246 9. MEASUREMENT
survey the shapes of mountains and rivers. Apparently the Emperor had drainage
canals dug to channel floods out of the valleys and into the Yangtze and Yellow
Rivers.
The third-century commentary on the Zhou Bi Suan Jing by Zhao Shuang
explains a method of surveying that was common in China, India, and the Mus-
lim world for centuries. The method is illustrated in Fig. 6, which assumes that
the height Ç of an inacessible object is to be determined. To determine H, it is
necessary to put two poles of a known height h vertically into the ground in line
with the object at a known distance D apart. The height h and the distance D are
theoretically arbitrary, but the larger they are, the more accurate the results will
be. After the poles are set up, the lengths of the shadows they would cast if the
Sun were at the inacessible object are measured as si and S2- Thus the lengths s\,
S2, h, and D are all known. A little trigonometry and algebra will show that
11 — ft, r ·
«2 - SiWe have given the result as a formula, but as a set of instructions it is very easy to
state in words: The required height is found by multiplying the height of the poles
by the distance between them, dividing by the difference of the shadow lengths, and
adding the height of the poles.
This method was expounded in more detail in a commentary on the Jiu Zhang
Suanshu written by Liu Hui in 263 CE. This commentary, along with the rest of the
material on right triangles in the Jiu Zhang Suanshu eventually became a separate
treatise, the Hai Dao Suan Jing (Sea Island Mathematical Manual, see Ang and
Swetz, 1986). Liu Hui mentioned that this method of surveying could be found
in the Zhou Bi Suan Jing and called it the double difference method (chong cha).
The name apparently arises because the difference Ç — h is obtained by dividing
Dh by the difference S2 - s\.
We have described the lengths si and S2 as shadow lengths here because that
is the problem used by Zhao Shuang to illustrate the method of surveying. He
attempts to calculate the height of the Sun, given that at the summer solstice a
stake 8 chi high casts a shadow 6 chi long and that the shadow length decreases
by 1 fen for every 100 li that the stake is moved south, casting no shadow at
all when moved 60,000 li to the south. This model assumes a flat Earth, under
which the shadow length is proportional to the distance from the pole to the foot
of the perpendicular from the Sun to the plane of the Earth. Even granting this
assumption, as we know, the Sun is so distant from the Earth that no lengthening
or shortening of shadows would be observed. To any observable precision the Sun's
rays are parallel at all points on the Earth's surface. The small change in shadow
length that we observe is due entirely to the curvature of the Earth. But let us
continue, accepting Zhao Shuang's assumptions.
The data here are D = 1000 li, S2 - «i = 1 fen, h = 8 chi. One chi is about
25 centimeters, one fen is about 2.5 cm, and one li is 1800 chi, that is, about 450
meters. Because the pole height h is obviously insignificant in comparison with the
height of the Sun, we can neglect the first term in the formula we gave above, and
writeS2 - S\