Euclid’s Elements — Book XIII, Proposition 6

When the Golden Ratio Produces an Apotome

In Book XIII of Euclid’s Elements, Proposition 6 connects the golden ratio with a special type of irrational straight line called an apotome.

This result belongs to the deeper theory of irrational magnitudes developed in Book X and shows that when a rational straight line is divided in the extreme and mean ratio — the golden ratio — each of the resulting segments is irrational and belongs to the special class called apotome.

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Proposition Statement

If a rational straight line is cut in extreme and mean ratio, then each of the resulting segments is the irrational straight line called an apotome.

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Geometric Setup

• Let AB be a rational straight line.

• Cut it in extreme and mean ratio at C, so that:

  $$\frac{AB}{AC}=\frac{AC}{C\mathrm{B}}$$

  with AC being the greater segment.

• The goal is to prove:

  • Both AC and CB are apotomes.

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Key Result

Euclid proves that:

• The greater segment AC is an apotome.

• The lesser segment CB is also an apotome.

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Sketch of the Proof

This proof combines results from Book X (on irrational magnitudes) with properties of the golden ratio:

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1. Relationship from Proposition XIII.1

From Proposition XIII.1 we know that:

$$C{D}^2=5\cdot D{A}^2,,$$

where D is the midpoint of AB.

• Since the ratio $CD^2 : DA^2$ is a ratio of numbers, the squares are commensurable.

• However, the lengths CD and DA themselves are incommensurable.

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2. Commensurable in Square Only

Because:

• AB is rational,

• DA is half of AB and thus also rational,

• and CD is incommensurable in length with DA,

we conclude that CD and DA are commensurable in square only.
According to Book X, this means AC is an apotome.

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3. The Lesser Segment CB

From the property of the golden ratio:

$$AB\cdot BC=A{C}^2,$$

we deduce that when the square on an apotome (AC) is applied to the rational line AB, it produces BC as a breadth.

By Euclid X.97, the resulting breadth is also a first apotome.

Thus, CB is also an apotome.

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What Is an Apotome?

In Euclid’s classification of irrationals in Book X, an apotome is a type of irrational line obtained by subtracting from a rational line another rational line commensurable in square only with it.

Here, the golden ratio ensures that both AC and CB exhibit precisely this property.

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Summary

If a rational straight line is divided in the golden ratio,
then both segments are irrational apotomes.

• AC → an apotome

• CB → an apotome

This deepens the connection between golden ratio geometry and the classification of irrational magnitudes.

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Why This Matters

This proposition is important because it links rational geometry with irrational classifications in Greek mathematics.

The golden ratio doesn’t just produce irrational numbers — it produces a very specific kind of irrational, the apotome, which plays a central role in Euclid’s later work on the pentagon, decagon, and the icosahedron.