Source on Nicomachus

“His native place was Gerasa probably the modern Jerash about 56 miles northeast of Jerusalem. Two treatises bear the name of Nicomachus, the Introductionis arithmeticæ libri duo and the Enchiridion harmonikon. ...A third treatise called Theologoumena arithmetica is anonymous but is probably the work of Nicomachus.”

“In mathematical studies he is among the first to attempt a systematic treatment of Arithmetic distinct from Algebra.”

“Nicomachus is preparing for the answer, mathematical knowledge, and so he says, the knowledge which distinguishes accurately "the accidents of the Existents." Accident may refer either to a contingent or to a necessary attribute... Here it is the latter. To distinguish such accidents seems at first glance an incredibly difficult task. But Nicomachus, like his descendants, simplifies it by reducing the accidents to two: Magnitude and Multitude.”

“Nicomachus... mentions the customary Pythagorean divisions of quantum and the science that deals with each. Quantum is either discrete or continuous. Discrete quantum in itself considered, is the subject of Arithmetic; if in relation, the subject of Music. Continuous quantum, if immovable, is the subject of Geometry; if movable, of Spheric (Astronomy). These four sciences formed the quadrivium of the Pythagoreans. With the trivium (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire.”

“Nicomachus gives three definitions of number. The first, "determinate multitude," is by genus and difference, and is ascribed by Iamblichus to Eudoxus. The second, "a group (i.e., organized complex) of units," [Iamblichus] ascribes to Thales and the Egyptians. The third stream of quantity composed of units Philoponus explains as another attempt to distinguish the particular kind of quantum treated in Arithmetic. The Unit was conceived either mystically as an Idea whose "essence" passes in some way into concrete individuals and even into the Ideas to organize them, or spatially and temporally as the boundary of individuals. The former conception gave rise to fantastic speculations on the cosmic meaning of number, examples of which Nicomachus has... given us; the latter gave rise to the figurate numbers...”