Why are $X$ and $Y$ commonly used as mathematical placeholders?

Question

I realize that $X$ and $Y$ are relatively popular terms when wanting to use a placeholder for an unknown English or math term. What is the origin of this term, and why was it $X$ and $Y$; why not the other letters?

Answer 1

They became popular because of René Descartes’ usage in his La Géométrie. The letters at the end of the alphabet are chosen as the variables, while those at the beginning are constants. There is speculation about why this might have been done. It is likely to allow the largest number of sequential letters without overlap between the two sets.

Why x became the most common is unknown. Some sources attempt to draw a line from the Arabic word for unknown through the Greek letter chi (which resembles a capital X), but the claims are unsubstantiated (Arabic being the source of our numerals and Greek being the common letter set for variables).

The link from mathematics to common speech is likely just a simple repurposing of known concepts.

Answer 2

See Earliest uses of mathematical symbols, which quotes F. Cajori, A History of Mathematical Notations, 2 volumes (1928-29)

The use of z, y, x ... to represent unknowns is due to René Descartes, in his La géometrie (1637). Without comment, he introduces the use of the first letters of the alphabet to signify known quantities and the use of the last letters to signify unknown quantities.

Answer 3

Here is the original source : René Descartes, La Géométrie (1637), I, page 299, for $a,b$ used to denote parameters.

And see I, page 301 for $z$ and I, page 303 for $x,y$ respectively, used to refer to an unknown quantity.

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Letters was already used by François Viète (but the use of alphabetical variables to represent magnitudes is due to euclidean geometry).

See In artem analyticem isagoge (1591), Rule III :

Sunto duae magnitudines $A$ & $B$. [Let there be two magnitudes, $A$ and $B$.]

And also :

Oportet $A \dfrac {\text { plano }}{B}$ addere $Z$ [Suppose $Z$ is to be added to $A^p / B$].

But obviously the success of Descartes' "new geometry" explains the success of the new algebraic notation.