At the same time, however, and even earlier the so-called¹ Pythagoreans applied themselves to mathematics, and were the first to develop this science²; and through studying it they came to believe that its principles are the principles of everything. And since numbers are by nature first among these principles, and they fancied that they could detect in numbers, to a greater extent than in fire and earth and water, many analogues³ of what is and comes into being—such and such a property of number being justice ,⁴ and such and such soul or mind , another opportunity , and similarly, more or less, with all the rest—and since they saw further that the properties and ratios of the musical scales are based on numbers,⁵ and since it seemed clear that all other things have their whole nature modelled upon numbers, and that numbers are the ultimate things in the whole physical universe, 986athey assumed the elements of numbers to be the elements of everything, and the whole universe to be a proportion⁶ or number. Whatever analogues to the processes and parts of the heavens and to the whole order of the universe they could exhibit in numbers and proportions, these they collected and correlated; and if there was any deficiency anywhere, they made haste to supply it, in order to make their system a connected whole. For example, since the decad is considered to be a complete thing and to comprise the whole essential nature of the numerical system, they assert that the bodies which revolve in the heavens are ten; and there being only nine⁷ that are visible, they make the “antichthon”⁸ the tenth. We have treated this subject in greater detail elsewhere⁹; but the object of our present review is to discover from these thinkers too what causes they assume and how these coincide with our list of causes. Well, it is obvious that these thinkers too consider number to be a first principle, both as the material¹⁰ of things and as constituting their properties and states.¹¹ The elements of number, according to them, are the Even and the Odd. Of these the former is limited and the latter unlimited; Unity consists of both (since it is both odd and even)¹²; number is derived from Unity; and numbers, as we have said, compose the whole sensible universe. Others¹³ of this same school hold that there are ten principles, which they enunciate in a series of corresponding pairs: (1.) Limit and the Unlimited; (2.) Odd and Even; (3.) Unity and Plurality; (4.) Right and Left; (5.) Male and Female; (6.) Rest and Motion; (7.) Straight and Crooked; (8.) Light and Darkness; (9.) Good and Evil; (10.) Square and Oblong. Apparently Alcmaeon of Croton speculated along the same lines, and either he derived the theory from them or they from him; for [Alcmaeon was contemporary with the old age of Pythagoras, and]¹⁴ his doctrines were very similar to theirs.¹⁵ He says that the majority of things in the world of men are in pairs; but the contraries which he mentions are not, as in the case of the Pythagoreans, carefully defined, but are taken at random, e.g. white and black, sweet and bitter, good and bad, great and small. Thus Alcmaeon only threw out vague hints with regard to the other instances of contrariety, 986bbut the Pythagoreans pronounced how many and what the contraries are. Thus from both these authorities¹⁶ we can gather thus much, that the contraries are first principles of things; and from the former, how many and what the contraries are. How these can be referred to our list of causes is not definitely expressed by them, but they appear to reckon their elements as material; for they say that these are the original constituents of which Being is fashioned and composed. From this survey we can sufficiently understand the meaning of those ancients who taught that the elements of the natural world are a plurality. Others, however, theorized about the universe as though it were a single entity; but their doctrines are not all alike either in point of soundness or in respect of conformity with the facts of nature. For the purposes of our present inquiry an account of their teaching is quite irrelevant, since they do not, while assuming a unity, at the same time make out that Being is generated from the unity as from matter, as do some physicists, but give a different explanation; for the physicists assume motion also, at any rate when explaining the generation of the universe; but these thinkers hold that it is immovable. Nevertheless thus much is pertinent to our present inquiry. It appears that Parmenides conceived of the Unity as one in definition,¹⁷ but Melissus¹⁸ as materially one. Hence the former says that it is finite,¹⁹ and the latter that it is infinite.²⁰ But Xenophanes,²¹ the first exponent of the Unity (for Parmenides is said to have been his disciple), gave no definite teaching, nor does he seem to have grasped either of these conceptions of unity; but regarding the whole material universe he stated that the Unity is God. This school then, as we have said, may be disregarded for the purposes of our present inquiry; two of them, Xenophanes and Melissus, may be completely ignored, as being somewhat too crude in their views. Parmenides, however, seems to speak with rather more insight. For holding as he does that Not-being, as contrasted with Being, is nothing, he necessarily supposes that Being is one and that there is nothing else (we have discussed this point in greater detail in the Physics²²); but being compelled to accord with phenomena, and assuming that Being is one in definition but many in respect of sensation, he posits in his turn two causes, i.e. two first principles, Hot and Cold; or in other words, Fire and Earth. 987aOf these he ranks Hot under Being and the other under Not-being.²³ From the account just given, and from a consideration of those thinkers who have already debated this question, we have acquired the following information. From the earliest philosophers we have learned that the first principle is corporeal (since water and fire and the like are bodies); some of them assume one and others more than one corporeal principle, but both parties agree in making these principles material. Others assume in addition to this cause the source of motion, which some hold to be one and others two. Thus down to and apart from the Italian²⁴ philosophers the other thinkers have expressed themselves vaguely on the subject, except that, as we have said, they actually employ two causes, and one of these—the source of motion —some regard as one and others as two. The Pythagoreans, while they likewise spoke of two principles, made this further addition, which is peculiar to them: they believed, not that the Limited and the Unlimited are separate entities, like fire or water or some other such thing, but that the Unlimited itself and the One itself are the essence of those things of which they are predicated, and hence that number is the essence of all things. Such is the nature of their pronouncements on this subject. They also began to discuss and define the “what” of things; but their procedure was far too simple. They defined superficially, and supposed that the essence of a thing is that to which the term under consideration first applies—e.g. as if it were to be thought that “double” and “2” are the same, because 2 is the first number which is double another. But presumably “to be double a number” is not the same as “to be the number 2.” Otherwise, one thing will be many—a consequence which actually followed in their system.²⁵ This much, then, can be learned from other and earlier schools of thought.

The philosophies described above were succeeded by the system of Plato,²⁶ which in most respects accorded with them, but contained also certain peculiar features distinct from the philosophy of the Italians. In his youth Plato first became acquainted with Cratylus²⁷ and the Heraclitean doctrines—that the whole sensible world is always in a state of flux,²⁸ and that there is no scientific knowledge of it—and in after years he still held these opinions. 987bAnd when Socrates, disregarding the physical universe and confining his study to moral questions, sought in this sphere for the universal and was the first to concentrate upon definition, Plato followed him and assumed that the problem of definition is concerned not with any sensible thing but with entities of another kind; for the reason that there can be no general definition of sensible things which are always changing. These entities he called “Ideas,”²⁹ and held that all sensible things are named after³⁰ them sensible and in virtue of their relation to them; for the plurality of things which bear the same name as the Forms exist by participation in them. (With regard to the “participation,” it was only the term that he changed; for whereas the Pythagoreans say that things exist by imitation of numbers, Plato says that they exist by participation—merely a change of term. As to what this “participation” or “imitation” may be, they left this an open question.) Further, he states that besides sensible things and the Forms there exists an intermediate class, the objects of mathematics,³¹ which differ from sensible things in being eternal and immutable, and from the Forms in that there are many similar objects of mathematics, whereas each Form is itself unique. Now since the Forms are the causes of everything else, he supposed that their elements are the elements of all things. Accordingly the material principle is the “Great and Small,” and the essence 〈or formal principle〉 is the One, since the numbers are derived from the “Great and Small” by participation in the the One. In treating the One as a substance instead of a predicate of some other entity, his teaching resembles that of the Pythagoreans, and also agrees with it in stating that the numbers are the causes of Being in everything else; but it is peculiar to him to posit a duality instead of the single Unlimited, and to make the Unlimited consist of the “Great and Small.” He is also peculiar in regarding the numbers as distinct from sensible things, whereas they hold that things themselves are numbers, nor do they posit an intermediate class of mathematical objects. His distinction of the One and the numbers from ordinary things (in which he differed from the Pythagoreans) and his introduction of the Forms were due to his investigation of logic (the earlier thinkers were strangers to Dialectic)³²; his conception of the other principle as a duality to the belief that numbers other than primes³³ can be readily generated from it, as from a matrix.³⁴ 988aThe fact, however, is just the reverse, and the theory is illogical; for whereas the Platonists derive multiplicity from matter although their Form generates only once,³⁵ it is obvious that only one table can be made from one piece of timber, and yet he who imposes the form upon it, although he is but one, can make many tables. Such too is the relation of male to female: the female is impregnated in one coition, but one male can impregnate many females. And these relations are analogues of the principles referred to. This, then, is Plato’s verdict upon the question which we are investigating. From this account it is clear that he only employed two causes³⁶: that of the essence, and the material cause; for the Forms are the cause of the essence in everything else, and the One is the cause of it in the Forms. He also tells us what the material substrate is of which the Forms are predicated in the case of sensible things, and the One in that of the Forms—that it is this the duality, the “Great and Small.” Further, he assigned to these two elements respectively the causation of good³⁷ and of evil; a problem which, as we have said,³⁸ had also been considered by some of the earlier philosophers, e.g. Empedocles and Anaxagoras.