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In some places, garbage was burned or collected to be tossed over the city walls. ) In Italian cities, piazzas were kept clean, but in the backstreets, refuse simply lay around to be churned to a kind of black muck after rains. Farm animals living inside cities added to the mess, but when cities, with a view to cleanliness, banned their presence inside the walls, things got worse. It turned out that city pigs had been performing a major cleanup service. Occasionally the revival of ancient learning was absorbed, extended, and applied to a real problem, changing people’s conception of the world.

Much of the fighting was done by groups of undisciplined men who fought for hire or for booty. In Italy, they were led by condottieri, freelance commanders who were a law unto themselves. One of the most feared, Bartolomeo Colleoni, still throws a chill into tourists who run across his grim likeness, a huge equestrian statue by Andrea del Verrocchio that stands in a piazza in Venice. If passing troops did not rape, burn, and pillage, they lived off the land, stripping it so thoroughly that military commanders understood an army could never retreat along the same path it had advanced over.

Using Ptolemy’s distances, in addition to observations of another scholar, Pierre d’Ailly, who reasoned that between Spain and the Indies were only 135 degrees of open sea, Columbus cooked his figures to reduce the number of sea or land miles in each degree of longitude. By his reckoning, Japan lay only 2,760 miles from Spain, or roughly a quarter of what it is. Wild for financial backing, he once noted, “The end of Spain and the beginning of India are not far apart . . ” History has derided Spanish experts leery of Columbus’ estimates.