By the 15th century Samarkand in central Asia had become one of the greatest centres of civilisation. The Persian mathematician and astronomer Jamshid al-Kashani or al-Kashi (d. 1429) made extensive calculations with decimal fractions and proposed a notation for them, using a vertical line to separate the integer and fractional parts. A prodigious calculator, he determined pi to 16 decimal places and obtained a very precise value for the sine of 1°, from which many other trigonometrical values could be determined.

Al-Kashani’s patron was the Turkish astronomer Ulugh Beg (1394–1449), whose observatory contained a special 130-foot sextant, the largest of its type in the world. Using al-Kashani’s value for sin 1°, Ulugh Beg constructed extensive tables for the sine and tangent of every angle for each minute of arc to five sexagesimal places.

[Iran 1979; Russia 1987; Turkey 1983]