Part I on Nicomachus

“The philosophical arithmetic of the Greeks, αριθμητική, of which the arithmetic of Nicomachus is a specimen, corresponds in a measure to our number theory; the subject was designed for mature students as a preparation for the study of philosophy, and was not at all intended for children. Arithmetica is... the study of that which is implied in number.”

“The Introduction to Arithmetic of Nicomachus is but a restatement of facts which were common property... long before him, and that, except for the few unimportant propositions the discovery of which our author... claims for himself, the book is largely unoriginal. This naturally leads to the inference that the Introduction must be closely connected with other mathematical treatises, which served as the fountains... Because so little remains of this literature, [our inference] is difficult to demonstrate...”

“Iamblichus, when he refers to the Introduction as the... Art of Arithmetic, exactly describes it... The Introduction belongs... among the artes or... concise, practical descriptions and systematic expositions of the principles of various arts and sciences, a type of treatise exceedingly common in ancient times, and one which... made scant claim to originality. Designed for the use of students... they may best be compared to the modern school and college text-book. ...we must consider that the Introduction to Arithmetic differs from the great original treatises of Diophantus and Heron. Because in clearness, conciseness, compendiousness, orderly arrangement and adaptability for scholastic use, it satisfied the demands of seekers after education or general information, it remained the standard work of its class for many centuries.”

“Nicomachus planned his Introduction so as to explain the mathematical principles involved in the difficult Platonic passages concerning the world-soul in the Timaeus and the marriage-number in the Republic.”

“In geometry we begin with the point, which is indimensional. This is the beginning of the first dimensional form, the line, and by movement the point generates the line. Now Nicomachus had a similar idea of the nature of multitude and number; they form a series, as it were a moving stream, which proceeds out of unity, the monad. Just as the point is not part of the line (for it is indimensional, and the line is defined as that which has one dimension), but is potentially a line, so the monad is not a part of multitude nor of number, though it is the beginning of both, and potentially both. The monad is unity, absence of multitude, potentiality; out of it the dyad first separates itself and 'goes forward' and then in succession follow the other numbers.”