- CHINA 245
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FIGURE 5. Chinese illustration of the Pythagorean theorem.with astronomy and the applications of mathematics to the study of the heavens.
The title refers to the use of the sundial or gnomon in astronomy. This is the
physical model that led the Chinese to discover the Pythagorean theorem. Here is
a paraphrase of the discussion:
Cut a rectangle whose width is 3 units and whose length is 4 (units)
along its diagonal. After drawing a square on this diagonal, cover it
with half-rectangles identical to the piece of the original rectangle
that lies outside the square, so as to form a square of side 7. [See
Fig. 5.] Then the four outer half-rectangles, each of width 3 and
length 4 equal two of the original rectangle, and hence have area- When this amount is subtracted from the square of area 49,
the difference, which is the area of the square on the diagonal, is
seen to be 25. The length of the diagonal is therefore 25.
Although the proof is given only for the easily computable case of the 3-4-5
right triangle, it is obvious that the geometric method is perfectly general, lacking
only abstract symbols for unspecified numbers. In our terms, the author has proved
that the length of the diagonal of a rectangle whose width is a and whose length is
b is the square root of (a + b)^2 — 2ab. Note that this form of the theorem is not the
"a^2 + b^2 = c^2 " that we are familiar with.
The Zhou Bi Suan Jing contains three diagrams accompanying the discussion of
the Pythagorean theorem. According to Cullen (1996, p. 69), one of these diagrams
was apparently added in the third century by the commentator Zhao Shuang. This
diagram is shown in Fig. 5 for the special case of a 3 4-5 triangle. The other
two were probably added by later commentators in an attempt to elucidate Zhao
Shuang's commentary.
According to Li and Du (1987, p. 29), the vertical bar on a sundial was called
gu in Chinese, and its shadow on the sundial was called gou; for that reason the
Pythagorean theorem was known as the gougu theorem. Cullen (1996, p. 77) says
that gu means thigh and gou means hook. All authorities agree that the hypotenuse
was called xian (bowstring). The Zhou Bi Suan Jing says that the Emperor Yu
was able to bring order into the realm because he knew how to use this theorem to
compute distances. Zhao Shuang credited the Emperor Yu with saving his people
from floods and other great calamities, saying that in order to do so he had to