Book 10
23
[And in the same way as was explained in the case of rationals [Lemma following X. 18] it follows, as regards medials, that a straight line commensurable in length with a medial straight line is called medial and commensurable with it not only in length but in square also, since, in general, straight lines commensurable in length are always commensurable in square also.
23
But, if any straight line be commensurable in square with a medial straight line, then, if it is also commensurable in length with it, the straight lines are called, in this case too, medial and commensurable in length and in square, but, if in square only, they are called medial straight lines commensurable in square only.]
PROPOSITION 24.
24
The rectangle contained by medial straight lines commensurable in length is medial.
24
For let the rectangle AC be contained by the medial straight lines AB, BC which are commensurable in length; I say that AC is medial.
24
For on AB let the square AD be described; therefore AD is medial.
24
And, since AB is commensurable in length with BC, while AB is equal to BD, therefore DB is also commensurable in length with BC; so that DA is also commensurable with AC. [VI. 1, X. 11]
24
But DA is medial; therefore AC is also medial. [X. 23, Por.] Q. E. D.
PROPOSITION 25.
25
The rectangle contained by medial straight lines commensurable in square only is either rational or medial.
25
For let the rectangle AC be contained by the medial straight lines AB, BC which are commensurable in square only; I say that AC is either rational or medial.
25
For on AB, BC let the squares AD, BE be described; therefore each of the squares AD, BE is medial.
25
Let a rational straight line FG be set out, to FG let there be applied the rectangular parallelogram GH equal to AD, producing FH as breadth, to HM let there be applied the rectangular parallelogram MK equal to AC, producing HK as breadth, and further to KN let there be similarly applied NL equal to BE, producing KL as breadth; therefore FH, HK, KL are in a straight line.
