The story, and the caution it needs
The popular version goes like this: a member of the Pythagorean brotherhood discovered that the diagonal of a square cannot be expressed as a ratio of whole numbers, and was thrown from a boat and drowned for revealing a truth that threatened the school’s belief that everything is number. It is a vivid tale. It is also much less secure than it sounds. The mathematics is real and ancient; the details of who found it, when, and what happened to him come from writers who lived centuries later.
What incommensurability means
Two lengths are commensurable if some smaller length fits a whole number of times into each. The Greeks found that the side and the diagonal of a square have no such common measure. In modern language, the square root of 2 is irrational: it cannot be written as a fraction of two whole numbers. The sources in the Greek tradition speak of incommensurable magnitudes or of ratios that cannot be expressed, using the word “alogos”, roughly “without ratio” or “unspeakable”. The modern term “irrational” comes through Latin translation of that language, and is only loosely connected to reason or madness.
The argument behind the discovery
A standard argument, which Aristotle alludes to as a proof by contradiction using odd and even, runs as follows. Suppose the diagonal and side were in the ratio of two whole numbers, p to q, with the fraction reduced as far as possible. Squaring gives p squared equal to twice q squared, so p squared is even, and therefore p is even. Write p as twice some number; then q squared is also even, so q is even as well. But then p and q share the factor 2, contradicting the assumption that the fraction was reduced.
Whether the first discoverers argued exactly this way is unknown. Some historians have suggested the early discovery may have come from geometry, perhaps through repeated subtraction of a smaller length from a larger one without end, rather than from this arithmetic. That is a reconstruction, not a documented fact.
Who was Hippasus?
Hippasus of Metapontum, a town in southern Italy, is named by later writers as an early Pythagorean, active around the fifth century BCE. Beyond that, little is firmly known. Different ancient accounts credit him with different things: with work on the numerical basis of musical intervals, with a doctrine that differed from the school’s leadership, and with the discovery of the dodecahedron, one of the Platonic solids, or of the incommensurable. The accounts disagree about which of these brought about his fate.
How late are the sources?
The tale of drowning is told mainly in late antiquity. Iamblichus, writing around the late third or early fourth century CE, reports that someone who made the irrational or the dodecahedron public perished at sea, and some versions name Hippasus. That is roughly 700 years after the events supposedly took place. Another version appears in a commentary on Euclid’s Book X, preserved in Arabic and associated with the writer Pappus, that says the one who first revealed the irrational was punished by drowning, without clearly naming him.
None of these writers had access to contemporary records of the early Pythagoreans, whose own practices were secretive and whose teaching was largely oral. Scholars such as Walter Burkert have argued that the whole story reads as later legend, and that the connection between Hippasus and the irrational may be a retrospective guess. Other historians are somewhat more willing to accept a kernel of truth. The safest summary is that the sources do not establish the story.
A school built on whole numbers?
The legend depends on the idea that Pythagorean number philosophy held that everything could be expressed through whole-number ratios, so that incommensurability would be a scandal. There is evidence that ratios mattered to the Pythagoreans, notably in music, where simple ratios of string lengths give consonant intervals. But how much of the later picture of a doctrinal crisis reflects what the early school actually believed is disputed. Some scholars think the discovery was simply absorbed by mathematicians without any dramatic reaction.
What the Greek mathematical record does show
The solid evidence is mathematical rather than biographical. In Plato’s dialogue Theaetetus, Theodorus of Cyrene is described as showing that the square roots of several numbers, from 3 up to 17, are not commensurable with a unit, stopping at 17. The young Theaetetus then generalised the idea. Euclid’s Elements, compiled around 300 BCE, includes a long Book X classifying incommensurable lines, and Greek geometry as a whole adapted to the discovery by treating lengths as magnitudes rather than as numbers.
It is also worth remembering that Greek mathematicians did not treat “number” as including such quantities. For them numbers were collections of units, so a ratio between lines that could not be expressed as a ratio of numbers was a different kind of object. Our wider notion of real numbers developed much later.
Reading the legend
Taken together, the sources suggest a tidy moral tale grafted onto a genuine discovery. The mathematics shows that incommensurable magnitudes were known to Greek geometers by the later fifth or fourth century BCE. The detail of a drowning, the identity of the victim and the idea of a crisis in the school remain uncertain, and an honest account should leave them marked as such.