Aristotelis Opera (Bekker Edition, Vol. 2): Metaphysica, Ethica, Politica, Rhetorica, Poetica

PRELEGOMENA TO PHILOSOPHY BY DAVID.

Here the commentators, having found an opportunity, come and discuss the numbers of existing things up to the decade. They say that the monad original: "μονάς" is so called from the word "to remain" original: "μένειν". The monad, in whatever number it may appear, preserves the same species; for example, "once three" original: "ἅπαξ τρία"... since these types exist, we say that the monad is not a number, but the principle of number. If, therefore, the monad is one, and it forms a larger number when added to its opposite than when multiplied, it is clear that it is not a number; for "once one" is one, but "one plus one" is two. So much for the monad. The dyad original: "δυάς" is said to be so named from "to go in two" original: "δυέναι" or "to pass through," for the dyad was the first to separate itself from the monad. It must be known that the dyad is strictly a number; for, as we have learned, numbers multiplied by themselves form a larger number than when added to themselves, and the dyad forms the same number whether added or multiplied. The number three is named from "the being unshakable" original: "ἀταρὴς" and "indefatigable." It is also said to be so called because it cannot be divided into equal parts.

Section 17.

It must be known that the tetrad original: "τετράς" is said to be so named as if it were "firm" original: "ἑδρανής", that is to say, steady and permanent. Note that the tetrad forms a square shape, and a square shape is difficult to move because it has many parts on the ground. The pentad original: "πεντάς" is named as if "in all" original: "ἐν πᾶς"; for the number five is composed of the monad and the tetrad, and the tetrad is called "all" because there are four elements original: "σαχαῖα" from which all bodies are composed, or because the four is the number that, when added to those before it, completes the decade. The hexad original: "ἑξάς" is named as if "arranged" original: "ἑξεσσάς" from the arrangement of its own parts. The heptad original: "ἑπτάς" is named as if "venerable" original: "σεβαστά" from "reverence" original: "σεβασμόν" and "honorable." The number seven is honorable, whence there are seven days in the week, seven wandering stars note: planets, and seven vowels. The number seven is also called "the occasion" original: "καιρός" and "Athena." It is called "occasion" because at this time the changes occur; for in the seventh month infants grow teeth, and in the seventh year they lose their teeth. It is called "Athena" because, just as they mythologize Athena to be a virgin and motherless (having emerged from the head of Zeus, as they mythologize), in the same way the number seven alone among the numbers within the decade is neither born from the multiplication of another number, nor does it beget another number within the decade by multiplication. The ogdoad original: "ὀγδοάς" is named as if "leading the dyad" original: "ἀγοδυάς" because it leads two; for when bisected it proceeds down to the monad. The ennead original: "ἐννάς" is named from "in new" original: "ἐν νέον"; for it, when multiplied, brings a new number by the reduction of one monad from nine down to the monad, for instance, twice nine is eighteen... see how the nine, when multiplied, undergoes a decrease or subtraction of one monad. The decad original: "δεκάς" is named as if "the receiver" original: "δεχάς"; for the ten receives and contains within itself all the numbers before it.

Section 19.

It must be known that Plato divides the theoretical part differently than Aristotle. Plato subdivides the theoretical into the physiological and the theological, and he does not consider the mathematical

to be a part of philosophy, but rather a kind of preparatory exercise, just like grammar and rhetoric. Aristotle, however, subdivides the theoretical into the physiological, the mathematical, and the theological. The mathematical is the middle, being between the physiological and the theological, as it participates in both. It is "material" similar to the physiological, and "immaterial" similar to the theological; for, as has been said, the mathematical is material in its foundation but immaterial in its conception. The theological is necessarily last. For it is not possible to have the physiological, since one must not proceed immediately from things that are entirely material to things that are entirely immaterial, because we suffer the same fate as those who have spent a long time in darkness and, looking immediately at the sun, are blinded. Thus, one must not proceed immediately from the entirely material to the entirely immaterial. Thus, the poet note: Homer, Odyssey 11.314, Iliad 5.386 speaking poetically says this about Otus and Ephialtes: "They strove to set Ossa upon Olympus, and upon Ossa Pelion with its waving leaves, that heaven might be accessible." In the literal sense, he says that Otus and Ephialtes wanted to place mountains on top of each other, wishing to engineer an ascent into heaven; but understood allegorically, these things show that they wished to proceed immediately from physical and entirely material things to the knowledge of divine things. Plotinus also shows this, saying, "The mathematical sciences are to be handed down to the young for the assimilation of incorporeal nature, through which they know the incorporeal nature."

Furthermore, as the Pythagoreans say, it is for this reason alone that it is called "mathematical" original: "μαθηματικόν", because it has its existence in the mind original: "διάνοια"; for only the mind learns, whereas the intellect knows everything by a simple encounter.

Section 20.

It must be known that we have five chapters to say about mathematics. The first chapter is where we say how many and what kinds of mathematics there are; the second is where we say why there are so many; the third is where we speak of their order; the fourth is where we say whose discoveries they are; the fifth is where we say what is adjacent to these kinds.

It must be known that there are four kinds of mathematics: arithmetic, music, geometry, and astronomy. Arithmetic and music take precedence over geometry and astronomy because arithmetic and music deal with discrete quantity, while geometry and astronomy deal with continuous quantity, and discrete quantity is more honorable than continuous quantity. One can also state another reason thus: Arithmetic corresponds to the monad, music to the dyad, geometry to the triad, and astronomy to the tetrad. Therefore, it acquired this order in proportion to the order of the numbers. Arithmetic corresponds to the monad, dealing with quantity in itself, and quantity in itself is one thing. Music corresponds to the dyad; for it deals with quantity in relation, and a relation is taken as at least two.

Geometry corresponds to the triad; for geometry deals with plane figures, and the first figure is the triangle, because one line cannot form a figure. Astronomy...