The History of Mathematics: A Brief Course

QUESTIONS AND PROBLEMS 267

FIGURE 17. A disk cut into sectors and opened up.

are approximations for. How does this problem illustrate the claim that these
challenge problems were algebraic rather than geometric?

9.13. How is it possible that some Japanese mathematicians believed the area of
the sphere to be one-fourth the square of the circumference, that is, ÔÃ^2 Ã^2 rather
than the true value 4ðô·^2? Smith and Mikami [1914, p. 75) suggest a way in which
this belief might have appeared plausible. To explain it, we first need to see an
example in which the same line of reasoning really does work.
By imagining a circle sliced like a pie into a very large number of very thin
pieces, one can imagine it cut open and all the pieces laid out next to one another,
as shown in Fig. 17. Because these pieces are very thin, their bases are such short
segments of the circle that each base resembles a straight line. Neglecting a very tiny
error, we can say that if there are ç pieces, the base of each piece is a straight line
of length 2 -nr/n. The segments are then essentially triangles of height r (because
of their thinness), and hence area (1/2) · (2ðÃ^2 )/ç. Since there are ç of them, the
total area is nr^2. This heuristic argument gives the correct result. In fact, this very
figure appears in a Japanese work from 1698 (Smith and Mikami, 1914, Ñ· 131).
Now imagine a hemispherical bowl covering the pie. If the slices are extended
upward so as to slice the bowl into equally thin segments, and those segments are
then straightened out and arranged like the segments of the pie, they also will have
bases equal to but their height will be one-fourth of the circumference, in other
words, nr/2, giving a total area for the hemisphere of (1/2) · 7r^2 r^2. Since the area is
2ð7·^2 , this would imply that ð = 4. What is wrong with the argument? How much
error would there be in taking ð = 4?

9.14. What is the justification for the statement by the historian of mathematics

T. Murata that Japanese mathematics (wasan) was not a science but an art?

9.15. Show that Aryabhata's list of sine differences can be interpreted in our lan-

guage as the table whose nth entry is

Use a computer to generate this table for ç = 1,.. .,24, and compare the result

with Aryabhata's table.

9.16. If the recursive procedure described by Aryabhata is followed faithfully (as

a computer can do), the result is the following sequence.

225, 224, 222, 219, 215, 210, 204, 197, 189, 181, 172,

162, 151, 140, 128, 115, 102, 88, 74, 60, 45, 30, 15, 0