Expressible Lines

Proposition 24

An Expressible line emerges when the first pair of such completely incommensurable lines are added or subtracted

These lines are those described in Part 17 as fifth degree knowable.

They are necessarily Binomial and Apotome, see Euclid X, 112, 113, 114.

As when both the sum of the squares of a Binomial and of an Apotome, and their Rectangle, is Expressible, it is necessary that the individual Terms of the one should be commensurable with the individual Terms of the other, which is not the case for all Binomials and Apotomes.

Because two such lines which have the two required resultants necessarily form a Binomial and an Apotome.

This is proved in the same way as [Euclid] X, 33, except that for two pijrai<; Suvapsi povov [lines expressible only in the power] we use pqxai pqKEi [lines expressible in length] and for the word psaov [medial] we substitute pqxov [expressible]: and finally we use the definition of the Binomial and the Apotome.

By the addition and subtraction of a Binomial and an Apotome, with the required resultants, we get back to an Expressible line is seen as follows.

For when the sum of the squares is Expressible, and the Rectangle is Expressible; adding the lines together, the square [of the sum] will be composed of the square of each line, and twice the rectangle of the lines, that is, it is composed of two parts which are Expressible: so the whole square will be Expressible: thus so too will be the composite line, whose power is equal to the square.

Let:

• the Binomial be Xp, its square ko
• the apotome be XO, its square 6 k
• 6 k and ko be taken together be Expressible
• the Rectangle made of OX, Xp also be expressible

Two such rectangles Kp, k ^ complete the whole square of the composite line Op, which square is 0o.

For subtraction the proof is as follows.

If the line composed of OX, pX, that is Op, is expressible, half of it. On, will also be Expressible (as the larger term) and nX the smaller term, and the other half [of Ofj., namely] nii; take from it na equal to the line dX, and the remainder will also be Expressible, and also the complete line Xo, that is double the line na.

But Xo is the remainder after subtracting the Apotome fio from the Binomial Xfx. Thus the remainder is Expressible.

Part 25: The 12 kinds of Euclidiean Quantities

From the second pair of the sixth degree*, let us:

• add them to get a Mizon also called a Major
• subtract them to get an Elasson or Minor

*Part 18, consisting of lines completely incommensurable with one another the sum of whose squares is Expressible, while their Rectangle is Medial

From the third pair*, let us:

• add them to get the side of a square that is Expressible and Medial
• subtract them to make a Complete Medial with an Expressible

*where the sum of the squares is Medial and the Rectangle is Expressible

From a fourth pair of the seventh degree*, let us:

• add them to get the side of a square that is Bimedial
• subtract them [a quantity which is said] to Make with a Medial a Complete Medial.

*Section 19 where each resultant^’ is a Medial

Here is the Origin of the 12 kinds of quantities treated by Euclid and the explanation for their Number.

Euclid did not consider that he needed to go on to consider more remote kinds [of quantities] which as the sum of their squares, or as their common Rectangle, or both, go beyond the Expressible or the Medial to produce still lower kinds of quantity.