(6) And the numbers introduced are sixfold, as three hundred is six times fifty; and tenfold, as three hundred is ten times thirty; and containing one and two-thirds (epidimoiroi), for fifty is one and two-thirds of thirty.

(9) the Pure intelligence so called because it purifies the Numerations, it proves and corrects the designing of their representation, and disposes their unity with which they are combined without diminution or division.

(16) But here we may be questioned about these numbers which we describe as the primal and authentic:

"Where do you place these numbers, in what genus among Beings? To everyone they seem to come under Quantity and you have certainly brought Quantity in, where you say that discrete Quantity equally with the continuous holds place among Beings; but you go on to say that there are the numbers belonging to the Firsts and then talk of other numbers quite distinct, those of reckoning; tell us how you arrange all this, for there is difficulty here. And then, the unity in sense-things- is that a quantity or is quantity here just so many units brought together, the unity being the starting-point of quantity but not quantity itself? And, if the starting-point, is it a kindred thing or of another genus? All this you owe it to us to make clear."

Be it so; we begin by pointing out a distinction:

You take one thing with another- for we must first deal with objects of sense- a dog and a man, or two men; or you take a group and affirm ten, a decad of men: in this case the number affirmed is not a Reality, even as Reality goes in the sphere of sense, but is purely Quantity: similarly when you resolve into units, breaking up the decad, those units are your principle of Quantity since the single individual is not a unity absolute.

But the case is different when you consider one man in himself and affirm a certain number, duality, for example, in that he is at once living and reasoning.

By this analysis and totalling, you get quantity; but there are two objects under consideration and each of these is one; each of the unities contributes to the complete being and the oneness is inherent in each; this is another kind of number; number essential; even the duality so formed is no posterior; it does not signify a quantity apart from the thing but the quantity in the essence which holds the thing together. The number here is no mere result of your detailing; the things exist of themselves and are not brought together by your reckoning, but what has it to do with essential reality that you count one man in with another? There is here no resultant unity such as that of a choir- the decad is real only to you who count the ten; in the ten of your reckoning there cannot be a decad without a unitary basis; it is you that make the ten by your counting, by fixing that tenness down to quantity; in choir and army there is something more than that, something not of your placing.

But how do you come to have a number to place?

The Number inherent apart from any enumeration has its own manner of being, but the other, that resulting upon the appearance of an external to be appraised by the Number within yourself, is either an Act of these inherent numbers or an Act in accordance with them; in counting we produce number and so bring quantity into being just as in walking we bring a certain movement into being.

But what of that "Number within us having its own manner of being"?

It is the Number of our essence. "Our essence" we read "partakes of Number and harmony and, also, is Number and harmony." "Neither body nor magnitude," someone says: soul, then, is Number since it is essence. The number belonging to body is an essence of the order of body; the number belonging to soul constitutes the essences of souls.

In the Intellectuals, all, if the Absolute Living-Form, there is a multiple- a triad, let us say- that Triad of the Living-Form is of the nature of essence: and the Triad prior to any living thing, Triad in the realm of Being, is a principle of essence.

When you enumerate two things- say, animal and beauty- each of these remains one thing; the number is your production; it lay within yourself; it is you that elaborate quantity, here the dyad. But when you declare virtue to be a Tetrad, you are affirming a Tetrad which does actually exist; the parts, so to speak, make one thing; you are taking as the object of your act a Unity- Tetrad to which you accommodate the Tetrad within yourself.

(5) Another marked case is that of Zerah Colburn, the mathematical prodigy, whose feats attracted the attention of the scientific world during the last century. In this case, the child under eight years of age, without any previous knowledge of even the common rules of arithmetic, or even of the use and powers of the Arabic numerals, solved a great variety of arithmetical problems by a simple operation of the mind, and without the use of any visible symbols or contrivances. He could answer readily a question involving the statement of the exact number of minutes or seconds in any given period of time. He could also state with equal facility the exact product of the multiplication of any number containing two, three, or four figures by another number consisting of a like number of figures. He could state almost instantly all the factors composing a number of six or seven places of figures. He could likewise determine instantly questions concerning the extraction of the square and cube roots of any number proposed, and likewise whether it was a prime number incapable of division by any other number, for which there is no known general rule among mathematicians. Asked such questions in the midst of his ordinary childish play, he would answer them almost instantly and then proceed with his play.

(13) We turn to ask why Quantity is not included among the primary genera, and Quality also.

Quantity is not among the primaries, because these are permanently associated with Being. Motion is bound up with Actual Being , since it is its life; with Motion, Stability too gained its foothold in Reality; with these are associated Difference and Identity, so that they also are seen in conjunction with Being. But number is a posterior. It is posterior not only with regard to these genera but also within itself; in number the posterior is divided from the prior; this is a sequence in which the posteriors are latent in the priors . Number therefore cannot be included among the primary genera; whether it constitutes a genus at all remains to be examined.

Magnitude is in a still higher degree posterior and composite, for it contains within itself number, line and surface. Now if continuous magnitude derives its quantity from number, and number is not a genus, how can magnitude hold that status? Besides, magnitudes, like numbers, admit of priority and posteriority.

If, then, Quantity be constituted by a common element in both number and magnitude, we must ascertain the nature of this common element, and consider it, once discovered, as a posterior genus, not as one of the Primaries: thus failing of primary status, it must be related, directly or indirectly, to one of the Primaries.

We may take it as clear that it is the nature of Quantity to indicate a certain quantum, and to measure the quantum of the particular; Quantity is moreover, in a sense, itself a quantum. But if the quantum is the common element in number and magnitude, either we have number as a primary with magnitude derived from it, or else number must consist of a blending of Motion and Stability, while magnitude will be a form of Motion or will originate in Motion, Motion going forth to infinity and Stability creating the unit by checking that advance.

But the problem of the origin of number and magnitude, or rather of how they subsist and are conceived, must be held over. It may, thus, be found that number is among the primary genera, while magnitude is posterior and composite; or that number belongs to the genus Stability, while magnitude must be consigned to Motion. But we propose to discuss all this at a later stage.