5th to 8th Degree of Knowledge

Part 17: The fifth degree of knowledge

This is when we have 2 lines which:

• are not both Expressible, nor both Medial
• are completely incommensurable with one another
• make both the sum of their squares and their common rectangle an expressible quantity, no less than each of these is made by two lines Expressible in length, by Euclid X.20, or also by two lines expressible only in square, but commensurable with one another in length, by the same [proposition of Euclid ].

Thus, the side of the square [of area] 2 and the side of the square [of area] 8 are in double ratio, because the squares are in the ratio 4 to 1.

Thus, although the sides are Inexpressible in length, they are commensurable with one another. Their squares, 2 and 8, add up to 10, an Expressible area.

If they are multiplied one by the other (which is to form [them into] a Rectangle) they make a rectangle of [area] 4, also Expressible.

This [i.e. an expressible rectangle] I say is also made by two lines which are neither Expressible nor Medial, and further are completely incommensurable with one another: and for this reason they are not, like the earlier ones, to be assigned to the second or third degree of knowledge, but to the Fifth.

Note therefore that in this degree we shall measure not the lines themselves, nor their individual squares, but instead we shall measure both the Rectangle formed from them and the sum of their squares; so what is lacking in one square, making it less expressible, is exactly compensated by the other square that is associated with it.

Part 18: The sixth and lower degree of knowledge

This is when 2 lines are joined which are neither expressible, nor Medial, both together, and are also incommensurable with one another, and only one of the areas they make is Expressible, while the other is Medial.

There are two cases [for lines of this degree]; for either the sum of the squares is expressible and the Rectangle is Medial; or the former is Medial and the latter expressible.

In the former case, the lines are like two expressible lines commensurable only in square.

For both the powers, that is the Expressible squares, also have a sum that is Expressible in each case. In fact their rectangle is Medial, by Euclid X, 2 2 P

In the latter case, the lines are like two Medial lines commensurable only in square, whose ratio to one another is as that of two Expressible lines between which the first of the two Medials is a mean proportional, by Euclid X, 26 and 28.

For because they are commensurable in square: when added the powers give a sum commensurable with the parts [i.e. the powers]. But the parts are Medial, and anything commensurable with a Medial is itself Medial, by Euclid X, 24:

In this latter case, we are measuring the Rectangle formed by the 2 lines by the area of the square of the diameter, but we cannot also measure the sum of the squares of the lines: for, for that, we can only find two lines which form a rectangle equal to it, and the squares of these lines we measure by the square of the diameter.

Part 19: The seventh still lower degree of knowledge

This is when neither resultant of 2 mutually incommensurable lines is expressible, neither the sum of their squares nor their Rectangle: but each is however Medial.

In this case, the lines are like two Medials commensurable only in square, one of which is to the other as one of those commensurable lines (that is to say commensurable only in square), between which the Medial truly is a mean proportional, is to some third line, commensurable only in square, by Euclid X, 29.^^

Euclid is particularly concerned with finding these three pairs of lines, distinguished by making two kinds of area, because they contribute to the composition and structuring of following kinds.’^'^

Part 20: The eighth degree of knowledge

This is a continuation of what has gone before. It refers to individual lines which are made up of 2 terms, as:

• 2 combinations of the preceding combinations, or
• the subtraction of one, called the Epharmozusa [conjugate], from the other partner, to make a new kind of line.

So that for these we know or measure not complete lines, not the squares of complete lines, not pairs of terms taken one from each, but their combined squares and their Rectangle, as in sections XVIII and X IX above.

There are as many degrees of knowledge as there are kinds of line. The earlier degree is always higher than the later.

Yet because any addition or subtraction refers to its degree, and no operation of addition or subtraction gives rise to diversity, but all are equally related to their pair of Terms or Elements: on this account we shall make them only one degree: but let us recognize that it contains kinds of lines that differ in standing.