Proof
Commensurable-in-square-only means the square on each is rational
but the lengths are not in integer ratio. The rectangle is then in
a non-rational ratio to a rational area; its square root is the
medial straight line (Definition XIII.3).
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Depends on (3)
- X.9Proposition X.9The squares on straight lines commensurable in length have to one another the ratio which a square number has to a…
- X.3Definition X.3With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and…
- XIII.3Definition XIII.3A medial straight line is the mean proportional between two rational straight lines commensurable in square only.
Required by (dependents) (9)
- X.22Proposition X.22The square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line…
- X.23Proposition X.23A straight line commensurable with a medial straight line is medial.
- X.24Proposition X.24The rectangle contained by medial straight lines commensurable in length is medial.
- X.25Proposition X.25The rectangle contained by medial straight lines commensurable in square only is either rational or medial.
- X.26Proposition X.26A medial area does not exceed a medial area by a rational area.
- X.27Proposition X.27To find medial straight lines commensurable in square only which contain a rational rectangle.
- X.28Proposition X.28To find medial straight lines commensurable in square only which contain a medial rectangle.
- X.36Proposition X.36If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be…
- X.115Proposition X.115From a medial straight line there arise irrational straight lines infinite in number, and none of them is the same with…
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