The story of the circle's ratio

Compare the three values al-Khwarizmi mentions and then al-Kashi's, and see how many places agree. What changes as the precision grows, and what stays unknown?

Al-Khwarizmi says in his chapter on mensuration that taking the circumference as three and a seventh times the diameter is "a convention among people, not a necessity". He gives two other views of the geometers: multiply the diameter by itself and by ten and take the root, and the astronomers' rule, multiply the diameter by 62,832 and divide by 20,000; all of them, he says, are close to one another ². In the problem of a circle of diameter seven cubits he takes the circumference as twenty-two cubits ³. Centuries later, MacTutor dates al-Kashi's Treatise on the Circumference to 1424 and describes expressing twice the circumference-to-diameter ratio in sexagesimal and decimal forms with high precision ⁴.

The museum's records preserve no sentence in which al-Kashi says why he sought this precision. Iranica, as the record of his computation reports, describes concern about large instruments warping, and the record says geometric, numerical and material accuracy are related without being identical ⁵. The biographies emphasise his combination of geometry and computation ⁶. Al-Khwarizmi, by contrast, lists the measurement of land among the needs he wrote his book for ⁷, and calls the usual value a convention, not a necessity ².

Al-Khwarizmi and the editors of the Cairo edition do not weigh the three values in the same way. He says all of them are close to one another ². The editors comment that the astronomers' value is the closest to the truth and the root of ten the furthest, and quote the manuscript's gloss that this is "an approximation, not an exact determination", because the circumference is not a straight line ⁸. They also note that twenty-two cubits, in the problem of a circle of diameter seven cubits, is approximate ³.

Al-Kashi is sometimes credited with inventing decimal fractions. The record reports that MacTutor places decimals and root extraction within earlier work, including al-Karaji's school and al-Samawal, so "used fractions skilfully" must not become "invented all decimal fractions". A chronological comparison with later European results does not prove their authors read him: earlier occurrence and documented textual transmission are different claims ⁹.

Okten prints 6;16,59,28,1,34,51,46,15,50 in sexagesimal and 6.2831853071795865 in decimal. Checking the literal printed sequence, however, reveals a discrepancy: it converts to approximately 6.2831853071795924, not the accompanying decimal. The record flags the mismatch without guessing a correction to a historical digit, and since the original was not read, it cannot locate the error in transmission, typography or elsewhere ¹⁰. Okten names Luckey's edition and German translation, and the museum has not read the original treatise or the German edition ¹¹.

Al-Khwarizmi calls the usual value a convention and says the three values are close to one another, while the editors rank them by closeness to the truth. Take a position: does the difference among the three values matter when a builder measures a circle of diameter seven cubits? What is the strongest objection to your position, and how do you answer it?

• Mensuration (ilm al-misaha): the three values as the record reports them, and the editors' comment.
• Al-Kashi's high-precision circumference computation: the number in two systems, and the record's warning against confusing the ratio with twice the ratio.
• Treatise on the Circumference: what the museum has read of the treatise's description, and what it has not.
• Ghiyath al-Din al-Kashi: the record of the treatise's author.