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FIGURE 10. Rounding a square.the practitioners of wasan were not immediately attracted to Euclid when his work
arrived in Japan in the nineteenth century. According to Murata [1994, P- 109),
having seen Chinese translations of Euclid, they were repelled by the great amount
of fuss required to derive elementary facts. They may have taken just the ideas
that appealed to them out of the information reaching them through contacts with
the Chinese.
5. India
The Sulva Sutras contain many transformation-of-area constructions such as are
later found in Euclid. In particular, the Pythagorean theorem, and constructions
for finding the side of a square equal to a rectangle, or the sum or difference of two
other squares are given. This construction resembles the one found in Proposition 5
of Book 2 of Euclid rather than Euclid's construction of the mean proportional in
Book 6, both of which are discussed in Chapter 10. The Pythagorean theorem is
not given a name, but is stated as the fact that "the diagonal of a rectangle pro-
duces both [areas] which its length and breadth produce separately." Among other
transformation of area problems the Hindus considered in particular the problem
of squaring the circle. The Bodhayana Sutra states the converse problem of con-
structing a circle equal to a given square. The construction is shown in Fig. 10,
where LP = \LE.
In terms that we can appreciate, this construction gives a value for two-dimen-
sional ð of 18(3 - 2\/2), which is about 3.088.
5.1. Aryabhata I. Chapter 2 of Aryabhata's Aryabhatiya (Clark, 1930, pp. 21-
50) is called Ganitapada (Mathematics). In Stanza 6 of this chapter Aryabhata
gives the correct rule for area of a triangle, but declares that the volume of a
tetrahedron is half the product of the height and the area of the base. He says in
Stanza 7 that the area of a circle is half the diameter times half the circumference,
which is correct, and shows that he knew that one- and two-dimensional ð were
the same number. But he goes on to say that the volume of a sphere is the area
of a great circle times its own square root. This would be correct only if three-
dimensional ð equaled ø, very far from the truth! Yet Aryabhata knew a very
good approximation to one-dimensional ð. In Stanza 10 he writes:Add 4 to 100, multiply by 8, and add 62,000. The result is approxi-
mately the circumference of a circle of which the diameter is 20,000.