Treatise on the Circumference

Al-Kashi's Treatise on the Circumference computes the circumference-to-diameter ratio with high precision. MacTutor dates it to July1424 in the Samarkand scientific setting. It differs from the Key to Arithmetic and the Chord and Sine. The museum links it to the circumference contribution, not to an origin claim for all circle computation ¹.

Iranica describes inscribed and circumscribed polygons yielding a more precise approximation. Okten prints twice pi in sexagesimal and decimal forms, the latter6.2831853071795865. This approximates twice the ratio, not pi itself. Base60 positions differ from decimal places: conversion represents rounded output differently without creating additional information ^{2 3}.

The literal sexagesimal line on TDV's page does not match the decimal when converted: it gives approximately6.2831853071795924. The invention page explicitly flags this without guessing a historical correction. We have not read the original to locate the error and cannot attribute every web digit to the author's manuscript ³.

The contribution is careful numerical work, not invention of the circle or all decimal fractions. MacTutor identifies predecessors in decimals and roots, distinguishing them from al-Kashi's additions. Small bounds and recurrences on the invention page are independently checked modern examples, not literal translations of historical steps or proof of a huge polygon count not established from a read witness ⁴.

Okten names Paul Luckey's study and German translation, published by Adolf Siggel in Berlin1953, and a Russian translation. These are reported bibliography, not editions collated here. MacTutor separates the historical achievement from Europe's later awareness of it. A common mathematical target does not establish direct descent for every later calculation ^{5 6}.

We have not read the original manuscript or German edition. This record supplies no autograph page; the modern image linked to the invention illustrates only inscribed and circumscribed polygons. Letters, the arithmetic textbook and instruments illuminate other parts of the career, not interchangeable witnesses to this particular treatise ⁷.