The Construction Of Regular Figures

Proposition 21

From 2 lines commensurable with one another in length, nothing can be made which should be taken into account here, whether the lines are Expressible, or Medial, or of lower standing.

If they are commensurable in length, the whole built up from them will be commensurable with the parts. Now a line commensurable with an Expressible line is Expressible: by the definitions before Euclid X,20.

A line commensurable with a Medial line is a Medial by 24 of the same.

A line commensurable with any of the Inexpressible lines that follow the Medials is of the same kind as it is, by 66, 67, 68, 69, 70, 103, 104, 105, 106, 107 [of Euclid And so it is also with the other further kinds of line, not mentioned by Euclid, which make more remote degrees [of knowledge].

And even if it were not so for them, it does not matter to us. For they either come down to one of the kinds [of line] which we shall now constitute from incommensurable lines; and thus do not increase the number [of degrees]: or they make lower kinds of their own or another type; and thus they do not belong at this point, where we are setting out the degrees which are closest in rank to those already described.

Part 22 Lines commensurable in Area

If 2 such Expressible lines are combined, they form a Binomial.

If they are subtracted, the remainder is an Apotome. This has 6 subordinate kinds of each as stated by Propositions 48 and 85 of Book 10 of Euclid.

If we combine 2 such Medials, which either form an Expressible Rectangle or a Medial one, they will make by addition “Bimedials”

By subtraction Medial Apotomes, the former takes their name from the Binomials, the latter from the Apotomes.

Here we may not join up an Expressible line with a Medial one: for two such lines are simply incommensurable, a type that will be discussed in the following section.

Proposition 23

No pair of incommensurable lines can produce the required resultants

This is because they are both Medial or one is Medial and the other Expressible.

In the one case because the pair is of low standing, in the other because the natures of the two lines of the pair are different. See Euclid X, 71, 108, 109.

So no kind of combination can be called in here: we are left only with the lines of lower standing, having excluded the Expressible and the Medial.