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FIGURE 7. The double square umbrella.problem occurs in Problem 20, in which a circle is said to have area 35,000 square
bu, and its circumference is required. Since the area is taken as one-twelfth of the
square of the circumference, the author multiplies by 12, then takes the square root,
getting 648if§6 bu.
3.4. Liu Hui. Chinese mathematics was greatly enriched from the third through
the sixth centuries by a series of brilliant geometers, whose achievements deserve
to be remembered alongside those of Euclid, Archimedes, and Apollonius. We have
space to discuss only three of these, beginning with the third-century mathematician
Liu Hui (ca. 220-ca. 280). Liu Hui had a remarkable ability to visualize figures
in three dimensions. In his commentary on the Jiu Zhang Suanshu he asserted
that the circumference of a circle of diameter 100 is 314. In solid geometry he
provided dissections of many geometric figures into pieces that could be reassembled
to demonstrate their relative sizes beyond any doubt. As a result, real confidence
could be placed in the measurement formulas that he provided. He gave correct
procedures, based on such dissections, for finding the volumes enclosed by many
different kinds of polyhedra. But his greatest achievement is his work on the volume
of the sphere.
The Jiu Zhang Suanshu made what appears to be a very reasonable claim:
that the ratio of the volume enclosed by a sphere to the volume enclosed by the
circumscribed cylinder can be obtained by slicing the sphere and cylinder along the
axis of the cylinder and taking the ratio of the area enclosed by the circular cross
section of the sphere to the area enclosed by the square cross section of the cylinder.
In other words, it would seem that the ratio is ð : 4. This conjecture seems plausible,
since every such section produces exactly the same figure. It fails, however because
of what is called Pappus' principle: The volume of a solid of revolution equals
the area revolved about the axis times the distance traveled by the centroid of the
area. The half of the square that is being revolved to generate the cylinder has a
centroid that is farther away from the axis than the centroid of the semicircle inside
it whose revolution produces the sphere; hence when the two areas are multiplied
by the two distances, their ratios get changed. When a circle inscribed in a square