The Gregorian calendar was necessitated by the fact that a year is not an integral number of days. The tropical year is roughly 365.242191 days, or equivalently, 365 days, 5 hours, 48 minutes, and 45.96768 seconds. ("Nature, apparently, can make a gorgeous hexagon, but she cannot (or did not deign to) make a year with a nice ever number of days or lunations" [35, p. 137].) The Julian calendar starts off with 12 months of various lengths that add up to 365 days. It then makes a correction of imposing a leap day every fourth year, as a sequence of 365, 365, 365, 366 days, averaging out at 365.25 days per year. This is pretty close: it makes the year about 11 minutes and 154 seconds longer than it actually is. But 11 minutes a year can add up, and the civil calendar got more and more out of step from the solar calendar. By 1581, the vernal equinox was on April 2, rather than the accepted March 21. So Pope Gregory XIII appointed a committee, with the Jesuit mathematician Christopher Clavius as chair. His committee came up with two solutions, both imposed by Pope Gregory in a papal bull issued on February 24, 1582. First, to get the civil and solar...