Treatise on the Chord and Sine

The Treatise on the Chord and Sine is associated with al-Kashi's sine-one-degree computation, joining trigonometry and approximate iterative cubic calculation as described by scholarship. It is not another title for the Circumference or Key to Arithmetic, nor automatically identical to every sine work attributed to Qadi Zada ^{1 2}.

MacTutor relates the problem to angle trisection and a cubic, acknowledging al-Biruni's earlier approximation. Iranica reports seventeen digits in a decimal expansion, not a numerical witness we independently checked. Qadi Zada's biography summarizes accuracy at order10^-12; that differing formulation remains visible rather than silently corrected ^{1 3 4}.

The invention page uses a modern triple-angle relation, sin(3θ)=3*sin(θ)-4*sin(θ)^3, and a small-root iteration. This is neither original notation nor a verified original sequence. We have not read the text to identify historical input computation or rounding control. An explanatory example cannot fill those gaps by attributing modern work backward.

MacTutor calls this al-Kashi's final work, possibly unfinished at his death and completed by Qadi Zada. Iranica lists its title in the introduction to the1427 Key. Mentioning a title does not settle completion of all parts. Neither an exact composition year nor a copy/commentary date is supplied from the death year ^{1 5}.

TDV calls Qadi Zada's work a commentary, reporting Tehran1299 printing and Russian translation. His MacTutor biography instead describes a method different from al-Kashi's. We have not collated these texts to determine whether every report concerns the same work or stage. These descriptions do not automatically establish joint authorship, nor disappear for a simpler story ^{2 4}.

The evidence here is the specified actual articles, not original treatise, commentary, Dilgan or technical DSB pages they reference. We certify no autograph, copy count or unseen page. MacTutor's account of modern historical interest is not direct transmission to a particular European algorithm ^{6 7}.