of the boundaries he established for pi - 22 divided by 7 - is still a rule of thumb today. Last, but not least, he created the place-value system of number notation.