Euclid’s Elements — Book XIII, Proposition 11

The Side of a Pentagon Inscribed in a Circle with a Rational Diameter is an Irrational Straight Line Called “Minor”

In Book XIII of Euclid’s Elements, Proposition 11 explores a fascinating relationship between geometry and irrationality. It deals with the side of an equilateral pentagon inscribed in a circle whose diameter is rational and shows that this side is an irrational straight line known in Greek geometry as the “minor” (μείων γραμμή).

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Statement of the Proposition

If an equilateral pentagon is inscribed in a circle with a rational diameter, then the side of the pentagon is the irrational straight line called “minor.”

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Key Construction

• Consider a circle ABCDE with a rational diameter.

• Inscribe within it an equilateral pentagon ABCDE.

• Join various auxiliary lines and points within the circle to analyze relationships between sides, diagonals, and radii.

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Core Geometric Idea

Euclid uses a combination of results from earlier books, particularly from Book X (on irrational magnitudes) and Book XIII (on constructing regular solids), to establish the following chain of reasoning:

1. Rational Diameter → Rational Segments
   Since the diameter of the circle is rational, several key segments derived from it (such as radii and proportional subdivisions) are rational as well.

2. Extreme and Mean Ratio
   In a pentagon, the relationship between side and diagonal involves the golden ratio.
   Euclid uses this fact to set up proportions that ultimately lead to commensurability in square only — meaning two segments share a rational square ratio but are themselves incommensurable in length.

3. Classification of Irrationals
   Using Book X, Euclid identifies the type of irrationality involved. The side of the pentagon turns out to belong to the special class called a fourth apotome.

4. Conclusion
   By classical Greek terminology, the square root of a rectangle formed by a rational straight line and a fourth apotome is called minor.
   Therefore, the side of the pentagon is precisely this minor irrational straight line.

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Mathematical Significance

This proposition reveals the deep connection between:

• Regular pentagons

• Golden ratios

• Irrational magnitudes

It also shows how Greek geometers classified different kinds of irrational lengths, a precursor to modern number theory.

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Historical Note

The term “minor” originates from Euclid’s classification of irrationals in Book X. Unlike our modern real numbers, the Greeks viewed different irrational quantities as belonging to distinct “species,” and the minor is one of these specially defined types.

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Summary

If a circle has a rational diameter and an equilateral pentagon is inscribed within it,

then the side of the pentagon cannot be expressed as a rational length.

It is an irrational straight line — specifically, the one Euclid names “minor.”