Leap year

A leap year is the calendar’s correction for the difference between a year counted as 365 days and the longer time Earth takes to complete an orbit of the sun. In Jinzhen’s calculations of Mr Auslander’s lifespan, this familiar rule first exposes an error, then becomes the basis for a subtler question: whether correcting a count by adding calendar days makes it accurate when the calendar itself introduces discrepancies. The sequence turns a child’s painstaking arithmetic into an inquiry into the limits of measurement and the uneasy boundary between counted time and time that has actually passed.

Calendar Rule and the First Calculation

Duckling reaches his answer by adding the days year by year, because he does not know how to multiply. For a lifespan of eighty-nine years, he calculates 32,485 days, then subtracts 253, arriving at 32,232. Young Lillie recognizes that the method has ignored leap years: under the solar calendar, every four years includes a year of 366 days. The calculation is therefore not simply an exercise in addition; its result depends on what kind of year the arithmetic assumes.

Young Lillie nevertheless sees the care in Duckling’s work. Counting from Mr Auslander’s birthday, rather than beginning on January 1, avoids having to account for two partial years at the beginning and end of the count. He acknowledges that this choice has saved effort, even though the answer is not correct. The distinction matters to the episode: the method has a genuine practical advantage, but a more efficient procedure does not remove the need to examine its assumptions.

Error, Precision, and Reassurance

When Young Lillie starts to explain the error, Jinzhen breaks down in tears. He cannot be consoled by the people around him; he grips his thighs and digs in his nails. Young Lillie thumps the dining table and orders him to keep his hands above it. His sharpness interrupts the crying, but he then uses the calculation itself to help Jinzhen understand what being wrong means.

Young Lillie explains that the orbital year is 365 days, five hours, 48 minutes, and 46 seconds. The calendar’s four-year adjustment adds a full day, although four such ordinary years do not accumulate exactly 24 extra hours. For most practical purposes, the calendar’s remaining discrepancy is an acceptable error: without tolerating some error, it would be impossible to operate the solar calendar. Young Lillie therefore distinguishes between a calculation that has neglected leap years and a calculation judged within an accepted margin of error. For figures extending beyond four decimal places, he gives that margin as 0.01 percent.

Mr Auslander lived through 22 leap years during his 89 years. Adding one day for each produces a correction of 22 days, which Young Lillie explains is small relative to a total exceeding 30,000 days and within the stated margin. His purpose is not merely to teach calendar arithmetic. By first confronting Jinzhen and then showing why the mistake need not invalidate all of the work, he tries to help him forgive himself; Jinzhen eventually calms down.

Jinzhen Revises the Correction

The following evening, Jinzhen returns to the calculation with a qualification that initially sounds like another mistake. Although Mr Auslander lived through 22 leap years, Jinzhen argues that the original figure is out by only 21 days, not 22. Those present, including Mrs Lillie, think at first that he has lost his reason. His explanation shows that he is asking a different question from the one Young Lillie answered: not how many leap days the calendar inserted, but how much time the calendar’s adjustments represent against the astronomical year.

The gap between the actual orbital year and the simplified four-year rule is 11 minutes and 14 seconds per year. Across a four-year cycle, that discrepancy totals 44 minutes and 56 seconds. Over 22 leap-year cycles, the calendar has therefore added 16 hours, 28 minutes, and 32 seconds of what Jinzhen calls non-existent time to Mr Auslander’s life. The leap days are real entries in the calendar, but their relation to the earth’s movement is not exact.

Jinzhen then revisits the 112 days included in the original total. That total represents 88 years and 112 days, or 32,232 days; the additional 112 days had not been calibrated for leap-year effects. If each day is treated as exactly 24 hours, the omitted time over those 112 days amounts to 6,421 seconds, or one hour and 47 minutes. Subtracting this from the 16 hours, 28 minutes, and 32 seconds attributed to the leap-year cycles leaves 14 hours, 41 minutes, and 32 seconds.

The beginning and end of Mr Auslander’s life add another complication. Jinzhen has information that Mr Auslander was born at noon and died at nine in the evening, so the two endpoints together contribute at least ten hours of non-existent time to the count. Added to the 14 hours, 41 minutes, and 32 seconds accumulated through the calendar, this makes a full day’s worth of discrepancy. Jinzhen has not erased the leap-year correction; he has shown how the neglected fraction of a day can offset it. The apparent 22-day error is reduced by 24 hours.

From Lifespan to a General Calculation

A few days later, Master Rong’s father finds Jinzhen working through the problem and is astonished by the mathematical theorem he seems to have developed. Jinzhen expresses the calculation as a sequence: the 6,421 seconds omitted in the original count are set against the 2,696 seconds in the 44-minute-and-56-second discrepancy for each leap-year cycle. The successive terms begin with negative 6,421 seconds, then add one, two, and eventually 22 increments of 2,696 seconds.

The sequence converts a concrete puzzle about Mr Auslander into a method for tracking the accumulating difference between time omitted from the original day count and time introduced by the calendar’s four-year correction. Jinzhen works it out in 23 lines of elegant calculations. The text does not reduce his achievement to the discovery that leap years add days; it lies in keeping several imperfect measures in view at once: the orbital year, the calendar year, the 112-day remainder, and the precise times of birth and death. Leap year thus becomes both a practical rule and a test of how closely mathematical description can approach lived time.