The History of Mathematics: A Brief Course

242 9. MEASUREMENT FIGURE 3. Dissection of a frustum of a pyramid. a length to an area or multiplying two areas. For that reason ...

MESOPOTAMIA 243 led to join the midpoints of its sides in order, that is, to draw the diagonals of the four copies of the orig ...

244 9. MEASUREMENT The strongest area of Mesopotamian science that has been preserved is as- tronomy, and it is here that geomet ...

CHINA 245 × FIGURE 5. Chinese illustration of the Pythagorean theorem. with astronomy and the applications of mathematics to t ...

246 9. MEASUREMENT survey the shapes of mountains and rivers. Apparently the Emperor had drainage canals dug to channel floods o ...

CHINA 247 FIGURE 6. The double-difference method of surveying. When we insert the appropriate values, we find, as did Zhao Shu ...

248 9. MEASUREMENT approximating the area of a 192-sided polygon.^6 That is, he started with a hexagon and doubled the number of ...

CHINA 249 FIGURE 7. The double square umbrella. problem occurs in Problem 20, in which a circle is said to have area 35,000 sq ...

250 9. MEASUREMENT is rotated, the ratio of the volumes generated is 2:3, while that of the original areas is ð : 4. Liu Hui not ...

CHINA 251 r Ë Ê Ë 2\/á^2 - h^2 2á FIGURE 8. Sections of the cube, double square umbrella, and sphere at height h. Hence each d ...

252 9. MEASUREMENT 4. Japan The Wasanists mentioned in Chapter 3, whose work extended from 1600 to 1850, inherited a foundation ...

JAPAN 253 U FIGURE 9. Sawaguchi Kazuyuki's quadrilateral problem. It is given that z^3 -u^3 = 271 u^3 - v^3 = 217 v^3 -y^3 = 6 ...

254 9. MEASUREMENT In Japan these techniques were first applied in the area called yenri (circle theory),^11 a topic that had be ...

JAPAN 255 0.00000 00000 33333 35111 11225 39690 66667 28234 77694 79595 875+. But this value does not fit with the procedure f ...

256 9. MEASUREMENT see the recursive operation: multiplication by (h/d)\2n^2 /(n + l)(2n + 1)]. Putting these corrections togeth ...

INDIA 257 FIGURE 10. Rounding a square. the practitioners of wasan were not immediately attracted to Euclid when his work arri ...

FIGURE 11. The Hindu variant of the double-difference method of surveying. This procedure gives a value of one-dimensional ð equ ...

INDIA 259 FIGURE 12. For a quadrilateral inscribed in a circle, the product of the diagonals equals the sum of the products of ...

FIGURE 13. The "bowstring" diagram. The sine of the arc AR is the line AB. In Fig. 13 the arc AR can be measured by either line ...

INDIA^261 formula (f3438 - N/3438^2 - sin^2 è 2 We can therefore well understand why Aryabhata did not refine his table fur- t ...