See MULTIPLICATION TABLE.
TAIL of a distribution appears in the writings of Karl Pearson although the image probably presented itself to earlier workers: "... a frequency-curve calculated from low moments fits best near the centre; one calculated from high moments fits best near the tails of the observation-curve." ("Contributions to the Mathematical Theory of Evolution" Philosophical Transactions of the Royal Society. A, 185, (1894), p. 108.) The term tail area seems to have come into circulation in the 1920s: see e.g. Karl Pearson's "On a Method of Ascertaining Limits to the Actual Number of Marked Members in a Population of Given Size from a Sample," Biometrika, 20A, (1928) p. 171.
For "shorter" and "longer" tails see the entry on KURTOSIS. The terms "light" and "heavy" and "thin" and "thick" entered currency in the 1970s (JSTOR search.)
TANGENT (in geometry). Tangents to a circle are treated in the propositions of Book III of Euclid’s Elements.
The Latin līnea tangens (touching line) appears in Fincke’s Geometria rotundi (1583). In English the term “line tangent” appears in Thomas Blundevil’s Exercises (1597) “Our moderne Geometricians haue of late inuented two other right lines belonging to a Circle, called lines Tangent, and lines Secant.” Blundevil also uses “tangent” as a noun: “Of which Arke the line AD is the Tangent, and the line CD is the Secant thereof.” (The citations are from the OED.)
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')."
See Earliest Uses of Symbols of Trigonometry.
TANGRAM. The modern meaning of this term is a puzzle or square figure divided into five triangles, a square, and a rhomboid.
According to various dictionaries, the word may be derived from a Chinese word tang, or it may be derived from the obsolete English word trangam, meaning a trinket or a gimcrack.
Trangam is found in a 1658 dictionary. Merriam-Webster says the word is of unknown origin.
On June 1, 1809, the American Citzien reported, “Vast numbers of those ‘tangrams and gimcracks’ are piled up in the office, of every shape and size, making it a great toy shop. [Joel S. Berson]
A classified advertisement in the Franklin Gazette of Feb. 24, 1818, offers “Chinese Tangrams,” which were probably puzzles [Bill Mullins].
According to Wikipedia and this web page, the word tangram was coined by Dr. Thomas Hill in 1848 for his book Geometrical Puzzles for the Young. [Perhaps this is the first use of the word with its modern meaning.] According to the same web page, the device was invented between 1796 and 1802 in China by Yang-cho-chu-shih, who published the book Ch'i ch'iao t'u (Pictures using seven clever pieces).
The expression TAUBERIAN THEOREMS was introduced by G. H. Hardy and J. E. Littlewood in their "Contributions to the Arithmetic Theory of Series," Proc. London Math. Soc. (2) XI. (1913), p. 411: "The general character of the theorems which it [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." The reference is to Alfred Tauber "Ein Satz aus der Theorie der unendlichen Reihen" Monatsch. für Math., 8, (1897) 273-277. Norbert Wiener judged the work of Hardy and Littlewood such an advance on that of Tauber that, "I feel it would be far more appropriate to term these theorems Hardy-Littlewood theorems, were it not that usage has sanctioned the other appellation." ("Tauberian Theorems," Annals of Mathematics, 2nd Ser., 33, No. 1. (Jan., 1932), 1-100. See the article Tauberian theorems.
This entry was contributed by John Aldrich. See also HARDY-LITTLEWOOD THEOREM.
TAYLOR'S FORMULA is found in English in 1828 in An Elementary Treatise on the Differential and Integral Calculus by Jean-Louis Boucharlat and Ralph Blakelock. [Google print search]
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 Théorème 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 1819 in New Series of The Mathematical Repository by Thomas Leybourn: “Re-searches into some points of the theory of derivative functions, which lead to a new demonstration of TAYLOR'S series, and to the finite expression for the terms which are neglected, when the series stops at any term, by AMPERE.”
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 Théorème 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 was one of the family of terms introduced by William Rowan Hamilton (1805-1865) in his study of QUATERNIONS. VECTOR and SCALAR and VERSOR were among the others. The tensor is for quaternions what the MODULUS is for complex numbers. The term derives from the Latin tendĕre to stretch.
In 1846 Hamilton wrote 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 passage is reproduced in Section 19 of “On Quaternions”. This ‘article’ is a compilation of 18 short papers published in the Philosophical Magazine between 1844 and 1850 made by the editors of Hamilton’s Mathematical Papers. The editors concatenated them to form a seamless whole, with no indication as to how the material was distributed into the individual papers.
Tensor in Hamilton's sense is no longer used.
[Information for this article was provided by David Wilkins and Julio González Cabillón.]
TENSOR, TENSOR ANALYSIS, TENSOR CALCULUS, etc. are 20th century terms associated with the ABSOLUTE DIFFERENTIAL CALCULUS developed by Ricci-Curbastro in the 1880s and -90s on the basis of earlier work by Riemann, Christoffel, Bianchi and others. See Kline ch. 37 “The Differential Geometry of Gauss and Riemann” and ch. 48 “Tensor Analysis and Differential Geometry.”
Ricci's most influential publication was a substantial article written with his former student Levi-Civita. The article by the two Italians was written in French and appeared in the leading German mathematical journal: “Méthodes de calcul différentiel absolu et leurs applications,” Mathematische Annalen, 54 (1901), p. 125-201. The word tensor does not appear: Ricci and Levi-Civita write about systèmes. Tensor is due to the well-known Goettingen physicist Woldemar Voigt (1850-1919), who used it in his Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung of 1898 (OED and Julio González Cabillón).
Critically tensor was the term adopted by Einstein and Grossmann in their first publication on general relativity, Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation (Outline of a Generalized Theory of Relativity and of a Theory of Gravitation) (1913). Einstein made the subject fashionable. MacTutor relates that on a visit to Princeton in 1921 he commented on the large audience his lecture attracted: “I never realised that so many Americans were interested in tensor analysis.” See also MacTutor: General Relativity.
The OED reports that tensor analysis is found in English in 1922 in H. L. Brose's translation of Weyl's Space-Time-Matter (Raum, Zeit, Materie): “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.” The phrase tensor calculus appears in the same book. When Levi-Civita's Lezioni di calcolo differenziale assoluto was translated into English in 1926, its title included an explanation: The Absolute Differential Calculus (Calculus of Tensors).
[This entry was contributed by John Aldrich.]
The term TERAGON was coined by Mandelbrot, according to an Internet web page.
TERMINATING DECIMAL is found in 1850 in the fourth edition of Lectures on the philosophy of arithmetic and the adaptation of that science to the business purposes of life by Uriah Parke: "In order to determine whether a fraction will produce a terminating decimal, we may analyze the denominator, and see whether there is in it any prime factor but 2 and 5" [University of Michigan Digital Library].
Finite decimal appears in 1850 in Arithmetic on the productive system, accompanied by a key and cubical blocks by Roswell C. Smith: "17. REPEATING DECIMALS are also called INFINITE DECIMALS, those that terminate, or come to an end, FINITE DECIMALS" [University of Michigan Digital Library].
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. In Euclid 
is a proposition to be proved.
It comes from an older word meaning “to look at” and related modern English
words include “theatre” and “theory.” (From Schwartzman and the OED.)
The word was adopted in Late Latin and French before being taken into English.
The earliest citation given by the OED for the word in English is 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.” In the first English translation of Euclid, which appeared in 1570, Billingsley writes “A Theoreme, is a proposition, which requireth the searching out and demonstration of some propertie..of some figure.”
THEOREMA EGREGIUM is the name given to a theorem of Gauss to the effect that the measure of curvature does not change if the surface is transformed by a distance-preserving transformation. Gauss applied the adjective himself in the “Disquisitiones generales circa superficies curvas” (1827): see Gauss Werke 4, p. 237. The adjective is rendered as “remarkable” in General investigations of curved surfaces translated by Morehead and Hiltebeitel p. 20.
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 à noyau symétrique," by Emile Borel, Compt. Rend. Acad. Sci., (Dec. 19, 1921) pp. 1304-8.
The term Theorie der Gesellschaftsspiele appears in 1928 in the title, "Zur Theorie der Gesellschaftsspiele" by J. von Neumann, Math. Ann., 100, 295-320. 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.The English term, theory of games, appears in 1944 in the title Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. This work established the theory of games as a major subject. [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. See PROBABILITY.
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.]
TOPOLOGY and TOPOLOGICAL. These terms came into wide use only in the 1930s, although the name topology had been coined in the 19th century, topological results obtained in the 18th and the study projected in the 17th. See MacTutor History of Topology.
The earliest reference in Kline’s account of the history of topology (ch. 50, p. 1163) is to Leibniz’s Characteristica Geometrica of 1679. On Kline’s reading, "to the extent that he was at all clear, Leibniz envisioned what we now call combinatorial topology." On the other hand, M. J. Crowe A History of Vector Analysis interprets the same work as an envisioning of vector analysis! Leibniz called the study analysis situs or geometria situs.
Work from the 17th and 18th centuries, now be classified as topological, include the EULER-DESCARTES RELATION V + F - E = 2 for polyhedra and the KÖNIGSBERG BRIDGE PROBLEM. When Euler wrote about the latter he began by referring to Leibniz:
The branch of geometry that deals with magnitudes has been zealously studied throughout the past; but there is another branch that has been almost unknown up to now; Leibnitz spoke of it first, calling it the "geometry of position" (geometria situs). This branch of geometry deals with relations dependent on position alone, and investigates the properties of position; it does not take magnitudes into consideration, nor does it involve calculation with quantities.
(For publication details see the entry KÖNIGSBERG BRIDGE PROBLEM.)
The word 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. Listing wrote about the EULER-DESCARTES RELATION and about the MÖBIUS STRIP.
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).
Listing, however, was not a major contributor to the field he named and the major contributors of the 19th century favoured Leibniz’s term analysis situs. Riemann revived the term (see section 2 of his Theorie der Abel’schen Functionem (1857) Werke p. 92) and Poincaré, the most important contributor of the 19th century, also used it. Poincaré’s first big paper on the subject was "Analysis Situs," Journal de l’École Polytechnique, 1, (1892) 1-121; a short communication is available on-line, "Sur L’Analysis Situs," Comptes Rendus, 115, (1892), 633-636. Analysis situs was the title of Dehn and Heegaard’s survey in the Encyklopädie der mathematischen Wissenschaften (1907, p. 153) and of O. Veblen’s American Mathematical Society Colloquium lectures (1922).
In the first decades of the 20th century the adjective TOPOLOGICAL was more used than the noun. Thus the expression "topological point of view" appears in O. Veblen & J. W. Alexander "Manifolds of N Dimensions," Annals of Mathematics, 14, (1912 - 1913), p. 163 as a variation on "point of view of analysis situs." (JSTOR search.)
The term TOPOLOGICAL SPACE (topologischer Raum) was coined by Hausdorff and appears in his book on set theory (and analysis), Grundzüge der Mengenlehre (1914, p. 211). The same work discusses METRIC SPACES.
By 1930 the term analysis situs was dead and in 1930-5 major works appeared in English, French and German called TOPOLOGY: Lefschetz Topology (1930), Kuratowski Topologie I (1933) and Alexandroff & Hopf Topologie (1935).
The terms TOPOLOGICAL ALGEBRA and TOPOLOGICAL GROUP were coined by David van Dantzig. The first appears in the title of his 1931 Ph. D. dissertation “Studiën over topologische Algebra” (DSB) and the second in his “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].
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].
An early use of torus as a mathematical term in English is in 1860 in The Practical Draughtsman's Book of Industrial Design by William Johnson: “In geometry, the torus is a solid, generated by a circle, revolving about an axis, continuing constantly in the plane of this axis, in such a manner, that all sections made by planes passing through the axis are equal circles, and all sections by planes perpendicular to the axis will also be circles, but of variable diameters.” [Google print search]
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.
See Euler's φ function on Earliest Use of Symbols of Number Theory.
TOWER OF HANOI. The Tower of Hanoi puzzle was published by Edouard Lucas in 1883. Lucas published it under the name of M. Claus, an anagram of Lucas. See Paul K. Stockmeyer’s page.
TRACE (of a matrix) is a translation of the German die Spur (related to the English word "spoor.") In his 1922 translation of H. Weyl's Raum, Zeit, Materie (Space-Time-Matter) H. L. Brose writes: "the trace (spur) of a matrix." Some writers in English preferred the term "spur," e.g. A. C. Aitken Determinants and Matrices (9th edition 1956) writes "the spur or trace of A. We shall denote it by sp A."
The TRACTRIX was named by Christiaan Huygens (1629-1695) in 1692, according to the University of St. Andrews website.
The OED quotes the entry from Chambers Cyclopædia (1727-41), "Tractrix, in geometry, a curve line, called also catenaria." [Error: the tractrix is the involute of the catenary, not the catenary itself.]
In Webster's 1828 dictionary, the word is spelled tractatrix, with the middle syllable stressed.
TRANSCENDENTAL. Leibniz coined the word transcendental in mathematics, using transcendens in the fall of 1673 in Progressio figurae segmentorum circuli aut ei sygnotae.
Leibniz used curvae transcendentes and figurae transcendentes in December 1674 in De progressionibus et geometria arcana et methodo tangentium inversa.
Leibniz used aequatio transcendens in 1676 in Series convergentes duae.
[This information was provided by Siegmund Probst, who reports that these Leibniz papers
can be viewed at an archive of publications by Leibniz at
http://www.nlb-hannover.de/Leibniz/Leibnizarchiv/Veroeffentlichungen/].
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.
Transcendental equation is found in English in 1813 in Pantologia. A new cabinet cyclopædia, by John Mason Good, Olinthus Gilbert Gregory, and N. Bosworth [Google print search]
Transcendental quantity appears in 1820 in A Collection of Examples of the Applications of the Differential and Integral Calculus by George Peacock
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 π 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 π is also an irrational number.
Transcendental number appears (as transscendente Zahl)
in 1882 in "Ueber die Zahl 
" by F.
Lindemann.
Transcendental number appears in English in a lecture delivered on Sept. 4, 1893, "The Transcendency of the Numbers e and π," found in Lectures on Mathematics: Delivered from Aug. 28 to Sept. 9, 1893, Before Members of the Congress of Mathematics by Felix Klein: "The proof that π is a transcendental number will forever mark an epoch in mathematical science." [Google print search]
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."
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" Mathematische Annalen, 46, p. 481 The English word appears in Bertrand Russell's Principles of Mathematics. The OED quotes from a review of this work in Nature (1903) 3 Sept. 411/2 "To readers unacquainted with [Cantor's] 'Mengenlehre', the introduction of transfinite numbers must appear rather startling."
TRANSITIVE. (Of a binary relation) Bertrand Russell wrote in "On the Notion of Order," Mind, 10, (1901), p. 32. "When ARB and BRC imply ARC, I call R transitive .... This term was used in this sense by De Morgan ... It is now generally adopted." De Morgan wrote in "On the Symbols of Logic, the Theory of the Syllogism, and in particular of the Copula," Transactions of the Cambridge Philosophical Society, 9, (1850) p. 104: "The first is what I shall call transitiveness, symbolized in X—Y—Z = X—Z; meaning that if X stand in the relation denoted by — to Y, and Y to Z, X therefore stands in that relation to Z." (OED)
TRANSPOSE (noun, of a matrix). Transposed matrix appears in 1858 in A. Cayley "A Memoir on the Theory of Matrices" Phil. Trans. R. Soc. CXLVIII, p. 32: Coll Math Papers, I, 475-96 "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 manières 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 TREATMENT for any stimulus that is applied to observe the effect in an experiment was introduced, by R. A. Fisher in chapter VIII, section 48, Technique of Plot Experimentation, of his Statistical Methods for Research Workers (1925). This was the first presentation of Fisher’s theory of experimental design and several other terms were introduced there as well.
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).
TREND. According to the OED trend was originally a geological term for "the way something [e.g. a coast-line] trends or bends away" and then it acquired a figurative sense meaning "the general course, tendency, or drift of action, thought etc." Statistical writers used the word in this latter sense but R. H. Hooker seemed to be aware he was extending the meaning when he wrote, "The curve or line representing the successive instantaneous averages [moving-averages] I propose to call the trend." See his "Correlation of the Marriage-Rate with Trade," Journal of the Royal Statistical Society, 64, (1901), p. 486. W. I. King's Elements of Statistical Method (1912) took Hooker's construction and the term "trend" to a broader audience. The term was soon applied to curves or lines constructed on other principles.
See also MOVING AVERAGE.
TRIANGLE INEQUALITY appears in 1921 in Albert A. Bennett "Normalized Geometric Systems," Proc. National Acad. Sci. U.S.A. 7 p. 88. (JSTOR) Fréchet had drawn attention to this property of "l’écart" in his Sur quelques points du calcul fonctionnel," Rendiconti del Circolo matematico di Palermo, 22, (1906) p. 30 but he did not give it a name. In his Grundzüge der Mengenlehre (1914, p. 211) Hausdorff used the term "Dreiecksaxiom" (triangle axiom.)
See METRIC SPACE.
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 1855 in A treatise on plane and spherical trigonometry by William Chauvenet. In this book "trigonometric equations" is a chapter title but the term does not appear in the running text [University of Michigan Digital Library].
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 1843 in The Penny Cyclopædia of the Society for the Diffusion of Useful Knowledge. [Google print search]
Trigonometric series is found in English in 1855 in A treatise on plane and spherical trigonometry by William Chauvenet. In this book "trigonometric series" is a chapter title but the term does not appear in the running text [University of Michigan Digital Library].
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.
TRIMMING, TRIMMED MEAN, etc. are found in John W. Tukey's "The Future of Data Analysis," Annals of Mathematical Statistics, 33, (1962), pp. 1-67. Tukey writes that in problems of location "it is natural to attempt to eliminate the effects of "wild shots" by trimming each sample, by removing equal numbers of the lowest and highest observations, and then proceeding as if the trimmed sample were a complete sample." (p. 17) The same section contains a discussion of WINSORIZING.
(David 2001)
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].
TRIVIAL (deriving from trivium, for which see LOGIC) has become a term of art in mathematics.
In “Deuxième mémoire sur les fonctions doublement périodiques,” Journal de Mathématiques Pures et Appliquées, tom. xix. (1854), pp. 193-208, Cayley wrote, “Je suppose d'abord que 2k + 1 soit égal à l'unité, transformation que l'on peut nommer triviale.” [I first suppose that 2k + 1 may be equal to unity, a transformation that one could call “trivial.”]
In “A Second Memoir Upon Quantics,” Philosopohical Transactions of the Royal Society of London (1856), Cayley wrote in a footnote: “It is hardly necessary to remark, that the multipliers ... and generally any coefficients or quantities not expressly stated to contain the set {a, b, c, ...}, are considered as independent of the set, or to use a convenient word, are considered as ‘trivials.’”
The OED illustrates with the following quotations: R. D. Carmichael Diophantine Analysis (1915, p. 28) "We have thus established the fact that Eq. (2) has at least one integral solution which is not trivial." G. Birkhoff & S. MacLane Survey of Modern Algebra (1941, p. 135) "The reflexive property is trivial (every group is isomorphic to itself by the identity transformation)."
[John Aldrich, 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).
TRUNCATION (in statistics). See CENSORING.
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 appears in 1937 in A. Church's review of A. M.Turing's 1937 paper "On Computable Numbers with an Application to the Entscheidungsproblem": "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." (Journal of Symbolic Logic, 2, p. 42) (OED)
The phrase Turing test, referring to the procedure described in Turing's "Computing Machinery and Intelligence, " Mind, 79, (1950), 433-460, seems to have entered currency in the 1970s. A JSTOR search found it used in B. A. Farrell “On the Design of a Conscious Device,” Mind, 79, (1970), p. 345.
See DECISION PROBLEM.
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 Problem 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).