**Greek letters.** The use of letters to represent general numbers
goes back to Greek antiquity. Aristotle frequently used single
capital letters or two letters for the designation of magnitude or
number (Cajori vol. 2, page 1).

Diophantus (fl. about 250-275) used a Greek letter with an accent to represent an unknown. G. H. F. Nesselmann takes this symbol to be the final sigma and remarks that probably its selection was prompted by the fact that it was the only letter in the Greek alphabet which was not used in writing numbers. However, differing opinions exist (Cajori vol. 1, page 71).

In 1463, Benedetto of Florence used the Greek letter rho for an
unknown in *Trattato di praticha d'arismetrica.* (Franci and
Rigatelli, p. 314)

**Roman letters.** In Leonardo of Pisa's *Liber abbaci*
(1202) the representation of given numbers by small letters is found
(Cajori vol. 2, page 2).

Jordanus Nemorarius (1225-1260) used letters to replace numbers.

Christoff Rudolff used the letters *a, c,* and *d* to
represent numbers, although not in algebraic equations, in *Behend
vnnd Hubsch Rechnung* (1525) (Cajori vol. 1, page 136).

Michael Stifel used *q* (abbreviation for *quantita* (which
Cardan had already done) but he also used *A, B, C, D,* and
*F,* for unknowns in 1544 in *Arithmetica integra* (Cajori
vol. 1, page 140).

Girolamo Cardan (1501-1576) used the letters *a* and *b* to
designate known numbers in *De regula aliza* (1570) (Cajori vol.
1, page 120).

In 1575 Guilielmus Xylander translated the *Arithmetica* of
Diophantus from Greek into Latin and used *N* (*numerus*)
for unknowns in equations (Cajori vol. 1, page 380).

In 1591 Francois Vieta (1540-1603) was the first person to use letters
for unknowns and constants in algebraic equations. He used vowels for
unknowns and consonants for given numbers (all capital letters) in
*In artem analyticem isogoge.* Vieta wrote:

Quod oopus, ut arte aliqua juventur, symbolo constanti et perpetuo ac bene conspicuo date magnitudines ab incertis quaesititiis distinguantur ut [illegible in Cajori] magnitudines quaesititias elemento A aliave litera volcali,(Cajori vol. 1, page 183, and vol. 2, page 5).E, I, O, V, Y[illegible in Cajori] elementisB, G, D,aliisve consonis designando. [As one needs, in order that one may be aided by a particular device, some unvarying, fixed and clear symbol, the given magnitudes shall be distinguished from the unknown magnitudes with the letterAor with another vowelE, I, O, U, Y, the given ones with the lettersB, G, Dor other consonants.]

Thomas Harriot (1560-1621) in *Artis Analyticae Praxis, ad
Aequationes Algebraicas* used lower case vowels for unknowns and
lower case consonants for known quantities.

**Descartes' use of z, y, x.** The following is from
Cajori (vol 1, page 381):

The use ofz, y, x. . . to represent unknowns is due to René Descartes, in hisLa géometrie(1637). Without comment, he introduces the use of the first letters of the alphabet to signifyknownquantities and the use of the last letters to signifyunknownquantities. His own langauge is: "...l'autre,LN,est (1/2)ala moitié de l'autre quantité connue, qui estoit multipliée parz,que ie suppose estre la ligne inconnue." Again: "...ie considere ... Que le segment de la ligneAB,qui est entre les poinsAetB,soit nomméx,et quieBCsoit nomméy; ... la proportion qui est entre les costésABetBRest aussy donnée, et ie la pose comme dezab; de façon qu'ABestantx,RBserabx/z,et la touteCRseray=bx/z. ..." Later he says: "et pour ce queCBetBAsont deux quantités indeterminées et inconnuës, ie les nomme, l'uney; et l'autrex.Mais, affin de trouver le rapport de l'une a l'autre, ie considere aussy les quantités connuës qui determinent la description de cete ligne courbe: commeGAque je nommea,KLque je nommeb,etNL,parallele aGA,que ie nommeC." As co-ordinates he uses later onlyxandy.In equations, in the third book of theGéométrie,xpredominates. In manuscripts written in the interval 1629-40, the unknownzoccurs only once. In the other placesxandyoccur. In a paper on Cartesian ovals, prepared before 1629,xalone occurs as unknown,ybeing used as a parameter. This is the earliest place in which Descartes used one of the last letters of the alphabet to represent an unknown. A little later he usedx, y, zagain as known quantities.Some historical writers have focused their attention upon the

x,disregarding theyandz,and the other changes in notation made by Descartes; these wrtiers have endeavored to connect thisxwith older symbols or with Arabic words. Thus, J. Tropfke, P. Treutlein, and M. Curtze advanced the view that the symbol for the unknown used by early German writers, looked so much like anxthat it could easily have been taken as such, and that Descartes actually did interpret and use it as anx.But Descartes' mode of introducing the knownsa, b, c,etc., and the unknownsz, y, xmakes this hypothesis improbable. Moreover, G. Eneström has shown that in a letter of March 26, 1619, addressed to Isaac Beeckman, Descartes used the symbol as a symbol in form distinct fromx,hence later could not have mistaken it for an . At one time, before 1637, Descartes usedxalong the side of ; at that timex, y, zare still used by him as symbols for known quantities. German symbols including the forx,as they are found in the algebra of Clavius, occur regularly in a manuscript due to Descartes, theOpuscules de 1619-1621.All these facts caused Tropfke in 1921 to abandon his old view on the origin of

x,but he now argues with force that the resemblance ofxand , and Descartes' familiarity with , may account for the fact that in the latter part of Descartes'Géométriethexoccurs more frequently thanzandy.Eneström, on the other hand, inclines to the view that the predominance ofxoveryandzis due to typographical reasons, type forxbeing more plentiful because of the more frequent occurrence of the letterx,toyandz,in the French and Latin languages.

Descartes introduced the equation *ax* + *by* = *c,*
which is still used to describe the equation of a line (Johnson, page
145).

Johnson says (on page 145):

The predominant use of the letterThere are, however, other explanations for Descartes' use ofxto represent an unknown value came about in an interesting way. During the printing ofLa Geometrieand its appendix,Discours de La Methode,which introduced coordinate geometry, the printer reached a dilemma. While the text was being typeset, the printer began to run short of the last letters of the alphabet. He asked Descartes if it mattered whetherx, y,orzwas used in each of the book's many equations. Descartes replied that it made no difference which of the three letters was used to designate an unknown quantity. The printer selectedxfor most of the unknowns, since the lettersyandzare used in the French language more frequently than isx.

According to the *Oxford English Dictionary* (2nd ed.):

The introduction ofDescartes used letters to represent only positive numbers; a negative number could be represented as -x, y, zas symbols of unknown quantities is due to Descartes (Géométrie,1637), who, in order to provide symbols of unknowns corresponding to the symbolsa, b, cof knowns, took the last letter of the alphabet,z,for the first unknown and proceeded backwards toyandxfor the second and third respectively. There is no evidence in support of the hypothesis thatxis derived ultimately from the mediaeval transliterationxeiofshei"thing", used by the Arabs to denote the unknown quantity, or from the compendium for L.res"thing" orradix"root" (resembling a loosely-writtenx), used by mediaeval mathematicians.

John Hudde (1633-1704) was first to allow a letter to represent a
positive or negative number, in 1657 in *De reductione
aequationum,* published at the end of the first volume of F. Van
Schooten's second Latin edition of René Descartes'
*Géométrie* (Cajori vol. 2, page 5).

Jonas Moore wrote in *Arithmetic* (1660): "Note alwayes the
given quantities or numbers with Consonants, and those which are
sought with Vowels, or else the given quantities with the former
letters in the Alphabet, and the sought with the last sort of
letters, as *z y x,* &c. lest you make a confusion in your
work."

**Complex numbers.** The *a* + *bi* notation was
introduced by Leonhard Euler (1707-1783).