Read Online Fermat's Last Theorem by Simon Singh - Free Book Online Page A

Asia Minor in search of further volumes of knowledge. Even tourists to Alexandria could not escape the voracious appetite of the Library. Upon entering the city, their books were confiscated and taken to the scribes. The books were copied so that while the original was donated to the Library, a duplicate could graciously be given to the original owner. This meticulous replication service for ancient travellers gives today’s historians some hope that a copy of a great lost text will one day turn up in an attic somewhere in the world. In 1906 J.L. Heiberg discovered in Constantinople just such a manuscript,
The Method
, which contained some of Archimedes’ original writings.
    Ptolemy’s dream of building a treasure house of knowledge lived on after his death, and by the time a few more Ptolemys had ascended the throne the Library contained over 600,000 books. Mathematicians could learn everything in the known world by studying at Alexandria, and there to teach them were the most famous academics. The first head of the mathematics department was none other than Euclid.
    Euclid was born in about 330 BC . Like Pythagoras, Euclid believed in the search for mathematical truth for its own sake and did not look for applications in his work. One story tells of a student who questioned him about the use of the mathematics he was learning. Upon completing the lesson, Euclid turned to his slave and said, ‘Give the boy a penny since he desires to profit from all that he learns.’ The student was then expelled.
    Euclid devoted much of his life to writing the
Elements
, the most successful textbook in history. Until this century it was also the second best-selling book in the world after the Bible. The
Elements
consists of thirteen books, some of which are devoted to Euclid’s own work, and the remainder being a compilation of all the mathematical knowledge of the age, including two volumes devoted entirely to the works of the Pythagorean Brotherhood. In the centuries since Pythagoras, mathematicians had invented a variety of logical techniques which could be applied in different circumstances, and Euclid skilfully employed them all in the
Elements.
In particular Euclid exploited a logical weapon known as
reductio ad absurdum
, or proof by contradiction. The approach revolves around the perverse idea of trying to prove that a theorem is true by first assuming that the theorem is false. The mathematician then explores the logical consequences of the theorem being false. At some point along the chain of logic there is a contradiction (e.g. 2 + 2 = 5). Mathematics abhors a contradiction and therefore the original theorem cannot be false, i.e. it must be true.
    The English mathematician G.H. Hardy encapsulated the spirit of proof by contradiction in his book
A Mathematician’s Apology:
‘Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.’
    One of Euclid’s most famous proofs by contradiction established the existence of so-called
irrational numbers.
It is suspected that irrational numbers were originally discovered by the Pythagorean Brotherhood centuries earlier, but the concept was so abhorrent to Pythagoras that he denied their existence.
    When Pythagoras claimed that the universe is governed by numbers he meant whole numbers and ratios of whole numbers (fractions) together known as rational numbers. An irrational number is a number that is neither a whole number nor a fraction, and this is what made it so horrific to Pythagoras. In fact, irrational numbers are so strange that they cannot be written down as decimals, even recurring decimals. A recurring decimal such as 0.111111 … is in fact a fairly straightforward number, and is equivalent to the fraction 1 ⁄ 9 . The fact that the ‘1’