Ludolf's number in the Arab Empire
In the 7th century A.D., the world was astonished by the rapid development of the Arab Empire. The rulers of these lands — the caliphs — supported astronomy and mathematics. In Baghdad, they built an observatory with a library. Arab mathematicians had a very wide range of interests. Among other things, they dealt with the area of a circle and the volume of a sphere — that is, the problem of the number π.
al-Khwarizmi (780–850 A.D.) presented three values for the number π.
The work of al-Khwarizmi played a significant role in mathematics and its history. Through him, the decimal positional system, Indian numerals, and Arabic algebra reached Europe.
al-Buzjani (940–998) worked on creating tables of trigonometric functions, calculating higher roots, and spherical geometry. In his work “Book for Scribes,” negative numbers were used for the first time. He proposed his own value for π.
al-Biruni (937–1048), while studying the quadrature of the circle, found that the length of an arc on a unit circle corresponding to a central angle of 20 is l = 0.0349648.
al-Kashi (died 1429) based his determination of π on the perimeters of inscribed and circumscribed regular n-gons. He calculated π as the arithmetic mean of the perimeters of an inscribed and a circumscribed 805,306,368-gon and found:
This value is accurate to 16 decimal places.
From history, we know that some influential figures in Islamic states were hostile toward mathematical knowledge. According to legend, Caliph Umar is said to have ordered the destruction of scientific books with the reasoning: “If they contain what is in the Quran, they are useless — they must be destroyed. If they contain what is not in the Quran, they are dangerous — they must be destroyed.”