Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/38

 Liu Hui published in 263 an Arithmetic in nine sections which contains a determination of $$\pi$$. Starting with an inscribed regular hexagon, he proceeds to the inscribed dodecagon, 24agon, and so on, and finds the ratio of the circumference to the diameter to be 157: 50, which is equivalent to $$\pi = 3{\cdot}14$$.

By far the most interesting Chinese determination was that of the great Astronomer Tsu Ch'ung-chih (born 430 ). He found the two values $22⁄7$ and $355⁄113$ (= 3.1415929…). In fact he proved that $$10\pi$$ lies between 31.415927 and 31.415926, and deduced the value $355⁄113$.

The value $22⁄7$ which is that of Archimedes he spoke of as the "inaccurate" value, and $355⁄113$ as the "accurate value." This latter value was not obtained either by the Greeks or the Hindoos, and was only rediscovered in Europe more than a thousand years later, by Adriaen Anthonisz. The later Chinese Mathematicians employed for the most part the "inaccurate" value, but the "accurate" value was rediscovered by Chang Yu-chln, who employed an inscribed polygon with 214 sides.

In the middle ages a knowledge of Greek and Indian mathematics was introduced into Europe by the Arabs, largely by means of Arabic translations of Euclid's elements, Ptolemy's 🇬🇷, and treatises by Appollonius and Archimedes, including the treatise of Archimedes on the measurement of the circle.

The first Arabic Mathematician Muhammed ibn Mûsâ Alehwarizmi, at the beginning of the ninth century, gave the Greek value $$\textstyle \pi = 3\frac{1}{7}$$, and the Indian values $$\pi = \sqrt{10}$$, $$\textstyle \pi = \frac{62832}{20000}$$, which he states to be of Indian origin. He introduced the Indian system of numerals which was spread in Europe at the beginning of the 13th century by Leonardo Pisano, called Fibonacci.

The greatest Christian Mathematician of medieval times, Leonardo Pisano (born at Pisa at the end of the 12th century), wrote a work entitled Practica geometriae, in 1220, in which he improved on the results of Archimedes, using the same method of employing the in- and circum-scribed 96agons. His limits are $$\frac{1440}{458\frac{1}{5}} = 3{\cdot}1427$$ and