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were partially reunited, then divorced again; with results that will become apparent as the story unfolds.
    The Pythagorean synthesis would have been incomplete had it not also included precepts for a way of life.
    The Brotherhood was a religious order, but at the same time an academy of science, and a power in Italian politics. The ascetic rules of life seem to have anticipated the Essenes', which in turn served as a model to primitive Christian communities. They shared all property, led a communal existence, and gave equal status to women. They observed rites and abstinences, gave much time to contemplation and examinations of conscience. According to the degree of purification which a Brother achieved, he was gradually initiated into the higher mysteries of musical, mathematical and astronomical theoria . The secrecy surrounding these was partly due to the tradition of the older mystery cults, whose adepts had known that the Bacchic, and even the Orphic, ecstasies would cause havoc if offered to all and sundry. But the Pythagoreans also realized that similar dangers inhered in the orgies of reasoning. They apparently had an intuition of the hybris of science, and recognized it as a potential means both of man's liberation and destruction; hence their insistence that only those purified in body and spirit should be trusted with its secrets. In a word, they believed that scientists ought to be vegetarians, as Catholics believe that priests ought to live in celibacy.
    It may be thought that this interpretation of the Pythagorean insistence on secretiveness is far-fetched, or that it implies prophetic foresight on their part. The answer to this is that Pythagoras was, by personal experience, well aware of the immense technological potentialities of geometry. I have mentioned already that Polycrates, and the islanders he ruled, were devoted to engineering. Herodotus, who knew the island well, reports: 10
    "I have written thus at length of the Samians, because they are the makers of the three greatest works to be seen in any Greek land. First of these is the double-mouthed tunnel they pierced for an hundred and fifty fathoms through the base of a high hill ... through which the water, coming from an abundant spring, is carried by its pipes to the city of Samos".
    Herodotus is fond of telling tall stories, and his report was not taken very seriously, until, at the beginning of our century, the tunnel was actually found and excavated. It is no less than nine hundred yards long, complete with water-course and inspection-pathway, and its shape shows that it was begun from both ends. It further shows that the two digging parties, one working from the north, the other from the south, had met in the centre only a couple of feet apart. Having watched this fantastic feat being performed (by Eupalinos, who also built the second marvel mentioned by Herodotus, a huge mole to protect the Samian war-fleet), even a lesser genius than Pythagoras might have realized that Science may become a hymn to the creator or a Pandora's box, and that it should be trusted only to saints. It is said, incidentally, that Pythagoras, like St. Francis, preached to animals, which would seem rather odd behaviour in a modern mathematician; but in the Pythagorean view nothing could be more natural.
    5. Tragedy and greatness of the Pythagoreans
    Towards the end of the Master's life, or shortly after his death, two misfortunes befell the Pythagoreans, which would have meant the end of any sect or school with a less universal outlook. They triumphantly survived both.
    One
blow was the discovery of a type of numbers such as

    – the square root of 2 – which could not be fitted into any dot-diagram. And such numbers were common: they are, for instance, represented by the diagonal of any square. Let the side of the square be called a , and the diagonal d . It can be proved that if I assign to a any precise numerical value, then it becomes impossible to assign a precise