244 9. MEASUREMENT
The strongest area of Mesopotamian science that has been preserved is as-
tronomy, and it is here that geometry becomes most useful. The measurement of
angles—arcs of circles—is essential to observation of the Sun, Moon, stars, and
planets, since to the human eye they all appear to be attached to a large sphere
rotating overhead. The division of the circle into 360 degrees is one convention that
came from Mesopotamia, was embraced by the Greeks, and became an essential
part of applied geometry down to the present day. The reason for the number 360 is
the base-60 computational system used in Mesopotamia. The astronomers divided
all circles into 360 or 720 equal parts and the radius into 60 equal parts. In that
way, a unit of length along the radius was approximately equal to a unit of length
on the circle.
2.3. Volumes. The cuneiform tablets contain computations of some of the same
volumes as the Egyptian papyri. For example, the volume of a frustum of a square
pyramid is computed in an old Babylonian tablet (British Museum 85 194). This
volume is computed correctly in the Moscow Papyrus, but the Mesopotamian scribe
seems to have generalized incorrectly from the case of a trapezoid and reasoned that
the volume is the height times the average area of the upper and lower faces. This
rule overestimates the volume by twice the volume of the four corners cut out in
Fig. 3. There is, however, some disagreement as to the correct translation of the
tablet in question. Neugebauer (1935, Vol. 1, p. 187) claimed that the computation
was based on an algebraic formula that is geometrically correct. The square bases
are given as having sides 10 and 7 respectively, and the height is given as 18. The
incorrect rule wc are assuming would give a volume of 1341, which is 22,21 in
sexagesimal notation; but the actual text reads 22,30. The discrepancy could be
a simple misprint, with three ten-symbols carelessly written for two ten-symbols
and a one-symbol. The computation used is not entirely clear. The scribe first
took the average base side (10 + 7)/2 and squared it to get 1,12;15 in sexagesimal
notation (72.25). At this point there is apparently some obscurity in the tablet
itself. Neugebauer interpreted the next number as 0;45, which he assumed was
calculated as one-third of the square of (10 — 7)/2. The sum of these two numbers
is 1,13, which, multiplied by 18, yields 21,54 (that is, 1314), which is the correct
result. But it is difficult to see how this number could have been recorded incorrectly
as 22,30. If the number that Neugebauer interprets as 0;45 is actually 2;15 (which
is a stretch -three ten-symbols would have to become two one-symbols), it would
be exactly the square of (10 - 7)/2, and it would yield the same incorrect formula
as the assumption that the average of the areas of the two bases was being taken.
In any case, the same procedure is used to compute the volume of the frustum of a
cone (Neugebauer, 1935, p. 176), and in that case it definitely is the incorrect rule
stated here, taking the average of the two bases and multiplying by the height.
3. China
Three early Chinese documents contain a considerable amount of geometry, always
connected with the computation of areas and volumes. We shall discuss the geom-
etry in them in chronological order, omitting the parts that repeat procedures we
have already discussed in connection with Egyptian geometry.3.1. The Zhou Bi Suan Jing. As mentioned in Chapter 2, the earliest Chinese
mathematical document still in existence, the Zhou Bi Suan Jing, is concerned