Axis z is found in 1866 in the 8th edition of Elements of analytical mechanics by William Holms Chambers Bartlett: "Take the axis z as the axis of rotation; denote the angular velocity by φ, and the distance of the particle M from the axis z by r" [University of Michigan Digital Library].
The terms z and the z DISTRIBUTION were introduced by R. A. Fisher in "On a Distribution Yielding the Error Functions of Several well Known Statistics", Proceedings of the International Mathematics Congress, Toronto (1924). Fisher's development of the analysis of variance in this paper and in his book Statistical Methods for Research Workers (1925) was based on the z distribution. Fisher’s z is related to the modern F by z = ½ ln F [James A. Landau].
See also F DISTRIBUTION and VARIANCE.
ZENO’S PARADOXES are due to Zeno of Elea (c. 490 BC - c. 425 BC) but none of his works survive and the paradoxes are known through the writings of Aristotle and Simplicius, a 6th century AD commentator on Aristotle. The four paradoxes, relating to motion, the dichotomy, the Achilles, the arrow, and the stadium, are discussed in Book VI of Aristotle’s Physics. For a brief account see Kline (35-37) and, for a much more detailed one, Zeno's Paradoxes from the Stanford Encyclopedia of Philosophy.
The regular application of the word "paradox" is relatively recent; before the 20th century "fallacy" or "argument" were more common. Cajori’s 9-part "The History of Zeno's Arguments on Motion: Phases in the Development of the Theory of Limits," American Mathematical Monthly, 22, (1915) shows that for most of their history the arguments were held in low esteem but that in the 19th century their prestige rose. Bertrand Russell wrote of Zeno, "Having invented four arguments, all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms." Principles of Mathematics (1903, p. 347)
This entry was contributed by John Aldrich. See PARADOX.
ZERMELO-FRAENKEL SET THEORY is found in the title "Ein axiomatisches System der Mengenlehre nach Zermelo und Fraenkel," by Ernst-Jochen Thiele, Z. Math. Logik Grundlagen Math. (1955).
The term is also found in R. Montague, "Zermelo-Fraenkel set theory is not a finite extension of Zermelo set theory," Bull. Amer Math. Soc. 62 (1956).
Attributions to Ernst Zermelo " Untersuchungen über die Grundlagen der Mengenlehre. I" Math. Annalen (1908) occur in A. Fraenkel, " Zu den Grundlagen der Cantor-Zermeloschen Mengenlehre," Math. Annalen (1922) and "Über die Zermelosche Begründung der Mengenlehre," Jahresbericht der Deutschen Mathematiker-Vereinigung 30, 2nd section (1921) [James A. Landau].
ZERO. The Hindus called the symbol sunya, and the term passed over into Arabic as as-sifr or sifr (Smith vol. 2, page 71). Sunya, meaning "empty," was used around A. D. 400 to indicate the empty column on the abacus.
Dionysius Exiguus (died about 545) used the word nulla in his Easter tables. The first epact in each nineteen-year cycle is "nulla" (rather than thirty, as in those of his predecessors). A reference is Migne, Patrologiae Latinae, vol. 67, col. 493 [Christian Marinus Taisbak].
Abraham ben Meir ibn Ezra (1092-1167) used galgal for zero in a description he wrote of a decimal system of numeration.
Leonardo of Pisa (1180-1250) (or Fibonacci) used the word zephirum for this symbol in Liber Abaci: "...quod arabice zephirum appelatur."
According to Smith (vol. 2), some other old names for zero include sipos, tsiphron, tziphra, rota, omicron, circulus, theca, null, zeuero, ceuero, cifra, zepiro and figura nihili.
The OED shows a use of cipher in English in 1399 by William Langland in Richard the Redeles: "Than satte summe, as siphre doth in awgrym, That noteth a place, and no thing availith."
Other old names for this symbol are aught and naught.
According to Cajori (1919, page 128), the word zero "is found in some fourteenth century manuscripts."
Cajori also states that the first printed treatise containing the word zero is De arithmetrica opusculum, by Filippo Calandri, which was printed in Florence in 1491. Cajori attributes this information to Eneström. [Calandri's name is also spelled Philippus Calender and Philippus Calandrus.]
The earliest citation for zero in English in the OED2 is from Edward Grimstone's 1604 translation of The Natural and Moral History of the Indies by José de Acosta: "They accompted their weekes by thirteene dayes, marking the dayes with a Zero or cipher."
In 1702 A Mathematical Dictionary: Or; A Compendious Explication of All Mathematical Terms by Joseph Raphson and Jacques Ozanam has: "Zero, a word sometimes used (particularly among the French) for a Cypher, or (0.)"
In 1882 Complete Graded Arithmetic by James B. Thomson has: "The last one is called Naught, because when standing alone it has no value. It is also called Cipher or Zero."
For the history of symbols for zero (as opposed to words for zero), see the companion math symbols web page, linked from the front page of this website.
ZERO (of a function) is found in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "By a zero of P(x) is meant a point at which P(x) vanishes."
ZERO MATRIX appears in J. J. Sylvester "Note on the Theory of Simultaneous Linear Differential of Difference Equations with Constant Coefficients," American Journal of Mathematics, 4, (1881), 321-326, Coll Math Papers, III, pp. 551-6. A. Cayley "A Memoir on the Theory of Matrices" (1858) Coll Math Papers, I, p. 477 referred to the matrix zero.
ZERO-SUM GAME appears in 1944 in Theory of Games and Economic Behavior by J. von Neumann and O. Morgenstern (David, 1998).
ZEROTH is found in 1850 in The calculus of operations by John Paterson, A. M.: "To show inductively that this law persists, we have constructed the following table for ten successive units of time, in which the columns under φ''' and φ'' are obtained, the first, by multiplying AΦ''' by 3, 6 and 2 respectively for power of the second, first and zeroth order; and the second, by multiplying AΦ'' by 4 and by 2 for power of the first and of the zeroth order."
Zeroth is also found in 1893 in An elementary treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics with applications to problems in mathematical physics by William Elwood Byerly: "It is a finite sum terminating with the first power of x if m is odd, and with the zeroth power of x if m is even" [University of Michigan Digital Library, both citations].
The term ZETAIC MULTIPLICATION was coined by James Joseph Sylvester.
Zeta-ic multiplication was used in 1840 and is found in
Sylvester's Collected Mathematical Papers (1904) I. 47: "I use
the Greek letter 
to denote that the product of factors to
which it is prefixed is to be effected after a certain symbolical
manner. This I shall distinguish as the zeta-ic product. ... Rule for
zeta-ic multiplication. Note. An analogous interpretation may be
extended to any zeta-ic function whatever" (OED2).
ZORN's LEMMA, named after Max Zorn (1906-1993), has received plenty of attention from historians, perhaps because it illustrates the problems posed by eponymous terms. See EPONYMY.
The publication history is fairly straightforward. The "lemma" appears in "A remark on method in transfinite algebra," Bulletin of the American Mathematical Society, 41, (1935), 667-670, although Zorn calls it a "maximum principle" and refers to it as an "axiom." The phrase "Zorn's lemma" first appears in print in John W. Tukey's Convergence and Uniformity in Topology (1940, p. 7), although there are four different versions of it.
The historical context is discussed in Gregory Moore's Zermelo's Axiom of Choice: Its Origins, Development and Influence (1982) and Paul J. Campbell's "The Origins of 'Zorn's Lemma'," Historia Mathematica, 5, 1978, 77-89. Moore (p. 220) begins by saying, "The history of maximal principles, such as Zorn's Lemma, is strewn with multiple independent discoveries of fundamentally similar propositions. ... Unlike many other multiple independent discoveries, those involving maximal principles were in no sense simultaneous, since they extended over three decades." This work of Moore and Campbell prompts the question, why should Zorn's work be recognised and not the earlier work of Hausdorff and Kuratowski?
Who was responsible for the name "Zorn's lemma"? In 1976 Campbell asked Zorn and Tukey for their recollections. Tukey's reply indicates that the term was in circulation in Princeton before he wrote his book and he speculates that Lefschetz introduced it there. Zorn says the name was already in circulation before he presented his paper to the American Mathematical Society in October 1934! He thought he had probably first heard it from Řystein Ore.
Bill Dubuque, Julio González Cabillón and John Aldrich contributed to this entry. See AXIOM OF CHOICE.