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1. Commensurability

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   The rigorous investigation into commensurability is attributed to the Pythagoreans, most famously documented in Euclid’s Elements (Book X). The Pythagoreans initially sought to prove that all measurable quantities in the cosmos, particularly the sides and diagonals of geometric figures, must be commensurable, viewing this as a necessary foundation for numerical harmony.
   The discovery of the incommensurability of the side of a square and its diagonal ($\sqrt{2}$), often attributed to [Hippasus of M…

2. Irrational Number

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   Historical Context and Discovery
   The earliest recognized discovery of irrationality is attributed to the Pythagoreans, specifically Hippasus of Metapontum, around the 5th century BCE. Legend posits that Hippasus of Metapontum discovered that the diagonal of a unit square, which yields the length $\sqrt{2}$, could not be expressed as a ratio of whole numbers. This revelation reportedly caused significant philosophical distress among the [Pythagorea…

3. Pythagorean Cosmology

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   The Pythagorean faith in the primacy of integer ratios faced a profound crisis upon the discovery of incommensurable magnitudes. When the ratio of two quantities cannot be expressed as the ratio of two integers, the resulting value is termed irrational. The classic example, which reportedly caused significant philosophical distress among the Pythagoreans, is the side length of a unit square relative to its diagonal, $\sqrt{2}$.
   The d…