Proof
The sum with , rational and incommensurable in length
has square where is medial (X.21), so the
square on the sum is the sum of a rational and a medial: irrational.
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Depends on (2)
Required by (dependents) (9)
- X.37Proposition X.37If two medial straight lines commensurable in square only and containing a rational rectangle be added together, the…
- X.39Proposition X.39If two straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle…
- X.42Proposition X.42A binomial straight line is divided into its terms at one point only.
- X.48Proposition X.48To find the first binomial straight line.
- X.54Proposition X.54If an area be contained by a rational straight line and the first binomial, the side of the area is the irrational…
- X.66Proposition X.66A straight line commensurable in length with a binomial straight line is itself also binomial and the same in order.
- X.71Proposition X.71If a rational and a medial area be added together, four irrational straight lines arise, namely either a binomial, a…
- X.73Proposition X.73If from a rational straight line there be subtracted a rational straight line commensurable with the whole in square…
- X.111Proposition X.111The apotome is not the same as the binomial.
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