From Medial Straight Line arises Infinite Number of Irrational Straight Lines
Theorem
In the words of Euclid:
- From a medial straight line there arise irrational straight lines infinite in number, and none of them is the same as any of the preceding.
(The Elements: Book $\text{X}$: Proposition $115$)
Proof
Let $A$ be a medial straight line.
It is to be shown that there is an infinite number of irrational straight lines, and none of them are:
- Binomial
- First bimedial
- Second bimedial
- Major
- The side of a rational plus a medial area
- The side of the sum of two medial areas
- Apotome
- First apotome of a medial
- Second apotome of a medial
- Minor
- That which produces a medial whole with a rational area
- That which produces a medial whole with a medial area
Let the above set be called $\KK$.
Let $B$ be a rational straight line.
Let $C^2$ be equal to $A \cdot B$.
Therefore by:
and from:
it follows that:
- $C$ is irrational.
Consider a square on any of $\KK$.
None of them, when applied to a rational straight line, produces as breadth a medial straight line.
Let $D^2$ be equal to $B \cdot C$.
Therefore by:
and from:
it follows that:
- $D$ is irrational.
Consider a square on any of $\KK$.
None of them, when applied to a rational straight line, produces $C$ as breadth.
This process can be continued ad infinitum.
$\blacksquare$
Historical Note
This proof is Proposition $115$ 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