Harmonic Proportion: The Edition of Risāla Taʿlīfīyyah by Abu Isḥāq Kūbanānī

Harmonic mean, based on the concept of harmonic proportion, along with the arithmetic and geometric means are the three principal means discussed on ancient and medieval mathematics. This article intends to provide a survey of the concepts of harmonic mean and harmonic proportion in history of mathematics. In the first section the authors introduce the harmonic proportion and some relations obtained on its basis. In the second part a short history of this subjects is presented, while in the third part the monographs written in the Islamic period are introduced. In the two remaining sections, Abū Isḥāq Kūbanānī’s treatise on harmonic proportion, entitles Risālat Taʾlīfīyyah, has been introduced accompanied by a critical edition and mathematical commentary.