The History of Mathematics: A Brief Course

266 9. MEASUREMENT

0;9,36 from [0; 15], leaving 0;5,24. What is the square root of 0;5,24?

The lower end of the beam is [0;18] from the wall.

When the lower end is 0; 18 from the wall, how far has the top

slid down? Square 0;18, obtaining 0;5;24....

9.5. Show that the average of the areas of the two bases of a frustum of a square

pyramid is the sum of the squares of the average and semidifference of the sides of

the bases. Could this fact have led the Mesopotamian mathematicians astray in

their computation of the volume of the frustum? Could the analogy with the area

of a trapezoid have been another piece of misleading evidence pointing toward the

wrong conclusion?

9.6. The author of the Zhou Bi Suan Jing had a numerical method of finding the

length of the diagonal of a rectangle of width á and length 6, which can be described

as follows. Square the sum of width and length, subtract twice the area, then take

the square root. Should one conclude from this that the author knew that the

square on the hypotenuse was the sum of the squares on the legs?

9.7. What happens to the estimate of the Sun's altitude (36,000 km) given by Zhao

Shuang if the "corrected" figure for shadow lengthening (4 fen per 1000 li) is used

in place of the figure of 1 fen per 1000 Ιߺ

9.8. The gougu section of the Jiu Zhang Suanshu contains the following problem:

Under a tree 20 feet high and 3 in circumference there grows a vine,

which winds seven times the stem of the tree and just reaches its

top. How long is the vine?

Solve this problem.

9.9. Another right-triangle problem from the Jiu Zhang Suanshu is the following.
"There is a string hanging down from the top of a pole, and the last 3 feet of string
are lying flat on the ground. When the string is stretched, it reaches a point 8
feet from the pole. How long is the string?" Solve this problem. You can also, of
course, figure out how high the pole is from this information.

9.10. A frequently reprinted problem from the Jiu Zhang Suanshu is the "broken
bamboo" problem: A bamboo 10 feet high is broken and the top touches the ground
at a point 3 feet from the stem. What is the height of the break? Solve this problem,
which reappeared several centuries later in the writings of the Hindu mathematician
Brahmagupta.
9.11. The Jiu Zhang Suanshu implies that the diameter of a sphere is proportional
to the cube root of its volume. Since this fact is equivalent to saying that the
volume is proportional to the cube of the diameter, should we infer that the author
knew both proportions? More generally, if an author knows (or has proved) "fact
A," and fact A is logically equivalent to fact B, is it accurate to say that the author
knew or proved fact B? (See also Problem 9.6 above.)

9.12. Show that the solution to the quadrilateral problem of Sawaguchi Kazuyuki

is u = 9, í = 8, w = 5, χ = 4, y = yj(1213 + 69v^73)/40, æ = 10. (The

approximate value of y is 7.6698551.) From this result, explain how Sawaguchi

Kazuyuki must have invented the problem and what the two values 60.8 and 326.2