The History of Mathematics: A Brief Course

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QUESTIONS AND PROBLEMS 157

6.21. Compare the pond-filling problem (Problem 26 of Chapter 6) of the Jiu
Zhang Suanshu (discussed above) with the following problem from Greenleaf (1876,
p. 125): A cistern has three pipes; the first will fill it in 10 hours, the second in 15
hours, and the third in 16 hours. What time will it take them all to fill it? Is there
any real difference between the two problems?


6.22. The fair taxation problem from the Jiu Zhang Suanshu considered above
treats distances and population with equal weight. That is, if the population of one
county is double that of another, but that county is twice as far from the collection
center, the two counties will have exactly the same tax assessment in grain and
carts. Will this impose an equal burden on the taxpayers of the two counties? Is
there a direct proportionality between distance and population that makes them
interchangeable from the point of view of the taxpayers involved? Is the growing
of extra grain to pay the tax fairly compensated by a shorter journey?


6.23. Perform the division ^jjp following the method used by Brahmagupta.
6.24. Convert the sexagesimal number 5; 35, 10 to decimal form and the number
314.7 to sexagesimal form.
6.25. As mentioned in connection with the lunisolar calendar, 19 solar years equal
almost exactly 235 lunar months. (The difference is only about two hours.) In the
Julian calendar, which has a leap year every fourth year, there is a natural 28-year
cycle of calendars. The 28 years contain exactly seven leap-year days, giving a
total of exactly 1461 weeks. These facts conjoin to provide a natural 532-year cycle
(532 = 28-19) of calendars incorporating the phases of the Moon. In particular,
Easter, which is celebrated on the Sunday after the first full Moon of spring, has a
532-year cycle (spoiled only by the two-hour discrepancy between 19 years and 235
months). According to Simonov (1999), this 532-year cycle was known to Cyrus
(Kirik) of Novgorod when he wrote his "Method by which one may determine the
dates of all years" in the year 6644 from the creation of the world (1136 CE).
Describe how you would create a table of dates of Easter that could, in principle,
be used for all time, so that a user knowing the number of the current year could
look in the table and determine the date of Easter for that year. How many rows
and how many columns should such a table have, and how would it be used?

6.26. From 1901 through 2099 the Gregorian calendar behaves like the Julian
calendar, with a leap year every four years. Hence the 19-year lunar cycle and
28-year cycle of days interact in the same way during these two centuries. As
an example, we calculate the date of Easter in the year 2039. The procedure
is first to compute the remainder when 2039 is divided by 19. The result is 6
(2039 = 19 ÷ 107 + 6). This number tells us where the year 2039 occurs in the
19-year lunar cycle. In particular, by consulting the table below for year 6, we find
that the first full Moon of spring in 2039 will occur on April 8. (Before people
became familiar with the use of the number 0, it was customary to add 1 to this
remainder, getting what is still known in prayer books as the golden number. Thus
the golden number for the year 2039 is 7.)
We next determine by consulting the appropriate calendar in the 28-year cycle
which day of the week April 8 will be. In fact, it will be a Friday in 2039, so that
Easter will fall on April 10 in that year. The dates of the first full Moon in spring
for the years of the lunar cycle are as follows. The year numbers are computed as
above, by taking the remainder when the Gregorian year number is divided by 19.
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