The Brethren of Purity, the anonymous adepts of a tenth-century esoteric fraternity based in Basra and Baghdad, hold an eminent position in the history of science and philosophy in Islam due to the wide reception and assimilation of their monumental encyclopaedia, the Epistles of the Brethren of Purity. This compendium contains fifty-two epistles offering synoptic accounts of the classical sciences and philosophies of the age; divided into four classificatory parts, it treats themes in mathematics, logic, natural philosophy, psychology, metaphysics, and theology, in addition to didactic fables. Epistles 6 to 8 are from the first division of the Epistles, on the propaedeutical and mathematical sciences. Epistle 6 develops ideas concerning natural numbers and their arithmetic, geometric, and harmonic ratios, marked by the influence of Nicomachus of Gerasa and of Euclid. The Brethren here emphasize practical applications of proportionality in music, medicine, and alchemy. Epistle 7 addresses theoretical scientific knowledge as directed towards the spiritual realities of souls, the goal of which is to actualize human potential; this epistle also presents a remarkable classification of sciences. Epistle 8 surveys material cultures in the Islamic mediaeval milieu, embellished by a consideration of the effects of the heavenly bodies on the predisposition of individuals to follow specific trades. These three epistles are, of course, underpinned by the Brethren's perennial tropes of the microcosm/macrocosm analogy and the emanative hierarchy of existents.