Treatise on Equations — Sharaf al-Din al-Tusi

The equation work of Sharaf al-Din al-Tusi, called al-Muadalat in Ozel's entry, survives through an anonymous abridgement. It treats classes through degree three, distinguishing classes with positive solutions from those depending on conditions. The associated positive-root analysis is a particular contribution, not a claim that all algebra originated in this book ^{1 2}.

The accounts describe twelve lower-degree types, eight positive-soluble cubic types and five conditional ones. MacTutor illustrates x³+a=bx with positive parameters through a maximum condition. The symbols are modern exposition; the biography states that Sharaf used words. It also discusses numerical approximation and differing interpretations of the relation between maximum reasoning and derivatives ³.

The classification counts are not a modern inventory of complex roots. Nor does a Horner-method comparison document a transmission route. The invention article separates historical attribution from independently checked teaching calculations and explains positive roots, repeated roots and distinct-root counts ⁴.

The account summarized by the biographies says the abridger removed numerical tables. Ozel discusses comparison with another work to assess the abridgement; that is known through the reference, not a collation undertaken here. Missing tables must not be filled with new numbers labelled as the author's. MacTutor also questions the description “commentary” when reporting Sarton's characterization, distinguishing it from abridgement ^{1 5}.

Ozel records Rashed's two-volume1986 Paris critical edition and translation, and Fares's1998 Beirut Arabic translation. No page citation to these unread editions is presented as direct inspection, and manuscripts were not checked. Neither is1209 imposed as an exact composition date: the biography warns about a lifetime inference resulting from a misread erased manuscript date ⁶.

Ozel names later arithmetic and algebra works that discuss Sharaf's treatment. Modern historical accounts compare maxima, existence and approximation, and MacTutor records differing derivative interpretations. These are attributed through the read sources. The museum neither resolves that disagreement nor places modern symbolic notation on the author's twelfth-century pages. Describing a work through actual references is narrower than claiming every original listed in their bibliographies was inspected ^{7 8}.