Earliest Known Uses of Some of the Words of Mathematics
TANGENT (in trigonometry). Bradwardine and other writers used the term umbra versa.
Tangent was introduced by Thomas Fincke (1561-1656) in his Thomae Finkii Flenspurgensis Geometriae rotundi libri XIIII, Basileae: Per Sebastianum Henricpetri, 1583. He wrote tangens in Latin.
Vieta did not approve of the term tangent because it could be confused with the term in geometry. He used (c. 1593) sinus foecundarum (abridged to foecundus) and also amsinus and prosinus (Smith vol. 2, page 621).
According to the DSB, "Rheticus' Canon of the Doctrine of Triangles (Leipzig, 1551) was the first table to give all six trigonometric functions, including the first extensive table of tangents and secants (although such modern designations were eschewed by Rheticus as 'Saracenic barbarisms')."
TANGRAM is found in 1861 in Primary object lessons for a graduated course of development, a manual for teachers and parents with lessons for the proper training of the faculties of children by Norman Allison Calkins (1822-1895): "Among objects for illustrating form there should be a gonigraph and the Chinese tangram; and the child should also have provided for amusement at home little bricks--blocks made of some hard wood, as cherry or maple, four inches long, two inches wide, and one thick. ... The tangram may be made of metal, wood, or pasteboard."
The origin of the word is uncertain. Modern dictionaries suggest it may be derived from a Chinese word tang; an older dictionary suggests it may be a changed spelling from the obsolete English word trangam.
The term TAUBERIAN THEOREMS was coined by G. H. Hardy (Kramer, p. 504). The term was used by Hardy and Littlewood (DSB, article: "Wiener").
The term is found in 1913 in Hardy & Littlewood in Proc. London Math. Soc. XI. 411: "The general character of the theorems which it [sc. this paper] contains is 'Tauberian': they are theorems of the type whose first example was the beautiful converse of Abel's theorem originally proved by Tauber" (OED2).
TAYLOR'S FORMULA is found in English in 1855 in Elements of the differential and integral calculus by Albert Ensign Church [University of Michigan Digital Library].
The term TAYLOR'S SERIES "was probably first used by L'Huillier in 1786, although Condorcet used both the names of Taylor and d'Alembert in 1784" (DSB).
Lacroix used Théorčme de Taylor and la série de Taylor in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800).
Taylor's series appears in English appears in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young [James A. Landau].
TAYLOR'S THEOREM. Colin Maclaurin attributed the theorem to Taylor in his Treatise of Fluxions (1742): "This theorem was given by Dr. TAYLOR, method. increm." [Judith V. Grabiner].
Julio González Cabillón believes that Marie Jean Antoine Nicolas de Caritat Condorcet (1743-1794) used the term Taylor's theorem (in French) in 1784, in volume I of the Encyclopedie methodique (p. 104).
Lacroix used Théorčme de Taylor and la série de Taylor in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800).
Taylor's theorem appears in English in the 1816 translation of Lacroix's Differential and Integral Calculus: " This formula is called Taylor's Theorem, from the English geometer by whom it was discovered" (OED2).
TENSOR (in quaternions) was used by William Rowan Hamilton (1805-1865) in 1846 in The London, Edinburgh, and Dublin Philosophical Magazine XXIX. 27:
Since the square of a scalar is always positive, while the square of a vector is always negative, the algebraical excess of the former over the latter square is always a positive number; if then we make (TQ)2 = (SQ)2 - (VQ)2, and if we suppose TQ to be always a real and positive or absolute number, which we may call the tensor of the quaternion Q, we shall not thereby diminish the generality of that quaternion. This tensor is what was called in former articles the modulus.
The earliest use of tensor in the Proceedings of the Royal Irish Academy is on p. 282 of Volume 3, and is in the proceedings of the meeting held on July 20, 1846. The volume appeared in 1847. Hamilton writes:
Q = SQ + VQ = TQ [times] UQ
The factor TQ is always a positive, or rather an absolute (or signless) number; it is what was called by the author, in his first communication on this subject to the Academy, the modulus, but which he has since come to prefer to call it the TENSOR of the quaternion Q: and he calls the other factor UQ the VERSOR of the same quaternion. As the scalar of a sum is the sum of the scalars and the vector of the sum is the sum of the vectors, so that tensor of a product is the product of the tensors and the versor of a product is the product of the versors.
In other words, the tensor of a quaternion is simply its modulus.
In his paper "Researches respecting quaternions" (Transactions of the Royal Irish Academy, vol. 21 (1848) pp. 199-296), Hamilton uses the term "modulus," not "tensor." This paper purports to have been read on 13 November 1843 (i.e., at the same meeting as the short paper, or abstract, in the Proceedings of the RIA).
The terms vector, scalar, tensor and versor appear in the series of papers "On Quaternions" that appeared in the Philosophical Magazine (see pages 236-7 in vol III of "The Mathematical Papers of Sir William Rowan Hamilton," edited by H. Halberstam and R.E. Ingram). The editors have taken 18 short papers published in the Philosophical Magazine between 1844 and 1850, and concatenated them in the "Mathematical Papers" to form a seamless whole, with no indication as to how the material was distributed into the individual papers.
(Information for this article was provided by David Wilkins and Julio González Cabillón.)
TENSOR in its modern sense is due to the famous Goettingen Professor Woldemar Voigt (1850-1919), who in 1887 anticipated Lorentz transform to derive Doppler shift, in Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung, Leipzig: von Veit, 1898 (OED2 and Julio González Cabillón).
The term TENSOR ANALYSIS was introduced by Albert Einstein in 1916 (Kline, page 1123).
According to the University of St. Andrews website, Einstein is reported to have commented to the chairman at the lecture he gave in a large hall at Princeton which was overflowing with people:
I never realised that so many Americans were interested in tensor analysis.
Tensor analysis is found in English in 1922 in H. L. Brose's translation of Weyl's Space-Time-Matter: "Tensor analysis tells us how, by differentiating with respect to the space co-ordinates, a new tensor can be derived from the old one in a manner entirely independent of the co-ordinate system. This method, like tensor algebra, is of extreme simplicity" (OED2).
The term TERAGON was coined by Mandelbrot, according to an Internet web page.
TERMINATING DECIMAL appears in 1857 in Mathematical Dictionary and Cyclopedia of Mathematics: "Those decimal fractions which are expressed by a finite number of places of figures, are called terminating decimals."
Finite decimal appears in 1876 in Elementary arithmetic, with brief notices of its history by Robert Potts: "1. In what cases can an ordinary fraction be expressed by a finite decimal?" [University of Michigan Historical Math Collection].
TESSELLATION is found in 1660 in The History of the Propagation and Improvement of Vegetables by Robert Sharrock (1630-1684): "Yet they, instead of those elegant Tessellations, are beautified otherwise in their site with as great curiosity."
The OED2 shows numerous citations in the 1800s of the spellings tesselation, tesselated, and tesselate, and some modern U. S. dictionaries show these as alternate spellings.
TESSERACT was used in 1888 by Charles Howard Hinton (1853-1907) in A New Era of Thought (OED2). According to an Internet site, Hinton coined the term.
The term TEST OF INDIVIDUAL EQUIVALENCE RATIOS was coined by Anderson & Hauck (1990), according to an Internet web page by J. T. Gene Hwang.
TETRAHEDRON is found in English in Sir Henry Billingsley's 1570 translation of Euclid's Elements (OED2).
TETRATION, a term for repeated exponentiation, was introduced by Reuben Louis Goodstein. In "Transfinite ordinals in recursive number theory, " Journal of Symbolic Logic 12 (1947), he writes "... defines successive new processes (which we may call tetration, pentation, hexation, and so on)" [Samuel S. Kutler, Dave L. Renfro].
THEOREM appears in English in 1551 in The Pathwaie to Knowledge by Robert Recorde: "Argts., The Theoremes, (whiche maye be called approued truthes) seruing for the due knowledge and sure proofe of all conclusions...in Geometrye."
The term THEORY OF CLOSEDNESS was introduced in 1910 by Vladimir Andreevich Steklov (1864-1926) (DSB).
THEORY OF GAMES appears in the title "La théorie du jeu et les équations intégrales ŕ noyau symétrique," by Emile Borel, Compt. Rend. Acad. Sci., 173 (Dec. 19, 1921).
The term Theorie der Gesellschaftsspiele appears in 1928 in the title, "Zur Theorie der Gesellschaftsspiele" by John von Neumann, Math. Ann. 100. Gesellschaftsspiele is translated as "parlor games" by Kramer [James A. Landau].
Referring to the 1928 paper, von Neumann's collaborator Herman H. Goldstine wrote in The Computer from Pascal to von Neumann (1972):
This was his first venture in the field [of game theory], and while there had been other tentative approaches --- by Borel, Steinhaus, and Zermelo, among others --- his was the first to show the relations between games and economic behavior and to formulate and prove his now famous minimax theorem which assures the existence of good strategies for certain important classes of games.
Theory of games also appears in 1943 in the title Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern [James A. Landau].
Game theory appears in 1946-47 in Carl Kaysen, Review of Economic Studies XIV. 14: "It is extremely doubtful whether the degree of restriction of possible solutions offered by the 'solution' of game-theory will be great enough to be of much practical value in really complex cases" [Fred Shapiro].
THEORY OF PROBABILITY is found in the title Exposition de la Théorie des Chances et des Probabilités (1843) by A. A. Cournot [James A. Landau].
THEORY OF TYPES is found in Bertrand Russell, "Mathematical Logic as Based on the Theory of Types," American Journal of Mathematics 30 (1908) [James A. Landau].
The term THRACKLE was coined by John Horton Conway.
The use of the designations TIME DOMAIN and FREQUENCY DOMAIN to distinguish the correlation and the spectral approaches to filtering theory, and to time series analysis generally, seems to have originated in communication engineering.
"Frequency domain" appears in L. A. Zadeh's "Theory of Filtering" (Journal of the Society for Industrial and Applied Mathematics, 1, (1953), 35-51).
"Time domain" and "frequency domain" appear together in W. F. Trench's "A General Class of Discrete Time-Invariant Filters," Journal of the Society for Industrial and Applied Mathematics, 9, (1961), 405-421.
The terms soon became established in statistical time series analysis, see e.g. M. Rosenblatt & J. W. Van Ness's "Estimation of the Bispectrum," Annals of Mathematical Statistics, 36, (1965), 1120-1136 [John Aldrich].
TIME SERIES appears in W. M. Persons's "The Correlation of Economic Statistics," Publications of the American Statistical Association, 12, (1910), 287-322 [John Aldrich].
The phrase TIME SERIES ANALYSIS entered circulation at the end of 1920s, e.g. in S. Kuznets's "On the Analysis of Time Series," Journal of the American Statistical Association, 23, (1928), 398-410, although it only became really popular much later [John Aldrich].
The term TITANIC PRIME (a prime number with at least 1000 decimal digits) was coined in 1984 by Samuel D. Yates (died, 1991) of Delray Beach, Florida ["Sinkers of the Titanic", J. Recreational Math. 17, 1984/5, p268-274]. Yates also coined the term gigantic prime in the mid-1980s, referring to a prime number with at least 10,000 decimal digits. [The term megaprime refers to a prime of at least a million decimal digits.]
The term TOPOLOGICAL ALGEBRA was coined by David van Dantzig (1900-1959). The term appears in the title of his 1931 Ph. D. dissertation "Studiën over topologische Algebra" (DSB).
TOPOLOGICAL GROUP. David van Dantzig defines "eine topologische Gruppe" in "Ueber topologisch homogene Kontinua" in Fundamenta Mathematicae vol. 15 (1930) pages 102-125.
In a footnote van Dantzig states that this notion is essentially the same notion as that of a "limesgruppe" which is said to be introduced by Otto Schreier (1901-1929) in Abstrakte Kontinuierliech Gruppen (Abh. Math. Sem. Hambirg 4 (1925) 15-32) [Michael van Hartskamp].
TOPOLOGICAL SPACE. Felix Hausdorff used topologisch raum in Grundzüge der Mengenlehre (1914).
TOPOLOGY was introduced in German in 1847 by Johann Benedict Listing (1808-1882) in "Vorstudien zur Topologie," Vandenhoeck und Ruprecht, Göttingen, pp. 67, 1848. However, Listing had already used the word for ten years in correspondence. The term was introduced to replace the earlier name "analysis situs."
Topology is found in English in February 1883 in Nature: "The term Topology was introduced by Listing to distinguish what may be called qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated" (OED2).
According to several sources, topology was introduced in English by Solomon Lefschetz (1884-1972) in the title of a monograph written in the late 1920s. According to Encarta, the word topology was coined by Solomon Lefschetz in 1930.
TORSION. According to Howard Eves in A Survey of Geometry, vol. II (1965), "The name torsion was introduced by L. I. Valleé in 1825, replacing an older name flexion" [James A. Landau].
TORUS. Hero mentions a mathematician named Dionysodorus as the author of On the Tore, in which a formula for the volume of the torus is given [DSB]. The OED2 shows a use of torus in English by Cayley in 1870.
TOTIENT. E. Prouhet used indicateur (indicator) in 1846 in Nouv. Ann. de Math. V. 176.
Totient was introduced by Sylvester in "On Certain Ternary Cubic-Form Equations", Amer. J. Math 2 (1879) 280-285, 357-393, in Sylvester's Collected Mathematical Papers vol. III p. 321. He writes: "The so-called (phi) function of any number I shall here and hereafter designate as its (tau) function and call its Totient." This information was taken from a post in sci.math by Robert Israel.
TRACE (of a matrix) is found in 1938 in A. A. Albert, Modern Higher Algebra: "We call T(A) the trace of A" (OED2).
The TRACTRIX was named by Christiaan Huygens (1629-1695), according to the University of St. Andrews website.
In Webster's 1828 dictionary, the word is spelled tractatrix, with the middle syllable stressed.
TRANSCENDENTAL. Referring to curves, Gottfried Wilhelm Leibniz (1646-1716) used the terms algebraic and transcendental for Descartes' terms geometrical and mechanical in 1684 in Acta Eruditorum (Kline, page 312). Struik (page 276) writes, "This may be the first time that the term 'transcendental' in the sense of 'nonalgebraic' occurs in print.'" Leibniz also used phrases which are translated as "transcendental problems" and "transcendental relations."
According to S. Probst, the term transcendental was used by Leibniz in 1675.
According to Paulo Ribenboim in My Numbers, My Friends, "LEIBNIZ seems to be the first mathematician who employed the expression 'transcendental number' (1704)."
Euler used transcendental in his 1733 article in Nova Acta Eruditorum titled "Constructio aequationum quarundam differentialium quae indeterminatarum separationem non admittunt":
Now there are kinds of constructions, which can be called transcendental, which arise in solving differential equations and cannot be transformed into algebraic equations.
The above citation and translation were provided by Ed Sandifer.
Euler used a phrase which is translated transcendental quantities in 1745 in Introductio in analysin infinitorum [James A. Landau]. Euler wrote that these numbers "transcend the power of algebraic methods" (Burton, p. 603). He also used the term in the title "De plurimis quantitatibus transcendentibus, quas nullo modo per formulas integrales exprimere licet," which was presented in 1780 and published in 1784 in Acta Academiae Scientarum Imperialis Petropolitinae.
Transcendental function appears in 1809 in the title "Théorie d'un nouvelle fonction transcendente" by Soldner.
In 1828, in Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre), the term transcendental is not used in this passage:
It is probable that this number 
is not even included among algebraical irrational quantities, in other
words, that it cannot be the root of an algebraical equation having a finite
number of terms with rational co-efficients: but a rigorous demonstration of
this seems very difficult to find; we can only show that the square of is also an irrational number.
Transcendental quantities appears in English in Webster's 1828 dictionary: "Transcendental quantities, among geometricians, are indeterminate ones, or such as cannot be expressed or fixed to any constant equation."
Transcendental equation is found in English in 1857 in Mathematical Dictionary and Cyclopedia of Mathematics.
Transcendental number appears (as transscendente Zahl) in 1882 in "Ueber die Zahl " by F. Lindemann.
Transcendental irrational is found in 1902 in The Number-System of Algebra by Henry B. Fine (and may occur in the earlier 1891 edition): "This number e, the base of the Naperian system of logarithms, is a "transcendental" irrational, transcendental in the sense that there is no algebraic equation with integral coefficients of which it can be a root."
Transcendental number appears in English in "Transcendental numbers," American M. S. Bull. (1897).
In 1906 in History of Modern Mathematics, David Eugene Smith refers to transcendent numbers.
Webster's unabridged 1913 dictionary has: "In mathematics, a quantity is said to be transcendental relative to another quantity when it is expressed as a transcendental function of the latter; thus, ax, 102x, log x, sin x, tan x, etc., are transcendental relative to x.
TRANSFINITE. Georg Cantor (1845-1918) used this word in the title of a paper published in 1895, Beiträge zur Begründung der Transfiniten Mengenlehre.
TRANSPOSE (noun, of a matrix). Transposed matrix appears in 1858 in Phil. Trans. R. Soc. CXLVIII. 32: "A matrix compounded with the transposed matrix gives rise to a symmetrical matrix" (OED2).
Transpose is found 1937 in Mod. Higher Algebra by A. A. Albert: "Every square matrix is similar to its transpose" (OED2).
TRANSPOSITION (for a two-element cycle) is found in Cauchy's 1815 memoir "Sur le nombre des valeurs q'une fonction peut acquérir lorsqu'on permute de toutes les maničres possibles les quantités qu'elle renferme" (Journal de l'Ecole Polytechnique, Cahier XVII = Cauchy's Oeuvres, Second series, Vol. 13, pp. 64--96.) This usage was found by Roger Cooke, who believes this is the first use of the term.
TRANSVERSAL. In 1828 in Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre), the line is referred to as "a third line" and as "the secant line."
In Webster's dictionary of 1828, the term is "the cutting line."
Transversal is found in 1847 in Cayley, Camb. & Dubl. Math. Jrnl. II. 52: "When three conics have the same points of intersection, any transversal intersects the system in six points, which are said to be in involution."
TRAPEZIUM and TRAPEZOID. The early editions of Euclid 1482-1516 have the Arabic helmariphe; trapezium is in the Basle edition of 1546.
Both trapezium and trapezoid were used by Proclus (c. 410-485). From the time of Proclus until the end of the 18th century, a trapezium was a quadrilateral with two sides parallel and a trapezoid was a quadrilateral with no sides parallel. However, in 1795 a Mathematical and Philosophical Dictionary by Charles Hutton (1737-1823) appeared with the definitions of the two terms reversed:
Trapezium...a plane figure contained under four right lines, of which both the opposite pairs are not parallel. When this figure has two of its sides parallel to each other, it is sometimes called a trapezoid.
No previous use the words with Hutton's definitions is known. Nevertheless, the newer meanings of the two words now prevail in U. S. but not necessarily in Great Britain (OED2).
Some geometry textbooks define a trapezoid as a quadrilateral with at least one pair of parallel sides, so that a parallelogram is a type of trapezoid.
TRAVELING SALESMAN PROBLEM. The first use of this term "may have been in 1931 or 1932, when A. W. Tucker heard the term from Hassler Whitney of Princeton University." This information comes from an Internet web page, which refers to E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys, editors, The Traveling Salesman Problem (1985).
Other terms are knight's tour and the messenger problem.
The term TREE in graph theory was coined by James Joseph Sylvester, according to an Internet web site.
Tree was used by Arthur Cayley in 1857 and appears in his Mathematical Papers (1890) III. 242: "On the Theory of the analytical Forms called Trees" (OED2).
TRIANGLE INEQUALITY appears in 1941 in Survey of Modern Algebra by Birkhoff and MacLane (OED2).
TRIANGULAR NUMBER. Vieta used the terms triangular, pyramidal, triangulo-triangular, and triangulo-pyramidal number.
Triangular (as a noun) appears in English in 1706 in Synopsis Palmariorum Matheseos by William Jones (OED2).
The TRIDENT was named by Isaac Newton, according to John Harris in Lexicon Technicum.
Eves (page 279) has, "The locus is a cubic that Newton called a Cartesian parabola and that has also sometimes been called a trident; it appears frequently in La géometrie.
TRIDIMENSIONAL and UNIDIMENSIONAL appear in Sir William Rowan Hamilton, Lectures on Quaternions (London: Whittaker & Co, 1853) [James A. Landau].
Tridimensional appears in the following sentence: "But there was still another view of the whole subject, sketched not long afterwards in another communication to the R. I. Academy, on which it is unnecessary to say more than a few words in this place, because it is, in substance, the view adopted in the following Lectures, and developed with some fulness in them: namely, that view according to which a QUATERNION is considered as the QUOTIENT of two directed lines in tridimensional space."
Unidimensional appears in the following sentence: "It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception [3] of TIME, regarded here merely as an axis of continuous and uni-dimensional progression."
TRIGONOMETRIC EQUATION is found in English in 1857 in Mathematical Dictionary and Cyclopedia of Mathematics.
The term TRIGONOMETRIC FUNCTION was introduced in 1770 by Georg Simon Klügel (1739-1812), the author of a mathematical dictionary (Cajori 1919, page 234).
TRIGONOMETRIC LINE. Vincenzo Riccati (1707-1775) "for the first time used the term 'trigonometric lines' to indicate circular functions" in the three-volume Institutiones analyticae (1765-67), which he wrote in collaboration iwth Girolamo Saladini (DSB).
TRIGONOMETRIC SERIES. Trigonometrical series is found in English in 1857 in Mathematical Dictionary and Cyclopedia of Mathematics.
The term TRIGONOMETRY is due to Bartholomeo Pitiscus (1561-1613) and was first printed in his Trigonometria: sive de solutione triangulorum tractatus brevis et perspicuus, which was published as the final part of Abraham Scultetus' Sphaericorum libri tres methodicé conscripti et utilibus scholiis expositi (Heidelberg, 1595) (DSB).
The word first appears in English in 1614 in the English translation of the same work: Trigonometry: or The Doctrine of Triangles. First written in Latine, by B. Pitiscus..., and now Translated into English, by Ra. Handson.
TRINOMIAL was used in English in 1674 in Arith. (1696) Samuel Jeake (1623 - 1690): "If three Quantities be conjoyned, and but three, they are sometime called Trinomials" (OED2). [According to An Etymological Dictionary of the English Language (1879-1882), by Rev. Walter Skeat, "Not a good form; it should rather have been trinominal."]
TRISECTION appears in English in 1664 in Power, Exp. Philos.: "The Trisection of an Angle" (OED2).
TRIVARIATE appears in G. P. Steck, "A table for computing trivariate normal probabilities," Ann. Math. Statist. 29 (1958) [James A. Landau].
TROCHOID was coined by Gilles Persone de Roberval (1602-1675) (Smith vol. I, page 385; Cajori 1919, page 162).
The terms TRUNCATED CUBE, TRUNCATED OCTAHEDRON, TRUNCATED ICOSAHEDRON, and TRUNCATED DODECAHEDRON are all due to Johannes Kepler. He used cubus simus and dodekaedron simum in Harmonice Mundi (1619).
TRUTH SET is dated 1940 in MWCD10.
The term TRUTH TABLE was used by Emil Leon Post (1897-1954) in the title "Determination of all closed systems of truth tables" (abstract of a paper presented at the 24 April 1920 meeting of the American Mathematical Society), Bulletin of the American Meathematical Society 26 [James A. Landau].
Post also used the term in 1921 in the American Journal of Mathematics:
So corresponding to each of the 2n possible truth-configurations of the p's a definite truth-value of f is determined. The relation thus effected we shall call the truth-table of f.
TRUTH VALUE. Gottlob Frege (1848-1925) used the term Wahrheitswert in 1891 in Funktion, Begriff, Bedeutung (1975): "Ich sage nun: 'der Wert unserer Funktion ist ein Wahrheitswert' und unterscheide den Wahrheitswert des Wahren von dem des Falschen."
TSCHIRNHAUS' CUBIC appears in R. C. Archibald's paper written in 1600 where he attempted to classify curves, according to the University of St. Andrews website.
The term TURING MACHINE was used for the first time in 1937 by Stephen C. Kleene in the Journal of Symbolic Logic, according to an Internet website, which also states that the term Turing test seems to have appeared in the 1970s.
The OED2 shows a citation by A. Church in 1937 in Journal of Symbolic Logic: "[Abstract of Turing’s paper.] Certain further restrictions are imposed on the character of the machine, but these are of such a nature as obviously to cause no loss of generality - in particular, a human calculator, provided with pencil and paper and explicit instructions, can be regarded as a kind of Turing machine."
The term TWIN PRIME was coined in 1916 by Paul Gustav Stäckel (1862-1919) in "Die Darstellung der geraden Zahlen als Summen von zwei Primzahlen," Sitz. Heidelberger Akad. Wiss. (Mat.-Natur. Kl.) 7A (10) (1916), according to Algorithmic Number Theory by Bach and Shallit [Paul Pollack].
TYPE I ERROR and TYPE II ERROR. In their first joint paper "On the Use of Certain Test Criteria for Purposes of Statistical Inference, Part I," Biometrika, (1928) 20A, 175-240 Neyman and Pearson referred to "the first source of error" and "the second source of error" (David, 1995).
Errors of first and second kind is found in 1933 in J. Neyman and E. S. Pearson, "On the Problems of the Most Efficient Tests of Statistical Hypotheses," Philosophical Transactions of the Royal Society of London, Ser. A (1933), 289-337 (David, 1995).
Type I error and Type II error are found in 1933 in J. Neyman and E. S. Pearson, "The Testing of Statistical Hypotheses in Relation to Probabilities A Priori," Proceedings of the Cambridge Philosophical Society, 24, 492-510 (David, 1995).
The term ULTRAMETRIC was coined in 1944 by M. Krasner, according to an Internet web page.
The terms UMBRAL CALCULUS and UMBRAL NOTATION were coined by James Joseph Sylvester (1814-1897).
UNBIASED. See biased.
The term UNCONDITIONAL CONVERGENCE (of a continued fraction) was coined by Alfred Pringsheim (1850-1941).
UNDECAGON. Earlier terms for an 11-sided polygon, hendecagon and endecagon, are found in English in 1704 in Lexicon technicum, or an universal English dictionary of arts and sciences, by John Harris (OED2).
Undecagon is found in English in 1728 in Chambers' Cyclopedia.
UNDECIDABLE was used by Kurt Gödel (1906-1978) in 1931 in the title Uber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (On Formally Undecidable Propositions in Principia Mathematica and Related Systems).
UNGULA appears in 1710 in Lexicon technicum, or an universal English dictionary of arts and sciences, by John Harris: "Ungula, in Geometry, is the Section of a Cylinder cut off by a Plane, which passes obliquely thro' the Plane of the Basse, and part of the Cylindric Surface" (OED2).
UNIFORMLY DISTRIBUTED. Uniform distribution appears in 1937 in Introduction to Mathematical Probability by J. V. Uspensky. Page 237 reads, "A stochastic variable is said to have uniform distribution of probability if probabilities attached to two equal intervals are equal." This is a slight variant of the modern terminology, which would be "a variable is said to be uniformly distributed" or "a variable from the uniform distribution" [James A. Landau].
Uniformly distributed is found in H. Sakamoto, "On the distributions of the product and the quotient of the independent and uniformly distributed random variables," Tohoku Math. J. 49 (1943).
The phrase UNIFORMLY MOST POWERFUL occurs in R. A. Fisher, "Two New Properties of Mathematical Likelihood," Proceedings of the Royal Society, Series A, vol. 144 (1934) [James A. Landau].
UNIMODAL is found in 1904 in F. de Helguero, "Sui massimi delle curve dimorfiche," Biometrika, 3, 84-98 (David, 1995).
UNIT CIRCLE is found in 1895 in Plane and spherical trigonometry, surveying and tables by George Albert Wentworth: "The circle with radius equal to 1 is called a unit circle, AA' the horizontal, and BB' the vertical diameter" [University of Michigan Historic Math Collection].
UNIVARIATE is found in 1928 in Biometrika XXa. 32: "Various writers struggled with the problems that arise when samples are taken from uni-variate and bi-variate populations" (OED2).
The term UNIVERSAL ALGEBRA was first used by James Joseph Sylvester (1814-1897) for the theory of matrix algebras in a paper, "Lectures on the Principles of Universal Algebra," published in the American Journal of Mathematics, vol. 6, 1884, according to Whitehead [Encyclopaedia Britannica, article: Algebraic Structures].
Charles S. Peirce had objected to the term in a letter to Sylvester of Jan. 5, 1882: "I confess I cannot see why the system should be called 'universal.' It is a favorite epithet among inventors of all kinds in this country; but it seems to me best to restrict it to a very exact signification. A dual 'relative' is a collection of pairs, and this is an algebra of collections of pairs."
UNKNOWN. See root.
VANDERMONDE DETERMINANT. According to the University of St. Andrews website, this term was introduced by Henri Léon Lebesgue (1875-1941).
However, the DSB states that Lebesgue believed that the attribution of this determinant to Vandermonde was due to a misreading of his notation, implying Lebesgue did not introduce the term.
The term appears in Weill, "Sur une forme du déterminant de Vandermonde, Nouv. Ann. (1888).
The term VANISHING POINT was coined by Brook Taylor (1685-1731), according to Franceschetti (p. 500).
The term VARIABLE was introduced by Gottfried Wilhelm Leibniz (1646-1716) (Kline, page 340).
Variable is found in English as an adjective in 1710 in Lexicon Technicum by J. Harris: "Variable Quantities, in Fluxions, are such as are supposed to be continually increasing or decreasing; and so do by the motion of their said Increase or Decrease Generate Lines, Areas or Solidities" (OED2).
Variable is found in English as a noun in 1816 in a translation of Lacroix's Differential and Integral Calculus: "The limit of the ratio..will be obtained by dividing the differential of the function by that of the variable" (OED2).
VARIANCE. Edgeworth used fluctuation for the square of the standard deviation.
Variance was introduced by Ronald Aylmer Fisher in 1918 in "The Correlation Between Relatives on the Supposition of Mendelian Inheritance," Transactions of the Royal Society of Edinburgh, 52, 399-433: "It is ... desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance."
VARIATE appears in 1909 in Karl Pearson, "On a New Method of Determining Correlation...," Biometrika, 7, 96-105 (David, 1998).
VARIETY (as in modern algebraic geometry) was first used by E. Beltrami in 1869 [Joseph Rotman].
Birkhoff used the term equationally defined algebras in his AMS Colloquium Volume Lattice Theory in the first 1940, second 1948 and third 1967 edition.
Hanna Neumann (1914-1971) introduced the term variety in "On varieties of groups and their associated near-rings," Math. Zeits., 65, 36-69 (1956) and popularised the term in her 1967 book Varieties of Groups [Phill Schultz].
The word VECTOR in astronomy usually occurs as part of the term radius vector.
Vector appears in English in a 1704 dictionary; radius vector appears in English in a 1753 dictionary.
Laplace used rayon vecteur in his Méchanique Celeste (1799-1825).
Radius vector appears in a mathematical sense in 1831 in Elements of the Integral Calculus (1839) by J. R. Young: "...when the angle [omega] between the radius vector and fixed axis is taken for the independent variable, the formula is...."
William Rowan Hamilton used radius vector in article 14 of "On a General Method in Dynamics; by which the Study of the Motions of all free Systems of attracting or repelling Points is reduced to the Search and Differentiation of one central Relation, or characteristic Function." This paper was published in the Philosophical Transactions of the Royal Society of London in 1834.
VECTOR (in mathematics). Both the terms vector and scalar were introduced by William Rowan Hamilton (1805-1865).
Both terms appear in a paper presented by Hamilton at a meeting of the Royal Irish Academy on November 11, 1844. This paper adopts the convention of denoting a vector by a single (Greek) letter, and concludes with a discussion of formulae for applying rotations to vectors by conjugating with unit quaternions. It is on pages 1-16 in volume 3 of the Proceedings of the Royal Irish Academy, covering the years 1844-1847, and the volume is dated 1847. The following is from page 3:
On account of the facility with which this so called imaginary expression, or square root of a negative quantity, is constructed by a right line having direction in space, and having x, y, z for its three rectangular axes, he has been induced to call the trinomial expression itself, as well as the line which it represents, a VECTOR. A quaternion may thus be said to consist generally of a real part and a vector. The fixing a special attention on this last part, or element, of a quaternion, by giving it a special name, and denoting it in many calculations by a single and special sign, appears to the author to have been an improvement in his method of dealing with the subject: although the general notion of treating the constituents of the imaginary part as coordinates had occurred to him in his first researches.
The following is from page 8:
It is, however, a peculiarity of the calculus of quaternions, at least as lately modified by the author, and one which seems to him important, that it selects no one direction in space as eminent above another, but treats them as all equally related to that extra-spacial, or simply SCALAR direction, which has been recently called "Forward."
In Hamilton's time, radius-vector was an established term in astronomy to denote the distance of an astronomical object from the sun or the earth. Hamilton makes it clear that he is introducing the term vector with a new sense, involving both length and direction. He explains this in, for example, section 16 of Lecture I of his "Lectures on Quaternions." He expands on this distinction in article 17, which includes the following:
17. To illustrate more fully the distinction which was just now briefly mentioned, between the meanings of the "Vector" and the "Radius Vector" of a point, we may remark that the RADIUS-VECTOR, in astronomy, and indeed in geometry also, is usually understood to have only length; and therefore to be adequately expressed by a SINGLE NUMBER, denoting the magnitude (or length) of the straight line which is referred to by this usual name (radius-vector) as compared with the magnitude of some standard line, which has been assumed as the unit of length. Thus, in astronomy, the Geocentric Radius-Vector of the Sun is, in its mean value, nearly equal to ninety-five millions of miles: if, then, a million of miles be assumed as the standard or unit of length, the sun's geocentric radius-vector is equal (nearly) to, or is (approximately) expressible by, the number ninety-five: in such a manner that this single number, 95, with the unit here supposed, is (at certain seasons of the year) a full, complete and adequate representation or expression for that known radius vector of the sun. For it is usually the sun itself (or more fully the position of the sun's centre) and NOT the Sun's radius-vector, which is regarded as possessing also certain other (polar) coordinates of its own, namely, in general, some two angles, such as those which are called the Sun's geocentric right-ascension and declination; and which are merely associated with the radius-vector, but not inherent therein, nor belonging thereto...
But in the new mode of speaking which it is here proposed to introduce, and which is guarded from confusion with the older mode by the omission of the word "RADIUS," the VECTOR of the sun HAS (itself) DIRECTION, as well as length. It is, therefore NOT sufficiently characterized by ANY SINGLE NUMBER, such as 95 (were this even otherwise rigorous); but REQUIRES, for its COMPLETE NUMERICAL EXPRESSION, a SYSTEM OF THREE NUMBERS; such as the usual and well-known rectangular or polar co-ordinates of the Sun or other body or point whose place is to be examined...
A VECTOR is thus (as you will afterwards more clearly see) a sort of NATURAL TRIPLET (suggested by Geometry): and accordingly we shall find that QUATERNIONS offer an easy mode of symbolically representing every vector by a TRINOMIAL FORM (ix + jy + kz); which form brings the conception and expression of such a vector into the closest possible connexions with Cartesian and rectangular co-coordinates.
Hamilton, in his Lectures on Quaternions, is not satisfied with having introduced vector. Within a few pages we find vectum, vehend, revector, provector, provectum, transvehend, transvectum, etc., and identities such as
Provectum = Provector + Vector + Vehend.
Vector and scalar also appear in 1846 in a paper "On Symbolical Geometry" in the The Cambridge and Dublin Mathematical Journal vol. I:
If then we give the name of scalars to all numbers of the kind called usually real, because they are all contained on the one scale of progression of number from negative to positive infinity [...]
Next Hamilton goes on to tell us about another "chief class" of the "geometrical quotients," namely
the class in which the dividend is a line perpendicular to the divisor. A quotient of this latter class we shall call a vector, to mark its connection (which is closer than that of a scalar) with the conception of space [...]
David Wilkins believes that the paper "On quaternions" in the Proceedings of the Royal Irish Academy probably appeared earlier than the the CDMJ, probably some time in the first half of 1845.
The first occurrence of vector and scalar in the London, Edinburgh, and Dublin Philosophical Magazine is in volume XXIX (1846):
The separation of the real and imaginary parts of a quaternion is an operation of such frequent occurrence, and may be regarded as being so fundamental in this theory, that it is convenient to introduce symbols which shall denote concisely the two separate results of this operation. The algebraically real part may receive, according to the question in which it occurs, all values contained on the one scale of progression from number from negative to positive infinity; we shall call it therefore the scalar part, or simply the scalar of the quaternion, and shall form its symbol by prefixing, to the symbol of the quaternion, the characteristic Scal., or simply S., where no confusion seems likely to araise from using this last abbreviation. On the other hand, the algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion; and may be denoted by prefixing the characteristic Vect. or V...
(Information for this article was provided by David Wilkins and Julio González Cabillón.)
VECTOR ANALYSIS occurs in 1881 in the title Elements of Vector Analysis by J. W. Gibbs.
The OED2 shows an 1881 quotation from J. W. Gibbs Scientific Papers (1906): "An algebra or analytical method in which a single letter or other expression is used to specify a vector may be called a vector algebra or vector analysis."
VECTOR FIELD is found in "Natural Families of Trajectories: Conservative Fields of Force," Edward Kasner, Transactions of the American Mathematical Society, Vol. 10, No. 2. (Apr., 1909).
VECTOR PRODUCT and SCALAR PRODUCT are found in 1878 in Dynamic by William Kingdon Clifford (1845-1879) (OED2).
VECTOR SPACE. The notion of a vector space is due to Hermann Günter Grassmann (1844).
Peano's Geometrical Calculus (1888) defines the notion and presumably uses the term.
Vector space occurs in English in "On the Geometry of Planes in a Parabolic Space of Four Dimensions," Irving Stringham, Transactions of the American Mathematical Society, Vol. 2, No. 2. (Apr., 1901).
VECTOR TRIPLE PRODUCT occurs in 1901 in Gibbs and Wilson, Vector Analysis (OED2).
VENN DIAGRAM. Euler's scheme of notation is found in 1858 in Elements of logic by Henry Coppée (1821-1895): "Euler's scheme of notation is altogether the one best suited to our purpose, and we shall limit ourselves to the explanation of that. It is essentially an arrangement of three circles, to represent the three terms of a syllogism, and, by their combination, the three propositions" [University of Michigan Digital Library].
Euler's system of notation appears in 1863 in An outline of the necessary laws of thought: a treatise on pure and applied logic by William Thomson (University of Michigan Digital Library).
Euler's notation appears in about 1869 in The principles of logic, for high schools and colleges by Aaron Schuyler (University of Michigan Digital Library).
Euler's diagram appears in 1884 in Elementary Lessons in Logic by W. Stanley Jevons: "Euler's diagram for this proposition may be constructed in the same manner as for the proposition I as follows:..."
Euler's circles appears in 1893 in Logic by William Minto (1845-1893): "The relations between the terms in the four forms are represented by simple diagrams known as Euler's circles."
Euler's circles appears in October 1937 in George W. Hartmann, "Gestalt Psychology and Mathematical Insight," The Mathematics Teacher: "But in the case of 'Euler's circles' as used in elementary demonstrations of formal logic, one literally 'sees' how intimately syllogistic proof is linked to direct sensory perception of the basic pattern. It seems that the famous Swiss mathematician of the eighteenth century was once a tutor by correspondence to a dull-witted Russian princess and devised this method of convincing her of the reality and necessity of certain relations established deductively."
Venn diagram appears in 1918 in A Survey of Symbolic Logic by Clarence Irving Lewis: "This method resembles nothing so much as solution by means of the Venn diagrams" (OED2).
VERSED SINE. According to Smith (vol. 2, page 618), "This function, already occasionally mentioned in speaking of the sine, is first found in the Surya Siddhanta (c. 400) and, immediately following that work, in the writings of Aryabhata, who computed a table of these functions. A sine was called the jya; when it was turned through 90 degrees and was still limited by the arc, it became the turned (versed) sine, utkramajya or utramadjya."
Albategnius (al-Battani, c. 920) uses the expression "turned chord" (in some Latin translations chorda versa).
The Arabs spoke of the sahem, or arrow, and the word passed over into Latin as sagitta.
Boyer (page 278) seems to imply that sinus versus appears in 1145 in the Latin translation by Robert of Chester of al Khowarizmi's Algebra, although Boyer is unclear.
In Practica geomitrae, Fibonacci used the term sinus versus arcus. According to Smith (vol. 2), Fibonacci (1220) used sagitta.
Fincke used the term sinus secundus for the versed sine.
Regiomontanus (1436-1476) used sinus versus for the versed sine in De triangulis omnimodis (On triangles of all kinds; Nuremberg, 1533).
Maurolico (1558) used sinus versus major (Smith vol. 2).
The OED shows a use in 1596 in English of "versed signe" by W. Burrough in Variation of Compasse.
The term VERSIERA was coined by Luigi Guido Grandi (1671-1742) (DSB). See witch of Agnesi.
The term VERSOR was introduced by William Rowan Hamilton (1805-1865) (Julio González Cabillón.)
Versor appears about 1865 in Sir W. R. Hamilton, Elem. Quaternions ii. i. (1866) 133: "We shall now say that every Radial Quotient is a Versor. A Versor has thus, in general, a plane, an axis, and an angle" (OED2).
VERTEX occurs in English in 1570 in John Dee's preface to Billingsley's translation of Euclid (OED2).
In 1828, Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre) has:
In ordinary language, the word angle is often employed to designate the point situated at the vertex. This expression is inaccurate. It would be more correct and precise to use a particular name, such as that of vertices for designating the points at the corners of a polygon or of a polyedron. The denomination vertices of a polyedron, as employed by us, is to be understood in this sense.
VERTICAL ANGLE is found in English in 1571 in Thomas Digges, Pantometria.
VIGINTIANGULAR. The OED2 shows one citation, from 1822, for this term, meaning "having 20 angles." The word also appears in Webster's New International Dictionary, 2nd ed. (1934).
VINCULUM and VIRGULE. In the Middle Ages, the horizontal bar placed over Roman numerals was called a titulus. The term was used by Bernelinus. It was used more commonly to distinguish numerals from words, rather than to indicate multiplication by 1000.
Fibonacci used the Latin words virga and virgula for the horizontal fraction bar.
Tartaglia (1556) used virgoletta for the horizontal fraction bar (Smith vol. 2, page 220).
In 1594 Blundevil in Exerc. (1636) referred to the fraction bar as a "little line": "The Numerator is alwayes set above, and the Denominator beneath, having a little line drawne betwixt them thus 1/2 which signifieth one second or one halfe" (OED2).
In 1660 J. Moore in Arith. used separatrix for the line that was then placed after the units digit in decimals: "But the best and most distinct way of distinguishing them is by a rectangular line after the place of the unit, called Seperatrix. ... Therefore in writing of decimall parts let the seperatrix be always used" (OED2).
In 1696, Samuel Jeake referred to the fraction bar as "the intervening line" in his Arithmetick.
In 1771 separatrix was used for the fraction bar in Luckombe, Hist. Printing: "The Separatrix, or rule between the Numerator and Denominator [of fractions]" (OED2).
Leibniz, writing in Latin, used vinculum for the grouping symbol.
In mathematics, vinculum originally referred only to the grouping symbol, but some writers now use the word also to describe the horizontal fraction bar.
The term VON NEUMANN ALGEBRAS was used by Jacques Dixmier in 1957 in Algebras of operators in Hilbert space (von Neumann algebras). The term is named for John von Neumann (1903-1957), who had used the term "rings of operators." Another term is "W-algebras."
VULGAR FRACTION. In Latin, the term was fractiones vulgares, and the term originally was used to distinguish an ordinary fraction from a sexagesimal.
Trenchant (1566) used fraction vulgaire (Smith vol. 2, page 219).
Digges (1572) wrote "the vulgare or common Fractions."
Sylvester used the term in On the theory of vulgar fractions, Amer. J. Math. 3 (1880).
The term common fraction is now more widely used.
WELL-ORDERED. The term wohlgeordnet was used by Cantor in an extensive paper, "Über unendliche lineare Punctmannichfaltigkeiten," which appeared in Mathematische Annalen in six parts between 1879 and 1884. In part five, which appeared in vol. 21 (1883), he wrote (page 168):
By a well-ordered set we understand any well-defined set whose elements are related by a well-determined given succession according to which there is a first element in the set and for any element (if it is not the last one) there is a certain next following element. Furthermore, for any finite or infinite set of elements there is a certain element which is the next following one for all these elements (except for the case that such an element which is the next following one to these elements does not exist).
This translation was taken from Cantor's Philosophical Views by Walter Purkert.
WHITE NOISE. Originally the term referred to a form of sound or of electrical interference but it now also refers to a type of random process. "Inside the plane ... we hear all frequencies added together at once, producing a noise which is to sound what white light is to light." (L. D. Carson, W. R. Miles & S. S. Stevens, "Vision, Hearing and Aeronautical Design," Scientific Monthly, 56, (1943), 446-451). S. Goldman's book on radio engineering, Frequency Analysis, Modulation and Noise (1948), has a mathematical treatment of white noise.
By 1953 white noise had entered the stochastic process literature, as in "On the Fourier Expansion of Stationary Random Processes" by R. C. Davis (Proceedings of the American Mathematical Society, 4, 564-569) [John Aldrich].
WHOLE NUMBER. See integer.
WIENER PROCESS appears in M. Kac's "On Deviations Between Theoretical and Empirical Distributions," Proc. Nat. Acad. Sciences, 35, (1949), 252-257. The name recalls N. Wiener's analysis of "the Brownian movement" in "Differential-space" J. Math. and Phys. 2 (1923) 131-174. (See Brownian motion.) [John Aldrich]
WILSON'S THEOREM was given its name by Edward Waring (1734-1798) for his friend, John Wilson (1741-1793). The first published statement of the theorem was by Waring in his Meditationes algebraicae (1770), although manuscripts in the Hanover Library show that the result had been found by Leibniz.
WINSORIZED is found in 1960 in W. J. Dixon, "Simplified Estimation from Censored Normal Samples," The Annals of Mathematical Statistics, 31, 385-391 (David, 1998).
WITCH OF AGNESI. Luigi Guido Grandi (1671-1742) studied this curve in 1703 and is believed to have been the first to call it versiera or versoria in Latin, meaning "turning in every direction." According to Boyer in History of Analytic Geometry, Grandi coined the Italian word la versiera in 1718. The term appears in Father Guido Grandi's commentary on the Trattato del Galileo del moto naturalmente accelerato (Opere di G. Galilei, III, Firenze, 1718, p. 393): "...sarebbe quella curve, che io descrivo nel mio libro delle quadrature alla prop. 4, nata da seni versi, che da me suole chiamarsi la versiera in latino perň versoria..."
In 1748, Maria Gaetana Agnesi (1718-1799), in Istituzioni Analitiche, the first calculus book written by a woman, also called the curve la versiera, using the name twice.
The British mathematician John Colson (1680-1760), translating Agnesi's work into English, translated the Italian word versiera as "the Witch." He wrote, "...and therefore [equation] or [equation] will be the equation of the curve to be described, which is vulgarly called the Witch." He also wrote, "Let the curve to be described be that of Prob. III. n. 238, called the Witch, the equation of which is [equation]." Colson gave the name a third time, in a marginal note, "Another example of the curve called the Witch."
According to the translator's preface to the 1801 English edition of Analytical Institutions, Colson learned Italian for the sole purpose of translating this work.
Witch of Agnesi is found in English in 1875 in An elementary treatise on the integral calculus by Benjamin Williamson (1827-1916): "Find the area between the witch of Agnesi xy2 = 4a2 (2a - x) and its asymptote" (OED2).
WORKING HYPOTHESIS occurs in 1871 in R. H. Hutton, Ess. I. v. 112: "If it be only a working hypothesis, to keep us, while confined in the human, from blindly and unconsciously dashing ourselves against the laws of the divine" (OED2).
WORKING MATHEMATICIAN. In an article "The Ignorance of Bourbaki" (The Mathematical Intelligencer vol. 14, no 3, 1992), A. R. D. Mathias suggests that this phrase is due to Bourbaki. However, Carlos César de Araújo has found it in a paper by Eliakim Hastings Moore, "On the foundations of mathematics" (Bull. A. M. S., 1903, p. 406).
The term WRONSKIAN (for Höené Wronski) was coined by Thomas Muir (1844-1934) in 1881 (Cajori 1919, page 310).
X-AXIS. Axis of x appears in "On the Attractions of Homogeneous Ellipsoids" by James Ivory, Philosophical Transactions of the Royal Society of London, Vol. 99. (1809), pp. 345-372. [JSTOR].
X-axis appears in 1886 in W. B. Smith, Elem. Co-Ordinate Geom.: OX, OY, are called Co-ordinate Axes, or axes of X and Y, or X- and Y-axes" (OED2).
The terms X-COORDINATE, Y-COORDINATE, and Z-COORDINATE appear in a paper published by James Joseph Sylvester in 1863 [James A. Landau].
X-INTERCEPT is found in 1924 in Analytic Geometry by Arthur M. Harding and George W. Mullins: "If a curve cuts the x and y axes at the points A and B respectively, the segment OA is called the intercept on the x axis or the x intercept, and the segment OB is called the intercept on the y axis or the y intercept."
Y-AXIS. Axis of y appears in "On the Attractions of Homogeneous Ellipsoids" by James Ivory, Philosophical Transactions of the Royal Society of London, Vol. 99. (1809), pp. 345-372. [JSTOR].
Y-axis is dated 1875 in MWCD10.
Y-axis is found in "Law of Facility of Errors in Two Dimensions [Continued]," E. L. de Forest, The Analyst, Vol. 8, No. 3. (May, 1881), pp. 73-82 [JSTOR].
Y-COORDINATE. See x-coordinate.
Y-INTERCEPT. Isaac Todhunter refers to "the intercept on the axis of y" in the 7th ed. of A Treatise on Plane Co-ordinate Geometry (1881).
y-intercept is found in 1924 in Analytic Geometry by Arthur M. Harding and George W. Mullins: "Find the equation of the line ... whose slope is 2/3 and whose y intercept is -3."
The term YOUDEN SQUARE was used by R. A. Fisher and F. Yates in the introduction to their Statistical Tables (1938). The term was coined by Ronald Aylmer Fisher (1890-1962) and was named for William John Youden (1900-1971) (DSB, article: "Youden").
z-AXIS. Axis of z appears in "On the Attractions of Homogeneous Ellipsoids" by James Ivory, Philosophical Transactions of the Royal Society of London, Vol. 99. (1809), pp. 345-372. [JSTOR].
The terms z-STATISTIC and 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) [James A. Landau].
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 Zermelo 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.
Cipher is found in English as early as 1399. 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 1604: E. Grimstone, D'Acosta's Hist. Indies "They accompted their weekes by thirteene dayes, marking the dayes with a Zero or cipher."
A 1706 dictionary has: "Zero, a Word sometimes us'd especially among the French, for a Cipher or Nought (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-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 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 Historic Math Collection].
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. Gregory Moore, in his definitive book Zermelo's Axiom of Choice: Its Origins, Development and Influence, says, "By late in 1934, Zorn's principle had found users in the United States who dubbed it Zorn's Lemma." [Bill Dubuque] Julio González Cabillón believes the term may have been coined by John W. Tukey.