Lee Hartman, Southern Illinois University Carbondale
March 2016
The origin of “x the unknown” is not unknown. The Swiss-American historian of mathematics Florian Cajori effectively laid the question to rest in his 1919 article “How x Came to Stand for Unknown Quantity,” which he later elaborated and subsumed into his two-volume work on mathematical notation in general (Cajori 1919; 1928:379-384). Cajori convincingly credits René Descartes (La Géométrie, 1637) with the first published use of the symbol x for the principal unknown quantity in an algebra problem. The attribution to Descartes is generally accepted by historians of mathematics today (e.g. Bashmakova & Smirnova 2000:78; Boyer 1991:180; Burton 1995:319; Derbyshire 2006:93; Gow 1884:x; Hodgkin 2005:149; Katz & Parshall 2014:238; Kleiner 2007:10; Sasaki 2003:229; Sesiano 1999:125; and Swetz 2013:110).
Nevertheless, a rival theory, based on a putative Hispano-Arabic line of transmission, continues to circulate in popular culture today. This theory is one of several that Cajori (1928:382-383) examines, but finds lacking in support:
...
Nor is there historical evidence to support the statement found in
[the 1909 edition of] Noah Webster’s Dictionary, under
the letter x,
to the effect that “x
was used as an abbreviation of Ar. shei
‘a thing, something’, which, in the Middle Ages, was used to designate
the unknown, and was then prevailingly transcribed as xei.”
The purpose of this study is (1) to further document the absence of evidence in support of the Hispano-Arabic theory, and (2) to demonstrate how that theory nevertheless gained the popularity that it enjoys today.
The use of symbols to stand for unknown quantities is characteristic only of the most recent of three stages into which the history of algebra is traditionally, but somewhat arbitrarily, divided. In the earliest, the “rhetorical” stage, algebraic concepts were expressed entirely verbally, with the unknown quantity represented by words equivalent to “thing,” “heap,” “measure,” “part,” and others. The second stage, “syncopated” algebra, combined verbal expressions with abbreviations and symbols, as if in transition to the present, “symbolic” stage, in which all quantities and relationships are expressed in symbols, such as x, y, the root sign, the fraction bar, etc. Rhetorical algebra prevails from the time of the Babylonians and Egyptians (ca. 1650 B.C.) to approximately 200 A.D., when the mixed expression of syncopated algebra begins to appear. Fully symbolic algebra comes into its own after 1500. This tripartite division was originated by G. H. F. Nesselmann (1842) (but see Heeffer (2007) for a contrary view). It is necessary to be aware of this progression of stages in order to appreciate the reasoning that led from an Arabic word to its supposed transcription and abbreviation in the symbol x.
Through the history of algebra, the unknown has been represented variously by words including equivalents of “heap” (ancient Egypt); names of colors (India); accented Greek letters; a dot (India); names of coins or words meaning “thing”, “measure”, or “part” (Arabic sources); vowels of the Roman alphabet (vs. consonants for known quantities); uppercase letters (vs. lowercase for known quantities—or vice versa); and— in Descartes’s case—letters from the end of the alphabet (z, y, x...) for unknowns, vs. those from the beginning (a, b, c...) for knowns (Cajori 1928:379-381). At first Descartes used z for the first unknown, but later opted for x. Eneström (1905:317) suggested that this might have been because x was more abundant in the Latin and French typographical fonts of the time. Meanwhile Johnson (1994:xxx)—as quoted by Derbyshire (2006:93)— suggests, somewhat conversely, that it was because x was less in demand for printing words in French, and thus more available for algebraic expressions. (But according to Trost (2016), x has twice the frequency of z in modern French.)
Cajori reviews a handful of theories for the origin of algebraic x, including ones that we might call (1) the “German R” theory, (2) the “crossed numeral” theory, and (3) the Hispano-Arabic theory. But he rejects all three as undocumented, and opts instead for the Cartesian source.
A unique symbol for the unknown quantity (Figure 1, below) was used prior to Descartes’s Géométrie by some German and English mathematicians. It appears in Robert of Chester’s 1145 Latin translation of Al-Khwārizmī’s Algebra (at least in fifteenth-century manuscripts thereof). Bear in mind that Al-Khwārizmī (ninth century) wrote in the tradition of rhetorical algebra and used the Arabic words جذر jidhr ‘root’ and شيء shayɂ ‘thing’, rather than abbreviations or symbols, for the unknown (Cajori 1928:336; Katz & Parshall 2014:139). The symbol was also used by Christoff Rudolph in 1525, Michael Stifel in 1553, and others (Cajori 1919:698). It was interpreted by some German authors as a cursive letter R, for Latin radix ‘root’ or res ‘thing’. It was suggested (by Treutlein 1879:32; Tropfke 1902:150; and Curtze 1902:473) that Descartes might have misinterpreted this symbol as the letter x; but this conjecture was falsified by Eneström (1905:316-317 and 405-406), according to Cajori, with the evidence that Descartes, in a 1619 letter, had used the “German” symbol alongside, and distinct from, his own x (Cajori 1919:698; 1928:382). Pietro Cataldi (1548-1626) used the convention of a “crossed one”—the numeral 1 with an oblique strikethrough—to represent the first power of the unknown. Gustav Wertheim (as cited in Eneström 1905:317) speculated that this could be the origin of Descartes’s x; but Cajori, again, finds “nothing to support” that hypothesis (1919:699). The convention of crossed numerals “failed of adoption by other mathematicians” (Cajori 1928:381).
Figure 1: Symbol for unknown used by some early German writers
and interpreted by some as a cursive R.
Cajori (1919:699) concludes: “Any one [sic] reading Descartes’ Géométrie will see that his manner of introducing symbols for known and unknown quantities seems free from tradition and their choice purely arbitrary.”
But what about the Hispano-Arabic theory? What does it consist of; where did it come from; why was it adopted; and what is the evidence against it? The hypothesis relies on two assumptions: (1) that the Arabic word شيء shayɂ ‘thing’—which the Arab mathematicians of the rhetorical tradition did use to name the unknown quantity—was transcribed by early Spanish mathematicians as xei (and/or xai, xey, or xay); and (2) that the Spanish transcription was eventually abbreviated as x. The transcription of the Arabic letter shīn as x is quite plausible: Old Spanish had an “sh” sound in many native words, which was routinely spelled “x”. (In modern times the sound has migrated to the back of the throat, becoming phonetically a velar [x], and orthographically has been largely replaced by j.) We know about the medieval pronunciation, in part, because of Spanish words that were loaned to other languages, a well-known example being the name of the literary character Don Quixote. Even as late as the novel was published (1605), there was at least sufficient memory of the palatal “sh” sound for the French and Italian translations to reflect it in their spellings: Don Quichotte and Don Chisciotte, respectively. Additional testimony comes from sixteenth-century Spanish transcriptions of native Mexican languages (Canfield 1934).
For the Hispano-Arabic theory, Cajori gives no source earlier than the 1909 “Noah Webster’s Dictionary,” cited above, but its origin can be traced to 1882 and the German “orientalist” Paul de Lagarde. Around the same time (1883), Lagarde published his edition of De lingua arabica, a language manual in Spanish by Pedro de Alcalá, first published in 1505. In that book, Alcalá makes Arabic more accessible to Spanish-speaking readers by bypassing the Arabic script and transcribing Arabic words in the Roman alphabet. And indeed the glossary lists a number of phrases in which the Spanish word for ‘thing’, cosa, is equated with its Arabic counterpart شيء, transcribed as xei. This is good evidence of its transcription, but not of its use in algebra. Having argued the plausibility of the hypothesis, Lagarde says “Nothing was more natural than for 12 with [Arabic letter] shīn over it to be represented by 12x = xai in Latin letters” (“Nichts war also natürlicher als 12 mit darüber stehendem ﺵ= š = شيء lateinisch durch 12x = xai zu geben.”) (Lagarde 1882:411; 1884:136).
Others took the ball and ran with it. An obituary of Lagarde says
The use of symbols to stand for unknown quantities is characteristic only of the most recent of three stages into which the history of algebra is traditionally, but somewhat arbitrarily, divided. In the earliest, the “rhetorical” stage, algebraic concepts were expressed entirely verbally, with the unknown quantity represented by words equivalent to “thing,” “heap,” “measure,” “part,” and others. The second stage, “syncopated” algebra, combined verbal expressions with abbreviations and symbols, as if in transition to the present, “symbolic” stage, in which all quantities and relationships are expressed in symbols, such as x, y, the root sign, the fraction bar, etc. Rhetorical algebra prevails from the time of the Babylonians and Egyptians (ca. 1650 B.C.) to approximately 200 A.D., when the mixed expression of syncopated algebra begins to appear. Fully symbolic algebra comes into its own after 1500. This tripartite division was originated by G. H. F. Nesselmann (1842) (but see Heeffer (2007) for a contrary view). It is necessary to be aware of this progression of stages in order to appreciate the reasoning that led from an Arabic word to its supposed transcription and abbreviation in the symbol x.
Through the history of algebra, the unknown has been represented variously by words including equivalents of “heap” (ancient Egypt); names of colors (India); accented Greek letters; a dot (India); names of coins or words meaning “thing”, “measure”, or “part” (Arabic sources); vowels of the Roman alphabet (vs. consonants for known quantities); uppercase letters (vs. lowercase for known quantities—or vice versa); and— in Descartes’s case—letters from the end of the alphabet (z, y, x...) for unknowns, vs. those from the beginning (a, b, c...) for knowns (Cajori 1928:379-381). At first Descartes used z for the first unknown, but later opted for x. Eneström (1905:317) suggested that this might have been because x was more abundant in the Latin and French typographical fonts of the time. Meanwhile Johnson (1994:xxx)—as quoted by Derbyshire (2006:93)— suggests, somewhat conversely, that it was because x was less in demand for printing words in French, and thus more available for algebraic expressions. (But according to Trost (2016), x has twice the frequency of z in modern French.)
Cajori reviews a handful of theories for the origin of algebraic x, including ones that we might call (1) the “German R” theory, (2) the “crossed numeral” theory, and (3) the Hispano-Arabic theory. But he rejects all three as undocumented, and opts instead for the Cartesian source.
A unique symbol for the unknown quantity (Figure 1, below) was used prior to Descartes’s Géométrie by some German and English mathematicians. It appears in Robert of Chester’s 1145 Latin translation of Al-Khwārizmī’s Algebra (at least in fifteenth-century manuscripts thereof). Bear in mind that Al-Khwārizmī (ninth century) wrote in the tradition of rhetorical algebra and used the Arabic words جذر jidhr ‘root’ and شيء shayɂ ‘thing’, rather than abbreviations or symbols, for the unknown (Cajori 1928:336; Katz & Parshall 2014:139). The symbol was also used by Christoff Rudolph in 1525, Michael Stifel in 1553, and others (Cajori 1919:698). It was interpreted by some German authors as a cursive letter R, for Latin radix ‘root’ or res ‘thing’. It was suggested (by Treutlein 1879:32; Tropfke 1902:150; and Curtze 1902:473) that Descartes might have misinterpreted this symbol as the letter x; but this conjecture was falsified by Eneström (1905:316-317 and 405-406), according to Cajori, with the evidence that Descartes, in a 1619 letter, had used the “German” symbol alongside, and distinct from, his own x (Cajori 1919:698; 1928:382). Pietro Cataldi (1548-1626) used the convention of a “crossed one”—the numeral 1 with an oblique strikethrough—to represent the first power of the unknown. Gustav Wertheim (as cited in Eneström 1905:317) speculated that this could be the origin of Descartes’s x; but Cajori, again, finds “nothing to support” that hypothesis (1919:699). The convention of crossed numerals “failed of adoption by other mathematicians” (Cajori 1928:381).
Figure 1: Symbol for unknown used by some early German writers
and interpreted by some as a cursive R.
Cajori (1919:699) concludes: “Any one [sic] reading Descartes’ Géométrie will see that his manner of introducing symbols for known and unknown quantities seems free from tradition and their choice purely arbitrary.”
But what about the Hispano-Arabic theory? What does it consist of; where did it come from; why was it adopted; and what is the evidence against it? The hypothesis relies on two assumptions: (1) that the Arabic word شيء shayɂ ‘thing’—which the Arab mathematicians of the rhetorical tradition did use to name the unknown quantity—was transcribed by early Spanish mathematicians as xei (and/or xai, xey, or xay); and (2) that the Spanish transcription was eventually abbreviated as x. The transcription of the Arabic letter shīn as x is quite plausible: Old Spanish had an “sh” sound in many native words, which was routinely spelled “x”. (In modern times the sound has migrated to the back of the throat, becoming phonetically a velar [x], and orthographically has been largely replaced by j.) We know about the medieval pronunciation, in part, because of Spanish words that were loaned to other languages, a well-known example being the name of the literary character Don Quixote. Even as late as the novel was published (1605), there was at least sufficient memory of the palatal “sh” sound for the French and Italian translations to reflect it in their spellings: Don Quichotte and Don Chisciotte, respectively. Additional testimony comes from sixteenth-century Spanish transcriptions of native Mexican languages (Canfield 1934).
For the Hispano-Arabic theory, Cajori gives no source earlier than the 1909 “Noah Webster’s Dictionary,” cited above, but its origin can be traced to 1882 and the German “orientalist” Paul de Lagarde. Around the same time (1883), Lagarde published his edition of De lingua arabica, a language manual in Spanish by Pedro de Alcalá, first published in 1505. In that book, Alcalá makes Arabic more accessible to Spanish-speaking readers by bypassing the Arabic script and transcribing Arabic words in the Roman alphabet. And indeed the glossary lists a number of phrases in which the Spanish word for ‘thing’, cosa, is equated with its Arabic counterpart شيء, transcribed as xei. This is good evidence of its transcription, but not of its use in algebra. Having argued the plausibility of the hypothesis, Lagarde says “Nothing was more natural than for 12 with [Arabic letter] shīn over it to be represented by 12x = xai in Latin letters” (“Nichts war also natürlicher als 12 mit darüber stehendem ﺵ= š = شيء lateinisch durch 12x = xai zu geben.”) (Lagarde 1882:411; 1884:136).
Others took the ball and ran with it. An obituary of Lagarde says
Lagarde,
by the way, was the first to explain the mathematical symbol x
for the unknown quantity, which, observing that old Italian algebraic
writers speak of an unknown quantity as cosa, “thing,” he connected
through Spanish, which represented “sh” by x [...] with the
Arabic shai,
“thing.” Thus x
is an abbreviation of shai
= xai, and
has become the symbol for the unknown quantity. As soon as this origin
of x[ai] = ves [sic, for
res?] =
cosa “thing” [...] was forgotten, the other letters, y, z, v and w became natural
complements. (Muss-Arnolt 1892:64)
Lagarde,
as is well known, has proven that the X
of the mathematicians is an abbreviation of the Arabic word shei, ‘a
thing,’ ‘something,’ which as early as the eleventh century was used to
designate the unknown, and which in the then prevailing Western
transcription was rendered by
xei and still appears in this form in Pedro de Alcala.
(I would remind the reader that Alcalá was not writing about mathematics, and that xei was not a naturalized loanword in his Spanish, but rather an item in a foreign language vocabulary list.) Statements like these were followed by the etymology, cited by Cajori, in the prestigious “Webster’s Dictionary--Webster’s New International Dictionary, Merriam-Webster, first (1909) and second (1934) editions—with its claim that the Arabic word was “prevailingly” transcribed as xei. Of course Noah Webster (1758-1843) was not personally responsible for the etymology that was added to “his” dictionary by his heirs. But it seems clear that Cajori was referring to that dictionary when he wrote “Incorrect historical statements of the manner in which the letter x came to be used as a symbol for unknown quantity in mathematics are widely current” (Cajori 1919:698).
And the Hispano-Arabic theory continues to be “widely current” in the twenty-first century, with the aid of the Internet. Prominent among the vehicles for the idea today is the four-minute TED talk delivered by Terry Moore, director of the Radius Foundation (Moore 2012). The ideas of the talk are summarized in text in Cosmos, an online magazine (Moore 2015). The talk and the summary contain a number of erroneous assumptions and assertions about the Spanish, Arabic, and Greek languages. For Moore, the transcription of Arabic shayɂ ‘thing’ to Spanish xei is complicated by his belief that Spanish “has no sh sound”—which is true of the modern language, but entirely false with regard to medieval and early modern Spanish, as witness Don Quixote et al., above. This error leads Moore to assert that the Spanish translators, faced with the Arabic “sh” sound, “used the closest sound they could find instead: the ‘ch’ sound from classical Greek, which is represented by the letter Χ (chi).” (In reality, chi (χ) represented an aspirated velar stop, [kʰ ], in ancient Greek, which later developed into a velar fricative [x]. In either case, its similarity to the palatal fricative [ ʃ ] is slight. Learned Greek loanwords in Spanish render it consistently as the velar stop [k], as in Cristo, cristiano, carácter, etc.)
Moore’s errors with regard to Arabic grammar and pronunciation are less crucial to the argument about algebraic x, but they do reveal a dabbler’s grasp of the language, in spite of his touted six or seven years of study. His confusion of definite with indefinite articles is demonstrated as follows:
Sheen
is the first letter of the Arabic word shay-un,
which means “something” – an undetermined thing. In Arabic, we can make
that into a specific something by adding the indefinite [sic] article, al. So this al-shay-un,
this “specific something”, appears throughout mathematics texts to
represent the part of the equation that is not yet identified, and for
which the equation is to be solved.
In reality the prefixed al- is the definite article, while the suffixed -un is equivalent to the indefinite article, and thus the two are never used together. Moore’s pronunciation of “al-shay-un” in the video fails to assimilate the l of the article to the following “sun letter” shīn; a more faithful transcription might be “ash-shayʔ ” (with “ʔ” representing a glottal stop). (Arab grammarians classify consonants into “sun letters,” which trigger assimilation, and “moon letters,” which do not.) Students of Arabic usually encounter the use of the definite article and its assimilation to the following consonant in the first few lessons. With regard to his knowledge of Arabic sounds, Moore assures viewers of the video “Trust me.” In an email to me, Moore says “I find the vehemence of the scholarly debate fascinating, and more than a little hilarious” (7/8/2015).
The Hispano-Arabic conjecture is likewise presented as fact on a promotional webpage for the PBS documentary series Islam: Empire of Faith (2012):
When
al-Khwarizmi's work was translated in Spain, the Arabic word shay was
transcribed as xay,
since the letter x
was pronounced as sh
in Spain. In time this word was abbreviated as x, the universal
algebraic symbol for the unknown.
The claim does not appear in the documentary itself.
Webster’s New International Dictionary, whose first (1909) and second (1934) editions no doubt lent credibility to the Hispano-Arabic theory, discontinued that etymology in its third (1961) edition. And the Oxford English Dictionary (online and 1989 print edition) explicitly states:
The
introduction of x,
y, z as symbols of
unknown quantities is due to Descartes ( Géométrie, 1637),
who, in order to provide symbols of unknowns corresponding to the
symbols a,
b, c of knowns, took
the last letter of the alphabet, z,
for the first unknown and proceeded backwards to y and x for the second
and third respectively. There is no evidence in support of the
hypothesis that x
is derived ultimately from the mediæval transliteration xei of ﺵ [sic, error for شيء]
shei
‘thing’, used by the Arabs to denote the unknown quantity, or from the
compendium for Latin res
‘thing’ or radix
‘root’ (resembling a loosely written x), used by
mediæval mathematicians.
Medieval and early modern Spanish documents—manuscripts and published books—have been studied intensively by scholars, resulting in the compilation of concordances, glossaries, dictionaries, and, most recently, electronically searchable corpora. If a word such as xei (or xay, etc.) was used in a Spanish context in the thirteenth through the sixteenth century, we may presume that it would have been recorded in one or more of these lexical studies. However, a review of several major compiled historical vocabularies of Spanish yields no instance of xei or similar words. These works include Alonso (1986), Kasten & Cody (2001), and Oelschläger (1940), as well as the diachronic corpus of the Spanish Royal Academy (Academia, n.d.) and the Corpus del Español of Mark Davies (2002-).
And if the word is missing from general glossaries because it is a technical term used only in specialized works that were not catalogued, then we might look for it in specific early Spanish works on mathematics. Rider (1982) lists five treatises on algebra published in Spanish in the 16th century: Aurel (1552), Ortega (1552), Díez (1556), Pérez de Moya (1562), and Nunes (1567). All these works are grounded in the tradition of rhetorical algebra, and they do reflect the Arabic practice of naming the unknown as “thing.” However, their reflection of the Arabic shayɂ is not a transcription, but rather the Spanish translation—“cosa”—in all five works. Pérez de Moya and Nunes additionally abbreviate cosa as co., as does Andrés Puig in his Arithmetica especulativa (1672). Proponents of the Hispano-Arabic theory have never named a Spanish mathematician who used xei (etc.) or x prior to Descartes.
The Arabic etymology of algebraic x has logical plausibility in its favor, but no data to document its basis in reality.
Note
1. Sometime in the first half of June 2016, the online OED was corrected to
read “...the mediæval transliteration xei of šay' ‘thing’...
”—perhaps in response to my suggestion.
References
• Academia Española, Real. Corpus Diacrónico del Español.
• Alcalá Pedro de (1505). De lingua arabica. Granada. Edition by Paul de Lagarde (Göttingen: Arnold Hoyer, 1883).
• Derbyshire, John (2006). Unknown Quantity: A Real and Imaginary History of Algebra. Washington, DC: Joseph Henry Press.
•