On Nicomachus's Introduction to Arithmetic

IAMBLICHUS

IN NICOMACHI ARITHM. INTROD.

it is necessary. For once the necessary theorems in mathematics have been determined in advance, we are led by them toward the more perfect theories of numbers; for it is clear that those agree with this one. As for the conception of it, it is worthy to present it not as existing in mere notions, nor as something coming into existence later upon sensible things, nor as something stripping away and separating ghosts from sensible things, but as common notions capable of applying to all numbers that exist in any way whatsoever. It is now our task to speak regarding such a mathematical arithmetic.

We find indeed that Nicomachus, in accordance with the judgment of Pythagoras, has given an account of it in his Arithmetical Art. For the man is great in mathematics, and he had as guides regarding them those most experienced in mathematics. Without these, he transmits an admirable order and theory, along with an admirable proof of the scientific principles, precisely as a science. He knows how to construct an argument regarding them, and he transmits the theorems pure and genuine, in no way clouded by foreign opinions. Furthermore, he is versatile and multi-faceted, ordered and articulated in the knowledge of numbers. The universal and inventive aspect of knowledge is present in him in abundance; for he tracks the first constitution and the first generation of numbers. He also possesses the quality of omitting nothing; for he arrived commonly at all the genera and species of numbers, and within the finite...