The History of Mathematical Proof in Ancient Traditions

428 karine chemla

if such is the case, as a consequence, the diameter of the lower circle of the

solid to be considered is consequently equal to its height. Th e truncated

pyramid dealt with can thus be inscribed into a cube.

In the Classic, the outline of the problem is immediately followed by an

algorithm allowing the reader to rely on the data provided to determine the

desired volume. It reads as follows:

The circumferences of the upper and lower circles being multiplied by

one another, then multiplied each by itself, one adds these (the results);

one multiplies this by the height and divides by 36.

To expound the argument on proof that I have in view, I shall need to

make use of a representation of the algorithm as list of operations. To this

end, let us note, as on Figure 13.1 , C (^) s (resp. C i ) the circumference of the
upper (resp. lower) circle and h the height of the solid. With these nota-
tions, the algorithm can be represented in a synoptic way, as follows:
Figure 13.1 Th e truncated pyramid with circular base.
Cs
h
Ci
Multiplications Multiplication Division
sum by h by 36
C (^) i > C (^) i C (^) s + C (^) i 2 + C (^) s 2 > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /36
C (^) s
In what follows, I shall regularly employ such representations for lists of
operations.
Immediately aft er the statement of the algorithm as given by the Classic,
in the fi rst section of his exegesis, the commentator sets out to establish its
correctness within the framework of the hypothesis that Th e Nine Chapters
made use of a ratio between the circumference and the diameter of the
circle equivalent to taking π = 3. His proof proceeds along three interwoven