# Earliest Known Uses of Some of the Words of Mathematics (T)

**is a translation of the term**

*T*_{1}SPACE*T*

_{1}-

*Raum*introduced in P. Alexandroff and H. Hopf's

*Topologie*(1935). The letter is an abbreviation for

*Trennungsaxiom*(separation axiom) and Alexandroff and Hopf originally defined five such spaces

*T*

_{0},...,

*T*

_{4}corresponding to different axioms. They associated

*T*

_{0},

*T*

_{1}and

*T*

_{2}with Kolmogorov, Fréchet and Hausdorff respectively. See the

*Encyclopaedia of Mathematics*article.

**TABLE**in the sense of "an arrangement of numbers, words, or items of any kind, in a definite and compact form, so as to exhibit some set of facts or relations in a distinct and comprehensive way, for convenience of study, reference, or calculation." The OED's earliest quotation is from Chaucer's

*Canterbury Tales*c1386: "Hise tables Tolletanes forth he brought Ful wel corrected."

*Franklin's Tale*. The

*Toletan tables*were 'the astronomical tables composed by order of Alphonso X, king of Castile (1252-82), from their being adapted to the city of Toledo.' (OED)

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.

Viete 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. Merriam-Webster says the word is of unknown origin.

*Trangam*is found in a 1658 dictionary.

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

*Encyclopedia of Mathematics*article Tauberian theorems.

This entry was contributed by John Aldrich. See also HARDY-LITTLEWOOD THEOREM.

**TAYLOR'S FORMULA, SERIES, THEOREM etc.**are terms associated with propositions in Brooke Taylor's

*Methodus incrementorum directa et inversa*(1715), a book on what came to be called the CALCULUS OF FINITE DIFFERENCES. Taylor did not consider questions of convergence—that came much later—and the modern distinctions between formula, series and theorem were not registered.

Kline (pp. 441-2) states that Taylor's result was known to Gregory and Leibniz and John Bernoulli published "practically the same result in 1694." In the English literature the theorem was attributed to Taylor: Colin Maclaurin wrote in his

*Treatise of Fluxions*(1742): "This theorem was given by Dr. Taylor ..." French writers also attributed the result to Taylor, including Condorcet in 1784 and Lhuillier in 1786: the latter writes in his

*Exposition élémentaire des principes des calculs supérieurs*about "le beau théorème que Taylor a le premier développé" (p. 47) and refers to "le théorème de Taylor" (p. 207).

English writers of the early 19th century modelled their expositions on the French.

*Taylor's theorem*appears in English in the 1816 translation of

*Lacroix's Traité élémentaire de calcul différentiel et de calcul intégral*(1797-1800): "This formula is called Taylor's Theorem, from the English geometer by whom it was discovered." (

*OED*).

*Taylor's series*appears in 1819 in Thomas Leybourn's

*New Series of the Mathematical Repository*(p. 103): "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." (The book is available on Google Book Search.)

[John Aldrich, Judith V. Grabiner, Julio González Cabillón]

**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 20

^{th}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 Göttingen 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 1804 in

*The Mathematical Repository*by T. Leybourn: "It has been shown that a whole number, having no integral root, has no terminating decimal root...." [James A. Landau]

*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 and 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 Émile 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 NUMBERS.**See NUMBER THEORY.

**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].

**THRACKLE**and the

**THRACKLE CONJECTURE**date from the 1960s to speculations by John Horton Conway. For an explanation see the

*Wikipedia*entry Conway's thrackle conjecture.

The word

*thrackle*is not in the

*OED*and Dan Archdeacon's site quotes Conway's account of its origin:

When I was a teenager, on holiday with my parents in Scotland, we once stopped to ask directions of a man who was fishing by the side of a lake. He happened to mention that his line was thrackled. I'd previously called this kind of drawing a tangle, but since I'd just found a knot-theoretical use for that term, I changed this to thrackle. Several people have told me that they've searched in vain for this word in dialect dictionaries, but since I quizzed the fisherman about it, I'm sure I didn't mishear it; he really did use it.[This entry was contributed by John Aldrich.]

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 and J. W. Van Ness's "Estimation of the Bispectrum,"

*Annals of Mathematical Statistics,*

**36,**(1965), 1120-1136 [John Aldrich].

**TIME SERIES**and

**TIME SERIES ANALYSIS**. The term "time series" appears in W. M. Persons's "The Correlation of Economic Statistics,"

*Publications of the American Statistical Association,*

**12,**(1910), 287-322. 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, but it only became really popular much later [John Aldrich].

Entries on Time Series topics include ABBE-HELMERT CRITERION, AUTOCORRELATION, AUTOREGRESSION, BOX-JENKINS, CORRELOGRAM, MOVING AVERAGE, SAMPLE PATH, SERIAL CORRELATION, SPECTRUM, TIME DOMAIN, TREND. See also the full list of Probability and Statistics entries.

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 19

^{th}century, topological results obtained in the 18

^{th}and the study projected in the 17

^{th}. See MacTutor 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 17

^{th}and 18

^{th}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; Leibniz spoke of it first, calling it the "geometry of position" ((For publication details see the entry KÖNIGSBERG BRIDGE PROBLEM.)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.

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 19

^{th}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 19

^{th}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 20

^{th}century the adjective

**TOPOLOGICAL**was more used than the noun. Thus the expression "topological point of view" appears in O. Veblen and 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).

Hausdorff's definition, however, is not equivalent to what is now known as topological space; what he called

*Topologischer Raum*is, in present day terminology, a

*Hausdorff space*since he includes the separation axiom $T_{2}$ in his definition. In his later book

*Mengenlehre*(1927), Hausdorff discusses various separation axioms, adding that "...und wenigstens eins der Trennungsaxiome ist wohl das Minimum dessen, was man von einem topologischen Raum fordern muß, wenn er nicht gar zu abnorm ausfallen soll." (...at least one of these seems to be the minimum to be required so that a topological space shall not turn out too abnormal). In this book, he does not attempt to fix the concept in a definitive manner. The present usage of the term TOPOLOGICAL SPACE (

*topologischer Raum*) was firmly established in the German language textbook

*Topologie I*(1935) by Alexandroff and Hopf where they use what they call KURATOWSKI'S AXIOMS; they are equivalent to Hausdorff's axioms without axiom $T_{2}$ which is then discussed separately, along with other separation axioms. [Peter Flor]

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 "Über 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].

**TORSION**as used in group theory: an element of a group $G$ is a

*torsion element*if it generates a finite subgroup of $G$. An abelian group consisting entirely of torsion elements is called a

*torsion group.*In any abelian group, the torsion elements form a subgroup, frequently called the

*torsion subgroup*of $G$. An abelian group is

*torsion-free*if the neutral element is its only torsion element.

This terminology seems to have arisen around 1930. Its origin lies in algebraic (or combinatorial) topology. Poincaré (

*Second complément à l´Analysis Situs,*Proc. London Math. Soc, vol. 32 (1900), 277-308) defined torsion coefficients for manifolds (variétés), and he distinguished manifolds with and without torsion. In a later terminology, his torsion coefficients are structure constants of homology groups. In 1935, the textbook

*Topologie I*by Alexandroff-Hopf has the following concept of torsion: "The elements of finite order of the

*r*-th Betti group of $E$ form a subgroup called the

*r*-th torsion group of $E$." Here, the use of the word

*torsion group*is still tied to a topological context even though the concept of a torsion subgroup seems to be hinted at. In the group theoretical chapter of this book, the word torsion is not used. But in the same year 1935, the paper

*Countable torsion groups*by Leon Zippin (Annals of Math. 36, 86-99) contains the following definition: "A torsion group $T$ is a discrete countable abelian group, every element of which is of finite order" - quite the modern definition, except for the restriction to countable groups. Two years before, Ulm (

*Zur Theorie der abzählbar-unendlichen Abelschen Gruppen,*Math. Annalen 107, pp. 774-803) did not yet use these terms, writing instead "groups all of whose elements have finite exponents." The torsion terminology was slow in obtaining general acceptance; it was frequently used in research papers in the 1940s, and is applied consistently in Kaplansky's

*Infinite Abelian Groups*of 1954, while on the other hand, Marshall Hall's textbook on group theory, published in 1959, introduces the term "periodic group," explaining that "the term torsion group is used in certain applications." The influential book on topological groups,

*Abstract Harmonic Analysis I*by Hewitt and Ross (1963) uses the torsion terminology and may have been important in promoting it.

[This entry was contributed by Peter Flor.]

**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.

**TRAMCAR PROBLEM**also called

**NEWMAN'S TRAMCAR PROBLEM.**In the

*Theory of Probability*(1939, p. 186) Harold Jeffreys writes, "The following problem was suggested to me several years ago by Mr M. H. A. Newman. A man travelling in a foreign country has to change trains at a junction, and goes into the town, the existence of which he has only just heard. He has no idea of its size. The first thing that he sees is a tramcar numbered 100. What can he infer about the number of tramcars in the town? It may be assumed for the purpose that they are numbered consecutively from 1 upwards."

**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 curve*is found in English in 1811 in C. Hutton,

*Course Math. III.*: "Transcendental or mechanical curves, are such as cannot be ... expressed by a pure algebraical equation. Thus,

*y*= log

*x,*

*y*=

*A*sin

*x,*...

*y*=

*Ax,*are equations to transcendental curves." [OED]

*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 π" 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,

*a*

^{x}, 10

^{2x}, 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.**Three-sided figures are discussed in Book I of Euclid's

*Elements*; Definitions 20 and 21 describe different kinds of triangles. The English word "triangle" appears in Sir Henry Billingsley's translation of the

*Elements*(1570). However the

*OED*finds earlier appearances in English, including a passage from John de Trevisa's translation of

*De Proprietatibus Rerum*, "Some [nuts] ben distinguid in þe cop as it were with þe schap of a triangle [orig.

*per formam trianguli*]" (1398).

**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 with 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**(in Statistics) appears in the phrase "trivariate distributions" in S. J. Pretorius "Skew Bivariate Frequency Surfaces, Examined in the Light of Numerical Illustrations,"

*Biometrika*,

**22**, (1930), p. 138. (

*JSTOR*search)

See BIVARIATE, N-VARIATE, TRIVARIATE and UNIVARIATE.

**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 2

*k*+ 1 soit égal à l'unité, transformation que l'on peut nommer triviale." [I first suppose that 2

*k*+ 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 and 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).

**TROY WEIGHT.**This term is believed to derive from the name of the city of Troyes, France. The OED shows a use of "marc. de troye" in 1390-91 and

*troy weyght*in about 1500.

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 found in 1940 in

*Mathematical logic*by Willard Van Orman Quine: "A truth set S of φ will include..anywhere from all to none of the ultimate truth-functional components..of φ." [OED]

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 Mathematical Society*26.

Post also used the term in "Introduction to a General Theory of Elementary Propositions"

*American Journal of Mathematics*,

**43**, (1921), p. 167:

So corresponding to each of the $2^{n}$ 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.The same concept was introduced by Ludwig Wittgenstein (1889-1951) as

*Wahrheitstafel*in his

*Tractatus Logico-Philosophicus*(1921). In the facing English version the word is translated as

*truth table*.

This entry was contributed by James A. Landau and John Aldrich.

**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."

In English,

*truth value*is found in 1903 in

*Principles of Mathematics*by Bertrand Russell: "There are, we are told.., three elements in judgment: (1) the recognition of truth, (2) the Gedanke, (3) the truth-value (Wahrheitswerth)." [OED]

**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 term

**TURING TEST**, referring to the procedure described in Turing's "Computing Machinery and Intelligence,"

*Mind*,

**79**, (1950), 433-460, is apparently found in 1956 in

*Mental Health Research Institute Staff Publications,*University of Michigan, Mental Health Research Institute, based on a Google print search which provides only snippet view: "To check the validity of his claim he subjects it to the Turing test. This is a hypothetical test in which both the behavior of a human and the mechanical translator is presented to an impartial observer..."

The term 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].

*Twin-prime*is found in English in 1930 in

*Number*by T. Dantzig: " It has been shown that the so-called twin-primes, such as (3,5), (5,7),..(41,43), etc., become rarer and rarer as the numbers increase." [OED]

**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).