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Young people became bookkeepers by serving an apprenticeship. At first they would simply stand and watch the master at work, then they would graduate to standing alongside the expert, handing him the counters from bowls placed next to the counting board, and finally they would reach the stage when, under the watchful eye of the master, they could perform some of the computations themselves. Yet even in the hands of an expert, the counting board, like any form of abacus, was cumbersome and provided no permanent record of the calculation.

Many of his fellow passengers would have been pilgrims heading for the Holy Land. Sea journeys were always risky. In addition to the possibility of running into a storm—even in the summer—there was always a chance of being attacked by pirates. The Mediterranean swarmed with privateers, originating both from Muslim North Africa—the infamous Barbary Coast—as well as from the Italian ports of Venice, Genoa, and Pisa itself. Italian privateers were financed by groups of shareholders, just as the peaceful merchant vessels were.

He took pains to explain the concepts in a way that those highly practical men could understand, presenting many examples from everyday commercial life. CHAPTER 3 A Mathematical Journey FOR SOMEONE WHOSE INTELLECTUAL work was to change the course of history, Leonardo’s schooling would have been decidedly basic—for the simple reason that there was nothing else available. In the late twelfth century, education throughout Europe was in the hands of the monasteries and the cathedrals. The curriculum, if it can be so called, comprised little more than learning to read and write, and to write numbers using the Roman system.

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Analytic Number Theory: Proceedings by Knopp M.I. (ed.)

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