Al-Mahani's Treatise on Ratio

TDV lists a Treatise on Ratio by al-Mahani, with additional title forms transliterated as Risala fi al-shakl min amr al-nisba and fi al-mushkil min al-nisba. We give neither every form a separate book ID nor claim to have collated identical copies. Dold-Samplonius lists three ratio texts while questioning whether they differ. That uncertainty describes the witnesses, not permission to inflate a catalogue ^{1 3}.

The subject belongs to the ratio tradition around Euclid's fifth book. Dold-Samplonius describes a successive-quotient definition comparing the behaviour of magnitudes during a greatest-common-measure process. This does not mean the treatise invented the Euclidean algorithm or every continued fraction ².

In the historian's account these mathematicians accepted Euclid's definition as correct, but found that its synthetic presentation did not explain how ratio theory arose. The proposed account follows successive comparison quotients, making matching sequences a criterion for proportionality. Its pre-Eudoxian background and al-Mahani's reference to Thabit ibn Qurra remain visible: ratio was not created from nothing ⁴.

Dold-Samplonius says al-Mahani and al-Nayrizi did not establish the connection with Euclid's definition; Ibn al-Haytham did so later. Correct museum examples involving rational numbers do not remove that limitation. Our arithmetic is a modern explanation, not a complete proof recovered from an unread original ².

MacTutor's al-Nayrizi biography says he adopted concepts proposed by al-Mahani, who probably worked in Baghdad before him. That does not establish a personal meeting or transfer of a particular copy. A present library location is not automatically the place where the work was composed ^{5 1}.

We place the work within the author's ninth-century lifetime horizon without an exact composition date or invented Hijri conversion. His Sphere and Cylinder commentary remains a separate work: a shared author does not merge a ratio question with a cubic-equation problem ⁶.

We read TDV's list, the Dold-Samplonius biography and bibliography, and the al-Nayrizi biography. We did not inspect original manuscripts or read the Plooij and Becker studies listed in that bibliography during this session. Reports remain attributed to those historians; differing witness descriptions remain unresolved. This is not a new critical edition or a textual genealogy of every copy ^{7 1}.