A hair’s breadth
Al-Kashi chose his precision so that the circumference of a circle 600,000 times the Earth’s diameter would be out by less than the thickness of a horse’s hair.
39 / ANTIQUITY TO THE RENAISSANCE
Jamshid al-Kashi
How long is the rim of a circle of radius 1? Zoom in on the end of a cord that long, ten times closer each time, and every zoom gives one more digit.
Explore the mathematics ↓
01 / DECIMAL FRACTIONS
What does each decimal place pin down?
Find which tenth holds the end of the cord, then which tenth of that tenth, and so on. Each piece found is one digit.
Do the same in sixtieths, the fractions astronomers used, and compare how finely each pins the end down.
Count every piece in the smallest unit found: the decimal becomes a whole number of ten-thousandths.
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THE IDEADecimal places continue the place-value rule past the units: each place is a tenth of the one before, so each digit counts pieces ten times smaller. Counted in its smallest piece, a decimal fraction is a whole number, which is why al-Kashi could calculate with fractions as easily as with integers.
The person behind the ideas
- The leading astronomer and mathematician of Ulugh Beg’s scientific circle in Samarkand.
- In his Treatise on the Circumference (1424) he found 2π to nine sexagesimal places and turned it into sixteen decimal places.
- Completed The Key to Arithmetic in 1427.
More of the mathematics
The law of cosines
In France the law of cosines is called al-Kashi’s theorem, after the form in which he stated it for calculation.