- CHINA 247
FIGURE 6. The double-difference method of surveying.When we insert the appropriate values, we find, as did Zhao Shuang, that the Sun
is 80,000 li high, about 36,000 kilometers. Later Chinese commentators recognized
that this figure was inaccurate, and in the eighth century an expedition to survey
accurately a north-south line found the actual lengthening of the shadow to be
4 fen per thousand li. Notice that there seem to be two methods of computing
the height here. In the method just discussed, the fact that the Sun is directly
overhead at a distance of 60,000 li to the south is irrelevant to the computation. If
it is taken into account, one can immediately use the similar triangles to infer the
height of 80,000 li. This fact suggests that the original text was modified by later
commentators, but that not all the parts that became irrelevant as a result of the
modifications were removed.
3.2. The Jiu Zhang Suanshu. The Jiu Zhang Suanshu contains all the standard
formulas for the areas of squares, rectangles, triangles, and trapezoids, and also the
recognition of a relation between the circumference and the area of a circle, which
we could interpret as a connection between the one-dimensional ð and the two-
dimensional ð. The geometric formulas given in this treatise are more extensive
than those of the Ahmose Papyrus; for example, there are approximate formulas
for the volume of segment of a sphere and the area of a segment of a circle. It is
perhaps not fair to compare the two documents, since the Ahmose Papyrus was
written nearly two millennia earlier, and the Jiu Zhang Suan Shu was intended to
cover all the mathematics known at the time. The implied value of one-dimensional
ð, however, is ð = 3. It is surprising to find this value so late, since it is known
that the value 3.15147 had been obtained in China by the first century. According
to Li and Du (1987, p. 68), Liu Hui refined it to 3.14 + 64/62500 = 3.141024 by