261112 – Why Use x as the Unknown

by | Nov 12, 2026 | Physics | 0 comments

Why X Marks the Unknown

Thomas Lee Abshier, ND | 12 November 2026

A Small Letter with a Large Story

I recently came across a short piece asking why René Descartes chose the letter x for the unknown, and how that choice changed mathematics. It makes two claims worth testing. The first is that Descartes assigned letters on a deliberate system. The second is that the popular explanation, that x came from Arabic by way of a Spanish inability to say “sh,” is a myth.

I stand with the documented record. The convention of a, b, c for known quantities and x, y, z for unknowns traces to Descartes’ La Géométrie of 1637. The Spanish phonetic story, in the form usually told, does not survive examination, though a narrower version of it is not absurd. Why x came to dominate over y and z, no one knows for certain.

The letter is the small part of the story. The large part is the act the letter represents: giving a name to something you do not yet know, and then reasoning with it as rigorously as if you did. That act is the engine of algebra, and I will argue it is the engine of physics as well.

Before the Letter: Thing, Cosa, and Vowels

Algebra is far older than its letters. For most of its history, the unknown was written as a word. In the ninth century, al-Khwārizmī’s Kitāb al-jabr wa’l-muqābala, the book that gave us the word “algebra,” called the unknown shayʾ, meaning “thing.” Problems were stated in sentences: a thing and its square equal some number.

When Arabic algebra passed into Europe, the translators carried the meaning, not the sound. Latin writers called the unknown res, “thing.” Italian and Spanish algebraists called it cosa, also “thing,” and abbreviated it co. The word was so closely tied to the subject that German writers called algebra die Coss, the cossic art. Other traditions had their own devices: Indian mathematicians, when they needed several unknowns, named them by the first syllables of color words.

The decisive step toward symbols came from France. In 1591, François Viète published In artem analyticem isagoge, the first systematic letter notation in algebra. He used vowels for unknown quantities and consonants for known ones. This was a real revolution: for the first time, a general problem could be stated once, with letters standing for any given values. But Viète still wrote powers in words, quadratus and cubus, and his letters remained tied to geometric dimension, so that a length could not be added to an area.

What Descartes Did in 1637

Descartes published La Géométrie as an appendix to his Discourse on Method, printed in Leiden in 1637. In it he took the first letters of the alphabet, a, b, c, for known quantities. For unknowns he took the last letters, beginning with z and working backward through y and x. He also wrote powers as raised numbers, x³, much as we do today.

The pattern was not rigid in his private papers. In manuscripts written between 1629 and 1640, the unknown z appears only once, and x and y carry most of the work. In the third book of La Géométrie, the book on equations, x predominates. When he later used coordinates, he used only x and y.

The deeper change was not which letters he chose but how he used them. Descartes named every line in a problem, known or unknown, and wrote equations among them without distinction between the two kinds. He also chose a unit length, which let him treat the product of two lines as another line. That broke Viète’s rule that only like dimensions could be added. With that rule gone, x² + x was no longer an area plus a length; it was simply a quantity. Algebra had become a language of number, free of the pictures that had birthed it.

The Spanish “Sh” Story

The story runs like this. Arab mathematicians called the unknown al-shayʾ. Spanish scholars translating their texts had no “sh” sound, so they substituted the Greek chi, written Χ. Latin later turned Χ into x. The idea is old: a nineteenth-century German orientalist, Paul de Lagarde, proposed an Arabic origin in the 1880s, and the 1909–1916 edition of Webster’s Dictionary repeated a version of it. Terry Moore’s 2012 TED talk, with its punch line that x is unknown because you can’t say “sh” in Spanish, carried it around the world.

The story deserves its strongest form before it is judged. Old Spanish did have the “sh” sound, and it routinely spelled that sound with the letter x. That spelling survives in Don Quixote and in Mexican place names. A Spanish-language manual of Arabic printed around 1505, the work of Pedro de Alcalá, wrote the Arabic word for “thing” in Roman letters as xei. So a path from shayʾ to xei to x is linguistically possible.

That same evidence breaks the popular version. Spanish did not lack the sound; it had it and wrote it with x directly, so the detour through Greek chi is invented. More importantly, no one has produced a text in which an algebraist abbreviated xei as x. The Spanish algebraists who actually wrote equations wrote cosa, the Spanish word for “thing.” Descartes was French, began his unknowns with z, used x in private manuscripts by 1629, and never pointed to an Arabic source. Florian Cajori, whose History of Mathematical Notations (1929) remains the standard reference, found no historical evidence for the Arabic connection.

My verdict: Moore’s specific claim, that Spanish had no “sh,” is false on the evidence of Old Spanish spelling itself. The broader idea of an Arabic route is not disproven, only unsupported. A link no one has ever shown in writing is a conjecture, and the documented trail leads to Descartes.

The Printer Story

A second legend explains why x, rather than y or z, became the favorite. While La Géométrie was being typeset, the story goes, the printer ran short of type for the last letters of the alphabet. He asked Descartes whether it mattered which of the three he used, and Descartes said it did not. The printer then chose x most often, because French uses y and z more and he had more x type to spare.

This is a pleasing story with no document behind it. It also cannot explain the convention’s origin, since Descartes was already using x as an unknown in manuscripts years before the book went to press. At most it might help explain why x appears so often in the printed book. A German historian, Johannes Tropfke, offered another partial explanation: x resembles a sign that medieval writers used for the unknown, and Descartes knew that sign. Both ideas address frequency, not origin, and neither is proven.

The simplest account fits all the evidence: Descartes wanted one end of the alphabet for knowns and the other for unknowns, and x won out through use.

Why the Notation Transformed Mathematics

The post I am responding to is right that the notation unified algebra and geometry, and that is the heart of the matter. Once every line in a figure has a letter, a geometric condition becomes an equation. Once an equation relates two unknowns, x and y, it describes a curve: every pair of values that satisfies it marks a point. A circle, a parabola, or any curve can be studied by manipulating symbols, and any equation can be pictured as a shape.

Descartes was not alone. Pierre de Fermat reached the same union of algebra and geometry independently in the same years, writing in Viète’s vowel notation. But Descartes’ letters are the ones that spread. Frans van Schooten’s Latin editions carried La Géométrie across Europe, and it was among the handful of books the young Isaac Newton studied at Cambridge.

Three consequences followed:

  • Geometry became computable. Problems that had required a fresh construction each time could be solved by a general method.
  • Calculus became possible. Newton and Leibniz asked how a curve’s slope and area change, and they could only ask that of curves written as equations.
  • The limits of solvability became visible. In his third book, Descartes sorted problems by the degree of their equations and the curves needed to construct their solutions. Classifying problems by degree prepared the way for later proofs that some problems cannot be solved by certain means at all.

For physics, the result was decisive. Every law we write, from F = ma to the wave equation, is a relation among letters standing for quantities we measure or seek. Coordinates, the x and y axes, became the stage on which motion is described.

The Question Under the Question

Why does the origin of a single letter draw so much interest, and why does a charming false etymology travel faster than the true record? I think it is because x has come to mean mystery itself. When Wilhelm Röntgen found a ray in 1895 that he could not explain, he called it the X-ray. We sense that the letter holds something important, and we want its story to match.

The true story is better than the myth. The power of x is not in its shape or its sound. It is in the method it stands for, which the Greeks called analysis and which Viète deliberately revived: assume the thing you seek is already found, give it a name, and follow its relations to the things you know until the relations force its value. You reason about the unknown as a full citizen of the problem.

Physics has lived by this method. In 1846, Urbain Le Verrier treated an unseen planet as an unknown, reasoned from the irregularities in the orbit of Uranus, and predicted where it must be; Neptune was found within about a degree of his prediction. In 1930, Wolfgang Pauli proposed an unseen particle to keep energy conserved in radioactive beta decay; the neutrino was detected in 1956.

The method also carries a warning. Le Verrier later used the same reasoning to predict a planet, Vulcan, inside the orbit of Mercury. It was never found, because the unknown was not a missing object but a flaw in the framework, and general relativity resolved it in 1915. Naming an unknown does not make it real. The equation tells you what the unknown must be if your framework is right; observation decides whether it exists.

That is the discipline Descartes’ small letter carries: name what you do not know, reason with it honestly, and let the world answer.

Sources and Their Limits

This essay leans on the mainstream historical record, which for this question rests heavily on one scholar, Florian Cajori. I worked from published summaries of his findings, not from his volumes directly. I could not open the original post on MSN, so I responded to its text as shared with me. I did not read Lagarde’s nineteenth-century argument in the original. The case against the Arabic route is an absence of evidence in the archives historians have examined; an unexamined Spanish or Latin manuscript could still change the picture. The Le Verrier, Pauli, and Röntgen details are standard history of physics, stated without a separate citation.

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