| A straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order. | ||
| Let AB be an apotome of a medial straight line, and let CD be commensurable in length with AB.
I say that CD is also an apotome of a medial straight line and the same in order with AB. Since AB is an apotome of a medial straight line, let EB be the annex to it. | ||
| Then AE and EB are medial straight lines commensurable in square only. | X.74
X.75 | |
| Let it be contrived that AB is to CD as BE is to DF. Then AE is also commensurable with CF, and BE with DF. | VI.12
V.12 X.11 | |
| But AE and EB are medial straight lines commensurable in square only, therefore CF and FD are also medial straight lines commensurable in square only. | X.23
X.13 | |
| Therefore CD is an apotome of a medial straight line. | X.74
X.75 | |
| I say next that it is also the same in order with AB.
Since AE is to EB as CF is to FD, therefore the square on AE is to the rectangle AE by EB as the square on CF is to the rectangle CF by FD. | ||
| But the square on AE is commensurable with the square on CF, therefore the rectangle AE by EB is also commensurable with the rectangle CF by FD. | V.16
X.11 | |
| Therefore, if the rectangle AE by EB is rational, then the rectangle CF by FD is also rational, and if the rectangle AE by EB is medial, the rectangle CF by FD is also medial. | X.Def.4
X.23,Cor. | |
| Therefore CD is an apotome of a medial straight line and the same in order with AB. | X.74
X.75 | |
| Therefore, a straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order. | ||
| Q.E.D. | ||
Book X Introduction - Proposition X.103 - Proposition X.105.