Classification of Irrational Straight Lines derived from Binomial Straight Line
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Theorem
In the words of Euclid:
- The binomial straight line and the irrational straight lines after it are neither the same with the medial nor with one another.
(The Elements: Book $\text{X}$: Proposition $72$ : Summary)
Proof
From Proposition $22$ of Book $\text{X} $: Square on Medial Straight Line:
- the square on a medial, applied to a rational straight line, produces as breadth a rational straight line which is incommensurable in length with that to which it is applied.
- the square on a binomial, applied to a rational straight line, produces as breadth the first binomial.
- the square on a first bimedial, applied to a rational straight line, produces as breadth the second binomial.
- the square on a second bimedial, applied to a rational straight line, produces as breadth the third binomial.
- the square on the major, applied to a rational straight line, produces as breadth the fourth binomial.
- the square on the side of a rational plus medial area, applied to a rational straight line, produces as breadth the fifth binomial.
- the square on the side of the sum of two medial areas, applied to a rational straight line, produces as breadth the sixth binomial.
All of these breadths so produced differ from the first and from each other:
- from the first because it is rational
and:
- from each other because they are different in order.
$\blacksquare$
Historical Note
This proof is Proposition $72$ of Book $\text{X}$ of Euclid's The Elements.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{X}$. Propositions