Degrees of Knowledge

Part 12: Various degrees of knowledge

Some knolwedge are distant, some close.

The first and closest degree is when I know some line and can show that it is equal to the diameter or that a plane surface, which may be formed in another way, is equal to the square of the diameter.

Here the given measure perfectly, that is of itself and by one act, measures the thing that is knowable.

Part 13: The second degree of knowledge

This is when if the diameter is divided into a certain definite number of equal parts, or its square is similarly divided, then the line or plane surface we are given is equal to one or more such parts. Such a line is called in Greek ^TjXT| giiKei, expressible in length. Such an area is simply called pt|Tov, expressible. For number is the medium of expression for Geometers.’

We arrive at this degree of knowledge either by description and inscription; or alternatively by its relationship with some other quantity at which we arrive by those means.

On that account, this quality does not determine any quantity; for, as far as I know, it is not sufficient to determine it that I should know something which is compared with it in this way or that for the sake of measurement; I must also know how, that is by what number, it is expressible.

Part 14: The third degree of knowledge

This is when the line is inexpressible in length but its square is Expressible and belongs to the second degree. It is said to be ^>T|Tfi 6uvdgei, “Expressible in square.”

Part 15: The Inexpressible Knowledge

These follow the expressible knowledge.

Latin translators have rendered this as “Irrational” running a great risk of ambiguity and absurdity.

Let us stop this usage.

There are many lines which, although they are Inexpressible, are defined by the best computations.

Arithmeticians, by a similar translation, refer to deaf Numbers, that is numbers which cannot speak any more than a deaf man can hear: but under this name they include numbers Expressible only in square as well as inexpressible quantities.

Thus, the fourth degree in order, and the first in fact of inexpressible quantities, is when neither the line nor its square is Expressible; but never theless the Square can be transformed into a Rectangle such that its sides are Expressible at least in square.

This line is called Medial, because it is a mean proportional between two expressible lines commensurable only in square: as when one is Expressible in length and the other only in square; or if each is Expressible only in square, but the ratio between the squares is not that of one square number to another.'^

Such a line is not known or measured by the length of an aliquot part or parts of the diameter, nor is its square [measured by] the square of the diameter; but neither can the two lines between which the Medial is a mean proportional both together be measured by the Diameter; but as for the squares of these lines, these finally can be measured by the square of the diameter.

The square of a Medial [line] is also itself called Medial, whether it takes the form of a square or turns into a Rectangle: so we have this other type of Area, following the Expressible area: And the following kinds [of area] can be distinguished into these two types of area, the Expressible and the Medial.^*^

Part 16: Individual Lines

We now come to different individual lines, through the combination of pairs of lines which themselves also introduce new degrees of knowledge.

For let us cut either a diameter, or a line commensurable with the diameter at least in square and thus Expressible, or even a Medial line; I say let us cut if into two unequal parts, or let there be made up, from sections of any two such parts of any kind, either by the addition of parts, or by having their squares formed by addition, or subtraction, of such [parts], two lines, I say, that are of different types: they will either be commensurable with one another in length; or, although incommensurable in length, nevertheless commensurable in square.

Here, though the individual lines clearly have moved away from commensurability, yet when some of them are combined, either by their squares being put together, or by their being taken as sides to a Rectangle, they make up areas, [such as] those already described, no less than do those [lines] which are commensurable with one another.

Since the combination of two such completely incommensurable lines may take many forms, each sinking lower and lower, we shall not be able to assign every pair to a single degree.