Classification of Irrational Straight Lines derived from Binomial Straight Line

From ProofWiki
Jump to navigation Jump to search

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.


From Proposition $60$ of Book $\text{X} $: Square on Binomial Straight Line applied to Rational Straight Line:

the square on a binomial, applied to a rational straight line, produces as breadth the first binomial.

From Proposition $61$ of Book $\text{X} $: Square on First Bimedial Straight Line applied to Rational Straight Line:

the square on a first bimedial, applied to a rational straight line, produces as breadth the second binomial.

From Proposition $62$ of Book $\text{X} $: Square on Second Bimedial Straight Line applied to Rational Straight Line:

the square on a second bimedial, applied to a rational straight line, produces as breadth the third binomial.

From Proposition $63$ of Book $\text{X} $: Square on Major Straight Line applied to Rational Straight Line:

the square on the major, applied to a rational straight line, produces as breadth the fourth binomial.

From Proposition $64$ of Book $\text{X} $: Square on Side of Rational plus Medial Area applied to Rational Straight Line:

the square on the side of a rational plus medial area, applied to a rational straight line, produces as breadth the fifth binomial.

From Proposition $65$ of Book $\text{X} $: Square on Side of Sum of two Medial Area applied to Rational Straight Line:

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